Octave Manual

User Manual:
Open the PDF directly: View PDF .
Page Count: 1016
Download
Open PDF In Browser	View PDF
GNU Octave
A high-level interactive language for numerical computations
Edition 4 for Octave version 4.2.2
March 2018

Free Your Numbers

John W. Eaton
David Bateman
Søren Hauberg
Rik Wehbring

Copyright c 1996, 1997, 1999, 2000, 2001, 2002, 2005, 2006, 2007, 2011, 2013, 2015, 2016,
2017, 2018 John W. Eaton.
This is the fourth edition of the Octave documentation, and is consistent with version 4.2.2
of Octave.
Permission is granted to make and distribute verbatim copies of this manual provided the
copyright notice and this permission notice are preserved on all copies.
Permission is granted to copy and distribute modified versions of this manual under the conditions for verbatim copying, provided that the entire resulting derived work is distributed
under the terms of a permission notice identical to this one.
Permission is granted to copy and distribute translations of this manual into another language, under the same conditions as for modified versions.
Portions of this document have been adapted from the gawk, readline, gcc, and C library
manuals, published by the Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor,
Boston, MA 02110-1301–1307, USA.

i

Table of Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Citing Octave in Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
How You Can Contribute to Octave. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1

A Brief Introduction to Octave . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1
1.2

Running Octave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Simple Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.1 Elementary Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.2 Creating a Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.3 Matrix Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.4 Solving Systems of Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.5 Integrating Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.6 Producing Graphical Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2.7 Help and Documentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2.8 Editing What You Have Typed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3.1 Fonts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3.2 Evaluation Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3.3 Printing Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.4 Error Messages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.5 Format of Descriptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.5.1 A Sample Function Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.5.2 A Sample Command Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2

Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1

Invoking Octave from the Command Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1.1 Command Line Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1.2 Startup Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Quitting Octave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3 Commands for Getting Help . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.4 Command Line Editing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4.1 Cursor Motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4.2 Killing and Yanking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4.3 Commands for Changing Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4.4 Letting Readline Type for You . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4.5 Commands for Manipulating the History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.4.6 Customizing readline. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.4.7 Customizing the Prompt. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.4.8 Diary and Echo Commands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.5 How Octave Reports Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

ii

GNU Octave
2.6
2.7

3

Executable Octave Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Comments in Octave Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7.1 Single Line Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7.2 Block Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7.3 Comments and the Help System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36
37
37
38
38

Data Types. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.1

Built-in Data Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.1.1 Numeric Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.1.2 Missing Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.1.3 String Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.1.4 Data Structure Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.1.5 Cell Array Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2 User-defined Data Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.3 Object Sizes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4

Numeric Data Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.1

Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.1.1 Empty Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2 Ranges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.3 Single Precision Data Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.4 Integer Data Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.4.1 Integer Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.5 Bit Manipulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.6 Logical Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.7 Promotion and Demotion of Data Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.8 Predicates for Numeric Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5

Strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.1
5.2
5.3

Escape Sequences in String Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Character Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
Creating Strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.3.1 Concatenating Strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.3.2 Converting Numerical Data to Strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.4 Comparing Strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.5 Manipulating Strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.6 String Conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.7 Character Class Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

6

Data Containers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.1

Structures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.1.1 Basic Usage and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.1.2 Structure Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.1.3 Creating Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.1.4 Manipulating Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.1.5 Processing Data in Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

iii
6.2

Cell Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.2.1 Basic Usage of Cell Arrays. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.2.2 Creating Cell Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.2.3 Indexing Cell Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.2.4 Cell Arrays of Strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.2.5 Processing Data in Cell Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.3 Comma Separated Lists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6.3.1 Comma Separated Lists Generated from Cell Arrays . . . . . . . . . . . . . . . . . . . . 123
6.3.2 Comma Separated Lists Generated from Structure Arrays . . . . . . . . . . . . . . . 124

7

Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
7.1
7.2
7.3

8

Global Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
Persistent Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
Status of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
8.1

Index Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
8.1.1 Advanced Indexing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
8.2 Calling Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
8.2.1 Call by Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
8.2.2 Recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
8.3 Arithmetic Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
8.4 Comparison Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
8.5 Boolean Expressions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
8.5.1 Element-by-element Boolean Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
8.5.2 Short-circuit Boolean Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
8.6 Assignment Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
8.7 Increment Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
8.8 Operator Precedence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

9

Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
9.1
9.2

10

Calling a Function by its Name . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
Evaluation in a Different Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

10.1 The if Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
10.2 The switch Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
10.2.1 Notes for the C Programmer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
10.3 The while Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
10.4 The do-until Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
10.5 The for Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
10.5.1 Looping Over Structure Elements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
10.6 The break Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
10.7 The continue Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
10.8 The unwind protect Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
10.9 The try Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
10.10 Continuation Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

iv

GNU Octave

11

Functions and Scripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

11.1 Introduction to Function and Script Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
11.2 Defining Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
11.3 Multiple Return Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
11.4 Variable-length Argument Lists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
11.5 Ignoring Arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
11.6 Variable-length Return Lists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
11.7 Returning from a Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
11.8 Default Arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
11.9 Function Files. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
11.9.1 Manipulating the Load Path. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
11.9.2 Subfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
11.9.3 Private Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
11.9.4 Nested Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
11.9.5 Overloading and Autoloading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
11.9.6 Function Locking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
11.9.7 Function Precedence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
11.10 Script Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
11.10.1 Publish Octave Script Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
11.10.2 Publishing Markup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
11.10.2.1 Using Publishing Markup in Script Files . . . . . . . . . . . . . . . . . . . . . . . . . 207
11.10.2.2 Text Formatting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
11.10.2.3 Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
11.10.2.4 Preformatted Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
11.10.2.5 Preformatted Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
11.10.2.6 Bulleted Lists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
11.10.2.7 Numbered Lists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
11.10.2.8 Including File Content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
11.10.2.9 Including Graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
11.10.2.10 Including URLs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
11.10.2.11 Mathematical Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
11.10.2.12 HTML Markup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
11.10.2.13 LaTeX Markup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
11.11 Function Handles, Anonymous Functions, Inline Functions . . . . . . . . . . . . . . . . . 211
11.11.1 Function Handles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
11.11.2 Anonymous Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
11.11.3 Inline Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
11.12 Commands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
11.13 Organization of Functions Distributed with Octave . . . . . . . . . . . . . . . . . . . . . . . . 215

12

Errors and Warnings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

12.1 Handling Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
12.1.1 Raising Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
12.1.2 Catching Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
12.1.3 Recovering From Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
12.2 Handling Warnings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
12.2.1 Issuing Warnings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
12.2.2 Enabling and Disabling Warnings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

v

13

Debugging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

13.1
13.2
13.3
13.4
13.5
13.6
13.7

14

Entering Debug Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
Leaving Debug Mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
Breakpoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
Debug Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
Call Stack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
Profiling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
Profiler Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

Input and Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

14.1 Basic Input and Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
14.1.1 Terminal Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
14.1.1.1 Paging Screen Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
14.1.2 Terminal Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
14.1.3 Simple File I/O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
14.1.3.1 Saving Data on Unexpected Exits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
14.2 C-Style I/O Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
14.2.1 Opening and Closing Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
14.2.2 Simple Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
14.2.3 Line-Oriented Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
14.2.4 Formatted Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
14.2.5 Output Conversion for Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
14.2.6 Output Conversion Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
14.2.7 Table of Output Conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
14.2.8 Integer Conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
14.2.9 Floating-Point Conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
14.2.10 Other Output Conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
14.2.11 Formatted Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
14.2.12 Input Conversion Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
14.2.13 Table of Input Conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
14.2.14 Numeric Input Conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
14.2.15 String Input Conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
14.2.16 Binary I/O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
14.2.17 Temporary Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
14.2.18 End of File and Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
14.2.19 File Positioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

15

Plotting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

15.1 Introduction to Plotting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
15.2 High-Level Plotting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
15.2.1 Two-Dimensional Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
15.2.1.1 Axis Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
15.2.1.2 Two-dimensional Function Plotting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318
15.2.1.3 Two-dimensional Geometric Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
15.2.2 Three-Dimensional Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
15.2.2.1 Aspect Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344
15.2.2.2 Three-dimensional Function Plotting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345

vi

GNU Octave
15.2.2.3 Three-dimensional Geometric Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348
15.2.3 Plot Annotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
15.2.4 Multiple Plots on One Page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356
15.2.5 Multiple Plot Windows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
15.2.6 Manipulation of Plot Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358
15.2.7 Manipulation of Plot Windows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
15.2.8 Use of the interpreter Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363
15.2.8.1 Degree Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366
15.2.9 Printing and Saving Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366
15.2.10 Interacting with Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374
15.2.11 Test Plotting Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375
15.3 Graphics Data Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376
15.3.1 Introduction to Graphics Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376
15.3.2 Graphics Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
15.3.2.1 Creating Graphics Objects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378
15.3.2.2 Handle Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381
15.3.3 Graphics Object Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386
15.3.3.1 Root Figure Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386
15.3.3.2 Figure Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388
15.3.3.3 Axes Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393
15.3.3.4 Line Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399
15.3.3.5 Text Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401
15.3.3.6 Image Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403
15.3.3.7 Patch Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405
15.3.3.8 Surface Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408
15.3.3.9 Light Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411
15.3.3.10 Uimenu Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412
15.3.3.11 Uibuttongroup Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413
15.3.3.12 Uicontextmenu Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415
15.3.3.13 Uipanel Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416
15.3.3.14 Uicontrol Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417
15.3.3.15 Uitoolbar Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419
15.3.3.16 Uipushtool Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420
15.3.3.17 Uitoggletool Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421
15.3.4 Searching Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422
15.3.5 Managing Default Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424
15.4 Advanced Plotting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425
15.4.1 Colors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425
15.4.2 Line Styles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425
15.4.3 Marker Styles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425
15.4.4 Callbacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426
15.4.5 Application-defined Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427
15.4.6 Object Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428
15.4.6.1 Data Sources in Object Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433
15.4.6.2 Area Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433
15.4.6.3 Bar Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434
15.4.6.4 Contour Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435
15.4.6.5 Error Bar Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436

vii
15.4.6.6 Line Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437
15.4.6.7 Quiver Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437
15.4.6.8 Scatter Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438
15.4.6.9 Stair Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
15.4.6.10 Stem Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
15.4.6.11 Surface Group. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440
15.4.7 Graphics Toolkits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441
15.4.7.1 Customizing Toolkit Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441

16

Matrix Manipulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443

16.1
16.2
16.3
16.4

17

Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475

17.1
17.2
17.3
17.4
17.5
17.6
17.7
17.8
17.9

18

Exponents and Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475
Complex Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477
Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478
Sums and Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482
Utility Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484
Special Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492
Rational Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500
Coordinate Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500
Mathematical Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502

Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507

18.1
18.2
18.3
18.4
18.5

19

Finding Elements and Checking Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443
Rearranging Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447
Special Utility Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457
Famous Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465

Techniques Used for Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507
Basic Matrix Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507
Matrix Factorizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515
Functions of a Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527
Specialized Solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528

Vectorization and Faster Code Execution . . . . . . . . . . . . . 533

19.1 Basic Vectorization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533
19.2 Broadcasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535
19.2.1 Broadcasting and Legacy Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538
19.3 Function Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538
19.4 Accumulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543
19.5 JIT Compiler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545
19.6 Miscellaneous Techniques. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546
19.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548

20

Nonlinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549

20.1
20.2

Solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549
Minimizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552

viii

GNU Octave

21

Diagonal and Permutation Matrices . . . . . . . . . . . . . . . . . . . 555

21.1 Creating and Manipulating Diagonal/Permutation Matrices . . . . . . . . . . . . . . . . . 555
21.1.1 Creating Diagonal Matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556
21.1.2 Creating Permutation Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556
21.1.3 Explicit and Implicit Conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557
21.2 Linear Algebra with Diagonal/Permutation Matrices . . . . . . . . . . . . . . . . . . . . . . . . 558
21.2.1 Expressions Involving Diagonal Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558
21.2.2 Expressions Involving Permutation Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 559
21.3 Functions That Are Aware of These Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 560
21.3.1 Diagonal Matrix Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 560
21.3.2 Permutation Matrix Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 560
21.4 Examples of Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 560
21.5 Differences in Treatment of Zero Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561

22

Sparse Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563

22.1 Creation and Manipulation of Sparse Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563
22.1.1 Storage of Sparse Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563
22.1.2 Creating Sparse Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564
22.1.3 Finding Information about Sparse Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 570
22.1.4 Basic Operators and Functions on Sparse Matrices . . . . . . . . . . . . . . . . . . . . . 573
22.1.4.1 Sparse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574
22.1.4.2 Return Types of Operators and Functions . . . . . . . . . . . . . . . . . . . . . . . . 574
22.1.4.3 Mathematical Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576
22.2 Linear Algebra on Sparse Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584
22.3 Iterative Techniques Applied to Sparse Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593
22.4 Real Life Example using Sparse Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601

23

Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605

23.1
23.2
23.3

24

Functions of One Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605
Orthogonal Collocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612
Functions of Multiple Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613

Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617

24.1 Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617
24.1.1 Matlab-compatible solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619
24.2 Differential-Algebraic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624

25

Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633

25.1
25.2
25.3
25.4

Linear Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633
Quadratic Programming. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639
Nonlinear Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641
Linear Least Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643

ix

26

Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649

26.1
26.2
26.3
26.4
26.5
26.6
26.7

27

Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685

27.1

28

Evaluating Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 689
Finding Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 690
Products of Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 691
Derivatives / Integrals / Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694
Polynomial Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695
Miscellaneous Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704

Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707

29.1
29.2

30

Set Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685

Polynomial Manipulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 689

28.1
28.2
28.3
28.4
28.5
28.6

29

Descriptive Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649
Basic Statistical Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 656
Statistical Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659
Correlation and Regression Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 660
Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 662
Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 670
Random Number Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 678

One-dimensional Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707
Multi-dimensional Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 711

Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717

30.1 Delaunay Triangulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717
30.1.1 Plotting the Triangulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 719
30.1.2 Identifying Points in Triangulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 722
30.2 Voronoi Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724
30.3 Convex Hull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 728
30.4 Interpolation on Scattered Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 730

31

Signal Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733

32

Image Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 747

32.1
32.2
32.3
32.4
32.5

Loading and Saving Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 747
Displaying Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753
Representing Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755
Plotting on top of Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765
Color Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765

x

GNU Octave

33

Audio Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 767

33.1 Audio File Utilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 767
33.2 Audio Device Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 768
33.3 Audio Player . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 768
33.3.1 Playback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 769
33.3.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 769
33.4 Audio Recorder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 770
33.4.1 Recording . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 770
33.4.2 Data Retrieval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 771
33.4.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 771
33.5 Audio Data Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 771

34

Object Oriented Programming. . . . . . . . . . . . . . . . . . . . . . . . . 775

34.1 Creating a Class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775
34.2 Class Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777
34.3 Indexing Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 781
34.3.1 Defining Indexing And Indexed Assignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 781
34.3.2 Indexed Assignment Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784
34.4 Overloading Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785
34.4.1 Function Overloading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785
34.4.2 Operator Overloading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786
34.4.3 Precedence of Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 787
34.5 Inheritance and Aggregation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 789
34.6 classdef Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 793
34.6.1 Creating a classdef Class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 793
34.6.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795
34.6.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795
34.6.4 Inheritance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 797
34.6.5 Value Classes vs. Handle Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 797

35

GUI Development. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 801

35.1
35.2
35.3
35.4
35.5

36

I/O Dialogs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Progress Bar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
UI Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
GUI Utility Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
User-Defined Preferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

801
808
808
814
816

System Utilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 819

36.1 Timing Utilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 819
36.2 Filesystem Utilities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 830
36.3 File Archiving Utilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 839
36.4 Networking Utilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 842
36.4.1 FTP Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 842
36.4.2 URL Manipulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844
36.4.3 Base64 and Binary Data Transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845
36.5 Controlling Subprocesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845
36.6 Process, Group, and User IDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 853

xi
36.7 Environment Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854
36.8 Current Working Directory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854
36.9 Password Database Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 856
36.10 Group Database Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 857
36.11 System Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 858
36.12 Hashing Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 862

37

Packages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865

37.1 Installing and Removing Packages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865
37.2 Using Packages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 868
37.3 Administrating Packages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 869
37.4 Creating Packages. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 869
37.4.1 The DESCRIPTION File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 871
37.4.2 The INDEX File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 873
37.4.3 PKG ADD and PKG DEL Directives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874
37.4.4 Missing Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874

Appendix A

External Code Interface . . . . . . . . . . . . . . . . . . . . . 875

A.1 Oct-Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 876
A.1.1 Getting Started with Oct-Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 876
A.1.2 Matrices and Arrays in Oct-Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 879
A.1.3 Character Strings in Oct-Files. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 882
A.1.4 Cell Arrays in Oct-Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 883
A.1.5 Structures in Oct-Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 884
A.1.6 Sparse Matrices in Oct-Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885
A.1.6.1 Array and Sparse Class Differences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 886
A.1.6.2 Creating Sparse Matrices in Oct-Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 887
A.1.6.3 Using Sparse Matrices in Oct-Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 890
A.1.7 Accessing Global Variables in Oct-Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 890
A.1.8 Calling Octave Functions from Oct-Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 891
A.1.9 Calling External Code from Oct-Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 893
A.1.10 Allocating Local Memory in Oct-Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895
A.1.11 Input Parameter Checking in Oct-Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895
A.1.12 Exception and Error Handling in Oct-Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . 896
A.1.13 Documentation and Testing of Oct-Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 898
A.2 Mex-Files. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 899
A.2.1 Getting Started with Mex-Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 899
A.2.2 Working with Matrices and Arrays in Mex-Files . . . . . . . . . . . . . . . . . . . . . . . . 901
A.2.3 Character Strings in Mex-Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 903
A.2.4 Cell Arrays with Mex-Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 904
A.2.5 Structures with Mex-Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 905
A.2.6 Sparse Matrices with Mex-Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 906
A.2.7 Calling Other Functions in Mex-Files. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 910
A.3 Standalone Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 911
A.4 Java Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 913
A.4.1 Making Java Classes Available . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 914
A.4.2 How to use Java from within Octave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 915

xii

GNU Octave
A.4.3
A.4.4

Passing parameters to the JVM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 917
Java Interface Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 918

Appendix B
B.1
B.2

Test and Demo Functions . . . . . . . . . . . . . . . . . . . 925

Test Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 925
Demonstration Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 932

Appendix C

Obsolete Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 937

Appendix D Known Causes of Trouble . . . . . . . . . . . . . . . . . . . 941
D.1 Actual Bugs We Haven’t Fixed Yet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 941
D.2 Reporting Bugs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 941
D.2.1 Have You Found a Bug? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 941
D.2.2 Where to Report Bugs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 942
D.2.3 How to Report Bugs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 942
D.2.4 Sending Patches for Octave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 943
D.3 How To Get Help with Octave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 944
D.4 How to Distinguish Between Octave and Matlab . . . . . . . . . . . . . . . . . . . . . . . . . . . . 944

Appendix E

Installing Octave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 947

E.1 Build Dependencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 947
E.1.1 Obtaining the Dependencies Automatically . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 947
E.1.2 Build Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 947
E.1.3 External Packages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 948
E.2 Running Configure and Make . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 950
E.3 Compiling Octave with 64-bit Indexing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 954
E.4 Installation Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 957

Appendix F
F.1
F.2

Grammar and Parser . . . . . . . . . . . . . . . . . . . . . . . . 961

Keywords . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 961
Parser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 961

Appendix G GNU GENERAL PUBLIC LICENSE . . . . . . 963
Concept Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 975
Function Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 987
Operator Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1001

1

Preface
Octave was originally intended to be companion software for an undergraduate-level textbook on chemical reactor design being written by James B. Rawlings of the University of
Wisconsin-Madison and John G. Ekerdt of the University of Texas.
Clearly, Octave is now much more than just another ‘courseware’ package with limited
utility beyond the classroom. Although our initial goals were somewhat vague, we knew
that we wanted to create something that would enable students to solve realistic problems,
and that they could use for many things other than chemical reactor design problems. We
find that most students pick up the basics of Octave quickly, and are using it confidently in
just a few hours.
Although it was originally intended to be used to teach reactor design, it has been used in
several other undergraduate and graduate courses in the Chemical Engineering Department
at the University of Texas, and the math department at the University of Texas has been
using it for teaching differential equations and linear algebra as well. More recently, Octave
has been used as the primary computational tool for teaching Stanford’s online Machine
Learning class (ml-class . org) taught by Andrew Ng. Tens of thousands of students
participated in the course.
If you find Octave useful, please let us know. We are always interested to find out how
Octave is being used.
Virtually everyone thinks that the name Octave has something to do with music, but
it is actually the name of one of John W. Eaton’s former professors who wrote a famous
textbook on chemical reaction engineering, and who was also well known for his ability
to do quick ‘back of the envelope’ calculations. We hope that this software will make it
possible for many people to do more ambitious computations just as easily.
Everyone is encouraged to share this software with others under the terms of the GNU
General Public License (see Appendix G [Copying], page 963). You are also encouraged to
help make Octave more useful by writing and contributing additional functions for it, and
by reporting any problems you may have.

Acknowledgements
Many people have contributed to Octave’s development. The following people have helped
code parts of Octave or aided in various other ways (listed alphabetically).
Ben Abbott
Drew Abbot
Andy Adler
Adam H. Aitkenhead
Joakim Andén
Giles Anderson
Joel Andersson
Lachlan Andrew
Pedro Angelo
Damjan Angelovski
Muthiah Annamalai
Markus Appel
Branden Archer
Willem Atsma
Marco Atzeri
Ander Aurrekoetxea
Shai Ayal
Roger Banks
Ben Barrowes
Alexander Barth
David Bateman
Heinz Bauschke
Julien Bect
Stefan Beller
Roman Belov
Markus Bergholz
Karl Berry
Atri Bhattacharya
Ethan Biery
David Billinghurst
Don Bindner
Jakub Bogusz
Moritz Borgmann
Paul Boven
Richard Bovey
John Bradshaw

2

Marcus Brinkmann
Clemens Buchacher
Daniel Calvelo
Jean-Francois Cardoso
David Castelow
Albert Chin-A-Young
J. D. Cole
Michael Creel
Martin Dalecki
Carlo de Falco
Bill Denney
Pantxo Diribarne
David M. Doolin
John W. Eaton
Paul Eggert
Garrett Euler
Gunnar Farnebäck
Ramon Garcia Fernandez
Jose Daniel Munoz Frias
Eduardo Gallestey
Driss Ghaddab
Nicolo Giorgetti
Michael Goffioul
Keith Goodman
Steffen Groot
Kyle Guinn
Kai Habel
Jaroslav Hajek
Søren Hauberg
Daniel Heiserer
Stefan Hepp
Yozo Hida
A. Scottedward Hodel
David Hoover
Cyril Humbert
Alan W. Irwin
Vytautas Jančauskas
Robert Jenssen
Heikki Junes
Jarkko Kaleva
Lute Kamstra
Joel Keay
Lars Kindermann
Arno J. Klaassen
Heine Kolltveit
Daniel Kraft
Oyvind Kristiansen

GNU Octave

Max Brister
Ansgar Burchard
John C. Campbell
Joao Cardoso
Vincent Cautaerts
Carsten Clark
Jacopo Corno
Richard Crozier
Jacob Dawid
Thomas D. Dean
Fabian Deutsch
Vivek Dogra
Carnë Draug
Dirk Eddelbuettel
Stephen Eglen
Edmund Grimley Evans
Massimiliano Fasi
Torsten Finke
Brad Froehle
Walter Gautschi
Eugenio Gianniti
Arun Giridhar
Glenn Golden
Brian Gough
Etienne Grossmann
Vaibhav Gupta
Patrick Häcker
Benjamin Hall
Dave Hawthorne
Piotr Held
Martin Hepperle
Ryan Hinton
Richard Allan Holcombe
Kurt Hornik
John Hunt
Allan Jacobs
Nicholas R. Jankowski
Cai Jianming
Matthias Jüschke
Avinoam Kalma
Fotios Kasolis
Mumit Khan
Aaron A. King
Alexander Klein
Ken Kouno
Nir Krakauer
Artem Krosheninnikov

Remy Bruno
Marco Caliari
Juan Pablo Carbajal
Larrie Carr
Clinton Chee
Catalin Codreanu
Martin Costabel
Jeff Cunningham
Jorge Barros de Abreu
Philippe Defert
Christos Dimitrakakis
John Donoghue
Pascal A. Dupuis
Pieter Eendebak
Peter Ekberg
Rolf Fabian
Stephen Fegan
Colin Foster
Castor Fu
Klaus Gebhardt
Hartmut Gimpel
Michael D. Godfrey
Tomislav Goles
Michael C. Grant
David Grundberg
Peter Gustafson
William P. Y. Hadisoeseno
Kim Hansen
Oliver Heimlich
Martin Helm
Jordi Gutiérrez Hermoso
Roman Hodek
Tom Holroyd
Christopher Hulbert
Teemu Ikonen
Geoff Jacobsen
Mats Jansson
Steven G. Johnson
Atsushi Kajita
Mohamed Kamoun
Thomas Kasper
Paul Kienzle
Erik Kjellson
Geoffrey Knauth
Kacper Kowalik
Aravindh Krishnamoorthy
Piotr Krzyzanowski

Preface

Volker Kuhlmann
Philipp Kutin
Kai Labusch
Bill Lash
Friedrich Leisch
Jyh-miin Lin
Ross Lippert
Barbara Locsi
Massimo Lorenzin
Hoxide Ma
Jens-Uwe Mager
Alexander Mamonov
Axel Mathéi
Christoph Mayer
Ronald van der Meer
Thorsten Meyer
Mike Miller
Rafael Monteiro
Armin Müller
PrasannaKumar
Muralidharan
Carmen Navarrete
Al Niessner
Takuji Nishimura
Patrick Noffke
Michael O’Brien
Thorsten Ohl
Valentin Ortega-Clavero
Janne Olavi Paanajärvi
Jason Alan Palmer
Rolando Pereira
Jim Peterson
Elias Pipping
Sergey Plotnikov
Orion Poplawski
Francesco Potortı̀
Jarno Rajahalme
James B. Rawlings
Joshua Redstone
Ernst Reissner
Jason Riedy
Petter Risholm
Peter Rosin
Mark van Rossum
Kevin Ruland
Olli Saarela

3

Ilya Kurdyukov
Miroslaw Kwasniak
Claude Lacoursiere
Dirk Laurie
Johannes Leuschner
Timo Lindfors
Yu Liu
Sebastien Loisel
Emil Lucretiu
Colin Macdonald
Stefan Mahr
Ricardo Marranita
Makoto Matsumoto
Laurent Mazet
Júlio Hoffimann Mendes
Stefan Miereis
Serviscope Minor
Antoine Moreau
Hannes Müller
Iain Murray

Tetsuro Kurita
Rafael Laboissiere
Walter Landry
Maurice LeBrun
Thorsten Liebig
Benjamin Lindner
David Livings
Erik de Castro Lopo
Yi-Hong Lyu
James Macnicol
Rob Mahurin
Orestes Mas
Tatsuro Matsuoka
G. D. McBain
Ed Meyer
Petr Mikulik
Stefan Monnier
Kai P. Mueller
Victor Munoz
Markus Mützel

Todd Neal
Felipe G. Nievinski
Akira Noda
Eric Norum
Cillian O’Driscoll
Kai T. Ohlhus
Luis F. Ortiz
Scott Pakin
Gabriele Pannocchia
Per Persson
Danilo Piazzalunga
Robert Platt
Tom Poage
Ondrej Popp
Konstantinos Poulios
Eduardo Ramos
Eric S. Raymond
Lukas Reichlin
Jens Restemeier
E. Joshua Rigler
Matthew W. Roberts
Andrew Ross
Joe Rothweiler
Kristian Rumberg
Toni Saarela

Philip Nienhuis
Rick Niles
Kai Noda
Krzesimir Nowak
Peter O’Gorman
Arno Onken
Carl Osterwisch
José Luis Garcı́a Pallero
Sylvain Pelissier
Primozz Peterlin
Nicholas Piper
Hans Ekkehard Plesser
Nathan Podlich
Jef Poskanzer
Tejaswi D. Prakash
Pooja Rao
Balint Reczey
Michael Reifenberger
Anthony Richardson
Sander van Rijn
Dmitry Roshchin
Fabio Rossi
David Rörich
Ryan Rusaw
Juhani Saastamoinen

4

GNU Octave

Radek Salac
Aleksej Saushev
Julian Schnidder
Sebastian Schubert
Thomas L. Scofield
Vanya Sergeev
Baylis Shanks
Joseph P. Skudlarek
Shan G. Smith
Quentin H. Spencer
Richard Stallman
Doug Stewart
Thomas Stuart
John Swensen
Falk Tannhäuser
Kris Thielemans
Andrew Thornton
Thomas Treichl
Frederick Umminger
Peter Van Wieren
Gregory Vanuxem
Sébastien Villemot
Thomas Walter
Thomas Weber
Andreas Weingessel
David Wells
Sean Young
Michael Zeising

Mike Sander
Alois Schlögl
Sebastian Schöps
Lasse Schuirmann
Daniel J. Sebald
Marko Seric
Andriy Shinkarchuck
John Smith
Peter L. Sondergaard
Christoph Spiel
Russell Standish
Jonathan Stickel
Bernardo Sulzbach
Daisuke Takago
Duncan Temple Lang
Georg Thimm
Olaf Till
Karsten Trulsen
Utkarsh Upadhyay
James R. Van Zandt
Mihas Varantsou
Marco Vitetta
Andreas Weber
Rik Wehbring
Martin Weiser
Joachim Wiesemann
Michele Zaffalon
Federico Zenith

Ben Sapp
Michel D. Schmid
Nicol N. Schraudolph
Ludwig Schwardt
Dmitri A. Sergatskov
Ahsan Ali Shahid
Robert T. Short
Julius Smith
Joerg Specht
David Spies
Brett Stewart
Judd Storrs
Ivan Sutoris
Ariel Tankus
Matthew Tenny
Corey Thomasson
Christophe Tournery
David Turner
Stefan van der Walt
Risto Vanhanen
Ivana Varekova
Daniel Wagenaar
Olaf Weber
Bob Weigel
Michael Weitzel
Fook Fah Yap
Johannes Zarl
Alex Zvoleff

Special thanks to the following people and organizations for supporting the development
of Octave:
• The United States Department of Energy, through grant number DE-FG02-04ER25635.
• Ashok Krishnamurthy, David Hudak, Juan Carlos Chaves, and Stanley C. Ahalt of the
Ohio Supercomputer Center.

• The National Science Foundation, through grant numbers CTS-0105360, CTS-9708497,
CTS-9311420, CTS-8957123, and CNS-0540147.
• The industrial members of the Texas-Wisconsin Modeling and Control Consortium
(TWMCC).
• The Paul A. Elfers Endowed Chair in Chemical Engineering at the University of
Wisconsin-Madison.
• Digital Equipment Corporation, for an equipment grant as part of their External Research Program.
• Sun Microsystems, Inc., for an Academic Equipment grant.

• International Business Machines, Inc., for providing equipment as part of a grant to
the University of Texas College of Engineering.

Preface

5

• Texaco Chemical Company, for providing funding to continue the development of this
software.
• The University of Texas College of Engineering, for providing a Challenge for Excellence
Research Supplement, and for providing an Academic Development Funds grant.
• The State of Texas, for providing funding through the Texas Advanced Technology
Program under Grant No. 003658-078.
• Noel Bell, Senior Engineer, Texaco Chemical Company, Austin Texas.
• John A. Turner, Group Leader, Continuum Dynamics (CCS-2), Los Alamos National
Laboratory, for registering the octave.org domain name.
• James B. Rawlings, Professor, University of Wisconsin-Madison, Department of Chemical and Biological Engineering.
• Richard Stallman, for writing GNU.

This project would not have been possible without the GNU software used in and to
produce Octave.

Citing Octave in Publications
In view of the many contributions made by numerous developers over many years it is
common courtesy to cite Octave in publications when it has been used during the course of
research or the preparation of figures. The citation function can automatically generate
a recommended citation text for Octave or any of its packages. See the help text below on
how to use citation.

citation
citation package
Display instructions for citing GNU Octave or its packages in publications.
When called without an argument, display information on how to cite the core GNU
Octave system.
When given a package name package, display information on citing the specific named
package. Note that some packages may not yet have instructions on how to cite them.
The GNU Octave developers and its active community of package authors have invested a lot of time and effort in creating GNU Octave as it is today. Please give
credit where credit is due and cite GNU Octave and its packages when you use them.

How You Can Contribute to Octave
There are a number of ways that you can contribute to help make Octave a better system.
Perhaps the most important way to contribute is to write high-quality code for solving
new problems, and to make your code freely available for others to use. See http://www.
octave.org/get-involved.html for detailed information.
If you find Octave useful, consider providing additional funding to continue its development. Even a modest amount of additional funding could make a significant difference in
the amount of time that is available for development and support.
Donations supporting Octave development may be made on the web at https://my.
fsf.org/donate/working-together/octave. These donations also help to support the
Free Software Foundation

6

GNU Octave

If you’d prefer to pay by check or money order, you can do so by sending a check to the
FSF at the following address:
Free Software Foundation
51 Franklin Street, Suite 500
Boston, MA 02110-1335
USA
If you pay by check, please be sure to write “GNU Octave” in the memo field of your check.
If you cannot provide funding or contribute code, you can still help make Octave better
and more reliable by reporting any bugs you find and by offering suggestions for ways to
improve Octave. See Appendix D [Trouble], page 941, for tips on how to write useful bug
reports.

Distribution
Octave is free software. This means that everyone is free to use it and free to redistribute
it on certain conditions. Octave is not, however, in the public domain. It is copyrighted
and there are restrictions on its distribution, but the restrictions are designed to ensure
that others will have the same freedom to use and redistribute Octave that you have. The
precise conditions can be found in the GNU General Public License that comes with Octave
and that also appears in Appendix G [Copying], page 963.
To download a copy of Octave, please visit http://www.octave.org/download.html.

7

1 A Brief Introduction to Octave
GNU Octave is a high-level language primarily intended for numerical computations. It is
typically used for such problems as solving linear and nonlinear equations, numerical linear
algebra, statistical analysis, and for performing other numerical experiments. It may also
be used as a batch-oriented language for automated data processing.
The current version of Octave executes in a graphical user interface (GUI). The GUI
hosts an Integrated Development Environment (IDE) which includes a code editor with
syntax highlighting, built-in debugger, documentation browser, as well as the interpreter
for the language itself. A command-line interface for Octave is also available.
GNU Octave is freely redistributable software. You may redistribute it and/or modify
it under the terms of the GNU General Public License as published by the Free Software
Foundation. The GPL is included in this manual, see Appendix G [Copying], page 963.
This manual provides comprehensive documentation on how to install, run, use, and
extend GNU Octave. Additional chapters describe how to report bugs and help contribute
code.
This document corresponds to Octave version 4.2.2.

1.1 Running Octave
On most systems, Octave is started with the shell command ‘octave’. This starts the
graphical user interface. The central window in the GUI is the Octave command-line interface. In this window Octave displays an initial message and then a prompt indicating it is
ready to accept input. If you have chosen the traditional command-line interface then only
the command prompt appears in the same window that was running a shell. In either case,
you can immediately begin typing Octave commands.
If you get into trouble, you can usually interrupt Octave by typing Control-C (written
C-c for short). C-c gets its name from the fact that you type it by holding down CTRL and
then pressing c. Doing this will normally return you to Octave’s prompt.
To exit Octave, type quit or exit at the Octave prompt.
On systems that support job control, you can suspend Octave by sending it a SIGTSTP
signal, usually by typing C-z.

1.2 Simple Examples
The following chapters describe all of Octave’s features in detail, but before doing that, it
might be helpful to give a sampling of some of its capabilities.
If you are new to Octave, we recommend that you try these examples to begin learning
Octave by using it. Lines marked like so, ‘octave:13>’, are lines you type, ending each
with a carriage return. Octave will respond with an answer, or by displaying a graph.

1.2.1 Elementary Calculations
Octave can easily be used for basic numerical calculations. Octave knows about arithmetic
operations (+,-,*,/), exponentiation (^), natural logarithms/exponents (log, exp), and the
trigonometric functions (sin, cos, . . . ). Moreover, Octave calculations work on real or
imaginary numbers (i,j). In addition, some mathematical constants such as the base of

8

GNU Octave

the natural logarithm (e) and the ratio of a circle’s circumference to its diameter (pi) are
pre-defined.
For example, to verify Euler’s Identity,
eıπ = −1
type the following which will evaluate to -1 within the tolerance of the calculation.
octave:1> exp (i*pi)

1.2.2 Creating a Matrix
Vectors and matrices are the basic building blocks for numerical analysis. To create a new
matrix and store it in a variable so that you can refer to it later, type the command
octave:1> A = [ 1, 1, 2; 3, 5, 8; 13, 21, 34 ]
Octave will respond by printing the matrix in neatly aligned columns. Octave uses a comma
or space to separate entries in a row, and a semicolon or carriage return to separate one row
from the next. Ending a command with a semicolon tells Octave not to print the result of
the command. For example,
octave:2> B = rand (3, 2);
will create a 3 row, 2 column matrix with each element set to a random value between zero
and one.
To display the value of a variable, simply type the name of the variable at the prompt.
For example, to display the value stored in the matrix B, type the command
octave:3> B

1.2.3 Matrix Arithmetic
Octave uses standard mathematical notation with the advantage over low-level languages
that operators may act on scalars, vector, matrices, or N-dimensional arrays. For example,
to multiply the matrix A by a scalar value, type the command
octave:4> 2 * A
To multiply the two matrices A and B, type the command
octave:5> A * B
and to form the matrix product AT A, type the command
octave:6> A’ * A

1.2.4 Solving Systems of Linear Equations
Systems of linear equations are ubiquitous in numerical analysis. To solve the set of linear
equations Ax = b, use the left division operator, ‘\’:
x = A \ b
This is conceptually equivalent to A−1 b, but avoids computing the inverse of a matrix
directly.
If the coefficient matrix is singular, Octave will print a warning message and compute a
minimum norm solution.

Chapter 1: A Brief Introduction to Octave

9

A simple example comes from chemistry and the need to obtain balanced chemical
equations. Consider the burning of hydrogen and oxygen to produce water.
H2 + O 2 → H2 O
The equation above is not accurate. The Law of Conservation of Mass requires that the number of molecules of each type balance on the left- and right-hand sides of the equation. Writing the variable overall reaction with individual equations for hydrogen and oxygen one finds:
x1 H2 + x2 O2 → H2 O
H:

2x1 + 0x2 → 2

O:

0x1 + 2x2 → 1

The solution in Octave is found in just three steps.
octave:1> A = [ 2, 0; 0, 2 ];
octave:2> b = [ 2; 1 ];
octave:3> x = A \ b

1.2.5 Integrating Differential Equations
Octave has built-in functions for solving nonlinear differential equations of the form
dx
= f (x, t),
x(t = t0 ) = x0
dt
For Octave to integrate equations of this form, you must first provide a definition of the
function f (x, t). This is straightforward, and may be accomplished by entering the function
body directly on the command line. For example, the following commands define the righthand side function for an interesting pair of nonlinear differential equations. Note that
while you are entering a function, Octave responds with a different prompt, to indicate that
it is waiting for you to complete your input.
octave:1> function xdot = f (x, t)
>
> r = 0.25;
> k = 1.4;
> a = 1.5;
> b = 0.16;
> c = 0.9;
> d = 0.8;
>
> xdot(1) = r*x(1)*(1 - x(1)/k) - a*x(1)*x(2)/(1 + b*x(1));
> xdot(2) = c*a*x(1)*x(2)/(1 + b*x(1)) - d*x(2);
>
> endfunction
Given the initial condition
octave:2> x0 = [1; 2];
and the set of output times as a column vector (note that the first output time corresponds
to the initial condition given above)
octave:3> t = linspace (0, 50, 200)’;

10

GNU Octave

it is easy to integrate the set of differential equations:
octave:4> x = lsode ("f", x0, t);
The function lsode uses the Livermore Solver for Ordinary Differential Equations, described
in A. C. Hindmarsh, ODEPACK, a Systematized Collection of ODE Solvers, in: Scientific
Computing, R. S. Stepleman et al. (Eds.), North-Holland, Amsterdam, 1983, pages 55–64.

1.2.6 Producing Graphical Output
To display the solution of the previous example graphically, use the command
octave:1> plot (t, x)
Octave will automatically create a separate window to display the plot.
To save a plot once it has been displayed on the screen, use the print command. For
example,
print -dpdf foo.pdf
will create a file called foo.pdf that contains a rendering of the current plot in Portable
Document Format. The command
help print
explains more options for the print command and provides a list of additional output file
formats.

1.2.7 Help and Documentation
Octave has an extensive help facility. The same documentation that is available in printed
form is also available from the Octave prompt, because both forms of the documentation
are created from the same input file.
In order to get good help you first need to know the name of the command that you want
to use. The name of this function may not always be obvious, but a good place to start is to
type help --list. This will show you all the operators, keywords, built-in functions, and
loadable functions available in the current session of Octave. An alternative is to search
the documentation using the lookfor function (described in Section 2.3 [Getting Help],
page 20).
Once you know the name of the function you wish to use, you can get more help on the
function by simply including the name as an argument to help. For example,
help plot
will display the help text for the plot function.
Octave sends output that is too long to fit on one screen through a pager like less or
more. Type a RET to advance one line, a SPC to advance one page, and q to quit the pager.
The part of Octave’s help facility that allows you to read the complete text of the printed
manual from within Octave normally uses a separate program called Info. When you invoke
Info you will be put into a menu driven program that contains the entire Octave manual.
Help for using Info is provided in this manual, see Section 2.3 [Getting Help], page 20.

Chapter 1: A Brief Introduction to Octave

11

1.2.8 Editing What You Have Typed
At the Octave prompt, you can recall, edit, and reissue previous commands using Emacsor vi-style editing commands. The default keybindings use Emacs-style commands. For
example, to recall the previous command, press Control-p (written C-p for short). Doing
this will normally bring back the previous line of input. C-n will bring up the next line of
input, C-b will move the cursor backward on the line, C-f will move the cursor forward on
the line, etc.
A complete description of the command line editing capability is given in this manual,
see Section 2.4 [Command Line Editing], page 25.

1.3 Conventions
This section explains the notation conventions that are used in this manual. You may want
to skip this section and refer back to it later.

1.3.1 Fonts
Examples of Octave code appear in this font or form: svd (a). Names that represent
variables or function arguments appear in this font or form: first-number. Commands
that you type at the shell prompt appear in this font or form: ‘octave --no-init-file’.
Commands that you type at the Octave prompt sometimes appear in this font or form:
foo --bar --baz. Specific keys on your keyboard appear in this font or form: RET.

1.3.2 Evaluation Notation
In the examples in this manual, results from expressions that you evaluate are indicated
with ‘⇒’. For example:
sqrt (2)
⇒ 1.4142

You can read this as “sqrt (2) evaluates to 1.4142”.
In some cases, matrix values that are returned by expressions are displayed like this
[1, 2; 3, 4] == [1, 3; 2, 4]
⇒ [ 1, 0; 0, 1 ]

and in other cases, they are displayed like this
eye (3)
⇒

1
0
0

0
1
0

0
0
1

in order to clearly show the structure of the result.
Sometimes to help describe one expression, another expression is shown that produces
identical results. The exact equivalence of expressions is indicated with ‘ ≡ ’. For example:
rot90 ([1, 2; 3, 4], -1)
≡
rot90 ([1, 2; 3, 4], 3)
≡
rot90 ([1, 2; 3, 4], 7)

12

GNU Octave

1.3.3 Printing Notation
Many of the examples in this manual print text when they are evaluated. In this manual
the printed text resulting from an example is indicated by ‘ a ’. The value that is returned
by evaluating the expression is displayed with ‘⇒’ (1 in the next example) and follows on
a separate line.

printf ("foo %s\n", "bar")
a foo bar
⇒ 1

1.3.4 Error Messages
Some examples signal errors. This normally displays an error message on your terminal.
Error messages are shown on a line beginning with error:.

fieldnames ([1, 2; 3, 4])
error: fieldnames: Invalid input argument

1.3.5 Format of Descriptions
Functions and commands are described in this manual in a uniform format. The first line
of a description contains the name of the item followed by its arguments, if any. If there
are multiple ways to invoke the function then each allowable form is listed.
The description follows on succeeding lines, sometimes with examples.

1.3.5.1 A Sample Function Description
In a function description, the name of the function being described appears first. It is
followed on the same line by a list of parameters. The names used for the parameters are
also used in the body of the description.
After all of the calling forms have been enumerated, the next line is a concise one-sentence
summary of the function.
After the summary there may be documentation on the inputs and outputs, examples
of function usage, notes about the algorithm used, and references to related functions.
Here is a description of an imaginary function foo:

Chapter 1: A Brief Introduction to Octave

13

foo (x)
foo (x, y)
foo (x, y, . . . )
Subtract x from y, then add any remaining arguments to the result.
The input x must be a numeric scalar, vector, or array.
The optional input y defaults to 19 if it is not supplied.
Example:
foo (1, [3, 5], 3, 9)
⇒ [ 14, 16 ]
foo (5)
⇒ 14
More generally,
foo (w, x, y, ...)
≡
x - w + y + ...
See also: bar

Any parameter whose name contains the name of a type (e.g., integer or matrix) is
expected to be of that type. Parameters named object may be of any type. Parameters
with other sorts of names (e.g., new file) are discussed specifically in the description of
the function. In some sections, features common to parameters of several functions are
described at the beginning.

1.3.5.2 A Sample Command Description
Commands are functions that may be called without surrounding their arguments in parentheses. Command descriptions have a format similar to function descriptions. For example,
here is the description for Octave’s diary command:

14

GNU Octave

diary
diary on
diary off
diary filename
Record a list of all commands and the output they produce, mixed together just as
they appear on the terminal.
Valid options are:
on

Start recording a session in a file called diary in the current working
directory.

off

Stop recording the session in the diary file.

filename

Record the session in the file named filename.

With no arguments, diary toggles the current diary state.
See also: history, evalc.

15

2 Getting Started
This chapter explains some of Octave’s basic features, including how to start an Octave session, get help at the command prompt, edit the command line, and write Octave programs
that can be executed as commands from your shell.

2.1 Invoking Octave from the Command Line
Normally, Octave is used interactively by running the program ‘octave’ without any arguments. Once started, Octave reads commands from the terminal until you tell it to
exit.
You can also specify the name of a file on the command line, and Octave will read and
execute the commands from the named file and then exit when it is finished.
You can further control how Octave starts by using the command-line options described
in the next section, and Octave itself can remind you of the options available. Type ‘octave
--help’ to display all available options and briefly describe their use (‘octave -h’ is a shorter
equivalent).

2.1.1 Command Line Options
Here is a complete list of the command line options that Octave accepts.
--built-in-docstrings-file filename
Specify the name of the file containing documentation strings for the built-in
functions of Octave. This value is normally correct and should only need to
specified in extraordinary situations.
--debug
-d

Enter parser debugging mode. Using this option will cause Octave’s parser to
print a lot of information about the commands it reads, and is probably only
useful if you are actually trying to debug the parser.

--debug-jit
Enable JIT compiler debugging and tracing.
--doc-cache-file filename
Specify the name of the doc cache file to use. The value of filename specified
on the command line will override any value of OCTAVE_DOC_CACHE_FILE found
in the environment, but not any commands in the system or user startup files
that use the doc_cache_file function.
--echo-commands
-x
Echo commands as they are executed.
--eval code
Evaluate code and exit when finished unless --persist is also specified.
--exec-path path
Specify the path to search for programs to run. The value of path specified on
the command line will override any value of OCTAVE_EXEC_PATH found in the
environment, but not any commands in the system or user startup files that set
the built-in variable EXEC_PATH.

16

GNU Octave

--force-gui
Force the graphical user interface (GUI) to start.
--help
-h
-?

Print short help message and exit.

--image-path path
Add path to the head of the search path for images. The value of path specified
on the command line will override any value of OCTAVE_IMAGE_PATH found in
the environment, but not any commands in the system or user startup files that
set the built-in variable IMAGE_PATH.
--info-file filename
Specify the name of the info file to use. The value of filename specified on
the command line will override any value of OCTAVE_INFO_FILE found in the
environment, but not any commands in the system or user startup files that
use the info_file function.
--info-program program
Specify the name of the info program to use. The value of program specified
on the command line will override any value of OCTAVE_INFO_PROGRAM found
in the environment, but not any commands in the system or user startup files
that use the info_program function.
--interactive
-i
Force interactive behavior. This can be useful for running Octave via a remote
shell command or inside an Emacs shell buffer.
--jit-compiler
Enable the JIT compiler used for accelerating loops.
--line-editing
Force readline use for command-line editing.
--no-gui

Disable the graphical user interface (GUI) and use the command line interface
(CLI) instead.

--no-history
-H
Disable recording of command-line history.
--no-init-file
Don’t read the initialization files ~/.octaverc and .octaverc.
--no-init-path
Don’t initialize the search path for function files to include default locations.
--no-line-editing
Disable command-line editing.
--no-site-file
Don’t read the site-wide octaverc initialization files.
--no-window-system
-W
Disable use of a windowing system including graphics. This forces a strictly
terminal-only environment.

Chapter 2: Getting Started

--norc
-f

17

Don’t read any of the system or user initialization files at startup. This is
equivalent to using both of the options --no-init-file and --no-site-file.

--path path
-p path
Add path to the head of the search path for function files. The value of path
specified on the command line will override any value of OCTAVE_PATH found
in the environment, but not any commands in the system or user startup files
that set the internal load path through one of the path functions.
--persist
Go to interactive mode after --eval or reading from a file named on the command line.
--silent
--quiet
-q

Don’t print the usual greeting and version message at startup.

--texi-macros-file filename
Specify the name of the file containing Texinfo macros for use by makeinfo.
--traditional
--braindead
For compatibility with matlab, set initial values for user preferences to the
following values
PS1
PS2
beep_on_error
confirm_recursive_rmdir
crash_dumps_octave_core
disable_diagonal_matrix
disable_permutation_matrix
disable_range
fixed_point_format
history_timestamp_format_string
page_screen_output
print_empty_dimensions
save_default_options
struct_levels_to_print

=
=
=
=
=
=
=
=
=
=
=
=
=
=

">> "
""
true
false
false
true
true
true
true
"%%-- %D %I:%M %p --%%"
false
false
"-mat-binary"
0

and disable the following warnings
Octave:abbreviated-property-match
Octave:fopen-file-in-path
Octave:function-name-clash
Octave:load-file-in-path
Octave:possible-matlab-short-circuit-operator
Note that this does not enable the Octave:language-extension warning,
which you might want if you want to be told about writing code that works in
Octave but not matlab (see [warning], page 224, [warning ids], page 225).

18

GNU Octave

--verbose
-V
Turn on verbose output.
--version
-v
Print the program version number and exit.
Execute commands from file. Exit when done unless --persist is also specified.

file

Octave also includes several functions which return information about the command line,
including the number of arguments and all of the options.

argv ()
Return the command line arguments passed to Octave.
For example, if you invoked Octave using the command
octave --no-line-editing --silent
argv would return a cell array of strings with the elements --no-line-editing and
--silent.
If you write an executable Octave script, argv will return the list of arguments passed
to the script. See Section 2.6 [Executable Octave Programs], page 36, for an example
of how to create an executable Octave script.

program_name ()
Return the last component of the value returned by program_invocation_name.
See also: [program invocation name], page 18.

program_invocation_name ()
Return the name that was typed at the shell prompt to run Octave.
If executing a script from the command line (e.g., octave foo.m) or using an executable Octave script, the program name is set to the name of the script. See
Section 2.6 [Executable Octave Programs], page 36, for an example of how to create
an executable Octave script.
See also: [program name], page 18.
Here is an example of using these functions to reproduce the command line which invoked
Octave.
printf ("%s", program_name ());
arg_list = argv ();
for i = 1:nargin
printf (" %s", arg_list{i});
endfor
printf ("\n");
See Section 6.2.3 [Indexing Cell Arrays], page 118, for an explanation of how to retrieve
objects from cell arrays, and Section 11.2 [Defining Functions], page 175, for information
about the variable nargin.

Chapter 2: Getting Started

19

2.1.2 Startup Files
When Octave starts, it looks for commands to execute from the files in the following list.
These files may contain any valid Octave commands, including function definitions.
octave-home/share/octave/site/m/startup/octaverc
where octave-home is the directory in which Octave is installed (the default
is /usr/local). This file is provided so that changes to the default Octave
environment can be made globally for all users at your site for all versions of
Octave you have installed. Care should be taken when making changes to this
file since all users of Octave at your site will be affected. The default file may
be overridden by the environment variable OCTAVE_SITE_INITFILE.
octave-home/share/octave/version/m/startup/octaverc
where octave-home is the directory in which Octave is installed (the default
is /usr/local), and version is the version number of Octave. This file is provided so that changes to the default Octave environment can be made globally for all users of a particular version of Octave. Care should be taken
when making changes to this file since all users of Octave at your site will
be affected. The default file may be overridden by the environment variable
OCTAVE_VERSION_INITFILE.
~/.octaverc
This file is used to make personal changes to the default Octave environment.
.octaverc
This file can be used to make changes to the default Octave environment for a
particular project. Octave searches for this file in the current directory after it
reads ~/.octaverc. Any use of the cd command in the ~/.octaverc file will
affect the directory where Octave searches for .octaverc.
If you start Octave in your home directory, commands from the file
~/.octaverc will only be executed once.
startup.m
This file is used to make personal changes to the default Octave environment. It
is executed for matlab compatibility, but ~/.octaverc is the preferred location
for configuration changes.
A message will be displayed as each of the startup files is read if you invoke Octave with
the --verbose option but without the --silent option.

2.2 Quitting Octave
Shutdown is initiated with the exit or quit commands (they are equivalent). Similar
to startup, Octave has a shutdown process that can be customized by user script files.
During shutdown Octave will search for the script file finish.m in the function load path.
Commands to save all workspace variables or cleanup temporary files may be placed there.
Additional functions to execute on shutdown may be registered with atexit.

exit
exit (status)

20

GNU Octave

quit
quit (status)
Exit the current Octave session.
If the optional integer value status is supplied, pass that value to the operating system
as Octave’s exit status. The default value is zero.
When exiting, Octave will attempt to run the m-file finish.m if it exists. User
commands to save the workspace or clean up temporary files may be placed in that
file. Alternatively, another m-file may be scheduled to run using atexit.
See also: [atexit], page 20.

atexit (fcn)
atexit (fcn, flag)
Register a function to be called when Octave exits.
For example,
function last_words ()
disp ("Bye bye");
endfunction
atexit ("last_words");
will print the message "Bye bye" when Octave exits.
The additional argument flag will register or unregister fcn from the list of functions
to be called when Octave exits. If flag is true, the function is registered, and if flag
is false, it is unregistered. For example, after registering the function last_words
above,
atexit ("last_words", false);
will remove the function from the list and Octave will not call last_words when it
exits.
Note that atexit only removes the first occurrence of a function from the list, so if a
function was placed in the list multiple times with atexit, it must also be removed
from the list multiple times.
See also: [quit], page 19.

2.3 Commands for Getting Help
The entire text of this manual is available from the Octave prompt via the command doc.
In addition, the documentation for individual user-written functions and variables is also
available via the help command. This section describes the commands used for reading
the manual and the documentation strings for user-supplied functions and variables. See
Section 11.9 [Function Files], page 191, for more information about how to document the
functions you write.

help name
help –list
help .
help
Display the help text for name.

Chapter 2: Getting Started

21

For example, the command help help prints a short message describing the help
command.
Given the single argument --list, list all operators, keywords, built-in functions,
and loadable functions available in the current session of Octave.
Given the single argument ., list all operators available in the current session of
Octave.
If invoked without any arguments, help displays instructions on how to access help
from the command line.
The help command can provide information about most operators, but name must
be enclosed by single or double quotes to prevent the Octave interpreter from acting
on name. For example, help "+" displays help on the addition operator.
See also: [doc], page 21, [lookfor], page 21, [which], page 134, [info], page 22.

doc function_name
doc
Display documentation for the function function name directly from an online version
of the printed manual, using the GNU Info browser.
If invoked without an argument, the manual is shown from the beginning.
For example, the command doc rand starts the GNU Info browser at the rand node
in the online version of the manual.
Once the GNU Info browser is running, help for using it is available using the command C-h.
See also: [help], page 20.

lookfor str
lookfor -all str
[fcn, help1str] = lookfor (str)
[fcn, help1str] = lookfor ("-all", str)
Search for the string str in the documentation of all functions in the current function
search path.
By default, lookfor looks for str in just the first sentence of the help string for each
function found. The entire help text of each function can be searched by using the
"-all" argument. All searches are case insensitive.
When called with no output arguments, lookfor prints the list of matching functions
to the terminal. Otherwise, the output argument fcns contains the function names
and help1str contains the first sentence from the help string of each function.
Programming Note: The ability of lookfor to correctly identify the first sentence
of the help text is dependent on the format of the function’s help. All Octave core
functions are correctly formatted, but the same can not be guaranteed for external
packages and user-supplied functions. Therefore, the use of the "-all" argument
may be necessary to find related functions that are not a part of Octave.
The speed of lookup is greatly enhanced by having a cached documentation file. See
doc_cache_create for more information.
See also: [help], page 20, [doc], page 21, [which], page 134, [path], page 195,
[doc cache create], page 24.

22

GNU Octave

To see what is new in the current release of Octave, use the news function.

news
news package
Display the current NEWS file for Octave or an installed package.
When called without an argument, display the NEWS file for Octave.
When given a package name package, display the current NEWS file for that package.
See also: [ver], page 860, [pkg], page 865.

info ()
Display contact information for the GNU Octave community.

warranty ()
Describe the conditions for copying and distributing Octave.
The following functions can be used to change which programs are used for displaying
the documentation, and where the documentation can be found.

val = info_file ()
old_val = info_file (new_val)
info_file (new_val, "local")
Query or set the internal variable that specifies the name of the Octave info file.
The default value is octave-home/info/octave.info, in which octave-home is the
root directory of the Octave installation. The default value may be overridden by the
environment variable OCTAVE_INFO_FILE, or the command line argument --infofile FNAME.
When called from inside a function with the "local" option, the variable is changed
locally for the function and any subroutines it calls. The original variable value is
restored when exiting the function.
See also: [info program], page 22, [doc], page 21, [help], page 20, [makeinfo program],
page 23.

val = info_program ()
old_val = info_program (new_val)
info_program (new_val, "local")
Query or set the internal variable that specifies the name of the info program to run.
The default value is octave-home/libexec/octave/version/exec/arch/info
in which octave-home is the root directory of the Octave installation, version
is the Octave version number, and arch is the system type (for example,
i686-pc-linux-gnu). The default value may be overridden by the environment
variable OCTAVE_INFO_PROGRAM, or the command line argument --info-program
NAME.
When called from inside a function with the "local" option, the variable is changed
locally for the function and any subroutines it calls. The original variable value is
restored when exiting the function.
See also: [info file], page 22, [doc], page 21, [help], page 20, [makeinfo program],
page 23.

Chapter 2: Getting Started

23

val = makeinfo_program ()
old_val = makeinfo_program (new_val)
makeinfo_program (new_val, "local")
Query or set the internal variable that specifies the name of the program that Octave
runs to format help text containing Texinfo markup commands.
The default value is makeinfo.
When called from inside a function with the "local" option, the variable is changed
locally for the function and any subroutines it calls. The original variable value is
restored when exiting the function.
See also: [texi macros file], page 23, [info file], page 22, [info program], page 22, [doc],
page 21, [help], page 20.

val = texi_macros_file ()
old_val = texi_macros_file (new_val)
texi_macros_file (new_val, "local")
Query or set the internal variable that specifies the name of the file containing Texinfo macros that are prepended to documentation strings before they are passed to
makeinfo.
The default value is octave-home/share/octave/version/etc/macros.texi, in
which octave-home is the root directory of the Octave installation, and version
is the Octave version number. The default value may be overridden by the
environment variable OCTAVE_TEXI_MACROS_FILE, or the command line argument
--texi-macros-file FNAME.
When called from inside a function with the "local" option, the variable is changed
locally for the function and any subroutines it calls. The original variable value is
restored when exiting the function.
See also: [makeinfo program], page 23.

val = doc_cache_file ()
old_val = doc_cache_file (new_val)
doc_cache_file (new_val, "local")
Query or set the internal variable that specifies the name of the Octave documentation
cache file.
A cache file significantly improves the performance of the lookfor command. The
default value is octave-home/share/octave/version/etc/doc-cache, in which
octave-home is the root directory of the Octave installation, and version is the Octave
version number. The default value may be overridden by the environment variable
OCTAVE_DOC_CACHE_FILE, or the command line argument --doc-cache-file FNAME.
When called from inside a function with the "local" option, the variable is changed
locally for the function and any subroutines it calls. The original variable value is
restored when exiting the function.
See also: [doc cache create], page 24, [lookfor], page 21, [info program], page 22,
[doc], page 21, [help], page 20, [makeinfo program], page 23.
See also: [lookfor], page 21.

24

GNU Octave

val = built_in_docstrings_file ()
old_val = built_in_docstrings_file (new_val)
built_in_docstrings_file (new_val, "local")
Query or set the internal variable that specifies the name of the file containing docstrings for built-in Octave functions.
The default value is octave-home/share/octave/version/etc/built-indocstrings, in which octave-home is the root directory of the Octave installation,
and version is the Octave version number. The default value may be overridden by
the environment variable OCTAVE_BUILT_IN_DOCSTRINGS_FILE, or the command
line argument --built-in-docstrings-file FNAME.
Note: This variable is only used when Octave is initializing itself. Modifying it during
a running session of Octave will have no effect.

val = suppress_verbose_help_message ()
old_val = suppress_verbose_help_message (new_val)
suppress_verbose_help_message (new_val, "local")
Query or set the internal variable that controls whether Octave will add additional
help information to the end of the output from the help command and usage messages
for built-in commands.
When called from inside a function with the "local" option, the variable is changed
locally for the function and any subroutines it calls. The original variable value is
restored when exiting the function.
The following functions are principally used internally by Octave for generating the documentation. They are documented here for completeness and because they may occasionally
be useful for users.

doc_cache_create (out_file, directory)
doc_cache_create (out_file)
doc_cache_create ()
Generate documentation cache for all functions in directory.
A documentation cache is generated for all functions in directory which may be a
single string or a cell array of strings. The cache is used to speed up the function
lookfor.
The cache is saved in the file out file which defaults to the value doc-cache if not
given.
If no directory is given (or it is the empty matrix), a cache for built-in functions,
operators, and keywords is generated.
See also: [doc cache file], page 23, [lookfor], page 21, [path], page 195.

[text, format] = get_help_text (name)
Return the raw help text of function name.
The raw help text is returned in text and the format in format The format is a string
which is one of "texinfo", "html", or "plain text".
See also: [get help text from file], page 25.

Chapter 2: Getting Started

25

[text, format] = get_help_text_from_file (fname)
Return the raw help text from the file fname.
The raw help text is returned in text and the format in format The format is a string
which is one of "texinfo", "html", or "plain text".
See also: [get help text], page 24.

text = get_first_help_sentence (name)
text = get_first_help_sentence (name, max_len)
[text, status] = get_first_help_sentence ( . . . )
Return the first sentence of a function’s help text.
The first sentence is defined as the text after the function declaration until either the
first period (".") or the first appearance of two consecutive newlines ("\n\n"). The
text is truncated to a maximum length of max len, which defaults to 80.
The optional output argument status returns the status reported by makeinfo. If
only one output argument is requested, and status is nonzero, a warning is displayed.
As an example, the first sentence of this help text is
get_first_help_sentence ("get_first_help_sentence")
a ans = Return the first sentence of a function’s help text.

2.4 Command Line Editing
Octave uses the GNU Readline library to provide an extensive set of command-line editing
and history features. Only the most common features are described in this manual. In
addition, all of the editing functions can be bound to different key strokes at the user’s
discretion. This manual assumes no changes from the default Emacs bindings. See the
GNU Readline Library manual for more information on customizing Readline and for a
complete feature list.
To insert printing characters (letters, digits, symbols, etc.), simply type the character.
Octave will insert the character at the cursor and advance the cursor forward.
Many of the command-line editing functions operate using control characters. For example, the character Control-a moves the cursor to the beginning of the line. To type
C-a, hold down CTRL and then press a. In the following sections, control characters such as
Control-a are written as C-a.
Another set of command-line editing functions use Meta characters. To type M-u, hold
down the META key and press u. Depending on the keyboard, the META key may be labeled
ALT or even WINDOWS. If your terminal does not have a META key, you can still type Meta
characters using two-character sequences starting with ESC. Thus, to enter M-u, you would
type ESC u. The ESC character sequences are also allowed on terminals with real Meta keys.
In the following sections, Meta characters such as Meta-u are written as M-u.

2.4.1 Cursor Motion
The following commands allow you to position the cursor.
C-b

Move back one character.

C-f

Move forward one character.

26

GNU Octave

BACKSPACE
Delete the character to the left of the cursor.
DEL

Delete the character underneath the cursor.

C-d

Delete the character underneath the cursor.

M-f

Move forward a word.

M-b

Move backward a word.

C-a

Move to the start of the line.

C-e

Move to the end of the line.

C-l

Clear the screen, reprinting the current line at the top.

C-_
C-/

Undo the last action. You can undo all the way back to an empty line.
Undo all changes made to this line. This is like typing the ‘undo’ command
enough times to get back to the beginning.

M-r

The above table describes the most basic possible keystrokes that you need in order to
do editing of the input line. On most terminals, you can also use the left and right arrow
keys in place of C-f and C-b to move forward and backward.
Notice how C-f moves forward a character, while M-f moves forward a word. It is a loose
convention that control keystrokes operate on characters while meta keystrokes operate on
words.
The function clc will allow you to clear the screen from within Octave programs.

clc ()
home ()
Clear the terminal screen and move the cursor to the upper left corner.

2.4.2 Killing and Yanking
Killing text means to delete the text from the line, but to save it away for later use, usually
by yanking it back into the line. If the description for a command says that it ‘kills’ text,
then you can be sure that you can get the text back in a different (or the same) place later.
Here is the list of commands for killing text.
C-k

Kill the text from the current cursor position to the end of the line.

M-d

Kill from the cursor to the end of the current word, or if between words, to the
end of the next word.

M-DEL

Kill from the cursor to the start of the previous word, or if between words, to
the start of the previous word.

C-w

Kill from the cursor to the previous whitespace. This is different than M-DEL
because the word boundaries differ.

And, here is how to yank the text back into the line. Yanking means to copy the
most-recently-killed text from the kill buffer.
C-y

Yank the most recently killed text back into the buffer at the cursor.

Chapter 2: Getting Started

M-y

27

Rotate the kill-ring, and yank the new top. You can only do this if the prior
command is C-y or M-y.

When you use a kill command, the text is saved in a kill-ring. Any number of consecutive
kills save all of the killed text together, so that when you yank it back, you get it in one
clean sweep. The kill ring is not line specific; the text that you killed on a previously typed
line is available to be yanked back later, when you are typing another line.

2.4.3 Commands for Changing Text
The following commands can be used for entering characters that would otherwise have a
special meaning (e.g., TAB, C-q, etc.), or for quickly correcting typing mistakes.
C-q
C-v

Add the next character that you type to the line verbatim. This is how to insert
things like C-q for example.

M-TAB

Insert a tab character.

C-t

Drag the character before the cursor forward over the character at the cursor,
also moving the cursor forward. If the cursor is at the end of the line, then
transpose the two characters before it.

M-t

Drag the word behind the cursor past the word in front of the cursor moving
the cursor over that word as well.

M-u

Uppercase the characters following the cursor to the end of the current (or
following) word, moving the cursor to the end of the word.

M-l

Lowercase the characters following the cursor to the end of the current (or
following) word, moving the cursor to the end of the word.

M-c

Uppercase the character following the cursor (or the beginning of the next word
if the cursor is between words), moving the cursor to the end of the word.

2.4.4 Letting Readline Type for You
The following commands allow Octave to complete command and variable names for you.
TAB

Attempt to do completion on the text before the cursor. Octave can complete
the names of commands and variables.

M-?

List the possible completions of the text before the cursor.

val = completion_append_char ()
old_val = completion_append_char (new_val)
completion_append_char (new_val, "local")
Query or set the internal character variable that is appended to successful commandline completion attempts.
The default value is " " (a single space).
When called from inside a function with the "local" option, the variable is changed
locally for the function and any subroutines it calls. The original variable value is
restored when exiting the function.

28

GNU Octave

completion_matches (hint)
Generate possible completions given hint.
This function is provided for the benefit of programs like Emacs which might be
controlling Octave and handling user input. The current command number is not
incremented when this function is called. This is a feature, not a bug.

2.4.5 Commands for Manipulating the History
Octave normally keeps track of the commands you type so that you can recall previous
commands to edit or execute them again. When you exit Octave, the most recent commands
you have typed, up to the number specified by the variable history_size, are saved in a
file. When Octave starts, it loads an initial list of commands from the file named by the
variable history_file.
Here are the commands for simple browsing and searching the history list.
LFD
RET

Accept the current line regardless of where the cursor is. If the line is nonempty, add it to the history list. If the line was a history line, then restore the
history line to its original state.

C-p

Move ‘up’ through the history list.

C-n

Move ‘down’ through the history list.

M-<

Move to the first line in the history.

M->

Move to the end of the input history, i.e., the line you are entering!

C-r

Search backward starting at the current line and moving ‘up’ through the history as necessary. This is an incremental search.

C-s

Search forward starting at the current line and moving ‘down’ through the
history as necessary.

On most terminals, you can also use the up and down arrow keys in place of C-p and
C-n to move through the history list.
In addition to the keyboard commands for moving through the history list, Octave
provides three functions for viewing, editing, and re-running chunks of commands from the
history list.

history
history opt1 . . .
h = history ()
h = history (opt1, . . . )
If invoked with no arguments, history displays a list of commands that you have
executed.
Valid options are:
n
-n

Display only the most recent n lines of history.

-c

Clear the history list.

Chapter 2: Getting Started

29

-q

Don’t number the displayed lines of history. This is useful for cutting and
pasting commands using the X Window System.

-r file

Read the file file, appending its contents to the current history list. If the
name is omitted, use the default history file (normally ~/.octave_hist).

-w file

Write the current history to the file file. If the name is omitted, use the
default history file (normally ~/.octave_hist).

For example, to display the five most recent commands that you have typed without
displaying line numbers, use the command history -q 5.
If invoked with a single output argument, the history will be saved to that argument
as a cell string and will not be output to screen.
See also: [edit history], page 29, [run history], page 29.

edit_history
edit_history cmd_number
edit_history first last
Edit the history list using the editor named by the variable EDITOR.
The commands to be edited are first copied to a temporary file. When you exit
the editor, Octave executes the commands that remain in the file. It is often more
convenient to use edit_history to define functions rather than attempting to enter
them directly on the command line. The block of commands is executed as soon as
you exit the editor. To avoid executing any commands, simply delete all the lines
from the buffer before leaving the editor.
When invoked with no arguments, edit the previously executed command; With one
argument, edit the specified command cmd number; With two arguments, edit the
list of commands between first and last. Command number specifiers may also be
negative where -1 refers to the most recently executed command. The following are
equivalent and edit the most recently executed command.
edit_history
edit_history -1
When using ranges, specifying a larger number for the first command than the last
command reverses the list of commands before they are placed in the buffer to be
edited.
See also: [run history], page 29, [history], page 28.

run_history
run_history cmd_number
run_history first last
Run commands from the history list.
When invoked with no arguments, run the previously executed command;
With one argument, run the specified command cmd number;
With two arguments, run the list of commands between first and last. Command
number specifiers may also be negative where -1 refers to the most recently executed
command. For example, the command

30

GNU Octave

run_history
OR
run_history -1
executes the most recent command again. The command
run_history 13 169
executes commands 13 through 169.
Specifying a larger number for the first command than the last command reverses the
list of commands before executing them. For example:
disp (1)
disp (2)
run_history -1 -2
⇒
2
1
See also: [edit history], page 29, [history], page 28.
Octave also allows you customize the details of when, where, and how history is saved.

val = history_save ()
old_val = history_save (new_val)
history_save (new_val, "local")
Query or set the internal variable that controls whether commands entered on the
command line are saved in the history file.
When called from inside a function with the "local" option, the variable is changed
locally for the function and any subroutines it calls. The original variable value is
restored when exiting the function.
See also: [history control], page 30, [history file], page 31, [history size], page 31,
[history timestamp format string], page 31.

val = history_control ()
old_val = history_control (new_val)
Query or set the internal variable that specifies how commands are saved to the
history list.
The default value is an empty character string, but may be overridden by the environment variable OCTAVE_HISTCONTROL.
The value of history_control is a colon-separated list of values controlling how
commands are saved on the history list. If the list of values includes ignorespace,
lines which begin with a space character are not saved in the history list. A value of
ignoredups causes lines matching the previous history entry to not be saved. A value
of ignoreboth is shorthand for ignorespace and ignoredups. A value of erasedups
causes all previous lines matching the current line to be removed from the history list
before that line is saved. Any value not in the above list is ignored. If history_
control is the empty string, all commands are saved on the history list, subject to
the value of history_save.
See also: [history file], page 31, [history size], page 31, [history timestamp format string],
page 31, [history save], page 30.

Chapter 2: Getting Started

31

val = history_file ()
old_val = history_file (new_val)
Query or set the internal variable that specifies the name of the file used to store
command history.
The default value is ~/.octave_hist, but may be overridden by the environment
variable OCTAVE_HISTFILE.
See also: [history size], page 31, [history save], page 30, [history timestamp format string],
page 31.

val = history_size ()
old_val = history_size (new_val)
Query or set the internal variable that specifies how many entries to store in the
history file.
The default value is 1000, but may be overridden by the environment variable
OCTAVE_HISTSIZE.
See also: [history file], page 31, [history timestamp format string], page 31,
[history save], page 30.

val = history_timestamp_format_string ()
old_val = history_timestamp_format_string (new_val)
history_timestamp_format_string (new_val, "local")
Query or set the internal variable that specifies the format string for the comment
line that is written to the history file when Octave exits.
The format string is passed to strftime. The default value is
"# Octave VERSION, %a %b %d %H:%M:%S %Y %Z "
When called from inside a function with the "local" option, the variable is changed
locally for the function and any subroutines it calls. The original variable value is
restored when exiting the function.
See also: [strftime], page 821, [history file], page 31, [history size], page 31,
[history save], page 30.

val = EDITOR ()
old_val = EDITOR (new_val)
EDITOR (new_val, "local")
Query or set the internal variable that specifies the default text editor.
The default value is taken from the environment variable EDITOR when Octave starts.
If the environment variable is not initialized, EDITOR will be set to "emacs".
When called from inside a function with the "local" option, the variable is changed
locally for the function and any subroutines it calls. The original variable value is
restored when exiting the function.
See also: [edit], page 192, [edit history], page 29.

32

GNU Octave

2.4.6 Customizing readline
Octave uses the GNU Readline library for command-line editing and history features. Readline is very flexible and can be modified through a configuration file of commands (See the
GNU Readline library for the exact command syntax). The default configuration file is
normally ~/.inputrc.
Octave provides two commands for initializing Readline and thereby changing the command line behavior.

readline_read_init_file (file)
Read the readline library initialization file file.
If file is omitted, read the default initialization file (normally ~/.inputrc).
See Section “Readline Init File” in GNU Readline Library, for details.
See also: [readline re read init file], page 32.

readline_re_read_init_file ()
Re-read the last readline library initialization file that was read.
See Section “Readline Init File” in GNU Readline Library, for details.
See also: [readline read init file], page 32.

2.4.7 Customizing the Prompt
The following variables are available for customizing the appearance of the command-line
prompts. Octave allows the prompt to be customized by inserting a number of backslashescaped special characters that are decoded as follows:
‘\t’

The time.

‘\d’

The date.

‘\n’

Begins a new line by printing the equivalent of a carriage return followed by a
line feed.

‘\s’

The name of the program (usually just ‘octave’).

‘\w’

The current working directory.

‘\W’

The basename of the current working directory.

‘\u’

The username of the current user.

‘\h’

The hostname, up to the first ‘.’.

‘\H’

The hostname.

‘\#’

The command number of this command, counting from when Octave starts.

‘\!’

The history number of this command. This differs from ‘\#’ by the number of
commands in the history list when Octave starts.

‘\$’

If the effective UID is 0, a ‘#’, otherwise a ‘$’.

‘\nnn’

The character whose character code in octal is nnn.

‘\\’

A backslash.

Chapter 2: Getting Started

33

val = PS1 ()
old_val = PS1 (new_val)
PS1 (new_val, "local")
Query or set the primary prompt string.
When executing interactively, Octave displays the primary prompt when it is ready
to read a command.
The default value of the primary prompt string is ’octave:\#> ’. To change it, use
a command like
PS1 ("\\u@\\H> ")
which will result in the prompt ‘boris@kremvax> ’ for the user ‘boris’ logged in
on the host ‘kremvax.kgb.su’. Note that two backslashes are required to enter a
backslash into a double-quoted character string. See Chapter 5 [Strings], page 67.
You can also use ANSI escape sequences if your terminal supports them. This can be
useful for coloring the prompt. For example,
PS1 (’\[\033[01;31m\]\s:\#> \[\033[0m\]’)
will give the default Octave prompt a red coloring.
When called from inside a function with the "local" option, the variable is changed
locally for the function and any subroutines it calls. The original variable value is
restored when exiting the function.
See also: [PS2], page 33, [PS4], page 33.

val = PS2 ()
old_val = PS2 (new_val)
PS2 (new_val, "local")
Query or set the secondary prompt string.
The secondary prompt is printed when Octave is expecting additional input to complete a command. For example, if you are typing a for loop that spans several lines,
Octave will print the secondary prompt at the beginning of each line after the first.
The default value of the secondary prompt string is "> ".
When called from inside a function with the "local" option, the variable is changed
locally for the function and any subroutines it calls. The original variable value is
restored when exiting the function.
See also: [PS1], page 32, [PS4], page 33.

val = PS4 ()
old_val = PS4 (new_val)
PS4 (new_val, "local")
Query or set the character string used to prefix output produced when echoing commands is enabled.
The default value is "+ ". See Section 2.4.8 [Diary and Echo Commands], page 34,
for a description of echoing commands.
When called from inside a function with the "local" option, the variable is changed
locally for the function and any subroutines it calls. The original variable value is
restored when exiting the function.

34

GNU Octave

See also: [echo], page 34, [echo executing commands], page 34, [PS1], page 32, [PS2],
page 33.

2.4.8 Diary and Echo Commands
Octave’s diary feature allows you to keep a log of all or part of an interactive session by
recording the input you type and the output that Octave produces in a separate file.

diary
diary on
diary off
diary filename
Record a list of all commands and the output they produce, mixed together just as
they appear on the terminal.
Valid options are:
on

Start recording a session in a file called diary in the current working
directory.

off

Stop recording the session in the diary file.

filename

Record the session in the file named filename.

With no arguments, diary toggles the current diary state.
See also: [history], page 28, [evalc], page 159.
Sometimes it is useful to see the commands in a function or script as they are being
evaluated. This can be especially helpful for debugging some kinds of problems.

echo
echo
echo
echo
echo

on
off
on all
off all
Control whether commands are displayed as they are executed.
Valid options are:
on

Enable echoing of commands as they are executed in script files.

off

Disable echoing of commands as they are executed in script files.

on all

Enable echoing of commands as they are executed in script files and
functions.

off all

Disable echoing of commands as they are executed in script files and
functions.

With no arguments, echo toggles the current echo state.
See also: [echo executing commands], page 34.

val = echo_executing_commands ()
old_val = echo_executing_commands (new_val)

Chapter 2: Getting Started

35

echo_executing_commands (new_val, "local")
Query or set the internal variable that controls the echo state.
It may be the sum of the following values:
1

Echo commands read from script files.

2

Echo commands from functions.

4

Echo commands read from command line.

More than one state can be active at once. For example, a value of 3 is equivalent to
the command echo on all.
The value of echo_executing_commands may be set by the echo command or the
command line option --echo-commands.
When called from inside a function with the "local" option, the variable is changed
locally for the function and any subroutines it calls. The original variable value is
restored when exiting the function.
See also: [echo], page 34.

2.5 How Octave Reports Errors
Octave reports two kinds of errors for invalid programs.
A parse error occurs if Octave cannot understand something you have typed. For example, if you misspell a keyword,
octave:13> function y = f (x) y = x***2; endfunction
Octave will respond immediately with a message like this:
parse error:
syntax error
>>> function y = f (x) y = x***2; endfunction
^
For most parse errors, Octave uses a caret (‘^’) to mark the point on the line where it was
unable to make sense of your input. In this case, Octave generated an error message because
the keyword for exponentiation (**) was misspelled. It marked the error at the third ‘*’
because the code leading up to this was correct but the final ‘*’ was not understood.
Another class of error message occurs at evaluation time. These errors are called run-time
errors, or sometimes evaluation errors, because they occur when your program is being run,
or evaluated. For example, if after correcting the mistake in the previous function definition,
you type
octave:13> f ()
Octave will respond with
error: ‘x’ undefined near line 1 column 24
error: called from:
error:
f at line 1, column 22

36

GNU Octave

This error message has several parts, and gives quite a bit of information to help you locate
the source of the error. The messages are generated from the point of the innermost error,
and provide a traceback of enclosing expressions and function calls.
In the example above, the first line indicates that a variable named ‘x’ was found to be
undefined near line 1 and column 24 of some function or expression. For errors occurring
within functions, lines are counted from the beginning of the file containing the function
definition. For errors occurring outside of an enclosing function, the line number indicates
the input line number, which is usually displayed in the primary prompt string.
The second and third lines of the error message indicate that the error occurred within
the function f. If the function f had been called from within another function, for example,
g, the list of errors would have ended with one more line:
error:
g at line 1, column 17
These lists of function calls make it fairly easy to trace the path your program took
before the error occurred, and to correct the error before trying again.

2.6 Executable Octave Programs
Once you have learned Octave, you may want to write self-contained Octave scripts, using
the ‘#!’ script mechanism. You can do this on GNU systems and on many Unix systems1 .
Self-contained Octave scripts are useful when you want to write a program which users
can invoke without knowing that the program is written in the Octave language. Octave
scripts are also used for batch processing of data files. Once an algorithm has been developed
and tested in the interactive portion of Octave, it can be committed to an executable script
and used again and again on new data files.
As a trivial example of an executable Octave script, you might create a text file named
hello, containing the following lines:
#! octave-interpreter-name -qf
# a sample Octave program
printf ("Hello, world!\n");
(where octave-interpreter-name should be replaced with the full path and name of your
Octave binary). Note that this will only work if ‘#!’ appears at the very beginning of the
file. After making the file executable (with the chmod command on Unix systems), you can
simply type:
hello
at the shell, and the system will arrange to run Octave as if you had typed:
octave hello
The line beginning with ‘#!’ lists the full path and filename of an interpreter to be run,
and an optional initial command line argument to pass to that interpreter. The operating
system then runs the interpreter with the given argument and the full argument list of
the executed program. The first argument in the list is the full filename of the Octave
executable. The rest of the argument list will either be options to Octave, or data files, or
both. The ‘-qf’ options are usually specified in stand-alone Octave programs to prevent
1

The ‘#!’ mechanism works on Unix systems derived from Berkeley Unix, System V Release 4, and some
System V Release 3 systems.

Chapter 2: Getting Started

37

them from printing the normal startup message, and to keep them from behaving differently
depending on the contents of a particular user’s ~/.octaverc file. See Section 2.1 [Invoking
Octave from the Command Line], page 15.
Note that some operating systems may place a limit on the number of characters that
are recognized after ‘#!’. Also, the arguments appearing in a ‘#!’ line are parsed differently
by various shells/systems. The majority of them group all the arguments together in one
string and pass it to the interpreter as a single argument. In this case, the following script:
#! octave-interpreter-name -q -f # comment
is equivalent to typing at the command line:
octave "-q -f # comment"
which will produce an error message. Unfortunately, it is not possible for Octave to determine whether it has been called from the command line or from a ‘#!’ script, so some care
is needed when using the ‘#!’ mechanism.
Note that when Octave is started from an executable script, the built-in function argv
returns a cell array containing the command line arguments passed to the executable Octave
script, not the arguments passed to the Octave interpreter on the ‘#!’ line of the script. For
example, the following program will reproduce the command line that was used to execute
the script, not ‘-qf’.
#! /bin/octave -qf
printf ("%s", program_name ());
arg_list = argv ();
for i = 1:nargin
printf (" %s", arg_list{i});
endfor
printf ("\n");

2.7 Comments in Octave Programs
A comment is some text that is included in a program for the sake of human readers, and
which is NOT an executable part of the program. Comments can explain what the program
does, and how it works. Nearly all programming languages have provisions for comments,
because programs are typically hard to understand without them.

2.7.1 Single Line Comments
In the Octave language, a comment starts with either the sharp sign character, ‘#’, or the
percent symbol ‘%’ and continues to the end of the line. Any text following the sharp sign
or percent symbol is ignored by the Octave interpreter and not executed. The following
example shows whole line and partial line comments.
function countdown
# Count down for main rocket engines
disp (3);
disp (2);
disp (1);
disp ("Blast Off!"); # Rocket leaves pad
endfunction

38

GNU Octave

2.7.2 Block Comments
Entire blocks of code can be commented by enclosing the code between matching ‘#{’ and
‘#}’ or ‘%{’ and ‘%}’ markers. For example,
function quick_countdown
# Count down for main rocket engines
disp (3);
#{
disp (2);
disp (1);
#}
disp ("Blast Off!"); # Rocket leaves pad
endfunction
will produce a very quick countdown from ’3’ to "Blast Off" as the lines "disp (2);"
and "disp (1);" won’t be executed.
The block comment markers must appear alone as the only characters on a line (excepting
whitespace) in order to be parsed correctly.

2.7.3 Comments and the Help System
The help command (see Section 2.3 [Getting Help], page 20) is able to find the first block
of comments in a function and return those as a documentation string. This means that the
same commands used to get help on built-in functions are available for properly formatted
user-defined functions. For example, after defining the function f below,
function xdot = f (x, t)
#
#
#
#
#

usage: f (x, t)
This function defines the right-hand
side functions for a set of nonlinear
differential equations.

r = 0.25;
...
endfunction
the command help f produces the output
usage: f (x, t)
This function defines the right-hand
side functions for a set of nonlinear
differential equations.
Although it is possible to put comment lines into keyboard-composed, throw-away Octave programs, it usually isn’t very useful because the purpose of a comment is to help you
or another person understand the program at a later time.
The help parser currently only recognizes single line comments (see Section 2.7.1 [Single
Line Comments], page 37) and not block comments for the initial help text.

39

3 Data Types
All versions of Octave include a number of built-in data types, including real and complex
scalars and matrices, character strings, a data structure type, and an array that can contain
all data types.
It is also possible to define new specialized data types by writing a small amount of C++
code. On some systems, new data types can be loaded dynamically while Octave is running,
so it is not necessary to recompile all of Octave just to add a new type. See Appendix A
[External Code Interface], page 875, for more information about Octave’s dynamic linking
capabilities. Section 3.2 [User-defined Data Types], page 44, describes what you must do
to define a new data type for Octave.

typeinfo ()
typeinfo (expr)
Return the type of the expression expr, as a string.
If expr is omitted, return a cell array of strings containing all the currently installed
data types.
See also: [class], page 39, [isa], page 39.

3.1 Built-in Data Types
The standard built-in data types are real and complex scalars and matrices, ranges, character strings, a data structure type, and cell arrays. Additional built-in data types may
be added in future versions. If you need a specialized data type that is not currently provided as a built-in type, you are encouraged to write your own user-defined data type and
contribute it for distribution in a future release of Octave.
The data type of a variable can be determined and changed through the use of the
following functions.

classname = class (obj)
class (s, id)
class (s, id, p, . . . )
Return the class of the object obj, or create a class with fields from structure s and
name (string) id.
Additional arguments name a list of parent classes from which the new class is derived.
See also: [typeinfo], page 39, [isa], page 39.

isa (obj, classname)
Return true if obj is an object from the class classname.
classname may also be one of the following class categories:
"float"

Floating point value comprising classes "double" and "single".

"integer"
Integer value comprising classes (u)int8, (u)int16, (u)int32, (u)int64.
"numeric"
Numeric value comprising either a floating point or integer value.

40

GNU Octave

If classname is a cell array of string, a logical array of the same size is returned,
containing true for each class to which obj belongs to.
See also: [class], page 39, [typeinfo], page 39.

cast (val, "type")
Convert val to data type type.
Both val and type are typically one of the following built-in classes:
"double"
"single"
"logical"
"char"
"int8"
"int16"
"int32"
"int64"
"uint8"
"uint16"
"uint32"
"uint64"
The value val may be modified to fit within the range of the new type.
Examples:
cast (-5, "uint8")
⇒ 0
cast (300, "int8")
⇒ 127
Programming Note: This function relies on the object val having a conversion method
named type. User-defined classes may implement only a subset of the full list of types
shown above. In that case, it may be necessary to call cast twice in order to reach the
desired type. For example, the conversion to double is nearly always implemented,
but the conversion to uint8 might not be. In that case, the following code will work
cast (cast (user_defined_val, "double"), "uint8")
See also: [typecast], page 40, [int8], page 55, [uint8], page 55, [int16], page 55, [uint16],
page 55, [int32], page 55, [uint32], page 55, [int64], page 55, [uint64], page 55, [double],
page 47, [single], page 54, [logical], page 60, [char], page 71, [class], page 39, [typeinfo],
page 39.

y = typecast (x, "class")
Return a new array y resulting from interpreting the data of x in memory as data of
the numeric class class.
Both the class of x and class must be one of the built-in numeric classes:

Chapter 3: Data Types

41

"logical"
"char"
"int8"
"int16"
"int32"
"int64"
"uint8"
"uint16"
"uint32"
"uint64"
"double"
"single"
"double complex"
"single complex"
the last two are only used with class; they indicate that a complex-valued result is
requested. Complex arrays are stored in memory as consecutive pairs of real numbers.
The sizes of integer types are given by their bit counts. Both logical and char are
typically one byte wide; however, this is not guaranteed by C++. If your system is
IEEE conformant, single and double will be 4 bytes and 8 bytes wide, respectively.
"logical" is not allowed for class.
If the input is a row vector, the return value is a row vector, otherwise it is a column
vector.
If the bit length of x is not divisible by that of class, an error occurs.
An example of the use of typecast on a little-endian machine is
x = uint16 ([1, 65535]);
typecast (x, "uint8")
⇒ [
1,
0, 255, 255]
See also: [cast], page 40, [bitpack], page 41, [bitunpack], page 42, [swapbytes],
page 41.

swapbytes (x)
Swap the byte order on values, converting from little endian to big endian and vice
versa.
For example:
swapbytes (uint16 (1:4))
⇒ [
256
512
768 1024]
See also: [typecast], page 40, [cast], page 40.

y = bitpack (x, class)
Return a new array y resulting from interpreting the logical array x as raw bit patterns
for data of the numeric class class.
class must be one of the built-in numeric classes:

42

GNU Octave

"double"
"single"
"double complex"
"single complex"
"char"
"int8"
"int16"
"int32"
"int64"
"uint8"
"uint16"
"uint32"
"uint64"
The number of elements of x should be divisible by the bit length of class. If it is
not, excess bits are discarded. Bits come in increasing order of significance, i.e., x(1)
is bit 0, x(2) is bit 1, etc.
The result is a row vector if x is a row vector, otherwise it is a column vector.
See also: [bitunpack], page 42, [typecast], page 40.

y = bitunpack (x)
Return a logical array y corresponding to the raw bit patterns of x.
x must belong to one of the built-in numeric classes:
"double"
"single"
"char"
"int8"
"int16"
"int32"
"int64"
"uint8"
"uint16"
"uint32"
"uint64"
The result is a row vector if x is a row vector; otherwise, it is a column vector.
See also: [bitpack], page 41, [typecast], page 40.

3.1.1 Numeric Objects
Octave’s built-in numeric objects include real, complex, and integer scalars and matrices.
All built-in floating point numeric data is currently stored as double precision numbers.
On systems that use the IEEE floating point format, values in the range of approximately
2.2251 × 10−308 to 1.7977 × 10308 can be stored, and the relative precision is approximately
2.2204 × 10−16 . The exact values are given by the variables realmin, realmax, and eps,
respectively.
Matrix objects can be of any size, and can be dynamically reshaped and resized. It is
easy to extract individual rows, columns, or submatrices using a variety of powerful indexing
features. See Section 8.1 [Index Expressions], page 137.

Chapter 3: Data Types

43

See Chapter 4 [Numeric Data Types], page 47, for more information.

3.1.2 Missing Data
It is possible to represent missing data explicitly in Octave using NA (short for “Not Available”). Missing data can only be represented when data is represented as floating point
numbers. In this case missing data is represented as a special case of the representation of
NaN.

NA
NA
NA
NA
NA

(n)
(n, m)
(n, m, k, . . . )
( . . . , class)
Return a scalar, matrix, or N-dimensional array whose elements are all equal to the
special constant used to designate missing values.
Note that NA always compares not equal to NA (NA != NA). To find NA values, use
the isna function.
When called with no arguments, return a scalar with the value ‘NA’.
When called with a single argument, return a square matrix with the dimension
specified.
When called with more than one scalar argument the first two arguments are taken as
the number of rows and columns and any further arguments specify additional matrix
dimensions.
The optional argument class specifies the return type and may be either "double" or
"single".
See also: [isna], page 43.

isna (x)
Return a logical array which is true where the elements of x are NA (missing) values
and false where they are not.
For example:
isna ([13, Inf, NA, NaN])
⇒ [ 0, 0, 1, 0 ]

See also: [isnan], page 444, [isinf], page 444, [isfinite], page 445.

3.1.3 String Objects
A character string in Octave consists of a sequence of characters enclosed in either doublequote or single-quote marks. Internally, Octave currently stores strings as matrices of
characters. All the indexing operations that work for matrix objects also work for strings.
See Chapter 5 [Strings], page 67, for more information.

3.1.4 Data Structure Objects
Octave’s data structure type can help you to organize related objects of different types.
The current implementation uses an associative array with indices limited to strings, but
the syntax is more like C-style structures.
See Section 6.1 [Structures], page 101, for more information.

44

GNU Octave

3.1.5 Cell Array Objects
A Cell Array in Octave is general array that can hold any number of different data types.
See Section 6.2 [Cell Arrays], page 114, for more information.

3.2 User-defined Data Types
Someday I hope to expand this to include a complete description of Octave’s mechanism
for managing user-defined data types. Until this feature is documented here, you will have
to make do by reading the code in the ov.h, ops.h, and related files from Octave’s src
directory.

3.3 Object Sizes
The following functions allow you to determine the size of a variable or expression. These
functions are defined for all objects. They return −1 when the operation doesn’t make
sense. For example, Octave’s data structure type doesn’t have rows or columns, so the
rows and columns functions return −1 for structure arguments.

ndims (a)
Return the number of dimensions of a.
For any array, the result will always be greater than or equal to 2. Trailing singleton
dimensions are not counted.
ndims (ones (4, 1, 2, 1))
⇒ 3
See also: [size], page 45.

columns (a)
Return the number of columns of a.
See also: [rows], page 44, [size], page 45, [length], page 45, [numel], page 44, [isscalar],
page 63, [isvector], page 63, [ismatrix], page 63.

rows (a)
Return the number of rows of a.
See also: [columns], page 44, [size], page 45, [length], page 45, [numel], page 44,
[isscalar], page 63, [isvector], page 63, [ismatrix], page 63.

numel (a)
numel (a, idx1, idx2, . . . )
Return the number of elements in the object a.
Optionally, if indices idx1, idx2, . . . are supplied, return the number of elements that
would result from the indexing
a(idx1, idx2, ...)
Note that the indices do not have to be scalar numbers. For example,
a = 1;
b = ones (2, 3);
numel (a, b)

Chapter 3: Data Types

45

will return 6, as this is the number of ways to index with b. Or the index could be
the string ":" which represents the colon operator. For example,
a = ones (5, 3);
numel (a, 2, ":")
will return 3 as the second row has three column entries.
This method is also called when an object appears as lvalue with cs-list indexing, i.e.,
object{...} or object(...).field.
See also: [size], page 45, [length], page 45, [ndims], page 44.

length (a)
Return the length of the object a.
The length is 0 for empty objects, 1 for scalars, and the number of elements for
vectors. For matrix or N-dimensional objects, the length is the number of elements
along the largest dimension (equivalent to max (size (a))).
See also: [numel], page 44, [size], page 45.

sz = size (a)
dim_sz = size (a, dim)
[rows, cols, ..., dim_N_sz] = size ( . . . )
Return a row vector with the size (number of elements) of each dimension for the
object a.
When given a second argument, dim, return the size of the corresponding dimension.
With a single output argument, size returns a row vector. When called with multiple
output arguments, size returns the size of dimension N in the Nth argument. The
number of rows, dimension 1, is returned in the first argument, the number of columns,
dimension 2, is returned in the second argument, etc. If there are more dimensions
in a then there are output arguments, size returns the total number of elements in
the remaining dimensions in the final output argument.
Example 1: single row vector output
size ([1, 2; 3, 4; 5, 6])
⇒ [ 3, 2 ]
Example 2: number of elements in 2nd dimension (columns)
size ([1, 2; 3, 4; 5, 6], 2)
⇒ 2
Example 3: number of output arguments == number of dimensions
[nr, nc] = size ([1, 2; 3, 4; 5, 6])
⇒ nr = 3
⇒ nc = 2
Example 4: number of output arguments != number of dimensions
[nr, remainder] = size (ones (2, 3, 4, 5))
⇒ nr = 2
⇒ remainder = 60

See also: [numel], page 44, [ndims], page 44, [length], page 45, [rows], page 44,
[columns], page 44, [size equal], page 46, [common size], page 445.

46

GNU Octave

isempty (a)
Return true if a is an empty matrix (any one of its dimensions is zero).
See also: [isnull], page 46, [isa], page 39.

isnull (x)
Return true if x is a special null matrix, string, or single quoted string.
Indexed assignment with such a value on the right-hand side should delete array
elements. This function should be used when overloading indexed assignment for
user-defined classes instead of isempty, to distinguish the cases:
A(I) = [] This should delete elements if I is nonempty.
X = []; A(I) = X
This should give an error if I is nonempty.
See also: [isempty], page 46, [isindex], page 141.

sizeof (val)
Return the size of val in bytes.
See also: [whos], page 130.

size_equal (a, b, . . . )
Return true if the dimensions of all arguments agree.
Trailing singleton dimensions are ignored. When called with a single argument, or no
argument, size_equal returns true.
See also: [size], page 45, [numel], page 44, [ndims], page 44, [common size], page 445.

squeeze (x)
Remove singleton dimensions from x and return the result.
Note that for compatibility with matlab, all objects have a minimum of two dimensions and row vectors are left unchanged.
See also: [reshape], page 450.

47

4 Numeric Data Types
A numeric constant may be a scalar, a vector, or a matrix, and it may contain complex
values.
The simplest form of a numeric constant, a scalar, is a single number. Note that by
default numeric constants are represented within Octave by IEEE 754 double precision
(binary64) floating-point format (complex constants are stored as pairs of binary64 values).
It is, however, possible to represent real integers as described in Section 4.4 [Integer Data
Types], page 54.
If the numeric constant is a real integer, it can be defined in decimal, hexadecimal,
or binary notation. Hexadecimal notation starts with ‘0x’ or ‘0X’, binary notation starts
with ‘0b’ or ‘0B’, otherwise decimal notation is assumed. As a consequence, ‘0b’ is not a
hexadecimal number, in fact, it is not a valid number at all.
For better readability, digits may be partitioned by the underscore separator ‘_’, which is
ignored by the Octave interpreter. Here are some examples of real-valued integer constants,
which all represent the same value and are internally stored as binary64:
42
0x2A
0b101010
0b10_1010
round (42.1)

#
#
#
#
#

decimal notation
hexadecimal notation
binary notation
underscore notation
also binary64

In decimal notation, the numeric constant may be denoted as decimal fraction or even
in scientific (exponential) notation. Note that this is not possible for hexadecimal or binary
notation. Again, in the following example all numeric constants represent the same value:
.105
1.05e-1
.00105e+2
Unlike most programming languages, complex numeric constants are denoted as the
sum of real and imaginary parts. The imaginary part is denoted by a real-valued numeric
constant√followed immediately by a complex value indicator (‘i’, ‘j’, ‘I’, or ‘J’ which represents −1). No spaces are allowed between the numeric constant and the complex value
indicator. Some examples of complex numeric constants that all represent the same value:
3 + 42i
3 + 42j
3 + 42I
3 + 42J
3.0 + 42.0i
3.0 + 0x2Ai
3.0 + 0b10_1010i
0.3e1 + 420e-1i

double (x)
Convert x to double precision type.
See also: [single], page 54.

48

GNU Octave

complex (x)
complex (re, im)
Return a complex value from real arguments.
With 1 real argument x, return the complex result x + 0i.
With 2 real arguments, return the complex result re + imi. complex can often be
more convenient than expressions such as a + b*i. For example:
complex ([1, 2], [3, 4])
⇒ [ 1 + 3i
2 + 4i ]

See also: [real], page 478, [imag], page 477, [iscomplex], page 63, [abs], page 477, [arg],
page 477.

4.1 Matrices
It is easy to define a matrix of values in Octave. The size of the matrix is determined
automatically, so it is not necessary to explicitly state the dimensions. The expression
a = [1, 2; 3, 4]
results in the matrix


1 2
a=
3 4
Elements of a matrix may be arbitrary expressions, provided that the dimensions all
make sense when combining the various pieces. For example, given the above matrix, the
expression
[ a, a ]
produces the matrix
ans =
1 2 1 2
3 4 3 4
but the expression
[ a, 1 ]
produces the error
error: number of rows must match (1 != 2) near line 13, column 6
(assuming that this expression was entered as the first thing on line 13, of course).
Inside the square brackets that delimit a matrix expression, Octave looks at the surrounding context to determine whether spaces and newline characters should be converted
into element and row separators, or simply ignored, so an expression like
a = [ 1 2
3 4 ]
will work. However, some possible sources of confusion remain. For example, in the expression
[ 1 - 1 ]
the ‘-’ is treated as a binary operator and the result is the scalar 0, but in the expression
[ 1 -1 ]

Chapter 4: Numeric Data Types

49

the ‘-’ is treated as a unary operator and the result is the vector [ 1, -1 ]. Similarly, the
expression
[ sin (pi) ]
will be parsed as
[ sin, (pi) ]
and will result in an error since the sin function will be called with no arguments. To get
around this, you must omit the space between sin and the opening parenthesis, or enclose
the expression in a set of parentheses:
[ (sin (pi)) ]
Whitespace surrounding the single quote character (‘’’, used as a transpose operator
and for delimiting character strings) can also cause confusion. Given a = 1, the expression
[ 1 a’ ]
results in the single quote character being treated as a transpose operator and the result is
the vector [ 1, 1 ], but the expression
[ 1 a ’ ]
produces the error message
parse error:
syntax error
>>> [ 1 a ’ ]
^
because not doing so would cause trouble when parsing the valid expression
[ a ’foo’ ]
For clarity, it is probably best to always use commas and semicolons to separate matrix
elements and rows.
The maximum number of elements in a matrix is fixed when Octave is compiled. The
allowable number can be queried with the function sizemax. Note that other factors, such as
the amount of memory available on your machine, may limit the maximum size of matrices
to something smaller.

sizemax ()
Return the largest value allowed for the size of an array.
If Octave is compiled with 64-bit indexing, the result is of class int64, otherwise it is
of class int32. The maximum array size is slightly smaller than the maximum value
allowable for the relevant class as reported by intmax.
See also: [intmax], page 56.
When you type a matrix or the name of a variable whose value is a matrix, Octave
responds by printing the matrix in with neatly aligned rows and columns. If the rows of
the matrix are too large to fit on the screen, Octave splits the matrix and displays a header
before each section to indicate which columns are being displayed. You can use the following
variables to control the format of the output.

50

GNU Octave

val = output_max_field_width ()
old_val = output_max_field_width (new_val)
output_max_field_width (new_val, "local")
Query or set the internal variable that specifies the maximum width of a numeric
output field.
When called from inside a function with the "local" option, the variable is changed
locally for the function and any subroutines it calls. The original variable value is
restored when exiting the function.
See also:
page 50.

[format], page 246, [fixed point format], page 51, [output precision],

val = output_precision ()
old_val = output_precision (new_val)
output_precision (new_val, "local")
Query or set the internal variable that specifies the minimum number of significant
figures to display for numeric output.
When called from inside a function with the "local" option, the variable is changed
locally for the function and any subroutines it calls. The original variable value is
restored when exiting the function.
See also: [format], page 246, [fixed point format], page 51, [output max field width],
page 49.
It is possible to achieve a wide range of output styles by using different values of output_
precision and output_max_field_width. Reasonable combinations can be set using the
format function. See Section 14.1 [Basic Input and Output], page 245.

val = split_long_rows ()
old_val = split_long_rows (new_val)
split_long_rows (new_val, "local")
Query or set the internal variable that controls whether rows of a matrix may be split
when displayed to a terminal window.
If the rows are split, Octave will display the matrix in a series of smaller pieces, each
of which can fit within the limits of your terminal width and each set of rows is labeled
so that you can easily see which columns are currently being displayed. For example:
octave:13> rand (2,10)
ans =
Columns 1 through 6:
0.75883
0.75697

0.93290
0.51942

0.40064
0.40031

0.43818
0.61784

Columns 7 through 10:
0.90174
0.44672

0.11854
0.94303

0.72313
0.56564

0.73326
0.82150

0.94958
0.92309

0.16467
0.40201

Chapter 4: Numeric Data Types

51

When called from inside a function with the "local" option, the variable is changed
locally for the function and any subroutines it calls. The original variable value is
restored when exiting the function.
See also: [format], page 246.
Octave automatically switches to scientific notation when values become very large or
very small. This guarantees that you will see several significant figures for every value in
a matrix. If you would prefer to see all values in a matrix printed in a fixed point format,
you can set the built-in variable fixed_point_format to a nonzero value. But doing so is
not recommended, because it can produce output that can easily be misinterpreted.

val = fixed_point_format ()
old_val = fixed_point_format (new_val)
fixed_point_format (new_val, "local")
Query or set the internal variable that controls whether Octave will use a scaled
format to print matrix values.
The scaled format prints a scaling factor on the first line of output chosen such that
the largest matrix element can be written with a single leading digit. For example:
logspace (1, 7, 5)’
ans =
1.0e+07

*

0.00000
0.00003
0.00100
0.03162
1.00000
Notice that the first value appears to be 0 when it is actually 1. Because of the
possibility for confusion you should be careful about enabling fixed_point_format.
When called from inside a function with the "local" option, the variable is changed
locally for the function and any subroutines it calls. The original variable value is
restored when exiting the function.
See also: [format], page 246, [output max field width], page 49, [output precision],
page 50.

4.1.1 Empty Matrices
A matrix may have one or both dimensions zero, and operations on empty matrices are
handled as described by Carl de Boor in An Empty Exercise, SIGNUM, Volume 25, pages
2-6, 1990 and C. N. Nett and W. M. Haddad, in A System-Theoretic Appropriate Realization of the Empty Matrix Concept, IEEE Transactions on Automatic Control, Volume 38,
Number 5, May 1993. Briefly, given a scalar s, an m × n matrix Mm×n , and an m × n empty

52

GNU Octave

matrix [ ]m×n (with either one or both dimensions equal to zero), the following are true:
s · [ ]m×n = [ ]m×n · s = [ ]m×n
[ ]m×n + [ ]m×n = [ ]m×n
[ ]0×m · Mm×n = [ ]0×n
Mm×n · [ ]n×0 = [ ]m×0
[ ]m×0 · [ ]0×n = 0m×n
By default, dimensions of the empty matrix are printed along with the empty matrix
symbol, ‘[]’. The built-in variable print_empty_dimensions controls this behavior.

val = print_empty_dimensions ()
old_val = print_empty_dimensions (new_val)
print_empty_dimensions (new_val, "local")
Query or set the internal variable that controls whether the dimensions of empty
matrices are printed along with the empty matrix symbol, ‘[]’.
For example, the expression
zeros (3, 0)
will print
ans = [](3x0)
When called from inside a function with the "local" option, the variable is changed
locally for the function and any subroutines it calls. The original variable value is
restored when exiting the function.
See also: [format], page 246.
Empty matrices may also be used in assignment statements as a convenient way to delete
rows or columns of matrices. See Section 8.6 [Assignment Expressions], page 153.
When Octave parses a matrix expression, it examines the elements of the list to determine
whether they are all constants. If they are, it replaces the list with a single matrix constant.

4.2 Ranges
A range is a convenient way to write a row vector with evenly spaced elements. A range
expression is defined by the value of the first element in the range, an optional value for the
increment between elements, and a maximum value which the elements of the range will
not exceed. The base, increment, and limit are separated by colons (the ‘:’ character) and
may contain any arithmetic expressions and function calls. If the increment is omitted, it
is assumed to be 1. For example, the range
1 : 5
defines the set of values [ 1, 2, 3, 4, 5 ], and the range
1 : 3 : 5
defines the set of values [ 1, 4 ].
Although a range constant specifies a row vector, Octave does not normally convert range
constants to vectors unless it is necessary to do so. This allows you to write a constant like
1 : 10000 without using 80,000 bytes of storage on a typical 32-bit workstation.

Chapter 4: Numeric Data Types

53

A common example of when it does become necessary to convert ranges into vectors
occurs when they appear within a vector (i.e., inside square brackets). For instance, whereas
x = 0 : 0.1 : 1;
defines x to be a variable of type range and occupies 24 bytes of memory, the expression
y = [ 0 : 0.1 : 1];
defines y to be of type matrix and occupies 88 bytes of memory.
This space saving optimization may be disabled using the function disable range.

val = disable_range ()
old_val = disable_range (new_val)
disable_range (new_val, "local")
Query or set the internal variable that controls whether ranges are stored in a special
space-efficient format.
The default value is true. If this option is disabled Octave will store ranges as full
matrices.
When called from inside a function with the "local" option, the variable is changed
locally for the function and any subroutines it calls. The original variable value is
restored when exiting the function.
See also:
page 555.

[disable diagonal matrix],

page 555,

[disable permutation matrix],

Note that the upper (or lower, if the increment is negative) bound on the range is not
always included in the set of values, and that ranges defined by floating point values can
produce surprising results because Octave uses floating point arithmetic to compute the
values in the range. If it is important to include the endpoints of a range and the number of
elements is known, you should use the linspace function instead (see Section 16.3 [Special
Utility Matrices], page 457).
When adding a scalar to a range, subtracting a scalar from it (or subtracting a range
from a scalar) and multiplying by scalar, Octave will attempt to avoid unpacking the range
and keep the result as a range, too, if it can determine that it is safe to do so. For instance,
doing
a = 2*(1:1e7) - 1;
will produce the same result as 1:2:2e7-1, but without ever forming a vector with ten
million elements.
Using zero as an increment in the colon notation, as 1:0:1 is not allowed, because a
division by zero would occur in determining the number of range elements. However, ranges
with zero increment (i.e., all elements equal) are useful, especially in indexing, and Octave
allows them to be constructed using the built-in function ones. Note that because a range
must be a row vector, ones (1, 10) produces a range, while ones (10, 1) does not.
When Octave parses a range expression, it examines the elements of the expression to
determine whether they are all constants. If they are, it replaces the range expression with
a single range constant.

54

GNU Octave

4.3 Single Precision Data Types
Octave includes support for single precision data types, and most of the functions in Octave
accept single precision values and return single precision answers. A single precision variable
is created with the single function.

single (x)
Convert x to single precision type.
See also: [double], page 47.
for example:
sngl = single (rand (2, 2))
⇒ sngl =
0.37569
0.92982
0.11962
0.50876
class (sngl)
⇒ single

Many functions can also return single precision values directly. For example
ones (2, 2, "single")
zeros (2, 2, "single")
eye (2, 2, "single")
rand (2, 2, "single")
NaN (2, 2, "single")
NA (2, 2, "single")
Inf (2, 2, "single")
will all return single precision matrices.

4.4 Integer Data Types
Octave supports integer matrices as an alternative to using double precision. It is possible
to use both signed and unsigned integers represented by 8, 16, 32, or 64 bits. It should be
noted that most computations require floating point data, meaning that integers will often
change type when involved in numeric computations. For this reason integers are most
often used to store data, and not for calculations.
In general most integer matrices are created by casting existing matrices to integers.
The following example shows how to cast a matrix into 32 bit integers.
float = rand (2, 2)
⇒ float = 0.37569
0.11962
integer = int32 (float)
⇒ integer = 0 1
0 1

0.92982
0.50876

As can be seen, floating point values are rounded to the nearest integer when converted.

isinteger (x)
Return true if x is an integer object (int8, uint8, int16, etc.).

Chapter 4: Numeric Data Types

55

Note that isinteger (14) is false because numeric constants in Octave are double
precision floating point values.
See also: [isfloat], page 62, [ischar], page 68, [islogical], page 62, [isnumeric], page 62,
[isa], page 39.

int8 (x)
Convert x to 8-bit integer type.
See also: [uint8], page 55, [int16], page 55, [uint16], page 55, [int32], page 55, [uint32],
page 55, [int64], page 55, [uint64], page 55.

uint8 (x)
Convert x to unsigned 8-bit integer type.
See also: [int8], page 55, [int16], page 55, [uint16], page 55, [int32], page 55, [uint32],
page 55, [int64], page 55, [uint64], page 55.

int16 (x)
Convert x to 16-bit integer type.
See also: [int8], page 55, [uint8], page 55, [uint16], page 55, [int32], page 55, [uint32],
page 55, [int64], page 55, [uint64], page 55.

uint16 (x)
Convert x to unsigned 16-bit integer type.
See also: [int8], page 55, [uint8], page 55, [int16], page 55, [int32], page 55, [uint32],
page 55, [int64], page 55, [uint64], page 55.

int32 (x)
Convert x to 32-bit integer type.
See also: [int8], page 55, [uint8], page 55, [int16], page 55, [uint16], page 55, [uint32],
page 55, [int64], page 55, [uint64], page 55.

uint32 (x)
Convert x to unsigned 32-bit integer type.
See also: [int8], page 55, [uint8], page 55, [int16], page 55, [uint16], page 55, [int32],
page 55, [int64], page 55, [uint64], page 55.

int64 (x)
Convert x to 64-bit integer type.
See also: [int8], page 55, [uint8], page 55, [int16], page 55, [uint16], page 55, [int32],
page 55, [uint32], page 55, [uint64], page 55.

uint64 (x)
Convert x to unsigned 64-bit integer type.
See also: [int8], page 55, [uint8], page 55, [int16], page 55, [uint16], page 55, [int32],
page 55, [uint32], page 55, [int64], page 55.

56

GNU Octave

intmax (type)
Return the largest integer that can be represented in an integer type.
The variable type can be
int8

signed 8-bit integer.

int16

signed 16-bit integer.

int32

signed 32-bit integer.

int64

signed 64-bit integer.

uint8

unsigned 8-bit integer.

uint16

unsigned 16-bit integer.

uint32

unsigned 32-bit integer.

uint64

unsigned 64-bit integer.

The default for type is int32.
See also: [intmin], page 56, [flintmax], page 56.

intmin (type)
Return the smallest integer that can be represented in an integer type.
The variable type can be
int8

signed 8-bit integer.

int16

signed 16-bit integer.

int32

signed 32-bit integer.

int64

signed 64-bit integer.

uint8

unsigned 8-bit integer.

uint16

unsigned 16-bit integer.

uint32

unsigned 32-bit integer.

uint64

unsigned 64-bit integer.

The default for type is int32.
See also: [intmax], page 56, [flintmax], page 56.

flintmax ()
flintmax ("double")
flintmax ("single")
Return the largest integer that can be represented consecutively in a floating point
value.
The default class is "double", but "single" is a valid option. On IEEE 754 compatible systems, flintmax is 253 for "double" and 224 for "single".
See also: [intmax], page 56, [realmax], page 504, [realmin], page 505.

Chapter 4: Numeric Data Types

57

4.4.1 Integer Arithmetic
While many numerical computations can’t be carried out in integers, Octave does support
basic operations like addition and multiplication on integers. The operators +, -, .*, and
./ work on integers of the same type. So, it is possible to add two 32 bit integers, but not
to add a 32 bit integer and a 16 bit integer.
When doing integer arithmetic one should consider the possibility of underflow and
overflow. This happens when the result of the computation can’t be represented using the
chosen integer type. As an example it is not possible to represent the result of 10 − 20
when using unsigned integers. Octave makes sure that the result of integer computations is
the integer that is closest to the true result. So, the result of 10 − 20 when using unsigned
integers is zero.
When doing integer division Octave will round the result to the nearest integer. This is
different from most programming languages, where the result is often floored to the nearest
integer. So, the result of int32 (5) ./ int32 (8) is 1.

idivide (x, y, op)
Integer division with different rounding rules.
The standard behavior of integer division such as a ./ b is to round the result to
the nearest integer. This is not always the desired behavior and idivide permits
integer element-by-element division to be performed with different treatment for the
fractional part of the division as determined by the op flag. op is a string with one
of the values:
"fix"

Calculate a ./ b with the fractional part rounded towards zero.

"round"

Calculate a ./ b with the fractional part rounded towards the nearest
integer.

"floor"

Calculate a ./ b with the fractional part rounded towards negative infinity.

"ceil"

Calculate a ./ b with the fractional part rounded towards positive infinity.

If op is not given it defaults to "fix". An example demonstrating these rounding
rules is
idivide (int8 ([-3,
⇒ int8 ([0, 0])
idivide (int8 ([-3,
⇒ int8 ([-1, 1])
idivide (int8 ([-3,
⇒ int8 ([-1, 0])
idivide (int8 ([-3,
⇒ int8 ([0, 1])

3]), int8 (4), "fix")
3]), int8 (4), "round")
3]), int8 (4), "floor")
3]), int8 (4), "ceil")

See also: [ldivide], page 146, [rdivide], page 147.

58

GNU Octave

4.5 Bit Manipulations
Octave provides a number of functions for the manipulation of numeric values on a bit by
bit basis. The basic functions to set and obtain the values of individual bits are bitset
and bitget.

C = bitset (A, n)
C = bitset (A, n, val)
Set or reset bit(s) n of the unsigned integers in A.
val = 0 resets and val = 1 sets the bits. The least significant bit is n = 1. All variables
must be the same size or scalars.
dec2bin (bitset (10, 1))
⇒ 1011

See also: [bitand], page 58, [bitor], page 59, [bitxor], page 59, [bitget], page 58,
[bitcmp], page 59, [bitshift], page 59, [intmax], page 56, [flintmax], page 56.

c = bitget (A, n)
Return the status of bit(s) n of the unsigned integers in A.
The least significant bit is n = 1.
bitget (100, 8:-1:1)
⇒ 0 1 1 0 0 1 0 0

See also: [bitand], page 58, [bitor], page 59, [bitxor], page 59, [bitset], page 58,
[bitcmp], page 59, [bitshift], page 59, [intmax], page 56, [flintmax], page 56.

The arguments to all of Octave’s bitwise operations can be scalar or arrays, except for
bitcmp, whose k argument must a scalar. In the case where more than one argument is an
array, then all arguments must have the same shape, and the bitwise operator is applied to
each of the elements of the argument individually. If at least one argument is a scalar and
one an array, then the scalar argument is duplicated. Therefore
bitget (100, 8:-1:1)
is the same as
bitget (100 * ones (1, 8), 8:-1:1)
It should be noted that all values passed to the bit manipulation functions of Octave
are treated as integers. Therefore, even though the example for bitset above passes the
floating point value 10, it is treated as the bits [1, 0, 1, 0] rather than the bits of the
native floating point format representation of 10.
As the maximum value that can be represented by a number is important for bit manipulation, particularly when forming masks, Octave supplies two utility functions: flintmax
for floating point integers, and intmax for integer objects (uint8, int64, etc.).
Octave also includes the basic bitwise ’and’, ’or’, and ’exclusive or’ operators.

bitand (x, y)
Return the bitwise AND of non-negative integers.
x, y must be in the range [0,intmax]
See also: [bitor], page 59, [bitxor], page 59, [bitset], page 58, [bitget], page 58,
[bitcmp], page 59, [bitshift], page 59, [intmax], page 56, [flintmax], page 56.

Chapter 4: Numeric Data Types

59

bitor (x, y)
Return the bitwise OR of non-negative integers x and y.
See also: [bitor], page 59, [bitxor], page 59, [bitset], page 58, [bitget], page 58,
[bitcmp], page 59, [bitshift], page 59, [intmax], page 56, [flintmax], page 56.

bitxor (x, y)
Return the bitwise XOR of non-negative integers x and y.
See also: [bitand], page 58, [bitor], page 59, [bitset], page 58, [bitget], page 58,
[bitcmp], page 59, [bitshift], page 59, [intmax], page 56, [flintmax], page 56.
The bitwise ’not’ operator is a unary operator that performs a logical negation of each
of the bits of the value. For this to make sense, the mask against which the value is negated
must be defined. Octave’s bitwise ’not’ operator is bitcmp.

bitcmp (A, k)
Return the k-bit complement of integers in A.
If k is omitted k = log2 (flintmax) + 1 is assumed.
bitcmp (7,4)
⇒ 8
dec2bin (11)
⇒ 1011
dec2bin (bitcmp (11, 6))
⇒ 110100
See also: [bitand], page 58, [bitor], page 59, [bitxor], page 59, [bitset], page 58, [bitget],
page 58, [bitcmp], page 59, [bitshift], page 59, [flintmax], page 56.
Octave also includes the ability to left-shift and right-shift values bitwise.

bitshift (a, k)
bitshift (a, k, n)
Return a k bit shift of n-digit unsigned integers in a.
A positive k leads to a left shift; A negative value to a right shift.
If n is omitted it defaults to 64. n must be in the range [1,64].
bitshift (eye (3), 1)
⇒
2 0 0
0 2 0
0 0 2
bitshift (10, [-2, -1, 0, 1, 2])
⇒ 2
5 10 20 40
See also: [bitand], page 58, [bitor], page 59, [bitxor], page 59, [bitset], page 58, [bitget],
page 58, [bitcmp], page 59, [intmax], page 56, [flintmax], page 56.

60

GNU Octave

Bits that are shifted out of either end of the value are lost. Octave also uses arithmetic
shifts, where the sign bit of the value is kept during a right shift. For example:
bitshift (-10, -1)
⇒ -5
bitshift (int8 (-1), -1)
⇒ -1

Note that bitshift (int8 (-1), -1) is -1 since the bit representation of -1 in the int8
data type is [1, 1, 1, 1, 1, 1, 1, 1].

4.6 Logical Values
Octave has built-in support for logical values, i.e., variables that are either true or false.
When comparing two variables, the result will be a logical value whose value depends on
whether or not the comparison is true.
The basic logical operations are &, |, and !, which correspond to “Logical And”, “Logical
Or”, and “Logical Negation”. These operations all follow the usual rules of logic.
It is also possible to use logical values as part of standard numerical calculations. In
this case true is converted to 1, and false to 0, both represented using double precision
floating point numbers. So, the result of true*22 - false/6 is 22.
Logical values can also be used to index matrices and cell arrays. When indexing with
a logical array the result will be a vector containing the values corresponding to true parts
of the logical array. The following example illustrates this.
data = [ 1, 2; 3, 4 ];
idx = (data <= 2);
data(idx)
⇒ ans = [ 1; 2 ]

Instead of creating the idx array it is possible to replace data(idx) with data( data <= 2 )
in the above code.
Logical values can also be constructed by casting numeric objects to logical values, or
by using the true or false functions.

logical (x)
Convert the numeric object x to logical type.
Any nonzero values will be converted to true (1) while zero values will be converted
to false (0). The non-numeric value NaN cannot be converted and will produce an
error.
Compatibility Note: Octave accepts complex values as input, whereas matlab issues
an error.
See also: [double], page 47, [single], page 54, [char], page 71.

true (x)
true (n, m)
true (n, m, k, . . . )
Return a matrix or N-dimensional array whose elements are all logical 1.

Chapter 4: Numeric Data Types

61

If invoked with a single scalar integer argument, return a square matrix of the specified
size.
If invoked with two or more scalar integer arguments, or a vector of integer values,
return an array with given dimensions.
See also: [false], page 61.

false (x)
false (n, m)
false (n, m, k, . . . )
Return a matrix or N-dimensional array whose elements are all logical 0.
If invoked with a single scalar integer argument, return a square matrix of the specified
size.
If invoked with two or more scalar integer arguments, or a vector of integer values,
return an array with given dimensions.
See also: [true], page 60.

4.7 Promotion and Demotion of Data Types
Many operators and functions can work with mixed data types. For example,
uint8 (1) + 1
⇒ 2

where the above operator works with an 8-bit integer and a double precision value and
returns an 8-bit integer value. Note that the type is demoted to an 8-bit integer, rather
than promoted to a double precision value as might be expected. The reason is that if
Octave promoted values in expressions like the above with all numerical constants would
need to be explicitly cast to the appropriate data type like
uint8 (1) + uint8 (1)
⇒ 2

which becomes difficult for the user to apply uniformly and might allow hard to find bugs
to be introduced. The same applies to single precision values where a mixed operation such
as
single (1) + 1
⇒ 2

returns a single precision value. The mixed operations that are valid and their returned
data types are
Mixed Operation
double OP single
double OP integer
double OP char
double OP logical
single OP integer
single OP char
single OP logical

Result
single
integer
double
double
integer
single
single

62

GNU Octave

The same logic applies to functions with mixed arguments such as
min (single (1), 0)
⇒ 0
where the returned value is single precision.
In the case of mixed type indexed assignments, the type is not changed. For example,
x = ones (2, 2);
x(1, 1) = single (2)
⇒ x = 2
1
1
1
where x remains of the double precision type.

4.8 Predicates for Numeric Objects
Since the type of a variable may change during the execution of a program, it can be
necessary to do type checking at run-time. Doing this also allows you to change the behavior
of a function depending on the type of the input. As an example, this naive implementation
of abs returns the absolute value of the input if it is a real number, and the length of the
input if it is a complex number.
function a = abs (x)
if (isreal (x))
a = sign (x) .* x;
elseif (iscomplex (x))
a = sqrt (real(x).^2 + imag(x).^2);
endif
endfunction
The following functions are available for determining the type of a variable.

isnumeric (x)
Return true if x is a numeric object, i.e., an integer, real, or complex array.
Logical and character arrays are not considered to be numeric.
See also: [isinteger], page 54, [isfloat], page 62, [isreal], page 63, [iscomplex], page 63,
[islogical], page 62, [ischar], page 68, [iscell], page 115, [isstruct], page 109, [isa],
page 39.

islogical (x)
isbool (x)
Return true if x is a logical object.
See also: [isfloat], page 62, [isinteger], page 54, [ischar], page 68, [isnumeric], page 62,
[isa], page 39.

isfloat (x)
Return true if x is a floating-point numeric object.
Objects of class double or single are floating-point objects.
See also: [isinteger], page 54, [ischar], page 68, [islogical], page 62, [isnumeric], page 62,
[isa], page 39.

Chapter 4: Numeric Data Types

63

isreal (x)
Return true if x is a non-complex matrix or scalar.
For compatibility with matlab, this includes logical and character matrices.
See also: [iscomplex], page 63, [isnumeric], page 62, [isa], page 39.

iscomplex (x)
Return true if x is a complex-valued numeric object.
See also: [isreal], page 63, [isnumeric], page 62, [islogical], page 62, [ischar], page 68,
[isfloat], page 62, [isa], page 39.

ismatrix (a)
Return true if a is a 2-D array.
See also: [isscalar], page 63, [isvector], page 63, [iscell], page 115, [isstruct], page 109,
[issparse], page 570, [isa], page 39.

isvector (x)
Return true if x is a vector.
A vector is a 2-D array where one of the dimensions is equal to 1. As a consequence
a 1x1 array, or scalar, is also a vector.
See also: [isscalar], page 63, [ismatrix], page 63, [size], page 45, [rows], page 44,
[columns], page 44, [length], page 45.

isrow (x)
Return true if x is a row vector 1xN with non-negative N.
See also: [iscolumn], page 63, [isscalar], page 63, [isvector], page 63, [ismatrix],
page 63.

iscolumn (x)
Return true if x is a column vector Nx1 with non-negative N.
See also: [isrow], page 63, [isscalar], page 63, [isvector], page 63, [ismatrix], page 63.

isscalar (x)
Return true if x is a scalar.
See also: [isvector], page 63, [ismatrix], page 63.

issquare (x)
Return true if x is a square matrix.
See also: [isscalar], page 63, [isvector], page 63, [ismatrix], page 63, [size], page 45.

issymmetric (A)
issymmetric (A, tol)
Return true if A is a symmetric matrix within the tolerance specified by tol.
The default tolerance is zero (uses faster code).
Matrix A is considered symmetric if norm (A - A.’, Inf) / norm (A, Inf) < tol.
See also: [ishermitian], page 64, [isdefinite], page 64.

64

GNU Octave

ishermitian (A)
ishermitian (A, tol)
Return true if A is Hermitian within the tolerance specified by tol.
The default tolerance is zero (uses faster code).
Matrix A is considered symmetric if norm (A - A’, Inf) / norm (A, Inf) < tol.
See also: [issymmetric], page 63, [isdefinite], page 64.

isdefinite (A)
isdefinite (A, tol)
Return 1 if A is symmetric positive definite within the tolerance specified by tol or 0
if A is symmetric positive semidefinite. Otherwise, return -1.
If tol is omitted, use a tolerance of 100 * eps * norm (A, "fro")
See also: [issymmetric], page 63, [ishermitian], page 64.

isbanded (A, lower, upper)
Return true if A is a matrix with entries confined between lower diagonals below the
main diagonal and upper diagonals above the main diagonal.
lower and upper must be non-negative integers.
See also: [isdiag], page 64, [istril], page 64, [istriu], page 64, [bandwidth], page 508.

isdiag (A)
Return true if A is a diagonal matrix.
See also: [isbanded], page 64, [istril], page 64, [istriu], page 64, [diag], page 456,
[bandwidth], page 508.

istril (A)
Return true if A is a lower triangular matrix.
A lower triangular matrix has nonzero entries only on the main diagonal and below.
See also: [istriu], page 64, [isbanded], page 64, [isdiag], page 64, [tril], page 454,
[bandwidth], page 508.

istriu (A)
Return true if A is an upper triangular matrix.
An upper triangular matrix has nonzero entries only on the main diagonal and above.
See also: [isdiag], page 64, [isbanded], page 64, [istril], page 64, [triu], page 454,
[bandwidth], page 508.

isprime (x)
Return a logical array which is true where the elements of x are prime numbers and
false where they are not.
A prime number is conventionally defined as a positive integer greater than 1 (e.g.,
2, 3, . . . ) which is divisible only by itself and 1. Octave extends this definition to
include both negative integers and complex values. A negative integer is prime if its
positive counterpart is prime. This is equivalent to isprime (abs (x)).

Chapter 4: Numeric Data Types

65

If class (x) is complex, then primality is tested in the domain of Gaussian integers
(http://en.wikipedia.org/wiki/Gaussian_integer). Some non-complex integers
are prime in the ordinary sense, but not in the domain of Gaussian integers. For
example, 5 = (1 + 2i) ∗ (1 − 2i) shows that 5 is not prime because it has a factor other
than itself and 1. Exercise caution when testing complex and real values together in
the same matrix.
Examples:
isprime (1:6)
⇒ [0, 1, 1, 0, 1, 0]
isprime ([i, 2, 3, 5])
⇒ [0, 0, 1, 0]
Programming Note: isprime is appropriate if the maximum value in x is not too
large (< 1e15). For larger values special purpose factorization code should be used.
Compatibility Note: matlab does not extend the definition of prime numbers and will
produce an error if given negative or complex inputs.
See also: [primes], page 491, [factor], page 490, [gcd], page 490, [lcm], page 490.
If instead of knowing properties of variables, you wish to know which variables are
defined and to gather other information about the workspace itself, see Section 7.3 [Status
of Variables], page 129.

67

5 Strings
A string constant consists of a sequence of characters enclosed in either double-quote or
single-quote marks. For example, both of the following expressions
"parrot"
’parrot’
represent the string whose contents are ‘parrot’. Strings in Octave can be of any length.
Since the single-quote mark is also used for the transpose operator (see Section 8.3
[Arithmetic Ops], page 144) but double-quote marks have no other purpose in Octave, it is
best to use double-quote marks to denote strings.
Strings can be concatenated using the notation for defining matrices. For example, the
expression
[ "foo" , "bar" , "baz" ]
produces the string whose contents are ‘foobarbaz’. See Chapter 4 [Numeric Data Types],
page 47, for more information about creating matrices.

5.1 Escape Sequences in String Constants
In double-quoted strings, the backslash character is used to introduce escape sequences that
represent other characters. For example, ‘\n’ embeds a newline character in a double-quoted
string and ‘\"’ embeds a double quote character. In single-quoted strings, backslash is not
a special character. Here is an example showing the difference:
toascii ("\n")
⇒ 10
toascii (’\n’)
⇒ [ 92 110 ]
Here is a table of all the escape sequences used in Octave (within double quoted strings).
They are the same as those used in the C programming language.
\\

Represents a literal backslash, ‘\’.

\"

Represents a literal double-quote character, ‘"’.

\’

Represents a literal single-quote character, ‘’’.

\0

Represents the null character, control-@, ASCII code 0.

\a

Represents the “alert” character, control-g, ASCII code 7.

\b

Represents a backspace, control-h, ASCII code 8.

\f

Represents a formfeed, control-l, ASCII code 12.

\n

Represents a newline, control-j, ASCII code 10.

\r

Represents a carriage return, control-m, ASCII code 13.

\t

Represents a horizontal tab, control-i, ASCII code 9.

\v

Represents a vertical tab, control-k, ASCII code 11.

\nnn

Represents the octal value nnn, where nnn are one to three digits between 0
and 7. For example, the code for the ASCII ESC (escape) character is ‘\033’.

68

GNU Octave

\xhh...

Represents the hexadecimal value hh, where hh are hexadecimal digits (‘0’
through ‘9’ and either ‘A’ through ‘F’ or ‘a’ through ‘f’). Like the same construct
in ansi C, the escape sequence continues until the first non-hexadecimal digit
is seen. However, using more than two hexadecimal digits produces undefined
results.

In a single-quoted string there is only one escape sequence: you may insert a single quote
character using two single quote characters in succession. For example,
’I can’’t escape’
⇒ I can’t escape
In scripts the two different string types can be distinguished if necessary by using is_
dq_string and is_sq_string.

is_dq_string (x)
Return true if x is a double-quoted character string.
See also: [is sq string], page 68, [ischar], page 68.

is_sq_string (x)
Return true if x is a single-quoted character string.
See also: [is dq string], page 68, [ischar], page 68.

5.2 Character Arrays
The string representation used by Octave is an array of characters, so internally the string
"dddddddddd" is actually a row vector of length 10 containing the value 100 in all places
(100 is the ASCII code of "d"). This lends itself to the obvious generalization to character
matrices. Using a matrix of characters, it is possible to represent a collection of same-length
strings in one variable. The convention used in Octave is that each row in a character matrix
is a separate string, but letting each column represent a string is equally possible.
The easiest way to create a character matrix is to put several strings together into a
matrix.
collection = [ "String #1"; "String #2" ];
This creates a 2-by-9 character matrix.
The function ischar can be used to test if an object is a character matrix.

ischar (x)
Return true if x is a character array.
See also: [isfloat], page 62, [isinteger], page 54, [islogical], page 62, [isnumeric], page 62,
[iscellstr], page 121, [isa], page 39.
To test if an object is a string (i.e., a character vector and not a character matrix) you
can use the ischar function in combination with the isvector function as in the following
example:

Chapter 5: Strings

69

ischar (collection)
⇒ 1
ischar (collection) && isvector (collection)
⇒ 0
ischar ("my string") && isvector ("my string")
⇒ 1
One relevant question is, what happens when a character matrix is created from strings
of different length. The answer is that Octave puts blank characters at the end of strings
shorter than the longest string. It is possible to use a different character than the blank
character using the string_fill_char function.

val = string_fill_char ()
old_val = string_fill_char (new_val)
string_fill_char (new_val, "local")
Query or set the internal variable used to pad all rows of a character matrix to the
same length.
The value must be a single character and the default is " " (a single space). For
example:
string_fill_char ("X");
[ "these"; "are"; "strings" ]
⇒ "theseXX"
"areXXXX"
"strings"
When called from inside a function with the "local" option, the variable is changed
locally for the function and any subroutines it calls. The original variable value is
restored when exiting the function.
This shows a problem with character matrices. It simply isn’t possible to represent
strings of different lengths. The solution is to use a cell array of strings, which is described
in Section 6.2.4 [Cell Arrays of Strings], page 120.

5.3 Creating Strings
The easiest way to create a string is, as illustrated in the introduction, to enclose a text
in double-quotes or single-quotes. It is however possible to create a string without actually
writing a text. The function blanks creates a string of a given length consisting only of
blank characters (ASCII code 32).

blanks (n)
Return a string of n blanks.
For example:

70

GNU Octave

blanks (10);
whos ans
⇒
Attr Name
==== ====
ans

Size
====
1x10

Bytes
=====
10

Class
=====
char

See also: [repmat], page 458.

5.3.1 Concatenating Strings
Strings can be concatenated using matrix notation (see Chapter 5 [Strings], page 67,
Section 5.2 [Character Arrays], page 68) which is often the most natural method. For
example:
fullname = [fname ".txt"];
email = ["<" user "@" domain ">"];
In each case it is easy to see what the final string will look like. This method is also the
most efficient. When using matrix concatenation the parser immediately begins joining the
strings without having to process the overhead of a function call and the input validation
of the associated function.
Nevertheless, there are several other functions for concatenating string objects which
can be useful in specific circumstances: char, strvcat, strcat, and cstrcat. Finally,
the general purpose concatenation functions can be used: see [cat], page 449, [horzcat],
page 449, and [vertcat], page 449.
• All string concatenation functions except cstrcat convert numerical input into character data by taking the corresponding ASCII character for each element, as in the
following example:
char ([98, 97, 110, 97, 110, 97])
⇒ banana

• char and strvcat concatenate vertically, while strcat and cstrcat concatenate horizontally. For example:
char ("an apple", "two pears")
⇒ an apple
two pears
strcat ("oc", "tave", " is", " good", " for you")
⇒ octave is good for you

• char generates an empty row in the output for each empty string in the input. strvcat,
on the other hand, eliminates empty strings.
char ("orange", "green", "", "red")
⇒ orange
green
red

Chapter 5: Strings

71

strvcat ("orange", "green", "", "red")
⇒ orange
green
red
• All string concatenation functions except cstrcat also accept cell array data (see
Section 6.2 [Cell Arrays], page 114). char and strvcat convert cell arrays into character arrays, while strcat concatenates within the cells of the cell arrays:
char ({"red", "green", "", "blue"})
⇒ red
green
blue
strcat ({"abc"; "ghi"}, {"def"; "jkl"})
⇒
{
[1,1] = abcdef
[2,1] = ghijkl
}
• strcat removes trailing white space in the arguments (except within cell arrays), while
cstrcat leaves white space untouched. Both kinds of behavior can be useful as can be
seen in the examples:
strcat (["dir1";"directory2"], ["/";"/"], ["file1";"file2"])
⇒ dir1/file1
directory2/file2
cstrcat (["thirteen apples"; "a banana"], [" 5$";" 1$"])
⇒ thirteen apples 5$
a banana
1$
Note that in the above example for cstrcat, the white space originates from the internal representation of the strings in a string array (see Section 5.2 [Character Arrays],
page 68).

char
char
char
char

(x)
(x, . . . )
(s1, s2, . . . )
(cell_array)
Create a string array from one or more numeric matrices, character matrices, or cell
arrays.
Arguments are concatenated vertically. The returned values are padded with blanks
as needed to make each row of the string array have the same length. Empty input
strings are significant and will concatenated in the output.
For numerical input, each element is converted to the corresponding ASCII character.
A range error results if an input is outside the ASCII range (0-255).
For cell arrays, each element is concatenated separately. Cell arrays converted through
char can mostly be converted back with cellstr. For example:

72

GNU Octave

char ([97, 98, 99], "", {"98", "99", 100}, "str1", ["ha", "lf"])
⇒ ["abc "
"
"
"98 "
"99 "
"d
"
"str1"
"half"]
See also: [strvcat], page 72, [cellstr], page 121.
(x)
(x, . . . )
(s1, s2, . . . )
(cell_array)
Create a character array from one or more numeric matrices, character matrices, or
cell arrays.

strvcat
strvcat
strvcat
strvcat

Arguments are concatenated vertically. The returned values are padded with blanks
as needed to make each row of the string array have the same length. Unlike char,
empty strings are removed and will not appear in the output.
For numerical input, each element is converted to the corresponding ASCII character.
A range error results if an input is outside the ASCII range (0-255).
For cell arrays, each element is concatenated separately. Cell arrays converted through
strvcat can mostly be converted back with cellstr. For example:
strvcat ([97, 98, 99], "", {"98", "99", 100}, "str1", ["ha", "lf"])
⇒ ["abc "
"98 "
"99 "
"d
"
"str1"
"half"]
See also: [char], page 71, [strcat], page 72, [cstrcat], page 73.

strcat (s1, s2, . . . )
Return a string containing all the arguments concatenated horizontally.
If the arguments are cell strings, strcat returns a cell string with the individual cells
concatenated. For numerical input, each element is converted to the corresponding
ASCII character. Trailing white space for any character string input is eliminated before the strings are concatenated. Note that cell string values do not have whitespace
trimmed.
For example:
strcat ("|", " leading space is preserved", "|")
⇒ | leading space is preserved|
strcat ("|", "trailing space is eliminated ", "|")
⇒ |trailing space is eliminated|

Chapter 5: Strings

73

strcat ("homogeneous space |", " ", "| is also eliminated")
⇒ homogeneous space || is also eliminated
s = [ "ab"; "cde" ];
strcat (s, s, s)
⇒
"ababab
"
"cdecdecde"
s = { "ab"; "cd " };
strcat (s, s, s)
⇒
{
[1,1] = ababab
[2,1] = cd cd cd
}
See also: [cstrcat], page 73, [char], page 71, [strvcat], page 72.

cstrcat (s1, s2, . . . )
Return a string containing all the arguments concatenated horizontally with trailing
white space preserved.
For example:
cstrcat ("ab
⇒ "ab

", "cd")
cd"

s = [ "ab"; "cde" ];
cstrcat (s, s, s)
⇒ "ab ab ab "
"cdecdecde"
See also: [strcat], page 72, [char], page 71, [strvcat], page 72.

5.3.2 Converting Numerical Data to Strings
Apart from the string concatenation functions (see Section 5.3.1 [Concatenating Strings],
page 70) which cast numerical data to the corresponding ASCII characters, there are several
functions that format numerical data as strings. mat2str and num2str convert real or
complex matrices, while int2str converts integer matrices. int2str takes the real part
of complex values and round fractional values to integer. A more flexible way to format
numerical data as strings is the sprintf function (see Section 14.2.4 [Formatted Output],
page 270, [sprintf], page 271).

s = mat2str (x, n)
s = mat2str (x, n, "class")
Format real, complex, and logical matrices as strings.
The returned string may be used to reconstruct the original matrix by using the eval
function.
The precision of the values is given by n. If n is a scalar then both real and imaginary
parts of the matrix are printed to the same precision. Otherwise n(1) defines the

74

GNU Octave

precision of the real part and n(2) defines the precision of the imaginary part. The
default for n is 15.
If the argument "class" is given then the class of x is included in the string in such
a way that eval will result in the construction of a matrix of the same class.
mat2str ([ -1/3 + i/7; 1/3 - i/7 ], [4 2])
⇒ "[-0.3333+0.14i;0.3333-0.14i]"
mat2str ([ -1/3 +i/7; 1/3 -i/7 ], [4 2])
⇒ "[-0.3333+0i 0+0.14i;0.3333+0i -0-0.14i]"
mat2str (int16 ([1 -1]), "class")
⇒ "int16([1 -1])"
mat2str (logical (eye (2)))
⇒ "[true false;false true]"
isequal (x, eval (mat2str (x)))
⇒ 1

See also: [sprintf], page 271, [num2str], page 74, [int2str], page 75.

num2str (x)
num2str (x, precision)
num2str (x, format)
Convert a number (or array) to a string (or a character array).
The optional second argument may either give the number of significant digits (precision) to be used in the output or a format template string (format) as in sprintf
(see Section 14.2.4 [Formatted Output], page 270). num2str can also process complex
numbers.
Examples:

Chapter 5: Strings

75

num2str (123.456)
⇒ "123.46"
num2str (123.456, 4)
⇒ "123.5"
s = num2str ([1, 1.34; 3, 3.56], "%5.1f")
⇒ s =
1.0 1.3
3.0 3.6
whos s
⇒
Attr Name
Size
==== ====
====
s
2x8

Bytes
=====
16

Class
=====
char

num2str (1.234 + 27.3i)
⇒ "1.234+27.3i"
The num2str function is not very flexible. For better control over the results, use
sprintf (see Section 14.2.4 [Formatted Output], page 270).
Programming Notes:
For matlab compatibility, leading spaces are stripped before returning the string.
Integers larger than flintmax may not be displayed correctly.
For complex x, the format string may only contain one output conversion specification
and nothing else. Otherwise, results will be unpredictable.
Any optional format specified by the programmer is used without modification. This
is in contrast to matlab which tampers with the format based on internal heuristics.
See also: [sprintf], page 271, [int2str], page 75, [mat2str], page 73.

int2str (n)
Convert an integer (or array of integers) to a string (or a character array).
int2str (123)
⇒ "123"
s = int2str ([1, 2, 3; 4, 5, 6])
⇒ s =
1 2 3
4 5 6
whos s
⇒
Attr Name
Size
Bytes
==== ====
====
=====
s
2x7
14
This function is not very flexible. For better control over the results,
(see Section 14.2.4 [Formatted Output], page 270).

Class
=====
char
use sprintf

76

GNU Octave

Programming Notes:
Non-integers are rounded to integers before display. Only the real part of complex
numbers is displayed.
See also: [sprintf], page 271, [num2str], page 74, [mat2str], page 73.

5.4 Comparing Strings
Since a string is a character array, comparisons between strings work element by element
as the following example shows:
GNU = "GNU’s Not UNIX";
spaces = (GNU == " ")
⇒ spaces =
0
0
0
0
0

1

0

0

0

1

0

0

0

0

To determine if two strings are identical it is necessary to use the strcmp function. It compares complete strings and is case sensitive. strncmp compares only the first N characters
(with N given as a parameter). strcmpi and strncmpi are the corresponding functions for
case-insensitive comparison.

strcmp (s1, s2)
Return 1 if the character strings s1 and s2 are the same, and 0 otherwise.
If either s1 or s2 is a cell array of strings, then an array of the same size is returned,
containing the values described above for every member of the cell array. The other
argument may also be a cell array of strings (of the same size or with only one
element), char matrix or character string.
Caution: For compatibility with matlab, Octave’s strcmp function returns 1 if the
character strings are equal, and 0 otherwise. This is just the opposite of the corresponding C library function.
See also: [strcmpi], page 77, [strncmp], page 76, [strncmpi], page 77.

strncmp (s1, s2, n)
Return 1 if the first n characters of strings s1 and s2 are the same, and 0 otherwise.
strncmp ("abce", "abcd", 3)
⇒ 1

If either s1 or s2 is a cell array of strings, then an array of the same size is returned,
containing the values described above for every member of the cell array. The other
argument may also be a cell array of strings (of the same size or with only one
element), char matrix or character string.
strncmp ("abce", {"abcd", "bca", "abc"}, 3)
⇒ [1, 0, 1]

Caution: For compatibility with matlab, Octave’s strncmp function returns 1 if
the character strings are equal, and 0 otherwise. This is just the opposite of the
corresponding C library function.
See also: [strncmpi], page 77, [strcmp], page 76, [strcmpi], page 77.

Chapter 5: Strings

77

strcmpi (s1, s2)
Return 1 if the character strings s1 and s2 are the same, disregarding case of alphabetic characters, and 0 otherwise.
If either s1 or s2 is a cell array of strings, then an array of the same size is returned,
containing the values described above for every member of the cell array. The other
argument may also be a cell array of strings (of the same size or with only one
element), char matrix or character string.
Caution: For compatibility with matlab, Octave’s strcmp function returns 1 if the
character strings are equal, and 0 otherwise. This is just the opposite of the corresponding C library function.
Caution: National alphabets are not supported.
See also: [strcmp], page 76, [strncmp], page 76, [strncmpi], page 77.

strncmpi (s1, s2, n)
Return 1 if the first n character of s1 and s2 are the same, disregarding case of
alphabetic characters, and 0 otherwise.
If either s1 or s2 is a cell array of strings, then an array of the same size is returned,
containing the values described above for every member of the cell array. The other
argument may also be a cell array of strings (of the same size or with only one
element), char matrix or character string.
Caution: For compatibility with matlab, Octave’s strncmpi function returns 1 if
the character strings are equal, and 0 otherwise. This is just the opposite of the
corresponding C library function.
Caution: National alphabets are not supported.
See also: [strncmp], page 76, [strcmp], page 76, [strcmpi], page 77.

5.5 Manipulating Strings
Octave supports a wide range of functions for manipulating strings. Since a string is just a
matrix, simple manipulations can be accomplished using standard operators. The following
example shows how to replace all blank characters with underscores.
quote = ...
"First things first, but not necessarily in that order";
quote( quote == " " ) = "_"
⇒ quote =
First_things_first,_but_not_necessarily_in_that_order
For more complex manipulations, such as searching, replacing, and general regular expressions, the following functions come with Octave.

deblank (s)
Remove trailing whitespace and nulls from s.
If s is a matrix, deblank trims each row to the length of longest string. If s is a cell
array of strings, operate recursively on each string element.

78

GNU Octave

Examples:
deblank ("
⇒ "

abc ")
abc"

deblank ([" abc
⇒ [" abc

"; "
" ; "

def
"])
def"]

See also: [strtrim], page 78.

strtrim (s)
Remove leading and trailing whitespace from s.
If s is a matrix, strtrim trims each row to the length of longest string. If s is a cell
array of strings, operate recursively on each string element.
For example:
strtrim ("
abc ")
⇒ "abc"
strtrim ([" abc
"; "
⇒ ["abc " ; "

def
def"]

"])

See also: [deblank], page 77.

strtrunc (s, n)
Truncate the character string s to length n.
If s is a character matrix, then the number of columns is adjusted.
If s is a cell array of strings, then the operation is performed on each cell element and
the new cell array is returned.

findstr (s, t)
findstr (s, t, overlap)
Return the vector of all positions in the longer of the two strings s and t where an
occurrence of the shorter of the two starts.
If the optional argument overlap is true (default), the returned vector can include
overlapping positions. For example:
findstr ("ababab", "a")
⇒ [1, 3, 5];
findstr ("abababa", "aba", 0)
⇒ [1, 5]
Caution: findstr is scheduled for deprecation. Use strfind in all new code.
See also: [strfind], page 79, [strmatch], page 80, [strcmp], page 76, [strncmp], page 76,
[strcmpi], page 77, [strncmpi], page 77, [find], page 445.

idx
idx
idx
[i,

= strchr (str, chars)
= strchr (str, chars, n)
= strchr (str, chars, n, direction)
j] = strchr ( . . . )
Search for the string str for occurrences of characters from the set chars.

Chapter 5: Strings

79

The return value(s), as well as the n and direction arguments behave identically as in
find.
This will be faster than using regexp in most cases.
See also: [find], page 445.

index (s, t)
index (s, t, direction)
Return the position of the first occurrence of the string t in the string s, or 0 if no
occurrence is found.
s may also be a string array or cell array of strings.
For example:
index ("Teststring", "t")
⇒ 4
If direction is "first", return the first element found. If direction is "last", return
the last element found.
See also: [find], page 445, [rindex], page 79.

rindex (s, t)
Return the position of the last occurrence of the character string t in the character
string s, or 0 if no occurrence is found.
s may also be a string array or cell array of strings.
For example:
rindex ("Teststring", "t")
⇒ 6
The rindex function is equivalent to index with direction set to "last".
See also: [find], page 445, [index], page 79.

idx = strfind (str, pattern)
idx = strfind (cellstr, pattern)
idx = strfind ( . . . , "overlaps", val)
Search for pattern in the string str and return the starting index of every such occurrence in the vector idx.
If there is no such occurrence, or if pattern is longer than str, or if pattern itself is
empty, then idx is the empty array [].
The optional argument "overlaps" determines whether the pattern can match at
every position in str (true), or only for unique occurrences of the complete pattern
(false). The default is true.
If a cell array of strings cellstr is specified then idx is a cell array of vectors, as
specified above.
Examples:

80

GNU Octave

strfind ("abababa", "aba")
⇒ [1, 3, 5]
strfind ("abababa", "aba", "overlaps", false)
⇒ [1, 5]
strfind ({"abababa", "bebebe", "ab"}, "aba")
⇒
{
[1,1] =
1

3

5

[1,2] = [](1x0)
[1,3] = [](1x0)
}
See also: [findstr], page 78, [strmatch], page 80, [regexp], page 87, [regexpi], page 90,
[find], page 445.

str = strjoin (cstr)
str = strjoin (cstr, delimiter)
Join the elements of the cell string array, cstr, into a single string.
If no delimiter is specified, the elements of cstr are separated by a space.
If delimiter is specified as a string, the cell string array is joined using the string.
Escape sequences are supported.
If delimiter is a cell string array whose length is one less than cstr, then the elements of
cstr are joined by interleaving the cell string elements of delimiter. Escape sequences
are not supported.
strjoin ({’Octave’,’Scilab’,’Lush’,’Yorick’}, ’*’)
⇒ ’Octave*Scilab*Lush*Yorick’
See also: [strsplit], page 81.

strmatch (s, A)
strmatch (s, A, "exact")
Return indices of entries of A which begin with the string s.
The second argument A must be a string, character matrix, or a cell array of strings.
If the third argument "exact" is not given, then s only needs to match A up to the
length of s. Trailing spaces and nulls in s and A are ignored when matching.
For example:

Chapter 5: Strings

81

strmatch ("apple", "apple juice")
⇒ 1
strmatch ("apple", ["apple
⇒ [1; 2]

"; "apple juice"; "an apple"])

strmatch ("apple", ["apple
⇒ [1]

"; "apple juice"; "an apple"], "exact")

Caution: strmatch is scheduled for deprecation. Use strncmp (normal case), or
strcmp ("exact" case), or regexp in all new code.
See also: [strfind], page 79, [findstr], page 78, [strcmp], page 76, [strncmp], page 76,
[strcmpi], page 77, [strncmpi], page 77, [find], page 445.

[tok, rem] = strtok (str)
[tok, rem] = strtok (str, delim)
Find all characters in the string str up to, but not including, the first character which
is in the string delim.
str may also be a cell array of strings in which case the function executes on every
individual string and returns a cell array of tokens and remainders.
Leading delimiters are ignored. If delim is not specified, whitespace is assumed.
If rem is requested, it contains the remainder of the string, starting at the first delimiter.
Examples:
strtok ("this is the life")
⇒ "this"
[tok, rem] = strtok ("14*27+31", "+-*/")
⇒
tok = 14
rem = *27+31
See also: [index], page 79, [strsplit], page 81, [strchr], page 78, [isspace], page 97.
(str)
(str, del)
( . . . , name, value)
strsplit ( . . . )
Split the string str using the delimiters specified by del and return a cell string array
of substrings.

[cstr]
[cstr]
[cstr]
[cstr,

= strsplit
= strsplit
= strsplit
matches] =

If a delimiter is not specified the string is split at whitespace {" ", "\f", "\n",
"\r", "\t", "\v"}. Otherwise, the delimiter, del must be a string or cell array of
strings. By default, consecutive delimiters in the input string s are collapsed into one
resulting in a single split.
Supported name/value pair arguments are:
• collapsedelimiters which may take the value of true (default) or false.

82

GNU Octave

• delimitertype which may take the value of "simple" (default) or
"regularexpression". A simple delimiter matches the text exactly as written.
Otherwise, the syntax for regular expressions outlined in regexp is used.
The optional second output, matches, returns the delimiters which were matched in
the original string.
Examples with simple delimiters:
strsplit ("a b c")
⇒
{
[1,1] = a
[1,2] = b
[1,3] = c
}
strsplit ("a,b,c", ",")
⇒
{
[1,1] = a
[1,2] = b
[1,3] = c
}
strsplit ("a foo b,bar c", {" ", ",", "foo", "bar"})
⇒
{
[1,1] = a
[1,2] = b
[1,3] = c
}
strsplit ("a,,b, c", {",", " "}, "collapsedelimiters", false)
⇒
{
[1,1] = a
[1,2] =
[1,3] = b
[1,4] =
[1,5] = c
}
Examples with regularexpression delimiters:
strsplit ("a foo b,bar c", ’,|\s|foo|bar’, "delimitertype", "regularexpression")
⇒
{
[1,1] = a
[1,2] = b
[1,3] = c

Chapter 5: Strings

83

}
strsplit ("a,,b, c", ’[, ]’, "collapsedelimiters", false, "delimitertype", "regularexpression")
⇒
{
[1,1] = a
[1,2] =
[1,3] = b
[1,4] =
[1,5] = c
}
strsplit ("a,\t,b, c", {’,’, ’\s’}, "delimitertype", "regularexpression")
⇒
{
[1,1] = a
[1,2] = b
[1,3] = c
}
strsplit ("a,\t,b, c", {’,’, ’ ’, ’\t’}, "collapsedelimiters", false)
⇒
{
[1,1] = a
[1,2] =
[1,3] =
[1,4] = b
[1,5] =
[1,6] = c
}

See also: [ostrsplit], page 83, [strjoin], page 80, [strtok], page 81, [regexp], page 87.

[cstr] = ostrsplit (s, sep)
[cstr] = ostrsplit (s, sep, strip_empty)
Split the string s using one or more separators sep and return a cell array of strings.

Consecutive separators and separators at boundaries result in empty strings, unless
strip empty is true. The default value of strip empty is false.

2-D character arrays are split at separators and at the original column boundaries.

Example:

84

GNU Octave

ostrsplit ("a,b,c",
⇒
{
[1,1] =
[1,2] =
[1,3] =
}
ostrsplit (["a,b"
⇒
{
[1,1]
[1,2]
[1,3]
}

",")

a
b
c

; "cde"], ",")

= a
= b
= cde

See also: [strsplit], page 81, [strtok], page 81.

[a,
[a,
[a,
[a,
[a,

(str)
(str, format)
(str, format, format_repeat)
(str, format, prop1, value1, . . . )
(str, format, format_repeat, prop1, value1, . . . )
Read data from a string.
The string str is split into words that are repeatedly matched to the specifiers in
format. The first word is matched to the first specifier, the second to the second
specifier and so forth. If there are more words than specifiers, the process is repeated
until all words have been processed.
The string format describes how the words in str should be parsed. It may contain
any combination of the following specifiers:

...]
...]
...]
...]
...]

=
=
=
=
=

strread
strread
strread
strread
strread

%s

The word is parsed as a string.

%f
%n

The word is parsed as a number and converted to double.

%d
%u

The word is parsed as a number and converted to int32.

%*
%*f
%*s

literals

The word is skipped.
For %s and %d, %f, %n, %u and the associated %*s . . . specifiers an
optional width can be specified as %Ns, etc. where N is an integer > 1.
For %f, format specifiers like %N.Mf are allowed.
In addition the format may contain literal character strings; these will be
skipped during reading.

Parsed word corresponding to the first specifier are returned in the first output argument and likewise for the rest of the specifiers.

Chapter 5: Strings

85

By default, format is "%f", meaning that numbers are read from str. This will do if
str contains only numeric fields.
For example, the string
str = "\
Bunny Bugs
5.5\n\
Duck Daffy -7.5e-5\n\
Penguin Tux
6"
can be read using
[a, b, c] = strread (str, "%s %s %f");
Optional numeric argument format repeat can be used for limiting the number of
items read:
-1

(default) read all of the string until the end.

N

Read N times nargout items. 0 (zero) is an acceptable value for format repeat.

The behavior of strread can be changed via property-value pairs. The following
properties are recognized:
"commentstyle"
Parts of str are considered comments and will be skipped. value is the
comment style and can be any of the following.
• "shell" Everything from # characters to the nearest end-of-line is
skipped.
• "c" Everything between /* and */ is skipped.
• "c++" Everything from // characters to the nearest end-of-line is
skipped.
• "matlab" Everything from % characters to the nearest end-of-line is
skipped.
• user-supplied. Two options: (1) One string, or 1x1 cell string: Skip
everything to the right of it; (2) 2x1 cell string array: Everything
between the left and right strings is skipped.
"delimiter"
Any character in value will be used to split str into words (default value
= any whitespace). Note that whitespace is implicitly added to the set
of delimiter characters unless a "%s" format conversion specifier is supplied; see "whitespace" parameter below. The set of delimiter characters
cannot be empty; if needed Octave substitutes a space as delimiter.
"emptyvalue"
Value to return for empty numeric values in non-whitespace delimited
data. The default is NaN. When the data type does not support NaN
(int32 for example), then default is zero.
"multipledelimsasone"
Treat a series of consecutive delimiters, without whitespace in between,
as a single delimiter. Consecutive delimiter series need not be vertically
"aligned".

86

GNU Octave

"treatasempty"
Treat single occurrences (surrounded by delimiters or whitespace) of the
string(s) in value as missing values.
"returnonerror"
If value true (1, default), ignore read errors and return normally. If false
(0), return an error.
"whitespace"
Any character in value will be interpreted as whitespace and trimmed; the
string defining whitespace must be enclosed in double quotes for proper
processing of special characters like "\t". In each data field, multiple
consecutive whitespace characters are collapsed into one space and leading
and trailing whitespace is removed. The default value for whitespace is
" \b\r\n\t" (note the space). Whitespace is always added to the set of
delimiter characters unless at least one "%s" format conversion specifier is
supplied; in that case only whitespace explicitly specified in "delimiter"
is retained as delimiter and removed from the set of whitespace characters.
If whitespace characters are to be kept as-is (in e.g., strings), specify an
empty value (i.e., "") for "whitespace"; obviously, whitespace cannot be
a delimiter then.
When the number of words in str doesn’t match an exact multiple of the number of
format conversion specifiers, strread’s behavior depends on the last character of str:
last character = "\n"
Data columns are padded with empty fields or Nan so that all columns
have equal length
last character is not "\n"
Data columns are not padded; strread returns columns of unequal length
See also: [textscan], page 260, [textread], page 258, [load], page 255, [dlmread],
page 258, [fscanf], page 276.

newstr = strrep (str, ptn, rep)
newstr = strrep (cellstr, ptn, rep)
newstr = strrep ( . . . , "overlaps", val)
Replace all occurrences of the pattern ptn in the string str with the string rep and
return the result.
The optional argument "overlaps" determines whether the pattern can match at
every position in str (true), or only for unique occurrences of the complete pattern
(false). The default is true.
s may also be a cell array of strings, in which case the replacement is done for each
element and a cell array is returned.
Example:
strrep ("This is a test string", "is", "&%$")
⇒ "Th&%$ &%$ a test string"

See also: [regexprep], page 90, [strfind], page 79, [findstr], page 78.

Chapter 5: Strings

87

substr (s, offset)
substr (s, offset, len)
Return the substring of s which starts at character number offset and is len characters
long.
Position numbering for offsets begins with 1. If offset is negative, extraction starts
that far from the end of the string.
If len is omitted, the substring extends to the end of s. A negative value for len
extracts to within len characters of the end of the string
Examples:
substr ("This is a test string", 6, 9)
⇒ "is a test"
substr ("This is a test string", -11)
⇒ "test string"
substr ("This is a test string", -11, -7)
⇒ "test"

This function is patterned after the equivalent function in Perl.

[s, e, te, m, t, nm, sp] = regexp (str, pat)
[...] = regexp (str, pat, "opt1", . . . )
Regular expression string matching.
Search for pat in str and return the positions and substrings of any matches, or empty
values if there are none.
The matched pattern pat can include any of the standard regex operators, including:
.

Match any character

* + ? {}

Repetition operators, representing
*

Match zero or more times

+

Match one or more times

?

Match zero or one times

{n}

Match exactly n times

{n,}

Match n or more times

{m,n}

Match between m and n times

[...] [^...]
List operators. The pattern will match any character listed between "["
and "]". If the first character is "^" then the pattern is inverted and any
character except those listed between brackets will match.
Escape sequences defined below can also be used inside list operators.
For example, a template for a floating point number might be [-+.\d]+.
() (?:)

Grouping operator. The first form, parentheses only, also creates a token.

|

Alternation operator. Match one of a choice of regular expressions. The
alternatives must be delimited by the grouping operator () above.

88

GNU Octave

^$

Anchoring operators. Requires pattern to occur at the start (^) or end
($) of the string.

In addition, the following escaped characters have special meaning.
\d

Match any digit

\D

Match any non-digit

\s

Match any whitespace character

\S

Match any non-whitespace character

\w

Match any word character

\W

Match any non-word character

\<

Match the beginning of a word

\>

Match the end of a word

\B

Match within a word

Implementation Note: For compatibility with matlab, escape sequences in pat (e.g.,
"\n" => newline) are expanded even when pat has been defined with single quotes.
To disable expansion use a second backslash before the escape sequence (e.g., "\\n")
or use the regexptranslate function.
The outputs of regexp default to the order given below
s

The start indices of each matching substring

e

The end indices of each matching substring

te

The extents of each matched token surrounded by (...) in pat

m

A cell array of the text of each match

t

A cell array of the text of each token matched

nm

A structure containing the text of each matched named token, with
the name being used as the fieldname. A named token is denoted by
(?...).

sp

A cell array of the text not returned by match, i.e., what remains if you
split the string based on pat.

Particular output arguments, or the order of the output arguments, can be selected
by additional opt arguments. These are strings and the correspondence between the
output arguments and the optional argument are
’start’
s
’end’
e
’tokenExtents’
te
’match’
m
’tokens’
t
’names’
nm
’split’
sp
Additional arguments are summarized below.
‘once’

Return only the first occurrence of the pattern.

Chapter 5: Strings

89

‘matchcase’
Make the matching case sensitive. (default)
Alternatively, use (?-i) in the pattern.
‘ignorecase’
Ignore case when matching the pattern to the string.
Alternatively, use (?i) in the pattern.
‘stringanchors’
Match the anchor characters at the beginning and end of the string.
(default)
Alternatively, use (?-m) in the pattern.
‘lineanchors’
Match the anchor characters at the beginning and end of the line.
Alternatively, use (?m) in the pattern.
‘dotall’

The pattern . matches all characters including the newline character.
(default)
Alternatively, use (?s) in the pattern.

‘dotexceptnewline’
The pattern . matches all characters except the newline character.
Alternatively, use (?-s) in the pattern.
‘literalspacing’
All characters in the pattern, including whitespace, are significant and
are used in pattern matching. (default)
Alternatively, use (?-x) in the pattern.
‘freespacing’
The pattern may include arbitrary whitespace and also comments beginning with the character ‘#’.
Alternatively, use (?x) in the pattern.
‘noemptymatch’
Zero-length matches are not returned. (default)
‘emptymatch’
Return zero-length matches.
regexp (’a’, ’b*’, ’emptymatch’) returns [1 2] because there are
zero or more ’b’ characters at positions 1 and end-of-string.
Stack Limitation Note: Pattern searches are done with a recursive function which can
overflow the program stack when there are a high number of matches. For example,
regexp (repmat (’a’, 1, 1e5), ’(a)+’)
may lead to a segfault. As an alternative, consider constructing pattern searches that
reduce the number of matches (e.g., by creatively using set complement), and then
further processing the return variables (now reduced in size) with successive regexp
searches.
See also: [regexpi], page 90, [strfind], page 79, [regexprep], page 90.

90

GNU Octave

[s, e, te, m, t, nm, sp] = regexpi (str, pat)
[...] = regexpi (str, pat, "opt1", . . . )
Case insensitive regular expression string matching.
Search for pat in str and return the positions and substrings of any matches, or empty
values if there are none. See [regexp], page 87, for details on the syntax of the search
pattern.
See also: [regexp], page 87.

outstr = regexprep (string, pat, repstr)
outstr = regexprep (string, pat, repstr, "opt1", . . . )
Replace occurrences of pattern pat in string with repstr.
The pattern is a regular expression as documented for regexp. See [regexp], page 87.
The replacement string may contain $i, which substitutes for the ith set of parentheses
in the match string. For example,
regexprep ("Bill Dunn", ’(\w+) (\w+)’, ’$2, $1’)
returns "Dunn, Bill"
Options in addition to those of regexp are
‘once’

Replace only the first occurrence of pat in the result.

‘warnings’
This option is present for compatibility but is ignored.
Implementation Note: For compatibility with matlab, escape sequences in pat (e.g.,
"\n" => newline) are expanded even when pat has been defined with single quotes.
To disable expansion use a second backslash before the escape sequence (e.g., "\\n")
or use the regexptranslate function.
See also: [regexp], page 87, [regexpi], page 90, [strrep], page 86.

regexptranslate (op, s)
Translate a string for use in a regular expression.
This may include either wildcard replacement or special character escaping.
The behavior is controlled by op which can take the following values
"wildcard"
The wildcard characters ., *, and ? are replaced with wildcards that are
appropriate for a regular expression. For example:
regexptranslate ("wildcard", "*.m")
⇒ ’.*\.m’
"escape"

The characters $.?[], that have special meaning for regular expressions
are escaped so that they are treated literally. For example:
regexptranslate ("escape", "12.5")
⇒ ’12\.5’

See also: [regexp], page 87, [regexpi], page 90, [regexprep], page 90.

Chapter 5: Strings

91

untabify (t)
untabify (t, tw)
untabify (t, tw, deblank)
Replace TAB characters in t with spaces.
The input, t, may be either a 2-D character array, or a cell array of character strings.
The output is the same class as the input.
The tab width is specified by tw, and defaults to eight.
If the optional argument deblank is true, then the spaces will be removed from the
end of the character data.
The following example reads a file and writes an untabified version of the same file
with trailing spaces stripped.
fid = fopen ("tabbed_script.m");
text = char (fread (fid, "uchar")’);
fclose (fid);
fid = fopen ("untabified_script.m", "w");
text = untabify (strsplit (text, "\n"), 8, true);
fprintf (fid, "%s\n", text{:});
fclose (fid);
See also: [strjust], page 94, [strsplit], page 81, [deblank], page 77.

5.6 String Conversions
Octave supports various kinds of conversions between strings and numbers. As an example,
it is possible to convert a string containing a hexadecimal number to a floating point number.
hex2dec ("FF")
⇒ 255

bin2dec (s)
Return the decimal number corresponding to the binary number represented by the
string s.
For example:
bin2dec ("1110")
⇒ 14

Spaces are ignored during conversion and may be used to make the binary number
more readable.
bin2dec ("1000 0001")
⇒ 129

If s is a string matrix, return a column vector with one converted number per row of
s; Invalid rows evaluate to NaN.
If s is a cell array of strings, return a column vector with one converted number per
cell element in s.
See also: [dec2bin], page 92, [base2dec], page 93, [hex2dec], page 92.

92

GNU Octave

dec2bin (d, len)
Return a binary number corresponding to the non-negative integer d, as a string of
ones and zeros.
For example:
dec2bin (14)
⇒ "1110"
If d is a matrix or cell array, return a string matrix with one row per element in d,
padded with leading zeros to the width of the largest value.
The optional second argument, len, specifies the minimum number of digits in the
result.
See also: [bin2dec], page 91, [dec2base], page 92, [dec2hex], page 92.

dec2hex (d, len)
Return the hexadecimal string corresponding to the non-negative integer d.
For example:
dec2hex (2748)
⇒ "ABC"
If d is a matrix or cell array, return a string matrix with one row per element in d,
padded with leading zeros to the width of the largest value.
The optional second argument, len, specifies the minimum number of digits in the
result.
See also: [hex2dec], page 92, [dec2base], page 92, [dec2bin], page 92.

hex2dec (s)
Return the integer corresponding to the hexadecimal number represented by the string
s.
For example:
hex2dec ("12B")
⇒ 299
hex2dec ("12b")
⇒ 299
If s is a string matrix, return a column vector with one converted number per row of
s; Invalid rows evaluate to NaN.
If s is a cell array of strings, return a column vector with one converted number per
cell element in s.
See also: [dec2hex], page 92, [base2dec], page 93, [bin2dec], page 91.

dec2base (d, base)
dec2base (d, base, len)
Return a string of symbols in base base corresponding to the non-negative integer d.
dec2base (123, 3)
⇒ "11120"
If d is a matrix or cell array, return a string matrix with one row per element in d,
padded with leading zeros to the width of the largest value.

Chapter 5: Strings

93

If base is a string then the characters of base are used as the symbols for the digits
of d. Space (’ ’) may not be used as a symbol.
dec2base (123, "aei")
⇒ "eeeia"
The optional third argument, len, specifies the minimum number of digits in the
result.
See also: [base2dec], page 93, [dec2bin], page 92, [dec2hex], page 92.

base2dec (s, base)
Convert s from a string of digits in base base to a decimal integer (base 10).
base2dec ("11120", 3)
⇒ 123
If s is a string matrix, return a column vector with one value per row of s. If a row
contains invalid symbols then the corresponding value will be NaN.
If s is a cell array of strings, return a column vector with one value per cell element
in s.
If base is a string, the characters of base are used as the symbols for the digits of s.
Space (’ ’) may not be used as a symbol.
base2dec ("yyyzx", "xyz")
⇒ 123
See also: [dec2base], page 92, [bin2dec], page 91, [hex2dec], page 92.

s = num2hex (n)
Typecast a double or single precision number or vector to a 8 or 16 character hexadecimal string of the IEEE 754 representation of the number.
For example:
num2hex ([-1, 1, e, Inf])
⇒ "bff0000000000000
3ff0000000000000
4005bf0a8b145769
7ff0000000000000"
If the argument n is a single precision number or vector, the returned string has a
length of 8. For example:
num2hex (single ([-1, 1, e, Inf]))
⇒ "bf800000
3f800000
402df854
7f800000"
See also: [hex2num], page 93, [hex2dec], page 92, [dec2hex], page 92.

n = hex2num (s)
n = hex2num (s, class)
Typecast the 16 character hexadecimal character string to an IEEE 754 double precision number.

94

GNU Octave

If fewer than 16 characters are given the strings are right padded with ’0’ characters.
Given a string matrix, hex2num treats each row as a separate number.
hex2num (["4005bf0a8b145769"; "4024000000000000"])
⇒ [2.7183; 10.000]
The optional argument class can be passed as the string "single" to specify that the
given string should be interpreted as a single precision number. In this case, s should
be an 8 character hexadecimal string. For example:
hex2num (["402df854"; "41200000"], "single")
⇒ [2.7183; 10.000]
See also: [num2hex], page 93, [hex2dec], page 92, [dec2hex], page 92.

str2double (s)
Convert a string to a real or complex number.
The string must be in one of the following formats where a and b are real numbers
and the complex unit is ’i’ or ’j’:
• a + bi
• a + b*i
• a + i*b
• bi + a
• b*i + a
• i*b + a
If present, a and/or b are of the form [+-]d[,.]d[[eE][+-]d] where the brackets indicate
optional arguments and ’d’ indicates zero or more digits. The special input values
Inf, NaN, and NA are also accepted.
s may be a character string, character matrix, or cell array. For character arrays
the conversion is repeated for every row, and a double or complex array is returned.
Empty rows in s are deleted and not returned in the numeric array. For cell arrays
each character string element is processed and a double or complex array of the same
dimensions as s is returned.
For unconvertible scalar or character string input str2double returns a NaN. Similarly, for character array input str2double returns a NaN for any row of s that could
not be converted. For a cell array, str2double returns a NaN for any element of s
for which conversion fails. Note that numeric elements in a mixed string/numeric cell
array are not strings and the conversion will fail for these elements and return NaN.
str2double can replace str2num, and it avoids the security risk of using eval on
unknown data.
See also: [str2num], page 95.

strjust (s)
strjust (s, pos)
Return the text, s, justified according to pos, which may be "left", "center", or
"right".
If pos is omitted it defaults to "right".

Chapter 5: Strings

95

Null characters are replaced by spaces. All other character data are treated as nonwhite space.
Example:
strjust (["a"; "ab"; "abc"; "abcd"])
⇒
"
a"
" ab"
" abc"
"abcd"
See also: [deblank], page 77, [strrep], page 86, [strtrim], page 78, [untabify], page 91.

x = str2num (s)
[x, state] = str2num (s)
Convert the string (or character array) s to a number (or an array).
Examples:
str2num ("3.141596")
⇒ 3.141596
str2num (["1, 2, 3"; "4, 5, 6"])
⇒ 1 2 3
4 5 6
The optional second output, state, is logically true when the conversion is successful.
If the conversion fails the numeric output, x, is empty and state is false.
Caution: As str2num uses the eval function to do the conversion, str2num will
execute any code contained in the string s. Use str2double for a safer and faster
conversion.
For cell array of strings use str2double.
See also: [str2double], page 94, [eval], page 159.

toascii (s)
Return ASCII representation of s in a matrix.
For example:
toascii ("ASCII")
⇒ [ 65, 83, 67, 73, 73 ]
See also: [char], page 71.

tolower (s)
lower (s)
Return a copy of the string or cell string s, with each uppercase character replaced
by the corresponding lowercase one; non-alphabetic characters are left unchanged.
For example:
tolower ("MiXeD cAsE 123")
⇒ "mixed case 123"

See also: [toupper], page 96.

96

GNU Octave

toupper (s)
upper (s)
Return a copy of the string or cell string s, with each lowercase character replaced by
the corresponding uppercase one; non-alphabetic characters are left unchanged.
For example:
toupper ("MiXeD cAsE 123")
⇒ "MIXED CASE 123"
See also: [tolower], page 95.

do_string_escapes (string)
Convert escape sequences in string to the characters they represent.
Escape sequences begin with a leading backslash (’\’) followed by 1–3 characters
(.e.g., "\n" => newline).
See also: [undo string escapes], page 96.

undo_string_escapes (s)
Convert special characters in strings back to their escaped forms.
For example, the expression
bell = "\a";
assigns the value of the alert character (control-g, ASCII code 7) to the string variable
bell. If this string is printed, the system will ring the terminal bell (if it is possible).
This is normally the desired outcome. However, sometimes it is useful to be able to
print the original representation of the string, with the special characters replaced by
their escape sequences. For example,
octave:13> undo_string_escapes (bell)
ans = \a
replaces the unprintable alert character with its printable representation.
See also: [do string escapes], page 96.

5.7 Character Class Functions
Octave also provides the following character class test functions patterned after the functions
in the standard C library. They all operate on string arrays and return matrices of zeros and
ones. Elements that are nonzero indicate that the condition was true for the corresponding
character in the string array. For example:
isalpha ("!Q@WERT^Y&")
⇒ [ 0, 1, 0, 1, 1, 1, 1, 0, 1, 0 ]

isalnum (s)
Return a logical array which is true where the elements of s are letters or digits and
false where they are not.
This is equivalent to (isalpha (s) | isdigit (s)).
See also: [isalpha], page 97, [isdigit], page 97, [ispunct], page 97, [isspace], page 97,
[iscntrl], page 98.

Chapter 5: Strings

97

isalpha (s)
Return a logical array which is true where the elements of s are letters and false where
they are not.
This is equivalent to (islower (s) | isupper (s)).
See also: [isdigit], page 97, [ispunct], page 97, [isspace], page 97, [iscntrl], page 98,
[isalnum], page 96, [islower], page 97, [isupper], page 97.

isletter (s)
Return a logical array which is true where the elements of s are letters and false where
they are not.
This is an alias for the isalpha function.
See also: [isalpha], page 97, [isdigit], page 97, [ispunct], page 97, [isspace], page 97,
[iscntrl], page 98, [isalnum], page 96.

islower (s)
Return a logical array which is true where the elements of s are lowercase letters and
false where they are not.
See also: [isupper], page 97, [isalpha], page 97, [isletter], page 97, [isalnum], page 96.

isupper (s)
Return a logical array which is true where the elements of s are uppercase letters and
false where they are not.
See also: [islower], page 97, [isalpha], page 97, [isletter], page 97, [isalnum], page 96.

isdigit (s)
Return a logical array which is true where the elements of s are decimal digits (0-9)
and false where they are not.
See also: [isxdigit], page 97, [isalpha], page 97, [isletter], page 97, [ispunct], page 97,
[isspace], page 97, [iscntrl], page 98.

isxdigit (s)
Return a logical array which is true where the elements of s are hexadecimal digits
(0-9 and a-fA-F).
See also: [isdigit], page 97.

ispunct (s)
Return a logical array which is true where the elements of s are punctuation characters
and false where they are not.
See also: [isalpha], page 97, [isdigit], page 97, [isspace], page 97, [iscntrl], page 98.

isspace (s)
Return a logical array which is true where the elements of s are whitespace characters
(space, formfeed, newline, carriage return, tab, and vertical tab) and false where they
are not.
See also: [iscntrl], page 98, [ispunct], page 97, [isalpha], page 97, [isdigit], page 97.

98

GNU Octave

iscntrl (s)
Return a logical array which is true where the elements of s are control characters
and false where they are not.
See also: [ispunct], page 97, [isspace], page 97, [isalpha], page 97, [isdigit], page 97.

isgraph (s)
Return a logical array which is true where the elements of s are printable characters
(but not the space character) and false where they are not.
See also: [isprint], page 98.

isprint (s)
Return a logical array which is true where the elements of s are printable characters
(including the space character) and false where they are not.
See also: [isgraph], page 98.

isascii (s)
Return a logical array which is true where the elements of s are ASCII characters (in
the range 0 to 127 decimal) and false where they are not.

isstrprop (str, prop)
Test character string properties.
For example:
isstrprop ("abc123", "alpha")
⇒ [1, 1, 1, 0, 0, 0]

If str is a cell array, isstrpop is applied recursively to each element of the cell array.
Numeric arrays are converted to character strings.
The second argument prop must be one of
"alpha"

True for characters that are alphabetic (letters).

"alnum"
"alphanum"
True for characters that are alphabetic or digits.
"lower"

True for lowercase letters.

"upper"

True for uppercase letters.

"digit"

True for decimal digits (0-9).

"xdigit"

True for hexadecimal digits (a-fA-F0-9).

"space"
"wspace"

True for whitespace characters (space, formfeed, newline, carriage return,
tab, vertical tab).

"punct"

True for punctuation characters (printing characters except space or letter
or digit).

"cntrl"

True for control characters.

Chapter 5: Strings

99

"graph"
"graphic"
True for printing characters except space.
"print"

True for printing characters including space.

"ascii"

True for characters that are in the range of ASCII encoding.

See also: [isalpha], page 97, [isalnum], page 96, [islower], page 97, [isupper], page 97,
[isdigit], page 97, [isxdigit], page 97, [isspace], page 97, [ispunct], page 97, [iscntrl],
page 98, [isgraph], page 98, [isprint], page 98, [isascii], page 98.

101

6 Data Containers

Octave includes support for two different mechanisms to contain arbitrary data types in
the same variable. Structures, which are C-like, and are indexed with named fields, and
cell arrays, where each element of the array can have a different data type and or shape.
Multiple input arguments and return values of functions are organized as another data
container, the comma separated list.

6.1 Structures
Octave includes support for organizing data in structures. The current implementation
uses an associative array with indices limited to strings, but the syntax is more like C-style
structures.

6.1.1 Basic Usage and Examples
Here are some examples of using data structures in Octave.
Elements of structures can be of any value type. For example, the three expressions
x.a = 1;
x.b = [1, 2; 3, 4];
x.c = "string";
create a structure with three elements. The ‘.’ character separates the structure name from
the field name and indicates to Octave that this variable is a structure. To print the value
of the structure you can type its name, just as for any other variable:
x
⇒ x =
{
a = 1
b =
1
3

2
4

c = string
}
Note that Octave may print the elements in any order.
Structures may be copied just like any other variable:

102

GNU Octave

y = x
⇒ y =
{
a = 1
b =
1
3

2
4

c = string
}

Since structures are themselves values, structure elements may reference other structures.
The following statements change the value of the element b of the structure x to be a data
structure containing the single element d, which has a value of 3.

x.b.d = 3;
x.b
⇒ ans =
{
d = 3
}
x
⇒ x =
{
a = 1
b =
{
d = 3
}
c = string
}

Note that when Octave prints the value of a structure that contains other structures,
only a few levels are displayed. For example:

Chapter 6: Data Containers

103

a.b.c.d.e = 1;
a
⇒ a =
{
b =
{
c =
{
1x1 struct array containing the fields:
d: 1x1 struct
}
}
}
This prevents long and confusing output from large deeply nested structures. The number
of levels to print for nested structures may be set with the function struct_levels_to_
print, and the function print_struct_array_contents may be used to enable printing
of the contents of structure arrays.

val = struct_levels_to_print ()
old_val = struct_levels_to_print (new_val)
struct_levels_to_print (new_val, "local")
Query or set the internal variable that specifies the number of structure levels to
display.
When called from inside a function with the "local" option, the variable is changed
locally for the function and any subroutines it calls. The original variable value is
restored when exiting the function.
See also: [print struct array contents], page 103.

val = print_struct_array_contents ()
old_val = print_struct_array_contents (new_val)
print_struct_array_contents (new_val, "local")
Query or set the internal variable that specifies whether to print struct array contents.
If true, values of struct array elements are printed. This variable does not affect scalar
structures whose elements are always printed. In both cases, however, printing will
be limited to the number of levels specified by struct levels to print.
When called from inside a function with the "local" option, the variable is changed
locally for the function and any subroutines it calls. The original variable value is
restored when exiting the function.
See also: [struct levels to print], page 103.
Functions can return structures. For example, the following function separates the real
and complex parts of a matrix and stores them in two elements of the same structure
variable.

104

GNU Octave

function y = f (x)
y.re = real (x);
y.im = imag (x);
endfunction
When called with a complex-valued argument, f returns the data structure containing
the real and imaginary parts of the original function argument.
f (rand (2) + rand (2) * I)
⇒ ans =
{
im =
0.26475
0.18436

0.14828
0.83669

re =
0.040239
0.238081

0.242160
0.402523

}
Function return lists can include structure elements, and they may be indexed like any
other variable. For example:
[ x.u, x.s(2:3,2:3), x.v ] = svd ([1, 2; 3, 4]);
x
⇒ x =
{
u =
-0.40455
-0.91451

-0.91451
0.40455

s =
0.00000
0.00000
0.00000

0.00000
5.46499
0.00000

0.00000
0.00000
0.36597

v =
-0.57605
-0.81742

0.81742
-0.57605

}
It is also possible to cycle through all the elements of a structure in a loop, using a
special form of the for statement (see Section 10.5.1 [Looping Over Structure Elements],
page 169).

Chapter 6: Data Containers

105

6.1.2 Structure Arrays
A structure array is a particular instance of a structure, where each of the fields of the
structure is represented by a cell array. Each of these cell arrays has the same dimensions.
Conceptually, a structure array can also be seen as an array of structures with identical
fields. An example of the creation of a structure array is
x(1).a
x(2).a
x(1).b
x(2).b

=
=
=
=

"string1";
"string2";
1;
2;

which creates a 1-by-2 structure array with two fields. Another way to create a structure
array is with the struct function (see Section 6.1.3 [Creating Structures], page 106). As
previously, to print the value of the structure array, you can type its name:
x
⇒ x =
{
1x2 struct array containing the fields:
a
b
}
Individual elements of the structure array can be returned by indexing the variable like
x(1), which returns a structure with two fields:
x(1)
⇒ ans =
{
a = string1
b = 1
}
Furthermore, the structure array can return a comma separated list of field values (see
Section 6.3 [Comma Separated Lists], page 122), if indexed by one of its own field names.
For example:
x.a
⇒

ans = string1
ans = string2

Here is another example, using this comma separated list on the left-hand side of an
assignment:
[x.a] = deal ("new string1", "new string2");
x(1).a
⇒ ans = new string1
x(2).a
⇒ ans = new string2

Just as for numerical arrays, it is possible to use vectors as indices (see Section 8.1 [Index
Expressions], page 137):

106

GNU Octave

x(3:4) = x(1:2);
[x([1,3]).a] = deal ("other string1", "other string2");
x.a
⇒
ans = other string1
ans = new string2
ans = other string2
ans = new string2
The function size will return the size of the structure. For the example above
size (x)
⇒ ans =
1
4
Elements can be deleted from a structure array in a similar manner to a numerical array,
by assigning the elements to an empty matrix. For example
in = struct ("call1", {x, Inf, "last"},
"call2", {x, Inf, "first"})
⇒ in =
{
1x3 struct array containing the fields:
call1
call2
}
in(1) = [];
in.call1
⇒
ans = Inf
ans = last

6.1.3 Creating Structures
Besides the index operator ".", Octave can use dynamic naming "(var)" or the struct
function to create structures. Dynamic naming uses the string value of a variable as the
field name. For example:
a = "field2";
x.a = 1;
x.(a) = 2;
x
⇒ x =
{
a = 1
field2 = 2
}
Dynamic indexing also allows you to use arbitrary strings, not merely valid Octave identifiers
(note that this does not work on matlab):

Chapter 6: Data Containers

a = "long field with spaces (and funny char$)";
x.a = 1;
x.(a) = 2;
x
⇒ x =
{
a = 1
long field with spaces (and funny char$) =
}

107

2

The warning id Octave:language-extension can be enabled to warn about this usage.
See [warning ids], page 225.
More realistically, all of the functions that operate on strings can be used to build the
correct field name before it is entered into the data structure.
names = ["Bill"; "Mary"; "John"];
ages = [37; 26; 31];
for i = 1:rows (names)
database.(names(i,:)) = ages(i);
endfor
database
⇒ database =
{
Bill = 37
Mary = 26
John = 31
}
The third way to create structures is the struct command. struct takes pairs of
arguments, where the first argument in the pair is the fieldname to include in the structure
and the second is a scalar or cell array, representing the values to include in the structure
or structure array. For example:
struct ("field1", 1, "field2", 2)
⇒ ans =
{
field1 = 1
field2 = 2
}
If the values passed to struct are a mix of scalar and cell arrays, then the scalar arguments are expanded to create a structure array with a consistent dimension. For example:

108

GNU Octave

s = struct ("field1", {1, "one"}, "field2", {2, "two"},
"field3", 3);
s.field1
⇒
ans = 1
ans = one
s.field2
⇒
ans = 2
ans = two
s.field3
⇒
ans =
ans =

3
3

If you want to create a struct which contains a cell array as an individual field, you must
wrap it in another cell array as shown in the following example:
struct ("field1", {{1, "one"}}, "field2", 2)
⇒ ans =
{
field1 =
{
[1,1] = 1
[1,2] = one
}
field2 =

2

}

s = struct ()
s = struct (field1, value1, field2, value2, . . . )
s = struct (obj)
Create a scalar or array structure and initialize its values.
The field1, field2, . . . variables are strings specifying the names of the fields and the
value1, value2, . . . variables can be of any type.
If the values are cell arrays, create a structure array and initialize its values. The
dimensions of each cell array of values must match. Singleton cells and non-cell values
are repeated so that they fill the entire array. If the cells are empty, create an empty
structure array with the specified field names.
If the argument is an object, return the underlying struct.
Observe that the syntax is optimized for struct arrays. Consider the following examples:

Chapter 6: Data Containers

109

struct ("foo", 1)
⇒ scalar structure containing the fields:
foo = 1
struct ("foo", {})
⇒ 0x0 struct array containing the fields:
foo
struct ("foo", { {} })
⇒ scalar structure containing the fields:
foo = {}(0x0)
struct ("foo", {1, 2, 3})
⇒ 1x3 struct array containing the fields:
foo
The first case is an ordinary scalar struct—one field, one value. The second produces
an empty struct array with one field and no values, since being passed an empty cell
array of struct array values. When the value is a cell array containing a single entry,
this becomes a scalar struct with that single entry as the value of the field. That
single entry happens to be an empty cell array.
Finally, if the value is a non-scalar cell array, then struct produces a struct array.
See also: [cell2struct], page 122, [fieldnames], page 109, [getfield], page 111, [setfield],
page 110, [rmfield], page 111, [isfield], page 110, [orderfields], page 111, [isstruct],
page 109, [structfun], page 542.
The function isstruct can be used to test if an object is a structure or a structure
array.

isstruct (x)
Return true if x is a structure or a structure array.
See also: [ismatrix], page 63, [iscell], page 115, [isa], page 39.

6.1.4 Manipulating Structures
Other functions that can manipulate the fields of a structure are given below.

numfields (s)
Return the number of fields of the structure s.
See also: [fieldnames], page 109.
(struct)
(obj)
(javaobj)
("javaclassname")
Return a cell array of strings with the names of the fields in the specified input.

names
names
names
names

=
=
=
=

fieldnames
fieldnames
fieldnames
fieldnames

When the input is a structure struct, the names are the elements of the structure.

110

GNU Octave

When the input is an Octave object obj, the names are the public properties of the
object.
When the input is a Java object javaobj or a string containing the name of a Java
class javaclassname, the names are the public fields (data members) of the object or
class.
See also: [numfields], page 109, [isfield], page 110, [orderfields], page 111, [struct],
page 108, [methods], page 777.

isfield (x, "name")
isfield (x, name)
Return true if the x is a structure and it includes an element named name.
If name is a cell array of strings then a logical array of equal dimension is returned.
See also: [fieldnames], page 109.

sout = setfield (s, field, val)
sout = setfield (s, sidx1, field1, fidx1, sidx2, field2, fidx2, . . . , val)
Return a copy of the structure s with the field member field set to the value val.
For example:
s = struct ();
s = setfield (s, "foo bar", 42);
This is equivalent to
s.("foo bar") = 42;
Note that ordinary structure syntax s.foo bar = 42 cannot be used here, as the field
name is not a valid Octave identifier because of the space character. Using arbitrary
strings for field names is incompatible with matlab, and this usage will emit a warning if the warning ID Octave:language-extension is enabled. See [warning ids],
page 225.
With the second calling form, set a field of a structure array. The input sidx selects
an element of the structure array, field specifies the field name of the selected element,
and fidx selects which element of the field (in the case of an array or cell array). The
sidx, field, and fidx inputs can be repeated to address nested structure array elements.
The structure array index and field element index must be cell arrays while the field
name must be a string.
For example:
s = struct ("baz", 42);
setfield (s, {1}, "foo", {1}, "bar", 54)
⇒
ans =
scalar structure containing the fields:
baz = 42
foo =
scalar structure containing the fields:
bar = 54
The example begins with an ordinary scalar structure to which a nested scalar structure is added. In all cases, if the structure index sidx is not specified it defaults to

Chapter 6: Data Containers

111

1 (scalar structure). Thus, the example above could be written more concisely as
setfield (s, "foo", "bar", 54)
Finally, an example with nested structure arrays:
sa.foo = 1;
sa = setfield (sa, {2}, "bar", {3}, "baz", {1, 4}, 5);
sa(2).bar(3)
⇒
ans =
scalar structure containing the fields:
baz = 0
0
0
5
Here sa is a structure array whose field at elements 1 and 2 is in turn another structure
array whose third element is a simple scalar structure. The terminal scalar structure
has a field which contains a matrix value.
Note that the same result as in the above example could be achieved by:
sa.foo = 1;
sa(2).bar(3).baz(1,4) = 5
See also: [getfield], page 111, [rmfield], page 111, [orderfields], page 111, [isfield],
page 110, [fieldnames], page 109, [isstruct], page 109, [struct], page 108.

val = getfield (s, field)
val = getfield (s, sidx1, field1, fidx1, . . . )
Get the value of the field named field from a structure or nested structure s.
If s is a structure array then sidx selects an element of the structure array, field
specifies the field name of the selected element, and fidx selects which element of
the field (in the case of an array or cell array). See setfield for a more complete
description of the syntax.
See also: [setfield], page 110, [rmfield], page 111, [orderfields], page 111, [isfield],
page 110, [fieldnames], page 109, [isstruct], page 109, [struct], page 108.

sout = rmfield (s, "f")
sout = rmfield (s, f)
Return a copy of the structure (array) s with the field f removed.
If f is a cell array of strings or a character array, remove each of the named fields.
See also: [orderfields], page 111, [fieldnames], page 109, [isfield], page 110.

sout =
sout =
sout =
sout =
[sout,

orderfields (s1)
orderfields (s1, s2)
orderfields (s1, {cellstr})
orderfields (s1, p)
p] = orderfields ( . . . )

Return a copy of s1 with fields arranged alphabetically, or as specified by the second
input.
Given one input struct s1, arrange field names alphabetically.
If a second struct argument is given, arrange field names in s1 as they appear in s2.
The second argument may also specify the order in a cell array of strings cellstr. The
second argument may also be a permutation vector.

112

GNU Octave

The optional second output argument p is the permutation vector which converts the
original name order to the new name order.
Examples:
s = struct ("d", 4, "b", 2, "a", 1, "c", 3);
t1 = orderfields (s)
⇒ t1 =
{
a = 1
b = 2
c = 3
d = 4
}
t = struct ("d", {}, "c", {}, "b", {}, "a", {});
t2 = orderfields (s, t)
⇒ t2 =
{
d = 4
c = 3
b = 2
a = 1
}
t3 = orderfields (s, [3, 2, 4, 1])
⇒ t3 =
{
a = 1
b = 2
c = 3
d = 4
}
[t4, p] = orderfields (s, {"d", "c", "b", "a"})
⇒ t4 =
{
d = 4
c = 3
b = 2
a = 1
}
p =
1
4
2
3
See also: [fieldnames], page 109, [getfield], page 111, [setfield], page 110, [rmfield],
page 111, [isfield], page 110, [isstruct], page 109, [struct], page 108.

Chapter 6: Data Containers

113

substruct (type, subs, . . . )
Create a subscript structure for use with subsref or subsasgn.
For example:

idx = substruct ("()", {3, ":"})
⇒
idx =
{
type = ()
subs =
{
[1,1] = 3
[1,2] = :
}
}
x = [1, 2, 3;
4, 5, 6;
7, 8, 9];
subsref (x, idx)
⇒ 7 8 9
See also: [subsref], page 781, [subsasgn], page 782.

6.1.5 Processing Data in Structures
The simplest way to process data in a structure is within a for loop (see Section 10.5.1
[Looping Over Structure Elements], page 169). A similar effect can be achieved with the
structfun function, where a user defined function is applied to each field of the structure.
See [structfun], page 542.
Alternatively, to process the data in a structure, the structure might be converted to
another type of container before being treated.

c = struct2cell (s)
Create a new cell array from the objects stored in the struct object.
If f is the number of fields in the structure, the resulting cell array will have a
dimension vector corresponding to [f size(s)]. For example:

114

GNU Octave

s = struct ("name", {"Peter", "Hannah", "Robert"},
"age", {23, 16, 3});
c = struct2cell (s)
⇒ c = {2x1x3 Cell Array}
c(1,1,:)(:)
⇒
{
[1,1] = Peter
[2,1] = Hannah
[3,1] = Robert
}
c(2,1,:)(:)
⇒
{
[1,1] = 23
[2,1] = 16
[3,1] = 3
}
See also: [cell2struct], page 122, [fieldnames], page 109.

6.2 Cell Arrays
It can be both necessary and convenient to store several variables of different size or type
in one variable. A cell array is a container class able to do just that. In general cell arrays
work just like N -dimensional arrays with the exception of the use of ‘{’ and ‘}’ as allocation
and indexing operators.

6.2.1 Basic Usage of Cell Arrays
As an example, the following code creates a cell array containing a string and a 2-by-2
random matrix
c = {"a string", rand(2, 2)};
To access the elements of a cell array, it can be indexed with the { and } operators. Thus,
the variable created in the previous example can be indexed like this:
c{1}
⇒ ans = a string

As with numerical arrays several elements of a cell array can be extracted by indexing with
a vector of indexes
c{1:2}
⇒ ans = a string
⇒ ans =
0.593993
0.377037

0.627732
0.033643

The indexing operators can also be used to insert or overwrite elements of a cell array.
The following code inserts the scalar 3 on the third place of the previously created cell array

Chapter 6: Data Containers

115

c{3} = 3
⇒ c =
{
[1,1] = a string
[1,2] =
0.593993
0.377037
[1,3] =

0.627732
0.033643

3

}
Details on indexing cell arrays are explained in Section 6.2.3 [Indexing Cell Arrays],
page 118.
In general nested cell arrays are displayed hierarchically as in the previous example.
In some circumstances it makes sense to reference them by their index, and this can be
performed by the celldisp function.

celldisp (c)
celldisp (c, name)
Recursively display the contents of a cell array.
By default the values are displayed with the name of the variable c. However, this
name can be replaced with the variable name. For example:
c = {1, 2, {31, 32}};
celldisp (c, "b")
⇒
b{1} =
1
b{2} =
2
b{3}{1} =
31
b{3}{2} =
32
See also: [disp], page 245.
To test if an object is a cell array, use the iscell function. For example:
iscell (c)
⇒ ans = 1
iscell (3)
⇒ ans = 0

iscell (x)
Return true if x is a cell array object.
See also: [ismatrix], page 63, [isstruct], page 109, [iscellstr], page 121, [isa], page 39.

116

GNU Octave

6.2.2 Creating Cell Arrays
The introductory example (see Section 6.2.1 [Basic Usage of Cell Arrays], page 114) showed
how to create a cell array containing currently available variables. In many situations,
however, it is useful to create a cell array and then fill it with data.
The cell function returns a cell array of a given size, containing empty matrices. This
function is similar to the zeros function for creating new numerical arrays. The following
example creates a 2-by-2 cell array containing empty matrices
c = cell (2,2)
⇒ c =
{
[1,1]
[2,1]
[1,2]
[2,2]

=
=
=
=

[](0x0)
[](0x0)
[](0x0)
[](0x0)

}
Just like numerical arrays, cell arrays can be multi-dimensional. The cell function
accepts any number of positive integers to describe the size of the returned cell array. It is
also possible to set the size of the cell array through a vector of positive integers. In the
following example two cell arrays of equal size are created, and the size of the first one is
displayed
c1 = cell (3, 4, 5);
c2 = cell ( [3, 4, 5] );
size (c1)
⇒ ans =
3
4
5
As can be seen, the [size], page 45, function also works for cell arrays. As do other functions
describing the size of an object, such as [length], page 45, [numel], page 44, [rows], page 44,
and [columns], page 44.

cell
cell
cell
cell

(n)
(m, n)
(m, n, k, . . . )
([m n . . . ])
Create a new cell array object.
If invoked with a single scalar integer argument, return a square NxN cell array. If
invoked with two or more scalar integer arguments, or a vector of integer values,
return an array with the given dimensions.
See also: [cellstr], page 121, [mat2cell], page 117, [num2cell], page 116, [struct2cell],
page 113.

As an alternative to creating empty cell arrays, and then filling them, it is possible to
convert numerical arrays into cell arrays using the num2cell, mat2cell and cellslices
functions.

Chapter 6: Data Containers

117

C = num2cell (A)
C = num2cell (A, dim)
Convert the numeric matrix A to a cell array.
If dim is defined, the value C is of dimension 1 in this dimension and the elements of
A are placed into C in slices. For example:
num2cell ([1,2;3,4])
⇒
{
[1,1] = 1
[2,1] = 3
[1,2] = 2
[2,2] = 4
}
num2cell ([1,2;3,4],1)
⇒
{
[1,1] =
1
3
[1,2] =
2
4
}
See also: [mat2cell], page 117.

C = mat2cell (A, m, n)
C = mat2cell (A, d1, d2, . . . )
C = mat2cell (A, r)
Convert the matrix A to a cell array.
If A is 2-D, then it is required that sum (m) == size (A, 1) and sum (n) == size (A,
2). Similarly, if A is multi-dimensional and the number of dimensional arguments is
equal to the dimensions of A, then it is required that sum (di) == size (A, i).
Given a single dimensional argument r, the other dimensional arguments are assumed
to equal size (A,i).
An example of the use of mat2cell is
mat2cell (reshape (1:16,4,4), [3,1], [3,1])
⇒
{
[1,1] =
1
2
3

5
6
7

[2,1] =

9
10
11

118

GNU Octave

4

8

12

[1,2] =
13
14
15
[2,2] = 16
}
See also: [num2cell], page 116, [cell2mat], page 121.

sl = cellslices (x, lb, ub, dim)
Given an array x, this function produces a cell array of slices from the array determined by the index vectors lb, ub, for lower and upper bounds, respectively.
In other words, it is equivalent to the following code:
n = length (lb);
sl = cell (1, n);
for i = 1:length (lb)
sl{i} = x(:,...,lb(i):ub(i),...,:);
endfor
The position of the index is determined by dim. If not specified, slicing is done along
the first non-singleton dimension.
See also: [cell2mat], page 121, [cellindexmat], page 120, [cellfun], page 540.

6.2.3 Indexing Cell Arrays
As shown in see Section 6.2.1 [Basic Usage of Cell Arrays], page 114, elements can be
extracted from cell arrays using the ‘{’ and ‘}’ operators. If you want to extract or access
subarrays which are still cell arrays, you need to use the ‘(’ and ‘)’ operators. The following
example illustrates the difference:
c = {"1", "2", "3"; "x", "y", "z"; "4", "5", "6"};
c{2,3}
⇒ ans = z
c(2,3)
⇒ ans =
{
[1,1] = z
}
So with ‘{}’ you access elements of a cell array, while with ‘()’ you access a sub array of a
cell array.
Using the ‘(’ and ‘)’ operators, indexing works for cell arrays like for multi-dimensional
arrays. As an example, all the rows of the first and third column of a cell array can be set
to 0 with the following command:

Chapter 6: Data Containers

119

c(:, [1, 3]) = {0}
⇒ =
{
[1,1] = 0
[2,1] = 0
[3,1] = 0
[1,2] = 2
[2,2] = y
[3,2] = 5
[1,3] = 0
[2,3] = 0
[3,3] = 0
}
Note, that the above can also be achieved like this:
c(:, [1, 3]) = 0;
Here, the scalar ‘0’ is automatically promoted to cell array ‘{0}’ and then assigned to the
subarray of c.
To give another example for indexing cell arrays with ‘()’, you can exchange the first
and the second row of a cell array as in the following command:
c = {1, 2, 3; 4, 5, 6};
c([1, 2], :) = c([2, 1], :)
⇒ =
{
[1,1] = 4
[2,1] = 1
[1,2] = 5
[2,2] = 2
[1,3] = 6
[2,3] = 3
}
Accessing multiple elements of a cell array with the ‘{’ and ‘}’ operators will result in
a comma-separated list of all the requested elements (see Section 6.3 [Comma Separated
Lists], page 122). Using the ‘{’ and ‘}’ operators the first two rows in the above example
can be swapped back like this:
[c{[1,2], :}] =
⇒ =
{
[1,1]
[2,1]
[1,2]
[2,2]
[1,3]
[2,3]
}

deal (c{[2, 1], :})

=
=
=
=
=
=

1
4
2
5
3
6

120

GNU Octave

As for struct arrays and numerical arrays, the empty matrix ‘[]’ can be used to delete
elements from a cell array:
x = {"1", "2"; "3", "4"};
x(1, :) = []
⇒ x =
{
[1,1] = 3
[1,2] = 4
}
The following example shows how to just remove the contents of cell array elements but
not delete the space for them:
x = {"1", "2"; "3", "4"};
x(1, :) = {[]}
⇒ x =
{
[1,1] = [](0x0)
[2,1] = 3
[1,2] = [](0x0)
[2,2] = 4
}
The indexing operations operate on the cell array and not on the objects within the cell
array. By contrast, cellindexmat applies matrix indexing to the objects within each cell
array entry and returns the requested values.

y = cellindexmat (x, varargin)
Perform indexing of matrices in a cell array.
Given a cell array of matrices x, this function computes
Y = cell (size (X));
for i = 1:numel (X)
Y{i} = X{i}(varargin{1}, varargin{2}, ..., varargin{N});
endfor
The indexing arguments may be scalar (2), arrays ([1, 3]), ranges (1:3), or the colon
operator (":"). However, the indexing keyword end is not available.
See also: [cellslices], page 118, [cellfun], page 540.

6.2.4 Cell Arrays of Strings
One common use of cell arrays is to store multiple strings in the same variable. It is also
possible to store multiple strings in a character matrix by letting each row be a string. This,
however, introduces the problem that all strings must be of equal length. Therefore, it is
recommended to use cell arrays to store multiple strings. For cases, where the character
matrix representation is required for an operation, there are several functions that convert
a cell array of strings to a character array and back. char and strvcat convert cell arrays
to a character array (see Section 5.3.1 [Concatenating Strings], page 70), while the function
cellstr converts a character array to a cell array of strings:

Chapter 6: Data Containers

121

a = ["hello"; "world"];
c = cellstr (a)
⇒ c =
{
[1,1] = hello
[2,1] = world
}

cstr = cellstr (strmat)
Create a new cell array object from the elements of the string array strmat.
Each row of strmat becomes an element of cstr. Any trailing spaces in a row are
deleted before conversion.
To convert back from a cellstr to a character array use char.
See also: [cell], page 116, [char], page 71.
One further advantage of using cell arrays to store multiple strings is that most functions
for string manipulations included with Octave support this representation. As an example,
it is possible to compare one string with many others using the strcmp function. If one
of the arguments to this function is a string and the other is a cell array of strings, each
element of the cell array will be compared to the string argument:
c = {"hello", "world"};
strcmp ("hello", c)
⇒ ans =
1
0
The following string functions support cell arrays of strings: char, strvcat, strcat (see
Section 5.3.1 [Concatenating Strings], page 70), strcmp, strncmp, strcmpi, strncmpi (see
Section 5.4 [Comparing Strings], page 76), str2double, deblank, strtrim, strtrunc,
strfind, strmatch, , regexp, regexpi (see Section 5.5 [Manipulating Strings], page 77)
and str2double (see Section 5.6 [String Conversions], page 91).
The function iscellstr can be used to test if an object is a cell array of strings.

iscellstr (cell)
Return true if every element of the cell array cell is a character string.
See also: [ischar], page 68.

6.2.5 Processing Data in Cell Arrays
Data that is stored in a cell array can be processed in several ways depending on the actual
data. The simplest way to process that data is to iterate through it using one or more
for loops. The same idea can be implemented more easily through the use of the cellfun
function that calls a user-specified function on all elements of a cell array. See [cellfun],
page 540.
An alternative is to convert the data to a different container, such as a matrix or a data
structure. Depending on the data this is possible using the cell2mat and cell2struct
functions.

122

GNU Octave

m = cell2mat (c)
Convert the cell array c into a matrix by concatenating all elements of c into a
hyperrectangle.
Elements of c must be numeric, logical, or char matrices; or cell arrays; or structs;
and cat must be able to concatenate them together.
See also: [mat2cell], page 117, [num2cell], page 116.

cell2struct (cell, fields)
cell2struct (cell, fields, dim)
Convert cell to a structure.
The number of fields in fields must match the number of elements in cell along dimension dim, that is numel (fields) == size (cell, dim). If dim is omitted, a value of
1 is assumed.
A = cell2struct ({"Peter", "Hannah", "Robert";
185, 170, 168},
{"Name","Height"}, 1);
A(1)
⇒
{
Name
= Peter
Height = 185
}
See also: [struct2cell], page 113, [cell2mat], page 121, [struct], page 108.

6.3 Comma Separated Lists
Comma separated lists1 are the basic argument type to all Octave functions - both for input
and return arguments. In the example
max (a, b)
‘a, b’ is a comma separated list. Comma separated lists can appear on both the right and
left hand side of an assignment. For example
x = [1 0 1 0 0 1 1; 0 0 0 0 0 0 7];
[i, j] = find (x, 2, "last");
Here, ‘x, 2, "last"’ is a comma separated list constituting the input arguments of find.
find returns a comma separated list of output arguments which is assigned element by
element to the comma separated list ‘i, j’.
Another example of where comma separated lists are used is in the creation of a new
array with [] (see Section 4.1 [Matrices], page 48) or the creation of a cell array with {}
(see Section 6.2.1 [Basic Usage of Cell Arrays], page 114). In the expressions
a = [1, 2, 3, 4];
c = {4, 5, 6, 7};
both ‘1, 2, 3, 4’ and ‘4, 5, 6, 7’ are comma separated lists.
1

Comma-separated lists are also sometimes informally referred to as cs-lists.

Chapter 6: Data Containers

123

Comma separated lists cannot be directly manipulated by the user. However, both
structure arrays and cell arrays can be converted into comma separated lists, and thus used
in place of explicitly written comma separated lists. This feature is useful in many ways,
as will be shown in the following subsections.

6.3.1 Comma Separated Lists Generated from Cell Arrays
As has been mentioned above (see Section 6.2.3 [Indexing Cell Arrays], page 118), elements
of a cell array can be extracted into a comma separated list with the { and } operators. By
surrounding this list with [ and ], it can be concatenated into an array. For example:
a = {1, [2, 3], 4, 5, 6};
b = [a{1:4}]
⇒ b =
1
2
3
4
5
Similarly, it is possible to create a new cell array containing cell elements selected with
{}. By surrounding the list with ‘{’ and ‘}’ a new cell array will be created, as the following
example illustrates:
a = {1, rand(2, 2), "three"};
b = { a{ [1, 3] } }
⇒ b =
{
[1,1] = 1
[1,2] = three
}
Furthermore, cell elements (accessed by {}) can be passed directly to a function. The
list of elements from the cell array will be passed as an argument list to a given function
as if it is called with the elements as individual arguments. The two calls to printf in the
following example are identical but the latter is simpler and can handle cell arrays of an
arbitrary size:
c = {"GNU",
printf ("%s
a GNU
printf ("%s
a GNU

"Octave", "is", "Free", "Software"};
", c{1}, c{2}, c{3}, c{4}, c{5});
Octave is Free Software
", c{:});
Octave is Free Software

If used on the left-hand side of an assignment, a comma separated list generated with
{} can be assigned to. An example is

124

GNU Octave

in{1} = [10, 20, 30];
in{2} = inf;
in{3} = "last";
in{4} = "first";
out = cell (4, 1);
[out{1:3}] = in{1 : 3};
[out{4:6}] = in{[1, 2, 4]})
⇒ out =
{
[1,1] =
10

20

30

[2,1] = Inf
[3,1] = last
[4,1] =
10

20

30

[5,1] = Inf
[6,1] = first
}

6.3.2 Comma Separated Lists Generated from Structure Arrays
Structure arrays can equally be used to create comma separated lists. This is done by
addressing one of the fields of a structure array. For example:
x = ceil (randn (10, 1));
in = struct ("call1", {x, 3, "last"},
"call2", {x, inf, "first"});
out = struct ("call1", cell (2, 1), "call2", cell (2, 1));
[out.call1] = find (in.call1);
[out.call2] = find (in.call2);

125

7 Variables
Variables let you give names to values and refer to them later. You have already seen
variables in many of the examples. The name of a variable must be a sequence of letters,
digits and underscores, but it may not begin with a digit. Octave does not enforce a limit
on the length of variable names, but it is seldom useful to have variables with names longer
than about 30 characters. The following are all valid variable names
x
x15
__foo_bar_baz__
fucnrdthsucngtagdjb
However, names like __foo_bar_baz__ that begin and end with two underscores are understood to be reserved for internal use by Octave. You should not use them in code you write,
except to access Octave’s documented internal variables and built-in symbolic constants.
Case is significant in variable names. The symbols a and A are distinct variables.
A variable name is a valid expression by itself. It represents the variable’s current value.
Variables are given new values with assignment operators and increment operators. See
Section 8.6 [Assignment Expressions], page 153.
There is one built-in variable with a special meaning. The ans variable always contains
the result of the last computation, where the output wasn’t assigned to any variable. The
code a = cos (pi) will assign the value -1 to the variable a, but will not change the value
of ans. However, the code cos (pi) will set the value of ans to -1.
Variables in Octave do not have fixed types, so it is possible to first store a numeric
value in a variable and then to later use the same name to hold a string value in the same
program. Variables may not be used before they have been given a value. Doing so results
in an error.

ans

[Automatic Variable]
The most recently computed result that was not explicitly assigned to a variable.
For example, after the expression
3^2 + 4^2
is evaluated, the value returned by ans is 25.

isvarname (name)
Return true if name is a valid variable name.
See also: [iskeyword], page 961, [exist], page 132, [who], page 130.

varname = genvarname (str)
varname = genvarname (str, exclusions)
Create valid unique variable name(s) from str.
If str is a cellstr, then a unique variable is created for each cell in str.
genvarname ({"foo", "foo"})
⇒
{
[1,1] = foo
[1,2] = foo1
}

126

GNU Octave

If exclusions is given, then the variable(s) will be unique to each other and to exclusions (exclusions may be either a string or a cellstr).
x = 3.141;
genvarname ("x", who ())
⇒ x1

Note that the result is a char array or cell array of strings, not the variables themselves.
To define a variable, eval() can be used. The following trivial example sets x to 42.
name = genvarname ("x");
eval ([name " = 42"]);
⇒ x = 42

This can be useful for creating unique struct field names.
x = struct ();
for i = 1:3
x.(genvarname ("a", fieldnames (x))) = i;
endfor
⇒ x =
{
a = 1
a1 = 2
a2 = 3
}
Since variable names may only contain letters, digits, and underscores, genvarname
will replace any sequence of disallowed characters with an underscore. Also, variables
may not begin with a digit; in this case an ‘x’ is added before the variable name.
Variable names beginning and ending with two underscores "__" are valid, but they
are used internally by Octave and should generally be avoided; therefore, genvarname
will not generate such names.
genvarname will also ensure that returned names do not clash with keywords such as
"for" and "if". A number will be appended if necessary. Note, however, that this
does not include function names such as "sin". Such names should be included in
exclusions if necessary.
See also: [isvarname], page 125, [iskeyword], page 961, [exist], page 132, [who],
page 130, [tempname], page 283, [eval], page 159.

namelengthmax ()
Return the matlab compatible maximum variable name length.
Octave is capable of storing strings up to 231 − 1 in length. However for matlab
compatibility all variable, function, and structure field names should be shorter than
the length returned by namelengthmax. In particular, variables stored to a matlab
file format (*.mat) will have their names truncated to this length.

7.1 Global Variables
A variable that has been declared global may be accessed from within a function body
without having to pass it as a formal parameter.

Chapter 7: Variables

127

A variable may be declared global using a global declaration statement. The following
statements are all global declarations.
global a
global a b
global c = 2
global d = 3 e f = 5
A global variable may only be initialized once in a global statement. For example, after
executing the following code
global gvar = 1
global gvar = 2
the value of the global variable gvar is 1, not 2. Issuing a ‘clear gvar’ command does not
change the above behavior, but ‘clear all’ does.
It is necessary declare a variable as global within a function body in order to access it.
For example,
global x
function f ()
x = 1;
endfunction
f ()
does not set the value of the global variable x to 1. In order to change the value of the
global variable x, you must also declare it to be global within the function body, like this
function f ()
global x;
x = 1;
endfunction
Passing a global variable in a function parameter list will make a local copy and not
modify the global value. For example, given the function
function f (x)
x = 0
endfunction
and the definition of x as a global variable at the top level,
global x = 13
the expression
f (x)
will display the value of x from inside the function as 0, but the value of x at the top level
remains unchanged, because the function works with a copy of its argument.

isglobal (name)
Return true if name is a globally visible variable.
For example:
global x
isglobal ("x")
⇒ 1
See also: [isvarname], page 125, [exist], page 132.

128

GNU Octave

7.2 Persistent Variables
A variable that has been declared persistent within a function will retain its contents in
memory between subsequent calls to the same function. The difference between persistent
variables and global variables is that persistent variables are local in scope to a particular
function and are not visible elsewhere.
The following example uses a persistent variable to create a function that prints the
number of times it has been called.
function count_calls ()
persistent calls = 0;
printf ("’count_calls’ has been called %d times\n",
++calls);
endfunction
for i = 1:3
count_calls ();
endfor
a ’count_calls’ has been called 1 times
a ’count_calls’ has been called 2 times
a ’count_calls’ has been called 3 times

As the example shows, a variable may be declared persistent using a persistent declaration statement. The following statements are all persistent declarations.
persistent
persistent
persistent
persistent

a
a b
c = 2
d = 3 e f = 5

The behavior of persistent variables is equivalent to the behavior of static variables in
C.
Like global variables, a persistent variable may only be initialized once. For example,
after executing the following code
persistent pvar = 1
persistent pvar = 2
the value of the persistent variable pvar is 1, not 2.
If a persistent variable is declared but not initialized to a specific value, it will contain an
empty matrix. So, it is also possible to initialize a persistent variable by checking whether
it is empty, as the following example illustrates.
function count_calls ()
persistent calls;
if (isempty (calls))
calls = 0;
endif
printf ("’count_calls’ has been called %d times\n",
++calls);
endfunction

Chapter 7: Variables

129

This implementation behaves in exactly the same way as the previous implementation of
count_calls.
The value of a persistent variable is kept in memory until it is explicitly cleared. Assuming that the implementation of count_calls is saved on disk, we get the following
behavior.
for i = 1:2
count_calls ();
endfor
a ’count_calls’ has been called 1 times
a ’count_calls’ has been called 2 times
clear
for i = 1:2
count_calls ();
endfor
a ’count_calls’ has been called 3 times
a ’count_calls’ has been called 4 times
clear all
for i = 1:2
count_calls ();
endfor
a ’count_calls’ has been called 1 times
a ’count_calls’ has been called 2 times
clear count_calls
for i = 1:2
count_calls ();
endfor
a ’count_calls’ has been called 1 times
a ’count_calls’ has been called 2 times
That is, the persistent variable is only removed from memory when the function containing
the variable is removed. Note that if the function definition is typed directly into the Octave
prompt, the persistent variable will be cleared by a simple clear command as the entire
function definition will be removed from memory. If you do not want a persistent variable to
be removed from memory even if the function is cleared, you should use the mlock function
(see Section 11.9.6 [Function Locking], page 201).

7.3 Status of Variables
When creating simple one-shot programs it can be very convenient to see which variables
are available at the prompt. The function who and its siblings whos and whos_line_format
will show different information about what is in memory, as the following shows.

130

GNU Octave

str = "A random string";
who
a Variables in the current scope:
a
a ans str

who
who pattern . . .
who option pattern . . .
C = who ("pattern", . . . )
List currently defined variables matching the given patterns.
Valid pattern syntax is the same as described for the clear command. If no patterns
are supplied, all variables are listed.
By default, only variables visible in the local scope are displayed.
The following are valid options, but may not be combined.
global

List variables in the global scope rather than the current scope.

-regexp

The patterns are considered to be regular expressions when matching the
variables to display. The same pattern syntax accepted by the regexp
function is used.

-file

The next argument is treated as a filename. All variables found within the
specified file are listed. No patterns are accepted when reading variables
from a file.

If called as a function, return a cell array of defined variable names matching the
given patterns.
See also: [whos], page 130, [isglobal], page 127, [isvarname], page 125, [exist], page 132,
[regexp], page 87.

whos
whos pattern . . .
whos option pattern . . .
S = whos ("pattern", . . . )
Provide detailed information on currently defined variables matching the given patterns.
Options and pattern syntax are the same as for the who command.
Extended information about each variable is summarized in a table with the following
default entries.
Attr

Attributes of the listed variable. Possible attributes are:
blank

Variable in local scope

a

Automatic variable. An automatic variable is one created by
the interpreter, for example argn.

c

Variable of complex type.

f

Formal parameter (function argument).

Chapter 7: Variables

131

g

Variable with global scope.

p

Persistent variable.

Name

The name of the variable.

Size

The logical size of the variable. A scalar is 1x1, a vector is 1xN or Nx1,
a 2-D matrix is MxN.

Bytes

The amount of memory currently used to store the variable.

Class

The class of the variable. Examples include double, single, char, uint16,
cell, and struct.

The table can be customized to display more or less information through the function
whos_line_format.
If whos is called as a function, return a struct array of defined variable names matching
the given patterns. Fields in the structure describing each variable are: name, size,
bytes, class, global, sparse, complex, nesting, persistent.
See also: [who], page 130, [whos line format], page 131.

val = whos_line_format ()
old_val = whos_line_format (new_val)
whos_line_format (new_val, "local")
Query or set the format string used by the command whos.
A full format string is:
%[modifier][:width[:left-min[:balance]]];

The following command sequences are available:
%a

Prints attributes of variables (g=global, p=persistent, f=formal parameter, a=automatic variable).

%b

Prints number of bytes occupied by variables.

%c

Prints class names of variables.

%e

Prints elements held by variables.

%n

Prints variable names.

%s

Prints dimensions of variables.

%t

Prints type names of variables.

Every command may also have an alignment modifier:
l

Left alignment.

r

Right alignment (default).

c

Column-aligned (only applicable to command %s).

The width parameter is a positive integer specifying the minimum number of columns
used for printing. No maximum is needed as the field will auto-expand as required.
The parameters left-min and balance are only available when the column-aligned
modifier is used with the command ‘%s’. balance specifies the column number within

132

GNU Octave

the field width which will be aligned between entries. Numbering starts from 0 which
indicates the leftmost column. left-min specifies the minimum field width to the
left of the specified balance column.
The default format is:
" %a:4; %ln:6; %cs:16:6:1; %rb:12; %lc:-1;\n"
When called from inside a function with the "local" option, the variable is changed
locally for the function and any subroutines it calls. The original variable value is
restored when exiting the function.
See also: [whos], page 130.
Instead of displaying which variables are in memory, it is possible to determine if a given
variable is available. That way it is possible to alter the behavior of a program depending
on the existence of a variable. The following example illustrates this.
if (! exist ("meaning", "var"))
disp ("The program has no ’meaning’");
endif

c = exist (name)
c = exist (name, type)
Check for the existence of name as a variable, function, file, directory, or class.
The return code c is one of
1

name is a variable.

2

name is an absolute filename, an ordinary file in Octave’s path, or (after
appending ‘.m’) a function file in Octave’s path.

3

name is a ‘.oct’ or ‘.mex’ file in Octave’s path.

5

name is a built-in function.

7

name is a directory.

103

name is a function not associated with a file (entered on the command
line).

0

name does not exist.

If the optional argument type is supplied, check only for symbols of the specified type.
Valid types are
"var"

Check only for variables.

"builtin"
Check only for built-in functions.
"dir"

Check only for directories.

"file"

Check only for files and directories.

"class"

Check only for classes. (Note: This option is accepted, but not currently
implemented)

Chapter 7: Variables

133

If no type is given, and there are multiple possible matches for name, exist will
return a code according to the following priority list: variable, built-in function, octfile, directory, file, class.
exist returns 2 if a regular file called name is present in Octave’s search path. If you
want information about other types of files not on the search path you should use
some combination of the functions file_in_path and stat instead.
Programming Note: If name is implemented by a buggy .oct/.mex file, calling exist
may cause Octave to crash. To maintain high performance, Octave trusts .oct/.mex
files instead of sandboxing them.
See also: [file in loadpath], page 196, [file in path], page 836, [dir in loadpath],
page 196, [stat], page 833.
Usually Octave will manage the memory, but sometimes it can be practical to remove
variables from memory manually. This is usually needed when working with large variables
that fill a substantial part of the memory. On a computer that uses the IEEE floating point
format, the following program allocates a matrix that requires around 128 MB memory.
large_matrix = zeros (4000, 4000);
Since having this variable in memory might slow down other computations, it can be necessary to remove it manually from memory. The clear function allows this.

clear [options] pattern . . .
Delete the names matching the given patterns from the symbol table.
The pattern may contain the following special characters:
?

Match any single character.

*

Match zero or more characters.

[ list ]

Match the list of characters specified by list. If the first character is
! or ^, match all characters except those specified by list. For example,
the pattern ‘[a-zA-Z]’ will match all lowercase and uppercase alphabetic
characters.

For example, the command
clear foo b*r
clears the name foo and all names that begin with the letter b and end with the letter
r.
If clear is called without any arguments, all user-defined variables (local and global)
are cleared from the symbol table.
If clear is called with at least one argument, only the visible names matching the arguments are cleared. For example, suppose you have defined a function foo, and then
hidden it by performing the assignment foo = 2. Executing the command clear foo
once will clear the variable definition and restore the definition of foo as a function.
Executing clear foo a second time will clear the function definition.
The following options are available in both long and short form
-all, -a

Clear all local and global user-defined variables and all functions from
the symbol table.

134

GNU Octave

-exclusive, -x
Clear the variables that don’t match the following pattern.
-functions, -f
Clear the function names and the built-in symbols names.
-global, -g
Clear global symbol names.
-variables, -v
Clear local variable names.
-classes, -c
Clears the class structure table and clears all objects.
-regexp, -r
The arguments are treated as regular expressions as any variables that
match will be cleared.
With the exception of exclusive, all long options can be used without the dash as
well.
See also: [who], page 130, [whos], page 130, [exist], page 132.

pack ()
Consolidate workspace memory in matlab.
This function is provided for compatibility, but does nothing in Octave.
See also: [clear], page 133.
Information about a function or variable such as its location in the file system can also be
acquired from within Octave. This is usually only useful during development of programs,
and not within a program.

type name . . .
type -q name . . .
text = type ("name", . . . )
Display the contents of name which may be a file, function (m-file), variable, operator,
or keyword.
type normally prepends a header line describing the category of name such as function
or variable; The -q option suppresses this behavior.
If no output variable is used the contents are displayed on screen. Otherwise, a cell
array of strings is returned, where each element corresponds to the contents of each
requested function.

which name . . .
Display the type of each name.
If name is defined from a function file, the full name of the file is also displayed.
See also: [help], page 20, [lookfor], page 21.

Chapter 7: Variables

135

what
what dir
w = what (dir)
List the Octave specific files in directory dir.
If dir is not specified then the current directory is used.
If a return argument is requested, the files found are returned in the structure w. The
structure contains the following fields:
path

Full path to directory dir

m

Cell array of m-files

mat

Cell array of mat files

mex

Cell array of mex files

oct

Cell array of oct files

mdl

Cell array of mdl files

slx

Cell array of slx files

p

Cell array of p-files

classes

Cell array of class directories (@classname/)

packages

Cell array of package directories (+pkgname/)

Compatibility Note: Octave does not support mdl, slx, and p files; nor does it support
package directories. what will always return an empty list for these categories.
See also: [which], page 134, [ls], page 855, [exist], page 132.

137

8 Expressions
Expressions are the basic building block of statements in Octave. An expression evaluates
to a value, which you can print, test, store in a variable, pass to a function, or assign a new
value to a variable with an assignment operator.
An expression can serve as a statement on its own. Most other kinds of statements
contain one or more expressions which specify data to be operated on. As in other languages,
expressions in Octave include variables, array references, constants, and function calls, as
well as combinations of these with various operators.

8.1 Index Expressions
An index expression allows you to reference or extract selected elements of a matrix or
vector.
Indices may be scalars, vectors, ranges, or the special operator ‘:’, which may be used
to select entire rows or columns.
Vectors are indexed using a single index expression. Matrices (2-D) and higher multidimensional arrays are indexed using either one index or N indices where N is the dimension
of the array. When using a single index expression to index 2-D or higher data the elements
of the array are taken in column-first order (like Fortran).
The output from indexing assumes the dimensions of the index expression. For example:
a(2)
a(1:2)
a([1; 2])

# result is a scalar
# result is a row vector
# result is a column vector

As a special case, when a colon is used as a single index, the output is a column vector
containing all the elements of the vector or matrix. For example:
a(:)
a(:)’

# result is a column vector
# result is a row vector

The above two code idioms are often used in place of reshape when a simple vector,
rather than an arbitrarily sized array, is needed.
Given the matrix
a = [1, 2; 3, 4]
all of the following expressions are equivalent and select the first row of the matrix.
a(1, [1, 2])
a(1, 1:2)
a(1, :)

# row 1, columns 1 and 2
# row 1, columns in range 1-2
# row 1, all columns

In index expressions the keyword end automatically refers to the last entry for a particular dimension. This magic index can also be used in ranges and typically eliminates the
needs to call size or length to gather array bounds before indexing. For example:

138

GNU Octave

a = [1, 2, 3, 4];
a(1:end/2)
a(end + 1) = 5;
a(end) = [];
a(1:2:end)
a(2:2:end)
a(end:-1:1)

#
#
#
#
#
#

first half of a => [1, 2]
append element
delete element
odd elements of a => [1, 3]
even elements of a => [2, 4]
reversal of a => [4, 3, 2 , 1]

8.1.1 Advanced Indexing
An array with ‘nd’ dimensions can be indexed by a vector idx which has from 1 to ‘nd’
elements. If any element of idx is not a scalar then the complete set of index tuples will be
generated from the Cartesian product of the index elements.
For the ordinary and most common case, the number of indices (nidx = numel (idx))
matches the number of dimensions ‘nd’. In this case, each element of idx corresponds to
its respective dimension, i.e., idx(1) refers to dimension 1, idx(2) refers to dimension 2,
etc. If nidx < nd, and every index is less than the size of the array in the ith dimension
(idx(i) < size (array, i)), then the index expression is padded with nd - nidx trailing
singleton dimensions. If nidx < nd but one of the indices idx(i) is outside the size of the
current array, then the last nd - nidx + 1 dimensions are folded into a single dimension
with an extent equal to the product of extents of the original dimensions. This is easiest to
understand with an example.
A = reshape (1:8, 2, 2, 2)
A =

# Create 3-D array

ans(:,:,1) =
1
2

3
4

ans(:,:,2) =
5
6

7
8

A(2,1,2);
A(2,1);
A(2,4);

#
#
#
#
#
#

Case (nidx == nd): ans = 6
Case (nidx < nd), idx within array:
equivalent to A(2,1,1), ans = 2
Case (nidx < nd), idx outside array:
Dimension 2 & 3 folded into new dimension of size 2x2 = 4
Select 2nd row, 4th element of [2, 4, 6, 8], ans = 8

One advanced use of indexing is to create arrays filled with a single value. This can be
done by using an index of ones on a scalar value. The result is an object with the dimensions
of the index expression and every element equal to the original scalar. For example, the
following statements

Chapter 8: Expressions

139

a = 13;
a(ones (1, 4))
produce a vector whose four elements are all equal to 13.
Similarly, by indexing a scalar with two vectors of ones it is possible to create a matrix.
The following statements
a = 13;
a(ones (1, 2), ones (1, 3))
create a 2x3 matrix with all elements equal to 13.
The last example could also be written as
13(ones (2, 3))
It is more efficient to use indexing rather than the code construction scalar * ones (N,
M, ...) because it avoids the unnecessary multiplication operation. Moreover, multiplication may not be defined for the object to be replicated whereas indexing an array is always
defined. The following code shows how to create a 2x3 cell array from a base unit which is
not itself a scalar.
{"Hello"}(ones (2, 3))
It should be, noted that ones (1, n) (a row vector of ones) results in a range (with zero
increment). A range is stored internally as a starting value, increment, end value, and total
number of values; hence, it is more efficient for storage than a vector or matrix of ones
whenever the number of elements is greater than 4. In particular, when ‘r’ is a row vector,
the expressions
r(ones (1, n), :)
r(ones (n, 1), :)
will produce identical results, but the first one will be significantly faster, at least for ‘r’
and ‘n’ large enough. In the first case the index is held in compressed form as a range which
allows Octave to choose a more efficient algorithm to handle the expression.
A general recommendation, for a user unaware of these subtleties, is to use the function
repmat for replicating smaller arrays into bigger ones.
A second use of indexing is to speed up code. Indexing is a fast operation and judicious
use of it can reduce the requirement for looping over individual array elements which is a
slow operation.
Consider√the following example which creates a 10-element row vector a containing the
values ai = i.
for i = 1:10
a(i) = sqrt (i);
endfor
It is quite inefficient to create a vector using a loop like this. In this case, it would have
been much more efficient to use the expression
a = sqrt (1:10);
which avoids the loop entirely.
In cases where a loop cannot be avoided, or a number of values must be combined to
form a larger matrix, it is generally faster to set the size of the matrix first (pre-allocate

140

GNU Octave

storage), and then insert elements using indexing commands. For example, given a matrix
a,
[nr, nc] = size (a);
x = zeros (nr, n * nc);
for i = 1:n
x(:,(i-1)*nc+1:i*nc) = a;
endfor
is considerably faster than
x = a;
for i = 1:n-1
x = [x, a];
endfor
because Octave does not have to repeatedly resize the intermediate result.

ind = sub2ind (dims, i, j)
ind = sub2ind (dims, s1, s2, . . . , sN)
Convert subscripts to linear indices.
The input dims is a dimension vector where each element is the size of the array in
the respective dimension (see [size], page 45). The remaining inputs are scalars or
vectors of subscripts to be converted.
The output vector ind contains the converted linear indices.
Background: Array elements can be specified either by a linear index which starts
at 1 and runs through the number of elements in the array, or they may be specified
with subscripts for the row, column, page, etc. The functions ind2sub and sub2ind
interconvert between the two forms.
The linear index traverses dimension 1 (rows), then dimension 2 (columns), then
dimension 3 (pages), etc. until it has numbered all of the elements. Consider the
following 3-by-3 matrices:
[(1,1), (1,2), (1,3)]
[1, 4, 7]
[(2,1), (2,2), (2,3)] ==> [2, 5, 8]
[(3,1), (3,2), (3,3)]
[3, 6, 9]
The left matrix contains the subscript tuples for each matrix element. The right
matrix shows the linear indices for the same matrix.
The following example shows how to convert the two-dimensional indices (2,1) and
(2,3) of a 3-by-3 matrix to linear indices with a single call to sub2ind.
s1 = [2, 2];
s2 = [1, 3];
ind = sub2ind ([3, 3], s1, s2)
⇒ ind = 2
8
See also: [ind2sub], page 140, [size], page 45.

[s1, s2, ..., sN] = ind2sub (dims, ind)
Convert linear indices to subscripts.

Chapter 8: Expressions

141

The input dims is a dimension vector where each element is the size of the array in
the respective dimension (see [size], page 45). The second input ind contains linear
indies to be converted.
The outputs s1, . . . , sN contain the converted subscripts.
Background: Array elements can be specified either by a linear index which starts
at 1 and runs through the number of elements in the array, or they may be specified
with subscripts for the row, column, page, etc. The functions ind2sub and sub2ind
interconvert between the two forms.
The linear index traverses dimension 1 (rows), then dimension 2 (columns), then
dimension 3 (pages), etc. until it has numbered all of the elements. Consider the
following 3-by-3 matrices:
[1, 4, 7]
[(1,1), (1,2), (1,3)]
[2, 5, 8] ==> [(2,1), (2,2), (2,3)]
[3, 6, 9]
[(3,1), (3,2), (3,3)]
The left matrix contains the linear indices for each matrix element. The right matrix
shows the subscript tuples for the same matrix.
The following example shows how to convert the two-dimensional indices (2,1) and
(2,3) of a 3-by-3 matrix to linear indices with a single call to sub2ind.
The following example shows how to convert the linear indices 2 and 8 in a 3-by-3
matrix into subscripts.
ind = [2, 8];
[r, c] = ind2sub ([3, 3], ind)
⇒ r = 2
2
⇒ c = 1
3
If the number of output subscripts exceeds the number of dimensions, the exceeded
dimensions are set to 1. On the other hand, if fewer subscripts than dimensions are
provided, the exceeding dimensions are merged into the final requested dimension.
For clarity, consider the following examples:
ind = [2, 8];
dims = [3, 3];
## same as dims = [3, 3, 1]
[r, c, s] = ind2sub (dims, ind)
⇒ r = 2
2
⇒ c = 1
3
⇒ s = 1
1
## same as dims = [9]
r = ind2sub (dims, ind)
⇒ r = 2
8
See also: [ind2sub], page 140, [size], page 45.

isindex (ind)
isindex (ind, n)
Return true if ind is a valid index.
Valid indices are either positive integers (although possibly of real data type), or
logical arrays.

142

GNU Octave

If present, n specifies the maximum extent of the dimension to be indexed. When
possible the internal result is cached so that subsequent indexing using ind will not
perform the check again.
Implementation Note: Strings are first converted to double values before the checks
for valid indices are made. Unless a string contains the NULL character "\0", it will
always be a valid index.

val = allow_noninteger_range_as_index ()
old_val = allow_noninteger_range_as_index (new_val)
allow_noninteger_range_as_index (new_val, "local")
Query or set the internal variable that controls whether non-integer ranges are allowed
as indices.
This might be useful for matlab compatibility; however, it is still not entirely compatible because matlab treats the range expression differently in different contexts.
When called from inside a function with the "local" option, the variable is changed
locally for the function and any subroutines it calls. The original variable value is
restored when exiting the function.

8.2 Calling Functions
A function is a name for a particular calculation. Because it has a name, you can ask for it
by name at any point in the program. For example, the function sqrt computes the square
root of a number.
A fixed set of functions are built-in, which means they are available in every Octave
program. The sqrt function is one of these. In addition, you can define your own functions.
See Chapter 11 [Functions and Scripts], page 175, for information about how to do this.
The way to use a function is with a function call expression, which consists of the function
name followed by a list of arguments in parentheses. The arguments are expressions which
give the raw materials for the calculation that the function will do. When there is more
than one argument, they are separated by commas. If there are no arguments, you can
omit the parentheses, but it is a good idea to include them anyway, to clearly indicate that
a function call was intended. Here are some examples:
sqrt (x^2 + y^2)
ones (n, m)
rand ()

# One argument
# Two arguments
# No arguments

Each function expects a particular number of arguments. For example, the sqrt function
must be called with a single argument, the number to take the square root of:
sqrt (argument)
Some of the built-in functions take a variable number of arguments, depending on the
particular usage, and their behavior is different depending on the number of arguments
supplied.
Like every other expression, the function call has a value, which is computed by the
function based on the arguments you give it. In this example, the value of sqrt (argument)
is the square root of the argument. A function can also have side effects, such as assigning
the values of certain variables or doing input or output operations.

Chapter 8: Expressions

143

Unlike most languages, functions in Octave may return multiple values. For example,
the following statement
[u, s, v] = svd (a)
computes the singular value decomposition of the matrix a and assigns the three result
matrices to u, s, and v.
The left side of a multiple assignment expression is itself a list of expressions, and is
allowed to be a list of variable names or index expressions. See also Section 8.1 [Index
Expressions], page 137, and Section 8.6 [Assignment Ops], page 153.

8.2.1 Call by Value
In Octave, unlike Fortran, function arguments are passed by value, which means that each
argument in a function call is evaluated and assigned to a temporary location in memory
before being passed to the function. There is currently no way to specify that a function
parameter should be passed by reference instead of by value. This means that it is impossible
to directly alter the value of a function parameter in the calling function. It can only change
the local copy within the function body. For example, the function
function f (x, n)
while (n-- > 0)
disp (x);
endwhile
endfunction
displays the value of the first argument n times. In this function, the variable n is used as a
temporary variable without having to worry that its value might also change in the calling
function. Call by value is also useful because it is always possible to pass constants for any
function parameter without first having to determine that the function will not attempt to
modify the parameter.
The caller may use a variable as the expression for the argument, but the called function
does not know this: it only knows what value the argument had. For example, given a
function called as
foo = "bar";
fcn (foo)
you should not think of the argument as being “the variable foo.” Instead, think of the
argument as the string value, "bar".
Even though Octave uses pass-by-value semantics for function arguments, values are not
copied unnecessarily. For example,
x = rand (1000);
f (x);
does not actually force two 1000 by 1000 element matrices to exist unless the function f
modifies the value of its argument. Then Octave must create a copy to avoid changing the
value outside the scope of the function f, or attempting (and probably failing!) to modify
the value of a constant or the value of a temporary result.

144

GNU Octave

8.2.2 Recursion
With some restrictions1 , recursive function calls are allowed. A recursive function is one
which calls itself, either directly or indirectly. For example, here is an inefficient2 way to
compute the factorial of a given integer:
function retval = fact (n)
if (n > 0)
retval = n * fact (n-1);
else
retval = 1;
endif
endfunction
This function is recursive because it calls itself directly. It eventually terminates because
each time it calls itself, it uses an argument that is one less than was used for the previous
call. Once the argument is no longer greater than zero, it does not call itself, and the
recursion ends.
The built-in variable max_recursion_depth specifies a limit to the recursion depth and
prevents Octave from recursing infinitely.

val = max_recursion_depth ()
old_val = max_recursion_depth (new_val)
max_recursion_depth (new_val, "local")
Query or set the internal limit on the number of times a function may be called
recursively.
If the limit is exceeded, an error message is printed and control returns to the top
level.
When called from inside a function with the "local" option, the variable is changed
locally for the function and any subroutines it calls. The original variable value is
restored when exiting the function.

8.3 Arithmetic Operators
The following arithmetic operators are available, and work on scalars and matrices. The
element-by-element operators and functions broadcast (see Section 19.2 [Broadcasting],
page 535).
x+y

Addition. If both operands are matrices, the number of rows and columns must
both agree, or they must be broadcastable to the same shape.

x .+ y

Element-by-element addition. This operator is equivalent to +.

x -y

Subtraction. If both operands are matrices, the number of rows and columns
of both must agree, or they must be broadcastable to the same shape.

1

2

Some of Octave’s functions are implemented in terms of functions that cannot be called recursively. For
example, the ODE solver lsode is ultimately implemented in a Fortran subroutine that cannot be called
recursively, so lsode should not be called either directly or indirectly from within the user-supplied
function that lsode requires. Doing so will result in an error.
It would be much better to use prod (1:n), or gamma (n+1) instead, after first checking to ensure that
the value n is actually a positive integer.

Chapter 8: Expressions

145

x .- y

Element-by-element subtraction. This operator is equivalent to -.

x*y

Matrix multiplication. The number of columns of x must agree with the number
of rows of y.

x .* y

Element-by-element multiplication. If both operands are matrices, the number
of rows and columns must both agree, or they must be broadcastable to the
same shape.

x /y

Right division. This is conceptually equivalent to the expression
(inverse (y’) * x’)’
but it is computed without forming the inverse of y’.
If the system is not square, or if the coefficient matrix is singular, a minimum
norm solution is computed.

x ./ y

Element-by-element right division.

x \y

Left division. This is conceptually equivalent to the expression
inverse (x) * y
but it is computed without forming the inverse of x.
If the system is not square, or if the coefficient matrix is singular, a minimum
norm solution is computed.

x .\ y

Element-by-element left division. Each element of y is divided by each corresponding element of x.

x ^y
x ** y

x .^ y
x .** y

Power operator. If x and y are both scalars, this operator returns x raised to
the power y. If x is a scalar and y is a square matrix, the result is computed
using an eigenvalue expansion. If x is a square matrix, the result is computed
by repeated multiplication if y is an integer, and by an eigenvalue expansion if
y is not an integer. An error results if both x and y are matrices.
The implementation of this operator needs to be improved.
Element-by-element power operator. If both operands are matrices, the number
of rows and columns must both agree, or they must be broadcastable to the
same shape. If several complex results are possible, the one with smallest nonnegative argument (angle) is taken. This rule may return a complex root even
when a real root is also possible. Use realpow, realsqrt, cbrt, or nthroot if
a real result is preferred.

-x

Negation.

+x

Unary plus. This operator has no effect on the operand.

x’

Complex conjugate transpose. For real arguments, this operator is the same as
the transpose operator. For complex arguments, this operator is equivalent to
the expression
conj (x.’)

x.’

Transpose.

146

GNU Octave

Note that because Octave’s element-by-element operators begin with a ‘.’, there is a
possible ambiguity for statements like
1./m
because the period could be interpreted either as part of the constant or as part of the
operator. To resolve this conflict, Octave treats the expression as if you had typed
(1) ./ m
and not
(1.) / m
Although this is inconsistent with the normal behavior of Octave’s lexer, which usually
prefers to break the input into tokens by preferring the longest possible match at any given
point, it is more useful in this case.

ctranspose (x)
Return the complex conjugate transpose of x.
This function and x’ are equivalent.
See also: [transpose], page 147.

ldivide (x, y)
Return the element-by-element left division of x and y.
This function and x .\ y are equivalent.
See also: [rdivide], page 147, [mldivide], page 146, [times], page 147, [plus], page 147.

minus (x, y)
This function and x - y are equivalent.
See also: [plus], page 147, [uminus], page 148.

mldivide (x, y)
Return the matrix left division of x and y.
This function and x \ y are equivalent.
See also: [mrdivide], page 146, [ldivide], page 146, [rdivide], page 147.

mpower (x, y)
Return the matrix power operation of x raised to the y power.
This function and x ^ y are equivalent.
See also: [power], page 147, [mtimes], page 147, [plus], page 147, [minus], page 146.

mrdivide (x, y)
Return the matrix right division of x and y.
This function and x / y are equivalent.
See also: [mldivide], page 146, [rdivide], page 147, [plus], page 147, [minus], page 146.

Chapter 8: Expressions

147

mtimes (x, y)
mtimes (x1, x2, . . . )
Return the matrix multiplication product of inputs.
This function and x * y are equivalent. If more arguments are given, the multiplication is applied cumulatively from left to right:
(...((x1 * x2) * x3) * ...)
At least one argument is required.
See also: [times], page 147, [plus], page 147, [minus], page 146, [rdivide], page 147,
[mrdivide], page 146, [mldivide], page 146, [mpower], page 146.

plus (x, y)
plus (x1, x2, . . . )
This function and x + y are equivalent.
If more arguments are given, the summation is applied cumulatively from left to right:
(...((x1 + x2) + x3) + ...)
At least one argument is required.
See also: [minus], page 146, [uplus], page 148.

power (x, y)
Return the element-by-element operation of x raised to the y power.
This function and x .^ y are equivalent.
If several complex results are possible, returns the one with smallest non-negative
argument (angle). Use realpow, realsqrt, cbrt, or nthroot if a real result is preferred.
See also: [mpower], page 146, [realpow], page 476, [realsqrt], page 476, [cbrt],
page 476, [nthroot], page 476.

rdivide (x, y)
Return the element-by-element right division of x and y.
This function and x ./ y are equivalent.
See also: [ldivide], page 146, [mrdivide], page 146, [times], page 147, [plus], page 147.

times (x, y)
times (x1, x2, . . . )
Return the element-by-element multiplication product of inputs.
This function and x .* y are equivalent. If more arguments are given, the multiplication is applied cumulatively from left to right:
(...((x1 .* x2) .* x3) .* ...)
At least one argument is required.
See also: [mtimes], page 147, [rdivide], page 147.

transpose (x)
Return the transpose of x.
This function and x.’ are equivalent.
See also: [ctranspose], page 146.

148

GNU Octave

uminus (x)
This function and - x are equivalent.
See also: [uplus], page 148, [minus], page 146.

uplus (x)
This function and + x are equivalent.
See also: [uminus], page 148, [plus], page 147, [minus], page 146.

8.4 Comparison Operators
Comparison operators compare numeric values for relationships such as equality. They are
written using relational operators.
All of Octave’s comparison operators return a value of 1 if the comparison is true, or 0
if it is false. For matrix values, they all work on an element-by-element basis. Broadcasting
rules apply. See Section 19.2 [Broadcasting], page 535. For example:
[1, 2; 3, 4] == [1, 3; 2, 4]
⇒ 1 0
0 1
According to broadcasting rules, if one operand is a scalar and the other is a matrix, the
scalar is compared to each element of the matrix in turn, and the result is the same size as
the matrix.
x= y

True if x is greater than or equal to y.

x>y

True if x is greater than y.

x != y
x ~= y

True if x is not equal to y.

For complex numbers, the following ordering is defined: z1 < z2 if and only if
abs (z1) < abs (z2)
|| (abs (z1) == abs (z2) && arg (z1) < arg (z2))
This is consistent with the ordering used by max, min and sort, but is not consistent
with matlab, which only compares the real parts.
String comparisons may also be performed with the strcmp function, not with the comparison operators listed above. See Chapter 5 [Strings], page 67.

eq (x, y)
Return true if the two inputs are equal.
This function is equivalent to x == y.
See also: [ne], page 149, [isequal], page 149, [le], page 149, [ge], page 149, [gt], page 149,
[ne], page 149, [lt], page 149.

Chapter 8: Expressions

149

ge (x, y)
This function is equivalent to x >= y.
See also: [le], page 149, [eq], page 148, [gt], page 149, [ne], page 149, [lt], page 149.

gt (x, y)
This function is equivalent to x > y.
See also: [le], page 149, [eq], page 148, [ge], page 149, [ne], page 149, [lt], page 149.

isequal (x1, x2, . . . )
Return true if all of x1, x2, . . . are equal.
See also: [isequaln], page 149.

isequaln (x1, x2, . . . )
Return true if all of x1, x2, . . . are equal under the additional assumption that NaN
== NaN (no comparison of NaN placeholders in dataset).
See also: [isequal], page 149.

le (x, y)
This function is equivalent to x <= y.
See also: [eq], page 148, [ge], page 149, [gt], page 149, [ne], page 149, [lt], page 149.

lt (x, y)
This function is equivalent to x < y.
See also: [le], page 149, [eq], page 148, [ge], page 149, [gt], page 149, [ne], page 149.

ne (x, y)
Return true if the two inputs are not equal.
This function is equivalent to x != y.
See also: [eq], page 148, [isequal], page 149, [le], page 149, [ge], page 149, [lt],
page 149.

8.5 Boolean Expressions
8.5.1 Element-by-element Boolean Operators
An element-by-element boolean expression is a combination of comparison expressions using
the boolean operators “or” (‘|’), “and” (‘&’), and “not” (‘!’), along with parentheses to
control nesting. The truth of the boolean expression is computed by combining the truth
values of the corresponding elements of the component expressions. A value is considered
to be false if it is zero, and true otherwise.
Element-by-element boolean expressions can be used wherever comparison expressions
can be used. They can be used in if and while statements. However, a matrix value used
as the condition in an if or while statement is only true if all of its elements are nonzero.
Like comparison operations, each element of an element-by-element boolean expression
also has a numeric value (1 if true, 0 if false) that comes into play if the result of the boolean
expression is stored in a variable, or used in arithmetic.

150

GNU Octave

Here are descriptions of the three element-by-element boolean operators.
boolean1 & boolean2
Elements of the result are true if both corresponding elements of boolean1 and
boolean2 are true.
boolean1 | boolean2
Elements of the result are true if either of the corresponding elements of
boolean1 or boolean2 is true.
! boolean
~ boolean Each element of the result is true if the corresponding element of boolean is
false.
These operators work on an element-by-element basis. For example, the expression
[1, 0; 0, 1] & [1, 0; 2, 3]
returns a two by two identity matrix.
For the binary operators, broadcasting rules apply. See Section 19.2 [Broadcasting],
page 535. In particular, if one of the operands is a scalar and the other a matrix, the
operator is applied to the scalar and each element of the matrix.
For the binary element-by-element boolean operators, both subexpressions boolean1 and
boolean2 are evaluated before computing the result. This can make a difference when the
expressions have side effects. For example, in the expression
a & b++
the value of the variable b is incremented even if the variable a is zero.
This behavior is necessary for the boolean operators to work as described for matrixvalued operands.

z = and (x, y)
z = and (x1, x2, . . . )
Return the logical AND of x and y.
This function is equivalent to the operator syntax x & y. If more than two arguments
are given, the logical AND is applied cumulatively from left to right:
(...((x1 & x2) & x3) & ...)
At least one argument is required.
See also: [or], page 150, [not], page 150, [xor], page 443.

z = not (x)
Return the logical NOT of x.
This function is equivalent to the operator syntax ! x.
See also: [and], page 150, [or], page 150, [xor], page 443.

z = or (x, y)
z = or (x1, x2, . . . )
Return the logical OR of x and y.

Chapter 8: Expressions

151

This function is equivalent to the operator syntax x | y. If more than two arguments
are given, the logical OR is applied cumulatively from left to right:
(...((x1 | x2) | x3) | ...)
At least one argument is required.
See also: [and], page 150, [not], page 150, [xor], page 443.

8.5.2 Short-circuit Boolean Operators
Combined with the implicit conversion to scalar values in if and while conditions, Octave’s element-by-element boolean operators are often sufficient for performing most logical
operations. However, it is sometimes desirable to stop evaluating a boolean expression as
soon as the overall truth value can be determined. Octave’s short-circuit boolean operators
work this way.
boolean1 && boolean2
The expression boolean1 is evaluated and converted to a scalar using the equivalent of the operation all (boolean1(:)). If it is false, the result of the overall
expression is 0. If it is true, the expression boolean2 is evaluated and converted
to a scalar using the equivalent of the operation all (boolean2(:)). If it is
true, the result of the overall expression is 1. Otherwise, the result of the overall
expression is 0.
Warning: there is one exception to the rule of evaluating all (boolean1(:)),
which is when boolean1 is the empty matrix. The truth value of an empty
matrix is always false so [] && true evaluates to false even though all
([]) is true.
boolean1 || boolean2
The expression boolean1 is evaluated and converted to a scalar using the equivalent of the operation all (boolean1(:)). If it is true, the result of the overall
expression is 1. If it is false, the expression boolean2 is evaluated and converted
to a scalar using the equivalent of the operation all (boolean2(:)). If it is
true, the result of the overall expression is 1. Otherwise, the result of the overall
expression is 0.
Warning: the truth value of an empty matrix is always false, see the previous
list item for details.
The fact that both operands may not be evaluated before determining the overall truth
value of the expression can be important. For example, in the expression
a && b++
the value of the variable b is only incremented if the variable a is nonzero.
This can be used to write somewhat more concise code. For example, it is possible write
function f (a, b, c)
if (nargin > 2 && ischar (c))
...
instead of having to use two if statements to avoid attempting to evaluate an argument
that doesn’t exist. For example, without the short-circuit feature, it would be necessary to
write

152

GNU Octave

function f (a, b, c)
if (nargin > 2)
if (ischar (c))
...
Writing
function f (a, b, c)
if (nargin > 2 & ischar (c))
...
would result in an error if f were called with one or two arguments because Octave would
be forced to try to evaluate both of the operands for the operator ‘&’.
matlab has special behavior that allows the operators ‘&’ and ‘|’ to short-circuit when
used in the truth expression for if and while statements. Octave also behaves the same way
by default, though the use of the ‘&’ and ‘|’ operators in this way is strongly discouraged.
Instead, you should use the ‘&&’ and ‘||’ operators that always have short-circuit behavior.

val = do_braindead_shortcircuit_evaluation ()
old_val = do_braindead_shortcircuit_evaluation (new_val)
do_braindead_shortcircuit_evaluation (new_val, "local")
Query or set the internal variable that controls whether Octave will do short-circuit
evaluation of ‘|’ and ‘&’ operators inside the conditions of if or while statements.
This feature is only provided for compatibility with matlab and should not be used
unless you are porting old code that relies on this feature.
To obtain short-circuit behavior for logical expressions in new programs, you should
always use the ‘&&’ and ‘||’ operators.
When called from inside a function with the "local" option, the variable is changed
locally for the function and any subroutines it calls. The original variable value is
restored when exiting the function.
Finally, the ternary operator (?:) is not supported in Octave. If short-circuiting is not
important, it can be replaced by the ifelse function.

merge (mask, tval, fval)
ifelse (mask, tval, fval)
Merge elements of true val and false val, depending on the value of mask.
If mask is a logical scalar, the other two arguments can be arbitrary values. Otherwise,
mask must be a logical array, and tval, fval should be arrays of matching class, or
cell arrays. In the scalar mask case, tval is returned if mask is true, otherwise fval is
returned.
In the array mask case, both tval and fval must be either scalars or arrays with
dimensions equal to mask. The result is constructed as follows:
result(mask) = tval(mask);
result(! mask) = fval(! mask);
mask can also be arbitrary numeric type, in which case it is first converted to logical.
See also: [logical], page 60, [diff], page 444.

Chapter 8: Expressions

153

8.6 Assignment Expressions
An assignment is an expression that stores a new value into a variable. For example, the
following expression assigns the value 1 to the variable z:
z = 1
After this expression is executed, the variable z has the value 1. Whatever old value z had
before the assignment is forgotten. The ‘=’ sign is called an assignment operator.
Assignments can store string values also. For example, the following expression would
store the value "this food is good" in the variable message:
thing = "food"
predicate = "good"
message = [ "this " , thing , " is " , predicate ]
(This also illustrates concatenation of strings.)
Most operators (addition, concatenation, and so on) have no effect except to compute
a value. If you ignore the value, you might as well not use the operator. An assignment
operator is different. It does produce a value, but even if you ignore the value, the assignment
still makes itself felt through the alteration of the variable. We call this a side effect.
The left-hand operand of an assignment need not be a variable (see Chapter 7 [Variables],
page 125). It can also be an element of a matrix (see Section 8.1 [Index Expressions],
page 137) or a list of return values (see Section 8.2 [Calling Functions], page 142). These
are all called lvalues, which means they can appear on the left-hand side of an assignment
operator. The right-hand operand may be any expression. It produces the new value which
the assignment stores in the specified variable, matrix element, or list of return values.
It is important to note that variables do not have permanent types. The type of a
variable is simply the type of whatever value it happens to hold at the moment. In the
following program fragment, the variable foo has a numeric value at first, and a string value
later on:
octave:13> foo = 1
foo = 1
octave:13> foo = "bar"
foo = bar
When the second assignment gives foo a string value, the fact that it previously had a
numeric value is forgotten.
Assignment of a scalar to an indexed matrix sets all of the elements that are referenced
by the indices to the scalar value. For example, if a is a matrix with at least two columns,
a(:, 2) = 5
sets all the elements in the second column of a to 5.
Assigning an empty matrix ‘[]’ works in most cases to allow you to delete rows or
columns of matrices and vectors. See Section 4.1.1 [Empty Matrices], page 51. For example,
given a 4 by 5 matrix A, the assignment
A (3, :) = []
deletes the third row of A, and the assignment
A (:, 1:2:5) = []

154

GNU Octave

deletes the first, third, and fifth columns.
An assignment is an expression, so it has a value. Thus, z = 1 as an expression has the
value 1. One consequence of this is that you can write multiple assignments together:
x = y = z = 0
stores the value 0 in all three variables. It does this because the value of z = 0, which is 0,
is stored into y, and then the value of y = z = 0, which is 0, is stored into x.
This is also true of assignments to lists of values, so the following is a valid expression
[a, b, c] = [u, s, v] = svd (a)
that is exactly equivalent to
[u,
a =
b =
c =

s, v] = svd (a)
u
s
v

In expressions like this, the number of values in each part of the expression need not
match. For example, the expression
[a, b] = [u, s, v] = svd (a)
is equivalent to
[u, s, v] = svd (a)
a = u
b = s
The number of values on the left side of the expression can, however, not exceed the number
of values on the right side. For example, the following will produce an error.
[a, b, c, d] = [u, s, v] = svd (a);
a error: element number 4 undefined in return list

The symbol ~ may be used as a placeholder in the list of lvalues, indicating that the
corresponding return value should be ignored and not stored anywhere:
[~, s, v] = svd (a);
This is cleaner and more memory efficient than using a dummy variable. The nargout
value for the right-hand side expression is not affected. If the assignment is used as an
expression, the return value is a comma-separated list with the ignored values dropped.
A very common programming pattern is to increment an existing variable with a given
value, like this
a = a + 2;
This can be written in a clearer and more condensed form using the += operator
a += 2;
Similar operators also exist for subtraction (-=), multiplication (*=), and division (/=). An
expression of the form
expr1 op= expr2
is evaluated as
expr1 = (expr1) op (expr2)

Chapter 8: Expressions

155

where op can be either +, -, *, or /, as long as expr2 is a simple expression with no side
effects. If expr2 also contains an assignment operator, then this expression is evaluated as
temp = expr2
expr1 = (expr1) op temp
where temp is a placeholder temporary value storing the computed result of evaluating
expr2. So, the expression
a *= b+1
is evaluated as
a = a * (b+1)
and not
a = a * b + 1
You can use an assignment anywhere an expression is called for. For example, it is valid
to write x != (y = 1) to set y to 1 and then test whether x equals 1. But this style tends
to make programs hard to read. Except in a one-shot program, you should rewrite it to get
rid of such nesting of assignments. This is never very hard.

8.7 Increment Operators
Increment operators increase or decrease the value of a variable by 1. The operator to
increment a variable is written as ‘++’. It may be used to increment a variable either before
or after taking its value.
For example, to pre-increment the variable x, you would write ++x. This would add one
to x and then return the new value of x as the result of the expression. It is exactly the
same as the expression x = x + 1.
To post-increment a variable x, you would write x++. This adds one to the variable x,
but returns the value that x had prior to incrementing it. For example, if x is equal to 2,
the result of the expression x++ is 2, and the new value of x is 3.
For matrix and vector arguments, the increment and decrement operators work on each
element of the operand.
Here is a list of all the increment and decrement expressions.
++x

This expression increments the variable x. The value of the expression is the
new value of x. It is equivalent to the expression x = x + 1.

--x

This expression decrements the variable x. The value of the expression is the
new value of x. It is equivalent to the expression x = x - 1.

x++

This expression causes the variable x to be incremented. The value of the
expression is the old value of x.

x--

This expression causes the variable x to be decremented. The value of the
expression is the old value of x.

156

GNU Octave

8.8 Operator Precedence
Operator precedence determines how operators are grouped, when different operators appear close by in one expression. For example, ‘*’ has higher precedence than ‘+’. Thus, the
expression a + b * c means to multiply b and c, and then add a to the product (i.e., a + (b
* c)).
You can overrule the precedence of the operators by using parentheses. You can think
of the precedence rules as saying where the parentheses are assumed if you do not write
parentheses yourself. In fact, it is wise to use parentheses whenever you have an unusual
combination of operators, because other people who read the program may not remember
what the precedence is in this case. You might forget as well, and then you too could make
a mistake. Explicit parentheses will help prevent any such mistake.
When operators of equal precedence are used together, the leftmost operator groups
first, except for the assignment operators, which group in the opposite order. Thus, the
expression a - b + c groups as (a - b) + c, but the expression a = b = c groups as a = (b =
c).
The precedence of prefix unary operators is important when another operator follows
the operand. For example, -x^2 means -(x^2), because ‘-’ has lower precedence than ‘^’.
Here is a table of the operators in Octave, in order of decreasing precedence. Unless
noted, all operators group left to right.
function call and array indexing, cell array indexing, and structure element
indexing
‘()’ ‘{}’ ‘.’
postfix increment, and postfix decrement
‘++’ ‘--’
These operators group right to left.
transpose and exponentiation
‘’’ ‘.’’ ‘^’ ‘**’ ‘.^’ ‘.**’
unary plus, unary minus, prefix increment, prefix decrement, and logical "not"
‘+’ ‘-’ ‘++’ ‘--’ ‘~’ ‘!’
multiply and divide
‘*’ ‘/’ ‘\’ ‘.\’ ‘.*’ ‘./’
add, subtract
‘+’ ‘-’
colon

‘:’

relational
‘<’ ‘<=’ ‘==’ ‘>=’ ‘>’ ‘!=’ ‘~=’
element-wise "and"
‘&’
element-wise "or"
‘|’

157

logical "and"
‘&&’
logical "or"
‘||’
assignment
‘=’ ‘+=’ ‘-=’ ‘*=’ ‘/=’ ‘\=’ ‘^=’ ‘.*=’ ‘./=’ ‘.\=’ ‘.^=’ ‘|=’ ‘&=’
These operators group right to left.

159

9 Evaluation
Normally, you evaluate expressions simply by typing them at the Octave prompt, or by
asking Octave to interpret commands that you have saved in a file.
Sometimes, you may find it necessary to evaluate an expression that has been computed
and stored in a string, which is exactly what the eval function lets you do.

eval (try)
eval (try, catch)
Parse the string try and evaluate it as if it were an Octave program.
If execution fails, evaluate the optional string catch.
The string try is evaluated in the current context, so any results remain available
after eval returns.
The following example creates the variable A with the approximate value of 3.1416
in the current workspace.
eval ("A = acos(-1);");
If an error occurs during the evaluation of try then the catch string is evaluated, as
the following example shows:
eval (’error ("This is a bad example");’,
’printf ("This error occurred:\n%s\n", lasterr ());’);
a This error occurred:
This is a bad example
Programming Note: if you are only using eval as an error-capturing mechanism,
rather than for the execution of arbitrary code strings, Consider using try/catch
blocks or unwind protect/unwind protect cleanup blocks instead. These techniques
have higher performance and don’t introduce the security considerations that the
evaluation of arbitrary code does.
See also: [evalin], page 162, [evalc], page 159, [assignin], page 162, [feval], page 160.
The evalc function additionally captures any console output produced by the evaluated
expression.

s = evalc (try)
s = evalc (try, catch)
Parse and evaluate the string try as if it were an Octave program, while capturing
the output into the return variable s.
If execution fails, evaluate the optional string catch.
This function behaves like eval, but any output or warning messages which would
normally be written to the console are captured and returned in the string s.
The diary is disabled during the execution of this function. When system is used,
any output produced by external programs is not captured, unless their output is
captured by the system function itself.
s = evalc ("t = 42"), t
⇒ s = t = 42
⇒ t =

42

160

GNU Octave

See also: [eval], page 159, [diary], page 34.

9.1 Calling a Function by its Name
The feval function allows you to call a function from a string containing its name. This
is useful when writing a function that needs to call user-supplied functions. The feval
function takes the name of the function to call as its first argument, and the remaining
arguments are given to the function.
The following example is a simple-minded function using feval that finds the root of a
user-supplied function of one variable using Newton’s method.
function result = newtroot (fname, x)
# usage: newtroot (fname, x)
#
#
fname : a string naming a function f(x).
#
x
: initial guess
delta = tol = sqrt (eps);
maxit = 200;
fx = feval (fname, x);
for i = 1:maxit
if (abs (fx) < tol)
result = x;
return;
else
fx_new = feval (fname, x + delta);
deriv = (fx_new - fx) / delta;
x = x - fx / deriv;
fx = fx_new;
endif
endfor
result = x;
endfunction
Note that this is only meant to be an example of calling user-supplied functions and
should not be taken too seriously. In addition to using a more robust algorithm, any serious
code would check the number and type of all the arguments, ensure that the supplied function really was a function, etc. See Section 4.8 [Predicates for Numeric Objects], page 62, for
a list of predicates for numeric objects, and see Section 7.3 [Status of Variables], page 129,
for a description of the exist function.

feval (name, . . . )
Evaluate the function named name.
Any arguments after the first are passed as inputs to the named function. For example,
feval ("acos", -1)
⇒ 3.1416

Chapter 9: Evaluation

161

calls the function acos with the argument ‘-1’.
The function feval can also be used with function handles of any sort (see
Section 11.11.1 [Function Handles], page 211). Historically, feval was the only way
to call user-supplied functions in strings, but function handles are now preferred due
to the cleaner syntax they offer. For example,
f = @exp;
feval (f, 1)
⇒ 2.7183
f (1)
⇒ 2.7183
are equivalent ways to call the function referred to by f. If it cannot be predicted
beforehand whether f is a function handle, function name in a string, or inline function
then feval can be used instead.
A similar function run exists for calling user script files, that are not necessarily on the
user path

run script
run ("script")
Run script in the current workspace.
Scripts which reside in directories specified in Octave’s load path, and which end with
the extension ".m", can be run simply by typing their name. For scripts not located
on the load path, use run.
The filename script can be a bare, fully qualified, or relative filename and with or
without a file extension. If no extension is specified, Octave will first search for a script
with the ".m" extension before falling back to the script name without an extension.
Implementation Note: If script includes a path component, then run first changes the
working directory to the directory where script is found. Next, the script is executed.
Finally, run returns to the original working directory unless script has specifically
changed directories.
See also: [path], page 195, [addpath], page 194, [source], page 204.

9.2 Evaluation in a Different Context
Before you evaluate an expression you need to substitute the values of the variables used in
the expression. These are stored in the symbol table. Whenever the interpreter starts a new
function it saves the current symbol table and creates a new one, initializing it with the list
of function parameters and a couple of predefined variables such as nargin. Expressions
inside the function use the new symbol table.
Sometimes you want to write a function so that when you call it, it modifies variables in
your own context. This allows you to use a pass-by-name style of function, which is similar
to using a pointer in programming languages such as C.
Consider how you might write save and load as m-files. For example:

162

GNU Octave

function create_data
x = linspace (0, 10, 10);
y = sin (x);
save mydata x y
endfunction
With evalin, you could write save as follows:
function save (file, name1, name2)
f = open_save_file (file);
save_var (f, name1, evalin ("caller", name1));
save_var (f, name2, evalin ("caller", name2));
endfunction
Here, ‘caller’ is the create_data function and name1 is the string "x", which evaluates
simply as the value of x.
You later want to load the values back from mydata in a different context:
function process_data
load mydata
... do work ...
endfunction
With assignin, you could write load as follows:
function load (file)
f = open_load_file (file);
[name, val] = load_var (f);
assignin ("caller", name, val);
[name, val] = load_var (f);
assignin ("caller", name, val);
endfunction
Here, ‘caller’ is the process_data function.
You can set and use variables at the command prompt using the context ‘base’ rather
than ‘caller’.
These functions are rarely used in practice. One example is the fail (‘code’,
‘pattern’) function which evaluates ‘code’ in the caller’s context and checks that the
error message it produces matches the given pattern. Other examples such as save and
load are written in C++ where all Octave variables are in the ‘caller’ context and evalin
is not needed.

evalin (context, try)
evalin (context, try, catch)
Like eval, except that the expressions are evaluated in the context context, which
may be either "caller" or "base".
See also: [eval], page 159, [assignin], page 162.

assignin (context, varname, value)
Assign value to varname in context context, which may be either "base" or "caller".
See also: [evalin], page 162.

163

10 Statements
Statements may be a simple constant expression or a complicated list of nested loops and
conditional statements.
Control statements such as if, while, and so on control the flow of execution in Octave
programs. All the control statements start with special keywords such as if and while,
to distinguish them from simple expressions. Many control statements contain other statements; for example, the if statement contains another statement which may or may not be
executed.
Each control statement has a corresponding end statement that marks the end of the
control statement. For example, the keyword endif marks the end of an if statement, and
endwhile marks the end of a while statement. You can use the keyword end anywhere a
more specific end keyword is expected, but using the more specific keywords is preferred
because if you use them, Octave is able to provide better diagnostics for mismatched or
missing end tokens.
The list of statements contained between keywords like if or while and the corresponding end statement is called the body of a control statement.

10.1 The if Statement
The if statement is Octave’s decision-making statement. There are three basic forms of an
if statement. In its simplest form, it looks like this:
if (condition)
then-body
endif
condition is an expression that controls what the rest of the statement will do. The thenbody is executed only if condition is true.
The condition in an if statement is considered true if its value is nonzero, and false if
its value is zero. If the value of the conditional expression in an if statement is a vector or
a matrix, it is considered true only if it is non-empty and all of the elements are nonzero.
The conceptually equivalent code when condition is a matrix is shown below.
if (matrix) ≡ if (all (matrix(:)))
The second form of an if statement looks like this:
if (condition)
then-body
else
else-body
endif
If condition is true, then-body is executed; otherwise, else-body is executed.
Here is an example:
if (rem (x, 2) == 0)
printf ("x is even\n");
else
printf ("x is odd\n");
endif

164

GNU Octave

In this example, if the expression rem (x, 2) == 0 is true (that is, the value of x is
divisible by 2), then the first printf statement is evaluated, otherwise the second printf
statement is evaluated.
The third and most general form of the if statement allows multiple decisions to be
combined in a single statement. It looks like this:
if (condition)
then-body
elseif (condition)
elseif-body
else
else-body
endif
Any number of elseif clauses may appear. Each condition is tested in turn, and if one is
found to be true, its corresponding body is executed. If none of the conditions are true and
the else clause is present, its body is executed. Only one else clause may appear, and it
must be the last part of the statement.
In the following example, if the first condition is true (that is, the value of x is divisible
by 2), then the first printf statement is executed. If it is false, then the second condition
is tested, and if it is true (that is, the value of x is divisible by 3), then the second printf
statement is executed. Otherwise, the third printf statement is performed.
if (rem (x, 2) == 0)
printf ("x is even\n");
elseif (rem (x, 3) == 0)
printf ("x is odd and divisible by 3\n");
else
printf ("x is odd\n");
endif
Note that the elseif keyword must not be spelled else if, as is allowed in Fortran. If
it is, the space between the else and if will tell Octave to treat this as a new if statement
within another if statement’s else clause. For example, if you write
if (c1)
body-1
else if (c2)
body-2
endif
Octave will expect additional input to complete the first if statement. If you are using
Octave interactively, it will continue to prompt you for additional input. If Octave is reading
this input from a file, it may complain about missing or mismatched end statements, or, if
you have not used the more specific end statements (endif, endfor, etc.), it may simply
produce incorrect results, without producing any warning messages.
It is much easier to see the error if we rewrite the statements above like this,

Chapter 10: Statements

165

if (c1)
body-1
else
if (c2)
body-2
endif
using the indentation to show how Octave groups the statements. See Chapter 11 [Functions
and Scripts], page 175.

10.2 The switch Statement
It is very common to take different actions depending on the value of one variable. This is
possible using the if statement in the following way
if (X == 1)
do_something ();
elseif (X == 2)
do_something_else ();
else
do_something_completely_different ();
endif
This kind of code can however be very cumbersome to both write and maintain. To overcome
this problem Octave supports the switch statement. Using this statement, the above
example becomes
switch (X)
case 1
do_something ();
case 2
do_something_else ();
otherwise
do_something_completely_different ();
endswitch
This code makes the repetitive structure of the problem more explicit, making the code
easier to read, and hence maintain. Also, if the variable X should change its name, only one
line would need changing compared to one line per case when if statements are used.
The general form of the switch statement is
switch (expression)
case label
command_list
case label
command_list
...
otherwise
command_list
endswitch

166

GNU Octave

where label can be any expression. However, duplicate label values are not detected, and
only the command list corresponding to the first match will be executed. For the switch
statement to be meaningful at least one case label command_list clause must be present,
while the otherwise command_list clause is optional.
If label is a cell array the corresponding command list is executed if any of the elements of
the cell array match expression. As an example, the following program will print ‘Variable
is either 6 or 7’.
A = 7;
switch (A)
case { 6, 7 }
printf ("variable is either 6 or 7\n");
otherwise
printf ("variable is neither 6 nor 7\n");
endswitch
As with all other specific end keywords, endswitch may be replaced by end, but you
can get better diagnostics if you use the specific forms.
One advantage of using the switch statement compared to using if statements is that
the labels can be strings. If an if statement is used it is not possible to write
if (X == "a string") # This is NOT valid
since a character-to-character comparison between X and the string will be made instead of
evaluating if the strings are equal. This special-case is handled by the switch statement,
and it is possible to write programs that look like this
switch (X)
case "a string"
do_something
...
endswitch

10.2.1 Notes for the C Programmer
The switch statement is also available in the widely used C programming language. There
are, however, some differences between the statement in Octave and C
• Cases are exclusive, so they don’t ‘fall through’ as do the cases in the switch statement
of the C language.
• The command list elements are not optional. Making the list optional would have
meant requiring a separator between the label and the command list. Otherwise,
things like
switch (foo)
case (1) -2
...
would produce surprising results, as would
switch (foo)
case (1)
case (2)
doit ();
...

Chapter 10: Statements

167

particularly for C programmers. If doit() should be executed if foo is either 1 or 2,
the above code should be written with a cell array like this
switch (foo)
case { 1, 2 }
doit ();
...

10.3 The while Statement
In programming, a loop means a part of a program that is (or at least can be) executed
two or more times in succession.
The while statement is the simplest looping statement in Octave. It repeatedly executes
a statement as long as a condition is true. As with the condition in an if statement, the
condition in a while statement is considered true if its value is nonzero, and false if its
value is zero. If the value of the conditional expression in a while statement is a vector or
a matrix, it is considered true only if it is non-empty and all of the elements are nonzero.
Octave’s while statement looks like this:
while (condition)
body
endwhile
Here body is a statement or list of statements that we call the body of the loop, and
condition is an expression that controls how long the loop keeps running.
The first thing the while statement does is test condition. If condition is true, it executes
the statement body. After body has been executed, condition is tested again, and if it is
still true, body is executed again. This process repeats until condition is no longer true. If
condition is initially false, the body of the loop is never executed.
This example creates a variable fib that contains the first ten elements of the Fibonacci
sequence.
fib = ones (1, 10);
i = 3;
while (i <= 10)
fib (i) = fib (i-1) + fib (i-2);
i++;
endwhile
Here the body of the loop contains two statements.
The loop works like this: first, the value of i is set to 3. Then, the while tests whether
i is less than or equal to 10. This is the case when i equals 3, so the value of the i-th
element of fib is set to the sum of the previous two values in the sequence. Then the i++
increments the value of i and the loop repeats. The loop terminates when i reaches 11.
A newline is not required between the condition and the body; but using one makes the
program clearer unless the body is very simple.

168

GNU Octave

10.4 The do-until Statement
The do-until statement is similar to the while statement, except that it repeatedly executes a statement until a condition becomes true, and the test of the condition is at the
end of the loop, so the body of the loop is always executed at least once. As with the
condition in an if statement, the condition in a do-until statement is considered true if
its value is nonzero, and false if its value is zero. If the value of the conditional expression
in a do-until statement is a vector or a matrix, it is considered true only if it is non-empty
and all of the elements are nonzero.
Octave’s do-until statement looks like this:
do
body
until (condition)
Here body is a statement or list of statements that we call the body of the loop, and
condition is an expression that controls how long the loop keeps running.
This example creates a variable fib that contains the first ten elements of the Fibonacci
sequence.
fib = ones (1, 10);
i = 2;
do
i++;
fib (i) = fib (i-1) + fib (i-2);
until (i == 10)
A newline is not required between the do keyword and the body; but using one makes
the program clearer unless the body is very simple.

10.5 The for Statement
The for statement makes it more convenient to count iterations of a loop. The general
form of the for statement looks like this:
for var = expression
body
endfor
where body stands for any statement or list of statements, expression is any valid expression,
and var may take several forms. Usually it is a simple variable name or an indexed variable.
If the value of expression is a structure, var may also be a vector with two elements. See
Section 10.5.1 [Looping Over Structure Elements], page 169, below.
The assignment expression in the for statement works a bit differently than Octave’s
normal assignment statement. Instead of assigning the complete result of the expression, it
assigns each column of the expression to var in turn. If expression is a range, a row vector,
or a scalar, the value of var will be a scalar each time the loop body is executed. If var is a
column vector or a matrix, var will be a column vector each time the loop body is executed.
The following example shows another way to create a vector containing the first ten
elements of the Fibonacci sequence, this time using the for statement:

Chapter 10: Statements

169

fib = ones (1, 10);
for i = 3:10
fib(i) = fib(i-1) + fib(i-2);
endfor
This code works by first evaluating the expression 3:10, to produce a range of values from 3
to 10 inclusive. Then the variable i is assigned the first element of the range and the body
of the loop is executed once. When the end of the loop body is reached, the next value in
the range is assigned to the variable i, and the loop body is executed again. This process
continues until there are no more elements to assign.
Within Octave is it also possible to iterate over matrices or cell arrays using the for
statement. For example consider
disp ("Loop over a matrix")
for i = [1,3;2,4]
i
endfor
disp ("Loop over a cell array")
for i = {1,"two";"three",4}
i
endfor
In this case the variable i takes on the value of the columns of the matrix or cell matrix.
So the first loop iterates twice, producing two column vectors [1;2], followed by [3;4],
and likewise for the loop over the cell array. This can be extended to loops over multidimensional arrays. For example:
a = [1,3;2,4]; c = cat (3, a, 2*a);
for i = c
i
endfor
In the above case, the multi-dimensional matrix c is reshaped to a two-dimensional matrix as
reshape (c, rows (c), prod (size (c)(2:end))) and then the same behavior as a loop
over a two-dimensional matrix is produced.
Although it is possible to rewrite all for loops as while loops, the Octave language has
both statements because often a for loop is both less work to type and more natural to
think of. Counting the number of iterations is very common in loops and it can be easier
to think of this counting as part of looping rather than as something to do inside the loop.

10.5.1 Looping Over Structure Elements
A special form of the for statement allows you to loop over all the elements of a structure:
for [ val, key ] = expression
body
endfor
In this form of the for statement, the value of expression must be a structure. If it is, key
and val are set to the name of the element and the corresponding value in turn, until there
are no more elements. For example:

170

GNU Octave

x.a = 1
x.b = [1, 2; 3, 4]
x.c = "string"
for [val, key] = x
key
val
endfor
a
a
a
a
a
a
a
a
a
a

key
val
key
val
1
3

= a
= 1
= b
=
2
4

key = c
val = string

The elements are not accessed in any particular order. If you need to cycle through
the list in a particular way, you will have to use the function fieldnames and sort the list
yourself.

10.6 The break Statement
The break statement jumps out of the innermost while, do-until, or for loop that encloses
it. The break statement may only be used within the body of a loop. The following example
finds the smallest divisor of a given integer, and also identifies prime numbers:
num = 103;
div = 2;
while (div*div <= num)
if (rem (num, div) == 0)
break;
endif
div++;
endwhile
if (rem (num, div) == 0)
printf ("Smallest divisor of %d is %d\n", num, div)
else
printf ("%d is prime\n", num);
endif
When the remainder is zero in the first while statement, Octave immediately breaks
out of the loop. This means that Octave proceeds immediately to the statement following
the loop and continues processing. (This is very different from the exit statement which
stops the entire Octave program.)
Here is another program equivalent to the previous one. It illustrates how the condition
of a while statement could just as well be replaced with a break inside an if:

Chapter 10: Statements

171

num = 103;
div = 2;
while (1)
if (rem (num, div) == 0)
printf ("Smallest divisor of %d is %d\n", num, div);
break;
endif
div++;
if (div*div > num)
printf ("%d is prime\n", num);
break;
endif
endwhile

10.7 The continue Statement
The continue statement, like break, is used only inside while, do-until, or for loops.
It skips over the rest of the loop body, causing the next cycle around the loop to begin
immediately. Contrast this with break, which jumps out of the loop altogether. Here is an
example:
# print elements of a vector of random
# integers that are even.
# first, create a row vector of 10 random
# integers with values between 0 and 100:
vec = round (rand (1, 10) * 100);
# print what we’re interested in:
for x = vec
if (rem (x, 2) != 0)
continue;
endif
printf ("%d\n", x);
endfor
If one of the elements of vec is an odd number, this example skips the print statement
for that element, and continues back to the first statement in the loop.
This is not a practical example of the continue statement, but it should give you a clear
understanding of how it works. Normally, one would probably write the loop like this:
for x = vec
if (rem (x, 2) == 0)
printf ("%d\n", x);
endif
endfor

172

GNU Octave

10.8 The unwind protect Statement
Octave supports a limited form of exception handling modeled after the unwind-protect
form of Lisp.
The general form of an unwind_protect block looks like this:
unwind_protect
body
unwind_protect_cleanup
cleanup
end_unwind_protect
where body and cleanup are both optional and may contain any Octave expressions or
commands. The statements in cleanup are guaranteed to be executed regardless of how
control exits body.
This is useful to protect temporary changes to global variables from possible errors. For
example, the following code will always restore the original value of the global variable
frobnosticate even if an error occurs in the first part of the unwind_protect block.
save_frobnosticate = frobnosticate;
unwind_protect
frobnosticate = true;
...
unwind_protect_cleanup
frobnosticate = save_frobnosticate;
end_unwind_protect
Without unwind_protect, the value of frobnosticate would not be restored if an error occurs
while evaluating the first part of the unwind_protect block because evaluation would stop
at the point of the error and the statement to restore the value would not be executed.
In addition to unwind protect, Octave supports another form of exception handling, the
try block.

10.9 The try Statement
The original form of a try block looks like this:
try
body
catch
cleanup
end_try_catch
where body and cleanup are both optional and may contain any Octave expressions or
commands. The statements in cleanup are only executed if an error occurs in body.
No warnings or error messages are printed while body is executing. If an error does
occur during the execution of body, cleanup can use the functions lasterr or lasterror
to access the text of the message that would have been printed, as well as its identifier. The
alternative form,

Chapter 10: Statements

173

try
body
catch err
cleanup
end_try_catch
will automatically store the output of lasterror in the structure err. See Chapter 12
[Errors and Warnings], page 217, for more information about the lasterr and lasterror
functions.

10.10 Continuation Lines
In the Octave language, most statements end with a newline character and you must tell
Octave to ignore the newline character in order to continue a statement from one line to
the next. Lines that end with the characters ... are joined with the following line before
they are divided into tokens by Octave’s parser. For example, the lines
x = long_variable_name ...
+ longer_variable_name ...
- 42
form a single statement.
Any text between the continuation marker and the newline character is ignored. For
example, the statement
x = long_variable_name ...
# comment one
+ longer_variable_name ...comment two
- 42
# last comment
is equivalent to the one shown above.
Inside double-quoted string constants, the character \ has to be used as continuation
marker. The \ must appear at the end of the line just before the newline character:
s = "This text starts in the first line \
and is continued in the second line."
Input that occurs inside parentheses can be continued to the next line without having to
use a continuation marker. For example, it is possible to write statements like
if (fine_dining_destination == on_a_boat
|| fine_dining_destination == on_a_train)
seuss (i, will, not, eat, them, sam, i, am, i,
will, not, eat, green, eggs, and, ham);
endif
without having to add to the clutter with continuation markers.

175

11 Functions and Scripts
Complicated Octave programs can often be simplified by defining functions. Functions can
be defined directly on the command line during interactive Octave sessions, or in external
files, and can be called just like built-in functions.

11.1 Introduction to Function and Script Files
There are seven different things covered in this section.
1. Typing in a function at the command prompt.
2. Storing a group of commands in a file — called a script file.
3. Storing a function in a file—called a function file.
4. Subfunctions in function files.
5. Multiple functions in one script file.
6. Private functions.
7. Nested functions.
Both function files and script files end with an extension of .m, for matlab compatibility.
If you want more than one independent functions in a file, it must be a script file (see
Section 11.10 [Script Files], page 203), and to use these functions you must execute the
script file before you can use the functions that are in the script file.

11.2 Defining Functions
In its simplest form, the definition of a function named name looks like this:
function name
body
endfunction
A valid function name is like a valid variable name: a sequence of letters, digits and underscores, not starting with a digit. Functions share the same pool of names as variables.
The function body consists of Octave statements. It is the most important part of the
definition, because it says what the function should actually do.
For example, here is a function that, when executed, will ring the bell on your terminal
(assuming that it is possible to do so):
function wakeup
printf ("\a");
endfunction
The printf statement (see Chapter 14 [Input and Output], page 245) simply tells Octave
to print the string "\a". The special character ‘\a’ stands for the alert character (ASCII
7). See Chapter 5 [Strings], page 67.
Once this function is defined, you can ask Octave to evaluate it by typing the name of
the function.
Normally, you will want to pass some information to the functions you define. The
syntax for passing parameters to a function in Octave is

176

GNU Octave

function name (arg-list)
body
endfunction
where arg-list is a comma-separated list of the function’s arguments. When the function is
called, the argument names are used to hold the argument values given in the call. The list
of arguments may be empty, in which case this form is equivalent to the one shown above.
To print a message along with ringing the bell, you might modify the wakeup to look
like this:
function wakeup (message)
printf ("\a%s\n", message);
endfunction
Calling this function using a statement like this
wakeup ("Rise and shine!");
will cause Octave to ring your terminal’s bell and print the message ‘Rise and shine!’,
followed by a newline character (the ‘\n’ in the first argument to the printf statement).
In most cases, you will also want to get some information back from the functions you
define. Here is the syntax for writing a function that returns a single value:
function ret-var = name (arg-list)
body
endfunction
The symbol ret-var is the name of the variable that will hold the value to be returned by
the function. This variable must be defined before the end of the function body in order
for the function to return a value.
Variables used in the body of a function are local to the function. Variables named
in arg-list and ret-var are also local to the function. See Section 7.1 [Global Variables],
page 126, for information about how to access global variables inside a function.
For example, here is a function that computes the average of the elements of a vector:
function retval = avg (v)
retval = sum (v) / length (v);
endfunction
If we had written avg like this instead,
function retval = avg (v)
if (isvector (v))
retval = sum (v) / length (v);
endif
endfunction
and then called the function with a matrix instead of a vector as the argument, Octave
would have printed an error message like this:
error: value on right hand side of assignment is undefined
because the body of the if statement was never executed, and retval was never defined.
To prevent obscure errors like this, it is a good idea to always make sure that the return

Chapter 11: Functions and Scripts

177

variables will always have values, and to produce meaningful error messages when problems
are encountered. For example, avg could have been written like this:
function retval = avg (v)
retval = 0;
if (isvector (v))
retval = sum (v) / length (v);
else
error ("avg: expecting vector argument");
endif
endfunction
There is still one additional problem with this function. What if it is called without an
argument? Without additional error checking, Octave will probably print an error message
that won’t really help you track down the source of the error. To allow you to catch errors
like this, Octave provides each function with an automatic variable called nargin. Each
time a function is called, nargin is automatically initialized to the number of arguments
that have actually been passed to the function. For example, we might rewrite the avg
function like this:
function retval = avg (v)
retval = 0;
if (nargin != 1)
usage ("avg (vector)");
endif
if (isvector (v))
retval = sum (v) / length (v);
else
error ("avg: expecting vector argument");
endif
endfunction
Although Octave does not automatically report an error if you call a function with more
arguments than expected, doing so probably indicates that something is wrong. Octave
also does not automatically report an error if a function is called with too few arguments,
but any attempt to use a variable that has not been given a value will result in an error.
To avoid such problems and to provide useful messages, we check for both possibilities and
issue our own error message.

nargin ()
nargin (fcn)
Report the number of input arguments to a function.
Called from within a function, return the number of arguments passed to the function.
At the top level, return the number of command line arguments passed to Octave.
If called with the optional argument fcn—a function name or handle—return the
declared number of arguments that the function can accept.
If the last argument to fcn is varargin the returned value is negative. For example,
the function union for sets is declared as

178

GNU Octave

function [y, ia, ib] = union (a, b, varargin)
and
nargin ("union")
⇒ -3

Programming Note: nargin does not work on compiled functions (.oct files) such as
built-in or dynamically loaded functions.
See also: [nargout], page 180, [narginchk], page 181, [varargin], page 186, [inputname],
page 178.

inputname (n)
Return the name of the n-th argument to the calling function.
If the argument is not a simple variable name, return an empty string. As an example,
a reference to a field in a structure such as s.field is not a simple name and will
return "".
inputname is only useful within a function. When used at the command line it always
returns an empty string.
See also: [nargin], page 177, [nthargout], page 179.

val = silent_functions ()
old_val = silent_functions (new_val)
silent_functions (new_val, "local")
Query or set the internal variable that controls whether internal output from a function is suppressed.
If this option is disabled, Octave will display the results produced by evaluating
expressions within a function body that are not terminated with a semicolon.
When called from inside a function with the "local" option, the variable is changed
locally for the function and any subroutines it calls. The original variable value is
restored when exiting the function.

11.3 Multiple Return Values
Unlike many other computer languages, Octave allows you to define functions that return
more than one value. The syntax for defining functions that return multiple values is
function [ret-list] = name (arg-list)
body
endfunction
where name, arg-list, and body have the same meaning as before, and ret-list is a commaseparated list of variable names that will hold the values returned from the function. The
list of return values must have at least one element. If ret-list has only one element, this
form of the function statement is equivalent to the form described in the previous section.
Here is an example of a function that returns two values, the maximum element of a
vector and the index of its first occurrence in the vector.

Chapter 11: Functions and Scripts

179

function [max, idx] = vmax (v)
idx = 1;
max = v (idx);
for i = 2:length (v)
if (v (i) > max)
max = v (i);
idx = i;
endif
endfor
endfunction
In this particular case, the two values could have been returned as elements of a single
array, but that is not always possible or convenient. The values to be returned may not
have compatible dimensions, and it is often desirable to give the individual return values
distinct names.
It is possible to use the nthargout function to obtain only some of the return values or
several at once in a cell array. See Section 3.1.5 [Cell Array Objects], page 44.

nthargout (n, func, . . . )
nthargout (n, ntot, func, . . . )
Return the nth output argument of the function specified by the function handle or
string func.
Any additional arguments are passed directly to func. The total number of arguments
to call func with can be passed in ntot; by default ntot is n. The input n can also be
a vector of indices of the output, in which case the output will be a cell array of the
requested output arguments.
The intended use nthargout is to avoid intermediate variables. For example, when
finding the indices of the maximum entry of a matrix, the following two compositions
of nthargout
m = magic (5);
cell2mat (nthargout ([1, 2], @ind2sub, size (m),
nthargout (2, @max, m(:))))
⇒ 5
3

are completely equivalent to the following lines:
m = magic (5);
[~, idx] = max (M(:));
[i, j] = ind2sub (size (m), idx);
[i, j]
⇒ 5
3

It can also be helpful to have all output arguments in a single cell in the following
manner:
USV = nthargout ([1:3], @svd, hilb (5));
See also: [nargin], page 177, [nargout], page 180, [varargin], page 186, [varargout],
page 186, [isargout], page 189.

180

GNU Octave

In addition to setting nargin each time a function is called, Octave also automatically
initializes nargout to the number of values that are expected to be returned. This allows
you to write functions that behave differently depending on the number of values that the
user of the function has requested. The implicit assignment to the built-in variable ans
does not figure in the count of output arguments, so the value of nargout may be zero.
The svd and lu functions are examples of built-in functions that behave differently
depending on the value of nargout.
It is possible to write functions that only set some return values. For example, calling
the function
function [x, y, z] = f ()
x = 1;
z = 2;
endfunction
as
[a, b, c] = f ()
produces:
a = 1
b = [](0x0)
c = 2
along with a warning.

nargout ()
nargout (fcn)
Report the number of output arguments from a function.
Called from within a function, return the number of values the caller expects to
receive. At the top level, nargout with no argument is undefined and will produce
an error.
If called with the optional argument fcn—a function name or handle—return the
number of declared output values that the function can produce.
If the final output argument is varargout the returned value is negative.
For example,
f ()
will cause nargout to return 0 inside the function f and
[s, t] = f ()
will cause nargout to return 2 inside the function f.
In the second usage,
nargout (@histc)

# or nargout ("histc") using a string input

will return 2, because histc has two outputs, whereas
nargout (@imread)
will return -2, because imread has two outputs and the second is varargout.

Chapter 11: Functions and Scripts

181

Programming Note. nargout does not work for built-in functions and returns -1 for
all anonymous functions.
See also: [nargin], page 177, [varargout], page 186, [isargout], page 189, [nthargout],
page 179.
It is good practice at the head of a function to verify that it has been called correctly.
In Octave the following idiom is seen frequently
if (nargin < min_#_inputs || nargin > max_#_inputs)
print_usage ();
endif
which stops the function execution and prints a message about the correct way to call the
function whenever the number of inputs is wrong.
For compatibility with matlab, narginchk and nargoutchk are available which provide
similar error checking.

narginchk (minargs, maxargs)
Check for correct number of input arguments.
Generate an error message if the number of arguments in the calling function is outside
the range minargs and maxargs. Otherwise, do nothing.
Both minargs and maxargs must be scalar numeric values. Zero, Inf, and negative
values are all allowed, and minargs and maxargs may be equal.
Note that this function evaluates nargin on the caller.
See also: [nargoutchk], page 181, [error], page 217, [nargout], page 180, [nargin],
page 177.

nargoutchk (minargs, maxargs)
msgstr = nargoutchk (minargs, maxargs, nargs)
msgstr = nargoutchk (minargs, maxargs, nargs, "string")
msgstruct = nargoutchk (minargs, maxargs, nargs, "struct")
Check for correct number of output arguments.
In the first form, return an error if the number of arguments is not between minargs
and maxargs. Otherwise, do nothing. Note that this function evaluates the value of
nargout on the caller so its value must have not been tampered with.
Both minargs and maxargs must be numeric scalars. Zero, Inf, and negative are all
valid, and they can have the same value.
For backwards compatibility, the other forms return an appropriate error message
string (or structure) if the number of outputs requested is invalid.
This is useful for checking to that the number of output arguments supplied to a
function is within an acceptable range.
See also: [narginchk], page 181, [error], page 217, [nargout], page 180, [nargin],
page 177.
Besides the number of arguments, inputs can be checked for various properties.
validatestring is used for string arguments and validateattributes for numeric
arguments.

182

GNU Octave

(str, strarray)
(str, strarray, funcname)
(str, strarray, funcname, varname)
( . . . , position)
Verify that str is an element, or substring of an element, in strarray.
When str is a character string to be tested, and strarray is a cellstr of valid values,
then validstr will be the validated form of str where validation is defined as str being
a member or substring of validstr. This is useful for both verifying and expanding
short options, such as "r", to their longer forms, such as "red". If str is a substring
of validstr, and there are multiple matches, the shortest match will be returned if all
matches are substrings of each other. Otherwise, an error will be raised because the
expansion of str is ambiguous. All comparisons are case insensitive.
The additional inputs funcname, varname, and position are optional and will make
any generated validation error message more specific.
Examples:

validstr
validstr
validstr
validstr

=
=
=
=

validatestring
validatestring
validatestring
validatestring

validatestring ("r", {"red", "green", "blue"})
⇒ "red"
validatestring ("b", {"red", "green", "blue", "black"})
⇒ error: validatestring: multiple unique matches were found for ’b’:
blue, black

See also: [strcmp], page 76, [strcmpi], page 77, [validateattributes], page 182,
[inputParser], page 184.

validateattributes
validateattributes
validateattributes
validateattributes
validateattributes
arg_idx)

(A,
(A,
(A,
(A,
(A,

classes,
classes,
classes,
classes,
classes,

attributes)
attributes, arg_idx)
attributes, func_name)
attributes, func_name, arg_name)
attributes, func_name, arg_name,

Check validity of input argument.
Confirms that the argument A is valid by belonging to one of classes, and holding
all of the attributes. If it does not, an error is thrown, with a message formatted
accordingly. The error message can be made further complete by the function name
fun name, the argument name arg name, and its position in the input arg idx.
classes must be a cell array of strings (an empty cell array is allowed) with the name
of classes (remember that a class name is case sensitive). In addition to the class
name, the following categories names are also valid:
"float"

Floating point value comprising classes "double" and "single".

"integer"
Integer value comprising classes (u)int8, (u)int16, (u)int32, (u)int64.
"numeric"
Numeric value comprising either a floating point or integer value.
attributes must be a cell array with names of checks for A. Some of them require an
additional value to be supplied right after the name (see details for each below).
"<="

All values are less than or equal to the following value in attributes.

Chapter 11: Functions and Scripts

183

"<"

All values are less than the following value in attributes.

">="

All values are greater than or equal to the following value in attributes.

">"

All values are greater than the following value in attributes.

"2d"

A 2-dimensional matrix. Note that vectors and empty matrices have 2
dimensions, one of them being of length 1, or both length 0.

"3d"

Has no more than 3 dimensions. A 2-dimensional matrix is a 3-D matrix
whose 3rd dimension is of length 1.

"binary"

All values are either 1 or 0.

"column"

Values are arranged in a single column.

"decreasing"
No value is NaN, and each is less than the preceding one.
"diag"

Value is a diagonal matrix.

"even"

All values are even numbers.

"finite"

All values are finite.

"increasing"
No value is NaN, and each is greater than the preceding one.
"integer"
All values are integer. This is different than using isinteger which only
checks its an integer type. This checks that each value in A is an integer
value, i.e., it has no decimal part.
"ncols"

Has exactly as many columns as the next value in attributes.

"ndims"

Has exactly as many dimensions as the next value in attributes.

"nondecreasing"
No value is NaN, and each is greater than or equal to the preceding one.
"nonempty"
It is not empty.
"nonincreasing"
No value is NaN, and each is less than or equal to the preceding one.
"nonnan"

No value is a NaN.

"nonnegative"
All values are non negative.
"nonsparse"
It is not a sparse matrix.
"nonzero"
No value is zero.
"nrows"

Has exactly as many rows as the next value in attributes.

"numel"

Has exactly as many elements as the next value in attributes.

184

GNU Octave

"odd"

All values are odd numbers.

"positive"
All values are positive.
"real"

It is a non-complex matrix.

"row"

Values are arranged in a single row.

"scalar"

It is a scalar.

"size"

Its size has length equal to the values of the next in attributes. The next
value must is an array with the length for each dimension. To ignore the
check for a certain dimension, the value of NaN can be used.

"square"

Is a square matrix.

"vector"

Values are arranged in a single vector (column or vector).

See also: [isa], page 39, [validatestring], page 181, [inputParser], page 184.
If none of the preceding functions is sufficient there is also the class inputParser which
can perform extremely complex input checking for functions.

p = inputParser ()
Create object p of the inputParser class.
This class is designed to allow easy parsing of function arguments. The class supports
four types of arguments:
1.
2.
3.
4.

mandatory (see addRequired);
optional (see addOptional);
named (see addParameter);
switch (see addSwitch).

After defining the function API with these methods, the supplied arguments can be
parsed with the parse method and the parsing results accessed with the Results
accessor.

inputParser.Parameters
Return list of parameter names already defined.

inputParser.Results
Return structure with argument names as fieldnames and corresponding values.

inputParser.Unmatched
Return structure similar to Results, but for unmatched parameters.
KeepUnmatched property.

See the

inputParser.UsingDefaults
Return cell array with the names of arguments that are using default values.

inputParser.CaseSensitive = boolean
Set whether matching of argument names should be case sensitive. Defaults to false.

Chapter 11: Functions and Scripts

185

inputParser.FunctionName = name
Set function name to be used in error messages; Defaults to empty string.

inputParser.KeepUnmatched = boolean
Set whether an error should be given for non-defined arguments. Defaults to false. If
set to true, the extra arguments can be accessed through Unmatched after the parse
method. Note that since Switch and Parameter arguments can be mixed, it is not
possible to know the unmatched type. If argument is found unmatched it is assumed
to be of the Parameter type and it is expected to be followed by a value.

inputParser.StructExpand = boolean
Set whether a structure can be passed to the function instead of parameter/value
pairs. Defaults to true.
The following example shows how to use this class:

function check (varargin)
p = inputParser ();
p.FunctionName = "check";
p.addRequired ("pack", @ischar);
p.addOptional ("path", pwd(), @ischar);

#
#
#
#

create object
set function name
mandatory argument
optional argument

## create a function handle to anonymous functions for validators
val_mat = @(x) isvector (x) && all (x <= 1) && all (x >= 0);
p.addOptional ("mat", [0 0], val_mat);
## create two arguments of type "Parameter"
val_type = @(x) any (strcmp (x, {"linear", "quadratic"}));
p.addParameter ("type", "linear", val_type);
val_verb = @(x) any (strcmp (x, {"low", "medium", "high"}));
p.addParameter ("tolerance", "low", val_verb);
## create a switch type of argument
p.addSwitch ("verbose");
p.parse (varargin{:});

# Run created parser on inputs

## the rest of the function can access inputs by using p.Results.
## for example, get the tolerance input with p.Results.tolerance
endfunction

186

GNU Octave

check
check
check
check

("mech");
();
(1);
("mech", "~/dev");

#
#
#
#

valid,
error,
error,
valid,

use defaults for other arguments
one argument is mandatory
since ! ischar
use defaults for other arguments

check ("mech", "~/dev", [0 1 0 0], "type", "linear");

# valid

## following is also valid. Note how the Switch argument type can
## be mixed into or before the Parameter argument type (but it
## must still appear after any Optional argument).
check ("mech", "~/dev", [0 1 0 0], "verbose", "tolerance", "high");
## following returns an error since not all optional arguments,
## ‘path’ and ‘mat’, were given before the named argument ‘type’.
check ("mech", "~/dev", "type", "linear");
Note 1 : A function can have any mixture of the four API types but they must appear
in a specific order. Required arguments must be first and can be followed by any
Optional arguments. Only the Parameter and Switch arguments may be mixed
together and they must appear at the end.
Note 2 : If both Optional and Parameter arguments are mixed in a function API
then once a string Optional argument fails to validate it will be considered the end
of the Optional arguments. The remaining arguments will be compared against any
Parameter or Switch arguments.
See also: [nargin], page 177, [validateattributes], page 182, [validatestring], page 181,
[varargin], page 186.

11.4 Variable-length Argument Lists
Sometimes the number of input arguments is not known when the function is defined. As
an example think of a function that returns the smallest of all its input arguments. For
example:
a = smallest (1, 2, 3);
b = smallest (1, 2, 3, 4);
In this example both a and b would be 1. One way to write the smallest function is
function val = smallest (arg1, arg2, arg3, arg4, arg5)
body
endfunction
and then use the value of nargin to determine which of the input arguments should be
considered. The problem with this approach is that it can only handle a limited number of
input arguments.
If the special parameter name varargin appears at the end of a function parameter list
it indicates that the function takes a variable number of input arguments. Using varargin
the function looks like this
function val = smallest (varargin)
body
endfunction

Chapter 11: Functions and Scripts

187

In the function body the input arguments can be accessed through the variable varargin.
This variable is a cell array containing all the input arguments. See Section 6.2 [Cell Arrays],
page 114, for details on working with cell arrays. The smallest function can now be defined
like this
function val = smallest (varargin)
val = min ([varargin{:}]);
endfunction
This implementation handles any number of input arguments, but it’s also a very simple
solution to the problem.
A slightly more complex example of varargin is a function print_arguments that prints
all input arguments. Such a function can be defined like this
function print_arguments (varargin)
for i = 1:length (varargin)
printf ("Input argument %d: ", i);
disp (varargin{i});
endfor
endfunction
This function produces output like this
print_arguments (1, "two", 3);
a Input argument 1: 1
a Input argument 2: two
a Input argument 3: 3

[reg, prop] = parseparams (params)
[reg, var1, ...] = parseparams (params, name1, default1, . . . )
Return in reg the cell elements of param up to the first string element and in prop
all remaining elements beginning with the first string element.
For example:
[reg, prop] = parseparams ({1, 2, "linewidth", 10})
reg =
{
[1,1] = 1
[1,2] = 2
}
prop =
{
[1,1] = linewidth
[1,2] = 10
}
The parseparams function may be used to separate regular numeric arguments from
additional arguments given as property/value pairs of the varargin cell array.
In the second form of the call, available options are specified directly with their default
values given as name-value pairs. If params do not form name-value pairs, or if an
option occurs that does not match any of the available options, an error occurs.

188

GNU Octave

When called from an m-file function, the error is prefixed with the name of the caller
function.
The matching of options is case-insensitive.
See also: [varargin], page 186, [inputParser], page 184.

11.5 Ignoring Arguments
In the formal argument list, it is possible to use the dummy placeholder ~ instead of a name.
This indicates that the corresponding argument value should be ignored and not stored to
any variable.
function val = pick2nd (~, arg2)
val = arg2;
endfunction
The value of nargin is not affected by using this declaration.
Return arguments can also be ignored using the same syntax. For example, the sort
function returns both the sorted values, and an index vector for the original input which
will result in a sorted output. Ignoring the second output is simple—don’t request more
than one output. But ignoring the first, and calculating just the second output, requires
the use of the ~ placeholder.
x = [2, 3, 1];
[s, i] = sort (x)
⇒
s =
1

2

3

1

2

i =
3

[~, i] = sort (x)
⇒
i =
3

1

2

When using the ~ placeholder, commas—not whitespace—must be used to separate
output arguments. Otherwise, the interpreter will view ~ as the logical not operator.
[~ i] = sort (x)
parse error:
invalid left hand side of assignment
Functions may take advantage of ignored outputs to reduce the number of calculations
performed. To do so, use the isargout function to query whether the output argument is
wanted. For example:

Chapter 11: Functions and Scripts

189

function [out1, out2] = long_function (x, y, z)
if (isargout (1))
## Long calculation
...
out1 = result;
endif
...
endfunction

isargout (k)
Within a function, return a logical value indicating whether the argument k will be
assigned to a variable on output.
If the result is false, the argument has been ignored during the function call through
the use of the tilde (~) special output argument. Functions can use isargout to avoid
performing unnecessary calculations for outputs which are unwanted.
If k is outside the range 1:max (nargout), the function returns false. k can also be
an array, in which case the function works element-by-element and a logical array is
returned. At the top level, isargout returns an error.
See also: [nargout], page 180, [varargout], page 186, [nthargout], page 179.

11.6 Variable-length Return Lists
It is possible to return a variable number of output arguments from a function using a
syntax that’s similar to the one used with the special varargin parameter name. To let a
function return a variable number of output arguments the special output parameter name
varargout is used. As with varargin, varargout is a cell array that will contain the
requested output arguments.
As an example the following function sets the first output argument to 1, the second to
2, and so on.
function varargout = one_to_n ()
for i = 1:nargout
varargout{i} = i;
endfor
endfunction
When called this function returns values like this
[a, b, c] = one_to_n ()
⇒ a = 1
⇒ b = 2
⇒ c = 3
If varargin (varargout) does not appear as the last element of the input (output)
parameter list, then it is not special, and is handled the same as any other parameter name.

[r1, r2, ..., rn] = deal (a)
[r1, r2, ..., rn] = deal (a1, a2, . . . , an)
Copy the input parameters into the corresponding output parameters.
If only a single input parameter is supplied, its value is copied to each of the outputs.

190

GNU Octave

For example,
[a, b, c] = deal (x, y, z);
is equivalent to
a = x;
b = y;
c = z;
and
[a, b, c] = deal (x);
is equivalent to
a = b = c = x;
Programming Note: deal is often used with comma separated lists derived from cell
arrays or structures. This is unnecessary as the interpreter can perform the same
action without the overhead of a function call. For example:
c = {[1 2], "Three", 4};
[x, y, z] = c{:}
⇒
x =
1

2

y = Three
z = 4
See also: [cell2struct], page 122, [struct2cell], page 113, [repmat], page 458.

11.7 Returning from a Function
The body of a user-defined function can contain a return statement. This statement returns
control to the rest of the Octave program. It looks like this:
return
Unlike the return statement in C, Octave’s return statement cannot be used to return
a value from a function. Instead, you must assign values to the list of return variables that
are part of the function statement. The return statement simply makes it easier to exit
a function from a deeply nested loop or conditional statement.
Here is an example of a function that checks to see if any elements of a vector are nonzero.
function retval = any_nonzero (v)
retval = 0;
for i = 1:length (v)
if (v (i) != 0)
retval = 1;
return;
endif
endfor
printf ("no nonzero elements found\n");
endfunction

Chapter 11: Functions and Scripts

191

Note that this function could not have been written using the break statement to exit
the loop once a nonzero value is found without adding extra logic to avoid printing the
message if the vector does contain a nonzero element.
[Keyword]
When Octave encounters the keyword return inside a function or script, it returns
control to the caller immediately. At the top level, the return statement is ignored.
A return statement is assumed at the end of every function definition.

return

11.8 Default Arguments
Since Octave supports variable number of input arguments, it is very useful to assign default
values to some input arguments. When an input argument is declared in the argument list
it is possible to assign a default value to the argument like this
function name (arg1 = val1, ...)
body
endfunction
If no value is assigned to arg1 by the user, it will have the value val1.
As an example, the following function implements a variant of the classic “Hello, World”
program.
function hello (who = "World")
printf ("Hello, %s!\n", who);
endfunction
When called without an input argument the function prints the following
hello ();
a Hello, World!
and when it’s called with an input argument it prints the following
hello ("Beautiful World of Free Software");
a Hello, Beautiful World of Free Software!
Sometimes it is useful to explicitly tell Octave to use the default value of an input
argument. This can be done writing a ‘:’ as the value of the input argument when calling
the function.
hello (:);
a Hello, World!

11.9 Function Files
Except for simple one-shot programs, it is not practical to have to define all the functions
you need each time you need them. Instead, you will normally want to save them in a file
so that you can easily edit them, and save them for use at a later time.
Octave does not require you to load function definitions from files before using them.
You simply need to put the function definitions in a place where Octave can find them.
When Octave encounters an identifier that is undefined, it first looks for variables or
functions that are already compiled and currently listed in its symbol table. If it fails to
find a definition there, it searches a list of directories (the path) for files ending in .m that

192

GNU Octave

have the same base name as the undefined identifier.1 Once Octave finds a file with a name
that matches, the contents of the file are read. If it defines a single function, it is compiled
and executed. See Section 11.10 [Script Files], page 203, for more information about how
you can define more than one function in a single file.
When Octave defines a function from a function file, it saves the full name of the file it
read and the time stamp on the file. If the time stamp on the file changes, Octave may reload
the file. When Octave is running interactively, time stamp checking normally happens at
most once each time Octave prints the prompt. Searching for new function definitions also
occurs if the current working directory changes.
Checking the time stamp allows you to edit the definition of a function while Octave is
running, and automatically use the new function definition without having to restart your
Octave session.
To avoid degrading performance unnecessarily by checking the time stamps on functions
that are not likely to change, Octave assumes that function files in the directory tree octavehome/share/octave/version/m will not change, so it doesn’t have to check their time
stamps every time the functions defined in those files are used. This is normally a very
good assumption and provides a significant improvement in performance for the function
files that are distributed with Octave.
If you know that your own function files will not change while you are running Octave,
you can improve performance by calling ignore_function_time_stamp ("all"), so that
Octave will ignore the time stamps for all function files. Passing "system" to this function
resets the default behavior.

edit name
edit field value
value = edit ("get", field)
value = edit ("get", "all")
Edit the named function, or change editor settings.
If edit is called with the name of a file or function as its argument it will be opened
in the text editor defined by EDITOR.
• If the function name is available in a file on your path and that file is modifiable,
then it will be edited in place. If it is a system function, then it will first be
copied to the directory HOME (see below) and then edited. If no file is found, then
the m-file variant, ending with ".m", will be considered. If still no file is found,
then variants with a leading "@" and then with both a leading "@" and trailing
".m" will be considered.
• If name is the name of a function defined in the interpreter but not in an m-file,
then an m-file will be created in HOME to contain that function along with its
current definition.
• If name.cc is specified, then it will search for name.cc in the path and try to
modify it, otherwise it will create a new .cc file in the current directory. If name
happens to be an m-file or interpreter defined function, then the text of that
function will be inserted into the .cc file as a comment.
1

The ‘.m’ suffix was chosen for compatibility with matlab.

Chapter 11: Functions and Scripts

193

• If name.ext is on your path then it will be edited, otherwise the editor will be
started with name.ext in the current directory as the filename. If name.ext is
not modifiable, it will be copied to HOME before editing.
Warning: You may need to clear name before the new definition is available. If
you are editing a .cc file, you will need to execute mkoctfile name.cc before the
definition will be available.
If edit is called with field and value variables, the value of the control field field will
be set to value.
If an output argument is requested and the first input argument is get then edit
will return the value of the control field field. If the control field does not exist, edit
will return a structure containing all fields and values. Thus, edit ("get", "all")
returns a complete control structure.
The following control fields are used:
‘home’

This is the location of user local m-files. Be sure it is in your path. The
default is ~/octave.

‘author’

This is the name to put after the "## Author:" field of new functions.
By default it guesses from the gecos field of the password database.

‘email’

This is the e-mail address to list after the name in the author field. By
default it guesses <$LOGNAME@$HOSTNAME>, and if $HOSTNAME is not defined it uses uname -n. You probably want to override this. Be sure to
use the format user@host.

‘license’
‘gpl’

GNU General Public License (default).

‘bsd’

BSD-style license without advertising clause.

‘pd’

Public domain.

‘"text"’

Your own default copyright and license.

Unless you specify ‘pd’, edit will prepend the copyright statement with
"Copyright (C) YYYY Author".
‘mode’

This value determines whether the editor should be started in async mode
(editor is started in the background and Octave continues) or sync mode
(Octave waits until the editor exits). Set it to "sync" to start the editor
in sync mode. The default is "async" (see [system], page 845).

‘editinplace’
Determines whether files should be edited in place, without regard to
whether they are modifiable or not. The default is false.

mfilename ()
mfilename ("fullpath")
mfilename ("fullpathext")
Return the name of the currently executing file.
When called from outside an m-file return the empty string.

194

GNU Octave

Given the argument "fullpath", include the directory part of the filename, but not
the extension.
Given the argument "fullpathext", include the directory part of the filename and
the extension.

val = ignore_function_time_stamp ()
old_val = ignore_function_time_stamp (new_val)
Query or set the internal variable that controls whether Octave checks the time stamp
on files each time it looks up functions defined in function files.
If the internal variable is set to "system", Octave will not automatically recompile
function files in subdirectories of octave-home/lib/version if they have changed
since they were last compiled, but will recompile other function files in the search
path if they change.
If set to "all", Octave will not recompile any function files unless their definitions
are removed with clear.
If set to "none", Octave will always check time stamps on files to determine whether
functions defined in function files need to recompiled.

11.9.1 Manipulating the Load Path
When a function is called, Octave searches a list of directories for a file that contains the
function declaration. This list of directories is known as the load path. By default the
load path contains a list of directories distributed with Octave plus the current working
directory. To see your current load path call the path function without any input or output
arguments.
It is possible to add or remove directories to or from the load path using addpath and
rmpath. As an example, the following code adds ‘~/Octave’ to the load path.
addpath ("~/Octave")
After this the directory ‘~/Octave’ will be searched for functions.

addpath (dir1, . . . )
addpath (dir1, . . . , option)
Add named directories to the function search path.
If option is "-begin" or 0 (the default), prepend the directory name to the current
path. If option is "-end" or 1, append the directory name to the current path.
Directories added to the path must exist.
In addition to accepting individual directory arguments, lists of directory names separated by pathsep are also accepted. For example:
addpath ("dir1:/dir2:~/dir3")
For each directory that is added, and that was not already in the path, addpath
checks for the existence of a file named PKG_ADD (note lack of .m extension) and runs
it if it exists.
See also: [path], page 195, [rmpath], page 195, [genpath], page 195, [pathdef],
page 196, [savepath], page 195, [pathsep], page 196.

Chapter 11: Functions and Scripts

195

genpath (dir)
genpath (dir, skip, . . . )
Return a path constructed from dir and all its subdirectories.
If additional string parameters are given, the resulting path will exclude directories
with those names.

rmpath (dir1, . . . )
Remove dir1, . . . from the current function search path.
In addition to accepting individual directory arguments, lists of directory names separated by pathsep are also accepted. For example:
rmpath ("dir1:/dir2:~/dir3")
For each directory that is removed, rmpath checks for the existence of a file named
PKG_DEL (note lack of .m extension) and runs it if it exists.
See also: [path], page 195, [addpath], page 194, [genpath], page 195, [pathdef],
page 196, [savepath], page 195, [pathsep], page 196.

savepath ()
savepath (file)
status = savepath ( . . . )
Save the unique portion of the current function search path that is not set during
Octave’s initialization process to file.
If file is omitted, Octave looks in the current directory for a project-specific .octaverc
file in which to save the path information. If no such file is present then the user’s
configuration file ~/.octaverc is used.
If successful, savepath returns 0.
The savepath function makes it simple to customize a user’s configuration file to
restore the working paths necessary for a particular instance of Octave. Assuming no
filename is specified, Octave will automatically restore the saved directory paths from
the appropriate .octaverc file when starting up. If a filename has been specified
then the paths may be restored manually by calling source file.
See also: [path], page 195, [addpath], page 194, [rmpath], page 195, [genpath],
page 195, [pathdef], page 196.

path ()
str = path ()
str = path (path1, . . . )
Modify or display Octave’s load path.
If nargin and nargout are zero, display the elements of Octave’s load path in an easy
to read format.
If nargin is zero and nargout is greater than zero, return the current load path.
If nargin is greater than zero, concatenate the arguments, separating them with
pathsep. Set the internal search path to the result and return it.
No checks are made for duplicate elements.
See also: [addpath], page 194, [rmpath], page 195, [genpath], page 195, [pathdef],
page 196, [savepath], page 195, [pathsep], page 196.

196

GNU Octave

val = pathdef ()
Return the default path for Octave.
The path information is extracted from one of four sources. The possible sources, in
order of preference, are:
1. .octaverc
2. ~/.octaverc
3. /...//m/startup/octaverc
4. Octave’s path prior to changes by any octaverc file.
See also: [path], page 195, [addpath], page 194, [rmpath], page 195, [genpath],
page 195, [savepath], page 195.

val = pathsep ()
old_val = pathsep (new_val)
Query or set the character used to separate directories in a path.
See also: [filesep], page 837.

rehash ()
Reinitialize Octave’s load path directory cache.

file_in_loadpath (file)
file_in_loadpath (file, "all")
Return the absolute name of file if it can be found in the list of directories specified
by path.
If no file is found, return an empty character string.
If the first argument is a cell array of strings, search each directory of the loadpath
for element of the cell array and return the first that matches.
If the second optional argument "all" is supplied, return a cell array containing the
list of all files that have the same name in the path. If no files are found, return an
empty cell array.
See also: [file in path], page 836, [dir in loadpath], page 196, [path], page 195.

restoredefaultpath ( . . . )
Restore Octave’s path to its initial state at startup.
See also: [path], page 195, [addpath], page 194, [rmpath], page 195, [genpath],
page 195, [pathdef], page 196, [savepath], page 195, [pathsep], page 196.

command_line_path ( . . . )
Return the command line path variable.
See also: [path], page 195, [addpath], page 194, [rmpath], page 195, [genpath],
page 195, [pathdef], page 196, [savepath], page 195, [pathsep], page 196.

dir_in_loadpath (dir)
dir_in_loadpath (dir, "all")
Return the full name of the path element matching dir.

Chapter 11: Functions and Scripts

197

The match is performed at the end of each path element. For example, if dir
is "foo/bar", it matches the path element "/some/dir/foo/bar", but not
"/some/dir/foo/bar/baz" "/some/dir/allfoo/bar".
If the optional second argument is supplied, return a cell array containing all name
matches rather than just the first.
See also: [file in path], page 836, [file in loadpath], page 196, [path], page 195.

11.9.2 Subfunctions
A function file may contain secondary functions called subfunctions. These secondary functions are only visible to the other functions in the same function file. For example, a file
f.m containing
function f ()
printf ("in f, calling g\n");
g ()
endfunction
function g ()
printf ("in g, calling h\n");
h ()
endfunction
function h ()
printf ("in h\n")
endfunction
defines a main function f and two subfunctions. The subfunctions g and h may only be
called from the main function f or from the other subfunctions, but not from outside the
file f.m.

localfunctions ()
Return a list of all local functions, i.e., subfunctions, within the current file.
The return value is a column cell array of function handles to all local functions
accessible from the function from which localfunctions is called. Nested functions
are not included in the list.
If the call is from the command line, an anonymous function, or a script, the return
value is an empty cell array.
Compatibility Note: Subfunctions which contain nested functions are not included in
the list. This is a known issue.

11.9.3 Private Functions
In many cases one function needs to access one or more helper functions. If the helper
function is limited to the scope of a single function, then subfunctions as discussed above
might be used. However, if a single helper function is used by more than one function,
then this is no longer possible. In this case the helper functions might be placed in a
subdirectory, called "private", of the directory in which the functions needing access to this
helper function are found.
As a simple example, consider a function func1, that calls a helper function func2 to
do much of the work. For example:

198

GNU Octave

function y = func1 (x)
y = func2 (x);
endfunction

Then if the path to func1 is /func1.m, and if func2 is found in the directory
/private/func2.m, then func2 is only available for use of the functions, like
func1, that are found in .

11.9.4 Nested Functions
Nested functions are similar to subfunctions in that only the main function is visible outside
the file. However, they also allow for child functions to access the local variables in their
parent function. This shared access mimics using a global variable to share information —
but a global variable which is not visible to the rest of Octave. As a programming strategy,
sharing data this way can create code which is difficult to maintain. It is recommended to
use subfunctions in place of nested functions when possible.

As a simple example, consider a parent function foo, that calls a nested child function
bar, with a shared variable x.

function y = foo ()
x = 10;
bar ();
y = x;
function bar ()
x = 20;
endfunction
endfunction
foo ()
⇒ 20
Notice that there is no special syntax for sharing x. This can lead to problems with accidental variable sharing between a parent function and its child. While normally variables
are inherited, child function parameters and return values are local to the child function.

Now consider the function foobar that uses variables x and y. foobar calls a nested
function foo which takes x as a parameter and returns y. foo then calls bat which does
some computation.

Chapter 11: Functions and Scripts

199

function z = foobar ()
x = 0;
y = 0;
z = foo (5);
z += x + y;
function y = foo (x)
y = x + bat ();
function z = bat ()
z = x;
endfunction
endfunction
endfunction
foobar ()
⇒ 10
It is important to note that the x and y in foobar remain zero, as in foo they are a return
value and parameter respectively. The x in bat refers to the x in foo.
Variable inheritance leads to a problem for eval and scripts. If a new variable is created
in a parent function, it is not clear what should happen in nested child functions. For
example, consider a parent function foo with a nested child function bar:
function y = foo (to_eval)
bar ();
eval (to_eval);
function bar ()
eval ("x = 100;");
eval ("y = x;");
endfunction
endfunction
foo ("x = 5;")
⇒ error: can not add variable "x" to a static workspace
foo ("y = 10;")
⇒ 10
foo ("")
⇒ 100
The parent function foo is unable to create a new variable x, but the child function bar
was successful. Furthermore, even in an eval statement y in bar is the same y as in its
parent function foo. The use of eval in conjunction with nested functions is best avoided.
As with subfunctions, only the first nested function in a file may be called from the
outside. Inside a function the rules are more complicated. In general a nested function may
call:

200

GNU Octave

0. Globally visible functions
1. Any function that the nested function’s parent can call
2. Sibling functions (functions that have the same parents)
3. Direct children
As a complex example consider a parent function ex_top with two child functions, ex_a
and ex_b. In addition, ex_a has two more child functions, ex_aa and ex_ab. For example:
function ex_top ()
## Can call: ex_top, ex_a, and ex_b
## Can NOT call: ex_aa and ex_ab
function ex_a ()
## Can call everything
function ex_aa ()
## Can call everything
endfunction
function ex_ab ()
## Can call everything
endfunction
endfunction
function ex_b ()
## Can call: ex_top, ex_a, and ex_b
## Can NOT call: ex_aa and ex_ab
endfunction
endfunction

11.9.5 Overloading and Autoloading
Functions can be overloaded to work with different input arguments. For example, the operator ’+’ has been overloaded in Octave to work with single, double, uint8, int32, and many
other arguments. The preferred way to overload functions is through classes and object
oriented programming (see Section 34.4.1 [Function Overloading], page 785). Occasionally,
however, one needs to undo user overloading and call the default function associated with
a specific type. The builtin function exists for this purpose.

[...] = builtin (f, . . . )
Call the base function f even if f is overloaded to another function for the given type
signature.
This is normally useful when doing object-oriented programming and there is a requirement to call one of Octave’s base functions rather than the overloaded one of a
new class.
A trivial example which redefines the sin function to be the cos function shows how
builtin works.

Chapter 11: Functions and Scripts

201

sin (0)
⇒ 0
function y = sin (x), y = cos (x); endfunction
sin (0)
⇒ 1
builtin ("sin", 0)
⇒ 0
A single dynamically linked file might define several functions. However, as Octave
searches for functions based on the functions filename, Octave needs a manner in which to
find each of the functions in the dynamically linked file. On operating systems that support
symbolic links, it is possible to create a symbolic link to the original file for each of the
functions which it contains.
However, there is at least one well known operating system that doesn’t support symbolic
links. Making copies of the original file for each of the functions is undesirable as it increases
the amount of disk space used by Octave. Instead Octave supplies the autoload function,
that permits the user to define in which file a certain function will be found.

autoload_map = autoload ()
autoload (function, file)
autoload ( . . . , "remove")
Define function to autoload from file.
The second argument, file, should be an absolute filename or a file name in the same
directory as the function or script from which the autoload command was run. file
should not depend on the Octave load path.
Normally, calls to autoload appear in PKG ADD script files that are evaluated when
a directory is added to Octave’s load path. To avoid having to hardcode directory
names in file, if file is in the same directory as the PKG ADD script then
autoload ("foo", "bar.oct");
will load the function foo from the file bar.oct. The above usage when bar.oct is
not in the same directory, or usages such as
autoload ("foo", file_in_loadpath ("bar.oct"))
are strongly discouraged, as their behavior may be unpredictable.
With no arguments, return a structure containing the current autoload map.
If a third argument "remove" is given, the function is cleared and not loaded anymore
during the current Octave session.
See also: [PKG ADD], page 870.

11.9.6 Function Locking
It is sometime desirable to lock a function into memory with the mlock function. This is
typically used for dynamically linked functions in Oct-files or mex-files that contain some
initialization, and it is desirable that calling clear does not remove this initialization.
As an example,
function my_function ()
mlock ();
...

202

GNU Octave

prevents my_function from being removed from memory after it is called, even if clear is
called. It is possible to determine if a function is locked into memory with the mislocked,
and to unlock a function with munlock, which the following illustrates.
my_function ();
mislocked ("my_function")
⇒ ans = 1
munlock ("my_function");
mislocked ("my_function")
⇒ ans = 0
A common use of mlock is to prevent persistent variables from being removed from
memory, as the following example shows:
function count_calls ()
mlock ();
persistent calls = 0;
printf ("’count_calls’ has been called %d times\n",
++calls);
endfunction
count_calls ();
a ’count_calls’ has been called 1 times
clear count_calls
count_calls ();
a ’count_calls’ has been called 2 times

mlock might equally be used to prevent changes to a function from having effect in
Octave, though a similar effect can be had with the ignore_function_time_stamp function.

mlock ()
Lock the current function into memory so that it can’t be cleared.
See also: [munlock], page 202, [mislocked], page 202, [persistent], page 128.

munlock ()
munlock (fcn)
Unlock the named function fcn.
If no function is named then unlock the current function.
See also: [mlock], page 202, [mislocked], page 202, [persistent], page 128.

mislocked ()
mislocked (fcn)
Return true if the named function fcn is locked.
If no function is named then return true if the current function is locked.
See also: [mlock], page 202, [munlock], page 202, [persistent], page 128.

Chapter 11: Functions and Scripts

203

11.9.7 Function Precedence
Given the numerous different ways that Octave can define a function, it is possible and even
likely that multiple versions of a function, might be defined within a particular scope. The
precedence of which function will be used within a particular scope is given by
1. Subfunction A subfunction with the required function name in the given scope.
2. Private function A function defined within a private directory of the directory which
contains the current function.
3. Class constructor A function that constructs a user class as defined in chapter
Chapter 34 [Object Oriented Programming], page 775.
4. Class method An overloaded function of a class as in chapter Chapter 34 [Object
Oriented Programming], page 775.
5. Command-line Function A function that has been defined on the command-line.
6. Autoload function A function that is marked as autoloaded with See [autoload],
page 201.
7. A Function on the Path A function that can be found on the users load-path. There can
also be Oct-file, mex-file or m-file versions of this function and the precedence between
these versions are in that order.
8. Built-in function A function that is a part of core Octave such as numel, size, etc.

11.10 Script Files
A script file is a file containing (almost) any sequence of Octave commands. It is read and
evaluated just as if you had typed each command at the Octave prompt, and provides a
convenient way to perform a sequence of commands that do not logically belong inside a
function.
Unlike a function file, a script file must not begin with the keyword function. If it does,
Octave will assume that it is a function file, and that it defines a single function that should
be evaluated as soon as it is defined.
A script file also differs from a function file in that the variables named in a script file
are not local variables, but are in the same scope as the other variables that are visible on
the command line.
Even though a script file may not begin with the function keyword, it is possible to
define more than one function in a single script file and load (but not execute) all of them
at once. To do this, the first token in the file (ignoring comments and other white space)
must be something other than function. If you have no other statements to evaluate, you
can use a statement that has no effect, like this:
# Prevent Octave from thinking that this
# is a function file:
1;
# Define function one:
function one ()
...

204

GNU Octave

To have Octave read and compile these functions into an internal form, you need to
make sure that the file is in Octave’s load path (accessible through the path function), then
simply type the base name of the file that contains the commands. (Octave uses the same
rules to search for script files as it does to search for function files.)
If the first token in a file (ignoring comments) is function, Octave will compile the function and try to execute it, printing a message warning about any non-whitespace characters
that appear after the function definition.
Note that Octave does not try to look up the definition of any identifier until it needs
to evaluate it. This means that Octave will compile the following statements if they appear
in a script file, or are typed at the command line,
# not a function file:
1;
function foo ()
do_something ();
endfunction
function do_something ()
do_something_else ();
endfunction
even though the function do_something is not defined before it is referenced in the function
foo. This is not an error because Octave does not need to resolve all symbols that are
referenced by a function until the function is actually evaluated.
Since Octave doesn’t look for definitions until they are needed, the following code will
always print ‘bar = 3’ whether it is typed directly on the command line, read from a script
file, or is part of a function body, even if there is a function or script file called bar.m in
Octave’s path.
eval ("bar = 3");
bar
Code like this appearing within a function body could fool Octave if definitions were
resolved as the function was being compiled. It would be virtually impossible to make
Octave clever enough to evaluate this code in a consistent fashion. The parser would have
to be able to perform the call to eval at compile time, and that would be impossible unless
all the references in the string to be evaluated could also be resolved, and requiring that
would be too restrictive (the string might come from user input, or depend on things that
are not known until the function is evaluated).
Although Octave normally executes commands from script files that have the name
file.m, you can use the function source to execute commands from any file.

source (file)
source (file, context)
Parse and execute the contents of file.
Without specifying context, this is equivalent to executing commands from a script
file, but without requiring the file to be named file.m or to be on the execution path.
Instead of the current context, the script may be executed in either the context of
the function that called the present function ("caller"), or the top-level context
("base").

Chapter 11: Functions and Scripts

205

See also: [run], page 161.

11.10.1 Publish Octave Script Files
The function publish provides a dynamic possibility to document your script file. Unlike
static documentation, publish runs the script file, saves any figures and output while
running the script, and presents them alongside static documentation in a desired output
format. The static documentation can make use of Section 11.10.2 [Publishing Markup],
page 207, to enhance and customize the output.

publish (filename)
publish (filename, output_format)
publish (filename, option1, value1, . . . )
publish (filename, options)
output_file = publish (filename, . . . )
Generate reports from Octave script files in several output formats.
The generated reports consider Publishing Markup in comments, which is explained
in detail in the GNU Octave manual. Assume the following example, using some
Publishing Markup, to be the content of a script file named ‘example.m’:
%% Headline title
%
% Some *bold*, _italic_, or |monospaced| Text with
% a .
%%
# "Real" Octave commands to be evaluated
sombrero ()
## Octave comment
#
# * Bulleted list
# * Bulleted list
#
# # Numbered list
# # Numbered list

style supported as well
item 1
item 2
item 1
item 2

To publish this script file, type publish ("example.m").
With only filename given, a HTML report is generated in a subdirectory ‘html’ relative to the current working directory. The Octave commands are evaluated in a
separate context and any figures created while executing the script file are included
in the report. All formatting syntax of filename is treated according to the specified
output format and included in the report.
Using publish (filename, output_format) is equivalent to the function call using
a structure
options.format = output_format;
publish (filename, options)
which is described below. The same holds for using option-value-pairs

206

GNU Octave

options.option1 = value1;
publish (filename, options)
The structure options can have the following field names. If a field name is not
specified, the default value is considered:
• ‘format’ — Output format of the published script file, one of
‘html’ (default), ‘doc’, ‘latex’, ‘ppt’, ‘xml’, or ‘pdf’.

The output formats ‘doc’, ‘ppt’, and ‘xml’ are currently not supported. To
generate a ‘doc’ report, open a generated ‘html’ report with your office suite.
• ‘outputDir’ — Full path string of a directory, where the generated report will be
located. If no directory is given, the report is generated in a subdirectory ‘html’
relative to the current working directory.
• ‘stylesheet’ — Not supported, only for matlab compatibility.

• ‘createThumbnail’ — Not supported, only for matlab compatibility.

• ‘figureSnapMethod’ — Not supported, only for matlab compatibility.

• ‘imageFormat’ — Desired format for images produced, while evaluating the code.
The allowed image formats depend on the output format:
• ‘html’ and ‘xml’ — ‘png’ (default), any other image format supported by
Octave
• ‘latex’ — ‘epsc2’ (default), any other image format supported by Octave

• ‘pdf’ — ‘jpg’ (default) or ‘bmp’, note matlab uses ‘bmp’ as default
• ‘doc’ or ‘ppt’ — ‘png’ (default), ‘jpg’, ‘bmp’, or ‘tiff’

• ‘maxHeight’ and ‘maxWidth’ — Maximum height (width) of the produced images
in pixels. An empty value means no restriction. Both values have to be set, to
work properly.
‘[]’ (default), integer value ≥ 0

• ‘useNewFigure’ — Use a new figure window for figures created by the evaluated
code. This avoids side effects with already opened figure windows.
‘true’ (default) or ‘false’
• ‘evalCode’ — Evaluate code of the Octave source file
‘true’ (default) or ‘false’

• ‘catchError’ — Catch errors while code evaluation and continue
‘true’ (default) or ‘false’

• ‘codeToEvaluate’ — Octave commands that should be evaluated prior to publishing the script file. These Octave commands do not appear in the generated
report.
• ‘maxOutputLines’ — Maximum number of shown output lines of the code evaluation
‘Inf’ (default) or integer value > 0
• ‘showCode’ — Show the evaluated Octave commands in the generated report
‘true’ (default) or ‘false’

Chapter 11: Functions and Scripts

207

The returned output file is a string with the path and file name of the generated
report.
See also: [grabcode], page 207.
The counterpart to publish is grabcode:

grabcode (url)
code_str = grabcode (url)
Grab by the publish function generated HTML reports from Octave script files.
The input parameter url must point to a local or remote HTML file with extension
‘.htm’ or ‘.html’ which was generated by the publish function. With any other
HTML file this will not work!
If no return value is given, the grabbed code is saved to a temporary file and opened
in the default editor.
NOTE: You have to save the file at another location with arbitrary name, otherwise
any grabbed code will be lost!
With a return value given, the grabbed code will be returned as string code str.
An example:
publish ("my_script.m");
grabcode ("html/my_script.html");
The example above publishes ‘my_script.m’ by default to ‘html/my_script.html’.
Afterwards this published Octave script is grabbed to edit its content in a new temporary file.
See also: [publish], page 205.

11.10.2 Publishing Markup
11.10.2.1 Using Publishing Markup in Script Files
To use Publishing Markup, start by typing ‘##’ or ‘%%’ at the beginning of a new line. For
matlab compatibility ‘%%’ is treated the same way as ‘##’.
The lines following ‘##’ or ‘%%’ start with one of either ‘#’ or ‘%’ followed by at least one
space. These lines are interpreted as section. A section ends at the first line not starting
with ‘#’ or ‘%’, or when the end of the document is reached.
A section starting in the first line of the document, followed by another start of a section
that might be empty, is interpreted as a document title and introduction text.
See the example below for clarity:

208

GNU Octave

%% Headline title
%
% Some *bold*, _italic_, or |monospaced| Text with
% a .
%%
# "Real" Octave commands to be evaluated
sombrero ()
## Octave comment
#
# * Bulleted list
# * Bulleted list
#
# # Numbered list
# # Numbered list

style supported as well
item 1
item 2
item 1
item 2

11.10.2.2 Text Formatting
Basic text formatting is supported inside sections, see the example given below:
##
# *bold*, _italic_, or |monospaced| Text
Additionally two trademark symbols are supported, just embrace the letters ‘TM’ or ‘R’.
##
# (TM) or (R)

11.10.2.3 Sections
A section is started by typing ‘##’ or ‘%%’ at the beginning of a new line. A section title can
be provided by writing it, separated by a space, in the first line after ‘##’ or ‘%%’. Without
a section title, the section is interpreted as a continuation of the previous section. For
matlab compatibility ‘%%’ is treated the same way as ‘%%’.
some_code ();
## Section 1
#
## Section 2
some_code ();
##
# Still in section 2
some_code ();
### Section 3
#
#

Chapter 11: Functions and Scripts

209

11.10.2.4 Preformatted Code
To write preformatted code inside a section, indent the code by three spaces after ‘#’ at the
beginning of each line and leave the lines above and below the code blank, except for ‘#’ at
the beginning of those lines.
##
# This is a syntax highlighted for-loop:
#
#
for i = 1:5
#
disp (i);
#
endfor
#
# And more usual text.

11.10.2.5 Preformatted Text
To write preformatted text inside a section, indent the code by two spaces after ‘#’ at the
beginning of each line and leave the lines above and below the preformatted text blank,
except for ‘#’ at the beginning of those lines.
##
# This following text is preformatted:
#
# "To be, or not to be: that is the question:
# Whether ’tis nobler in the mind to suffer
# The slings and arrows of outrageous fortune,
# Or to take arms against a sea of troubles,
# And by opposing end them? To die: to sleep;"
#
# --"Hamlet" by W. Shakespeare

11.10.2.6 Bulleted Lists
To create a bulleted list, type
##
#
# * Bulleted list item 1
# * Bulleted list item 2
#
to get output like
• Bulleted list item 1
• Bulleted list item 2
Notice the blank lines, except for the ‘#’ or ‘%’ before and after the bulleted list!

11.10.2.7 Numbered Lists
To create a numbered list, type

210

GNU Octave

##
#
# # Numbered list item 1
# # Numbered list item 2
#
to get output like
1. Numbered list item 1
2. Numbered list item 2
Notice the blank lines, except for the ‘#’ or ‘%’ before and after the numbered list!

11.10.2.8 Including File Content
To include the content of an external file, e.g., a file called ‘my_function.m’ at the same
location as the published Octave script, use the following syntax to include it with Octave
syntax highlighting.
Alternatively, you can write the full or relative path to the file.
##
#
# my_function.m
#
# /full/path/to/my_function.m
#
# ../relative/path/to/my_function.m
#

11.10.2.9 Including Graphics
To include external graphics, e.g., a graphic called ‘my_graphic.png’ at the same location
as the published Octave script, use the following syntax.
Alternatively, you can write the full path to the graphic.
##
#
# <>
#
# <>
#
# <<../relative/path/to/my_graphic.png>>
#

11.10.2.10 Including URLs
Basically, a URL is written between an opening ‘<’ and a closing ‘>’ angle.
##
# 
Text that is within these angles and separated by at least one space from the URL is a
displayed text for the link.
##
# 

Chapter 11: Functions and Scripts

211

A link starting with ‘

11.10.2.11 Mathematical Equations
One can insert LATEX inline math, surrounded by single ‘$’ signs, or displayed math, surrounded by double ‘$$’ signs, directly inside sections.
##
# Some shorter inline equation $e^{ix} = \cos x + i\sin x$.
#
# Or more complicated formulas as displayed math:
# $$e^x = \lim_{n\rightarrow\infty}\left(1+\dfrac{x}{n}\right)^{n}.$$

11.10.2.12 HTML Markup
If the published output is a HTML report, you can insert HTML markup, that is only
visible in this kind of output.
##
# 
# 
# 
# 
# 

11.10.2.13 LaTeX Markup
If the published output is a LATEX or PDF report, you can insert LATEX markup, that is
only visible in this kind of output.
##
# 
# Some output only visible in LaTeX or PDF reports.
# \begin{equation}
# e^x = \lim\limits_{n\rightarrow\infty}\left(1+\dfrac{x}{n}\right)^{n}
# \end{equation}
# 

11.11 Function Handles, Anonymous Functions, Inline
Functions
It can be very convenient store a function in a variable so that it can be passed to a different
function. For example, a function that performs numerical minimization needs access to
the function that should be minimized.

11.11.1 Function Handles
A function handle is a pointer to another function and is defined with the syntax
@function-name

212

GNU Octave

For example,
f = @sin;
creates a function handle called f that refers to the function sin.
Function handles are used to call other functions indirectly, or to pass a function as an
argument to another function like quad or fsolve. For example:
f = @sin;
quad (f, 0, pi)
⇒ 2
You may use feval to call a function using function handle, or simply write the name
of the function handle followed by an argument list. If there are no arguments, you must
use an empty argument list ‘()’. For example:
f = @sin;
feval (f, pi/4)
⇒ 0.70711
f (pi/4)
⇒ 0.70711

is_function_handle (x)
Return true if x is a function handle.
See also: [isa], page 39, [typeinfo], page 39, [class], page 39, [functions], page 212.

s = functions (fcn_handle)
Return a structure containing information about the function handle fcn handle.
The structure s always contains these three fields:
function

The function name. For an anonymous function (no name) this will be
the actual function definition.

type

Type of the function.
anonymous
The function is anonymous.
private

The function is private.

overloaded
The function overloads an existing function.
simple

The function is a built-in or m-file function.

subfunction
The function is a subfunction within an m-file.
file

The m-file that will be called to perform the function. This field is empty
for anonymous and built-in functions.

In addition, some function types may return more information in additional fields.
Warning: functions is provided for debugging purposes only. Its behavior may
change in the future and programs should not depend on any particular output format.
See also: [func2str], page 213, [str2func], page 213.

Chapter 11: Functions and Scripts

213

func2str (fcn_handle)
Return a string containing the name of the function referenced by the function handle
fcn handle.
See also: [str2func], page 213, [functions], page 212.

str2func (fcn_name)
str2func (fcn_name, "global")
Return a function handle constructed from the string fcn name.
If the optional "global" argument is passed, locally visible functions are ignored in
the lookup.
See also: [func2str], page 213, [inline], page 214, [functions], page 212.

11.11.2 Anonymous Functions
Anonymous functions are defined using the syntax
@(argument-list) expression
Any variables that are not found in the argument list are inherited from the enclosing scope.
Anonymous functions are useful for creating simple unnamed functions from expressions or
for wrapping calls to other functions to adapt them for use by functions like quad. For
example,
f = @(x) x.^2;
quad (f, 0, 10)
⇒ 333.33
creates a simple unnamed function from the expression x.^2 and passes it to quad,
quad (@(x) sin (x), 0, pi)
⇒ 2
wraps another function, and
a = 1;
b = 2;
quad (@(x) betainc (x, a, b), 0, 0.4)
⇒ 0.13867
adapts a function with several parameters to the form required by quad. In this example,
the values of a and b that are passed to betainc are inherited from the current environment.
Note that for performance reasons it is better to use handles to existing Octave functions,
rather than to define anonymous functions which wrap an existing function. The integration
of sin (x) is 5X faster if the code is written as
quad (@sin, 0, pi)
rather than using the anonymous function @(x) sin (x). There are many operators which
have functional equivalents that may be better choices than an anonymous function. Instead
of writing
f = @(x, y) x + y
this should be coded as
f = @plus
See Section 34.4.2 [Operator Overloading], page 786, for a list of operators which also
have a functional form.

214

GNU Octave

11.11.3 Inline Functions
An inline function is created from a string containing the function body using the inline
function. The following code defines the function f (x) = x2 + 2.
f = inline ("x^2 + 2");
After this it is possible to evaluate f at any x by writing f(x).
Caution: matlab has begun the process of deprecating inline functions. At some point
in the future support will be dropped and eventually Octave will follow matlab and also
remove inline functions. Use anonymous functions in all new code.

inline (str)
inline (str, arg1, . . . )
inline (str, n)
Create an inline function from the character string str.
If called with a single argument, the arguments of the generated function are extracted
from the function itself. The generated function arguments will then be in alphabetical
order. It should be noted that i and j are ignored as arguments due to the ambiguity
between their use as a variable or their use as an built-in constant. All arguments
followed by a parenthesis are considered to be functions. If no arguments are found,
a function taking a single argument named x will be created.
If the second and subsequent arguments are character strings, they are the names of
the arguments of the function.
If the second argument is an integer n, the arguments are "x", "P1", . . . , "PN".
Programming Note: The use of inline is discouraged and it may be removed from
a future version of Octave. The preferred way to create functions from strings is
through the use of anonymous functions (see Section 11.11.2 [Anonymous Functions],
page 213) or str2func.
See also: [argnames], page 214, [formula], page 214, [vectorize], page 534, [str2func],
page 213.

argnames (fun)
Return a cell array of character strings containing the names of the arguments of the
inline function fun.
See also: [inline], page 214, [formula], page 214, [vectorize], page 534.

formula (fun)
Return a character string representing the inline function fun.
Note that char (fun) is equivalent to formula (fun).
See also: [char], page 71, [argnames], page 214, [inline], page 214, [vectorize],
page 534.

vars = symvar (str)
Identify the symbolic variable names in the string str.
Common constant names such as i, j, pi, Inf and Octave functions such as sin or
plot are ignored.

Chapter 11: Functions and Scripts

215

Any names identified are returned in a cell array of strings. The array is empty if no
variables were found.
Example:
symvar ("x^2 + y^2 == 4")
⇒ {
[1,1] = x
[2,1] = y
}

11.12 Commands
Commands are a special class of functions that only accept string input arguments. A
command can be called as an ordinary function, but it can also be called without the
parentheses. For example,
my_command hello world
is equivalent to
my_command ("hello", "world")
The general form of a command call is
cmdname arg1 arg2 ...
which translates directly to
cmdname ("arg1", "arg2", ...)
Any regular function can be used as a command if it accepts string input arguments.
For example:
toupper lower_case_arg
⇒ ans = LOWER_CASE_ARG

One difficulty of commands occurs when one of the string input arguments is stored in a
variable. Because Octave can’t tell the difference between a variable name and an ordinary
string, it is not possible to pass a variable as input to a command. In such a situation a
command must be called as a function. For example:
strvar = "hello world";
toupper strvar
⇒ ans = STRVAR
toupper (strvar)
⇒ ans = HELLO WORLD

11.13 Organization of Functions Distributed with Octave
Many of Octave’s standard functions are distributed as function files. They are loosely
organized by topic, in subdirectories of octave-home/lib/octave/version/m, to make it
easier to find them.
The following is a list of all the function file subdirectories, and the types of functions
you will find there.
audio

Functions for playing and recording sounds.

216

GNU Octave

deprecated
Out-of-date functions which will eventually be removed from Octave.
elfun

Elementary functions, principally trigonometric.

@ftp

Class functions for the FTP object.

general

Miscellaneous matrix manipulations, like flipud, rot90, and triu, as well as
other basic functions, like ismatrix, narginchk, etc.

geometry

Functions related to Delaunay triangulation.

help

Functions for Octave’s built-in help system.

image

Image processing tools. These functions require the X Window System.

io

Input-output functions.

linear-algebra
Functions for linear algebra.
miscellaneous
Functions that don’t really belong anywhere else.
optimization
Functions related to minimization, optimization, and root finding.
path

Functions to manage the directory path Octave uses to find functions.

pkg

Package manager for installing external packages of functions in Octave.

plot

Functions for displaying and printing two- and three-dimensional graphs.

polynomial
Functions for manipulating polynomials.
prefs

Functions implementing user-defined preferences.

set

Functions for creating and manipulating sets of unique values.

signal

Functions for signal processing applications.

sparse

Functions for handling sparse matrices.

specfun

Special functions such as bessel or factor.

special-matrix
Functions that create special matrix forms such as Hilbert or Vandermonde
matrices.
startup

Octave’s system-wide startup file.

statistics
Statistical functions.
strings

Miscellaneous string-handling functions.

testfun

Functions for performing unit tests on other functions.

time

Functions related to time and date processing.

217

12 Errors and Warnings
Octave includes several functions for printing error and warning messages. When you write
functions that need to take special action when they encounter abnormal conditions, you
should print the error messages using the functions described in this chapter.
Since many of Octave’s functions use these functions, it is also useful to understand
them, so that errors and warnings can be handled.

12.1 Handling Errors
An error is something that occurs when a program is in a state where it doesn’t make sense
to continue. An example is when a function is called with too few input arguments. In this
situation the function should abort with an error message informing the user of the lacking
input arguments.
Since an error can occur during the evaluation of a program, it is very convenient to be
able to detect that an error occurred, so that the error can be fixed. This is possible with
the try statement described in Section 10.9 [The try Statement], page 172.

12.1.1 Raising Errors
The most common use of errors is for checking input arguments to functions. The following
example calls the error function if the function f is called without any input arguments.
function f (arg1)
if (nargin == 0)
error ("not enough input arguments");
endif
endfunction
When the error function is called, it prints the given message and returns to the Octave
prompt. This means that no code following a call to error will be executed.
It is also possible to assign an identification string to an error. If an error has such
an ID the user can catch this error as will be described in the next section. To assign
an ID to an error, simply call error with two string arguments, where the first is the
identification string, and the second is the actual error. Note that error IDs are in the format
"NAMESPACE:ERROR-NAME". The namespace "Octave" is used for Octave’s own errors. Any
other string is available as a namespace for user’s own errors.

error (template, . . . )
error (id, template, . . . )
Display an error message and stop m-file execution.
Format the optional arguments under the control of the template string template
using the same rules as the printf family of functions (see Section 14.2.4 [Formatted
Output], page 270) and print the resulting message on the stderr stream. The
message is prefixed by the character string ‘error: ’.
Calling error also sets Octave’s internal error state such that control will return to
the top level without evaluating any further commands. This is useful for aborting
from functions or scripts.

218

GNU Octave

If the error message does not end with a newline character, Octave will print a traceback of all the function calls leading to the error. For example, given the following
function definitions:
function f () g (); end
function g () h (); end
function h () nargin == 1 || error ("nargin != 1"); end
calling the function f will result in a list of messages that can help you to quickly find
the exact location of the error:
f ()
error: nargin != 1
error: called from:
error:
h at line 1, column 27
error:
g at line 1, column 15
error:
f at line 1, column 15
If the error message ends in a newline character, Octave will print the message but
will not display any traceback messages as it returns control to the top level. For
example, modifying the error message in the previous example to end in a newline
causes Octave to only print a single message:
function h () nargin == 1 || error ("nargin != 1\n"); end
f ()
error: nargin != 1
A null string ("") input to error will be ignored and the code will continue running
as if the statement were a NOP. This is for compatibility with matlab. It also makes
it possible to write code such as
err_msg = "";
if (CONDITION 1)
err_msg = "CONDITION 1 found";
elseif (CONDITION2)
err_msg = "CONDITION 2 found";
...
endif
error (err_msg);
which will only stop execution if an error has been found.
Implementation Note: For compatibility with matlab, escape sequences in template
(e.g., "\n" => newline) are processed regardless of whether template has been defined
with single quotes, as long as there are two or more input arguments. To disable escape
sequence expansion use a second backslash before the sequence (e.g., "\\n") or use
the regexptranslate function.
See also: [warning], page 224, [lasterror], page 220.
Since it is common to use errors when there is something wrong with the input to a
function, Octave supports functions to simplify such code. When the print_usage function
is called, it reads the help text of the function calling print_usage, and presents a useful
error. If the help text is written in Texinfo it is possible to present an error message that
only contains the function prototypes as described by the @deftypefn parts of the help

Chapter 12: Errors and Warnings

219

text. When the help text isn’t written in Texinfo, the error message contains the entire
help message.
Consider the following function.
## -*- texinfo -*## @deftypefn {} f (@var{arg1})
## Function help text goes here...
## @end deftypefn
function f (arg1)
if (nargin == 0)
print_usage ();
endif
endfunction
When it is called with no input arguments it produces the following error.
f ()
a
a
a
a
a
a
a
a
a
a
a
a

error: Invalid call to f.

Correct usage is:

-- f (ARG1)

Additional help for built-in functions and operators is
available in the online version of the manual. Use the command
’doc ’ to search the manual index.
Help and information about Octave is also available on the WWW
at http://www.octave.org and via the help@octave.org
mailing list.

print_usage ()
print_usage (name)
Print the usage message for the function name.
When called with no input arguments the print_usage function displays the usage
message of the currently executing function.
See also: [help], page 20.

beep ()
Produce a beep from the speaker (or visual bell).
This function sends the alarm character "\a" to the terminal. Depending on the
user’s configuration this may produce an audible beep, a visual bell, or nothing at all.
See also: [puts], page 269, [fputs], page 269, [printf], page 270, [fprintf], page 271.

val = beep_on_error ()
old_val = beep_on_error (new_val)
beep_on_error (new_val, "local")
Query or set the internal variable that controls whether Octave will try to ring the
terminal bell before printing an error message.

220

GNU Octave

When called from inside a function with the "local" option, the variable is changed
locally for the function and any subroutines it calls. The original variable value is
restored when exiting the function.

12.1.2 Catching Errors
When an error occurs, it can be detected and handled using the try statement as described
in Section 10.9 [The try Statement], page 172. As an example, the following piece of code
counts the number of errors that occurs during a for loop.
number_of_errors = 0;
for n = 1:100
try
...
catch
number_of_errors++;
end_try_catch
endfor
The above example treats all errors the same. In many situations it can however be
necessary to discriminate between errors, and take different actions depending on the error.
The lasterror function returns a structure containing information about the last error
that occurred. As an example, the code above could be changed to count the number of
errors related to the ‘*’ operator.
number_of_errors = 0;
for n = 1:100
try
...
catch
msg = lasterror.message;
if (strfind (msg, "operator *"))
number_of_errors++;
endif
end_try_catch
endfor
Alternatively, the output of the lasterror function can be found in a variable indicated
immediately after the catch keyword, as in the example below showing how to redirect an
error as a warning:
try
...
catch err
warning(err.identifier, err.message);
...
end_try_catch

lasterr = lasterror ()
lasterror (err)
lasterror ("reset")
Query or set the last error message structure.

Chapter 12: Errors and Warnings

221

When called without arguments, return a structure containing the last error message
and other information related to this error. The elements of the structure are:
message

The text of the last error message

identifier
The message identifier of this error message
stack

A structure containing information on where the message occurred. This
may be an empty structure if the information cannot be obtained. The
fields of the structure are:
file

The name of the file where the error occurred

name

The name of function in which the error occurred

line

The line number at which the error occurred

column

An optional field with the column number at which the error
occurred

The last error structure may be set by passing a scalar structure, err, as input. Any
fields of err that match those above are set while any unspecified fields are initialized
with default values.
If lasterror is called with the argument "reset", all fields are set to their default
values.
See also: [lasterr], page 221, [error], page 217, [lastwarn], page 225.

[msg, msgid] = lasterr ()
lasterr (msg)
lasterr (msg, msgid)
Query or set the last error message.
When called without input arguments, return the last error message and message
identifier.
With one argument, set the last error message to msg.
With two arguments, also set the last message identifier.
See also: [lasterror], page 220, [error], page 217, [lastwarn], page 225.
The next example counts indexing errors. The errors are caught using the field identifier
of the structure returned by the function lasterror.
number_of_errors = 0;
for n = 1:100
try
...
catch
id = lasterror.identifier;
if (strcmp (id, "Octave:invalid-indexing"))
number_of_errors++;
endif
end_try_catch
endfor

222

GNU Octave

The functions distributed with Octave can issue one of the following errors.
Octave:invalid-context
Indicates the error was generated by an operation that cannot be executed in
the scope from which it was called. For example, the function print_usage ()
when called from the Octave prompt raises this error.
Octave:invalid-input-arg
Indicates that a function was called with invalid input arguments.
Octave:invalid-fun-call
Indicates that a function was called in an incorrect way, e.g., wrong number of
input arguments.
Octave:invalid-indexing
Indicates that a data-type was indexed incorrectly, e.g., real-value index for
arrays, nonexistent field of a structure.
Octave:bad-alloc
Indicates that memory couldn’t be allocated.
Octave:undefined-function
Indicates a call to a function that is not defined. The function may exist but
Octave is unable to find it in the search path.
When an error has been handled it is possible to raise it again. This can be useful when
an error needs to be detected, but the program should still abort. This is possible using
the rethrow function. The previous example can now be changed to count the number of
errors related to the ‘*’ operator, but still abort if another kind of error occurs.
number_of_errors = 0;
for n = 1:100
try
...
catch
msg = lasterror.message;
if (strfind (msg, "operator *"))
number_of_errors++;
else
rethrow (lasterror);
endif
end_try_catch
endfor

rethrow (err)
Reissue a previous error as defined by err.
err is a structure that must contain at least the "message" and "identifier" fields.
err can also contain a field "stack" that gives information on the assumed location
of the error. Typically err is returned from lasterror.
See also: [lasterror], page 220, [lasterr], page 221, [error], page 217.

Chapter 12: Errors and Warnings

223

err = errno ()
err = errno (val)
err = errno (name)
Return the current value of the system-dependent variable errno, set its value to
val and return the previous value, or return the named error code given name as a
character string, or -1 if name is not found.
See also: [errno list], page 223.

errno_list ()
Return a structure containing the system-dependent errno values.
See also: [errno], page 223.

12.1.3 Recovering From Errors
Octave provides several ways of recovering from errors. There are try/catch blocks,
unwind_protect/unwind_protect_cleanup blocks, and finally the onCleanup command.
The onCleanup command associates an ordinary Octave variable (the trigger) with an
arbitrary function (the action). Whenever the Octave variable ceases to exist—whether
due to a function return, an error, or simply because the variable has been removed with
clear—then the assigned function is executed.
The function can do anything necessary for cleanup such as closing open file handles,
printing an error message, or restoring global variables to their initial values. The last
example is a very convenient idiom for Octave code. For example:
function rand42
old_state = rand ("state");
restore_state = onCleanup (@() rand ("state", old_state));
rand ("state", 42);
...
endfunction # rand generator state restored by onCleanup

obj = onCleanup (function)
Create a special object that executes a given function upon destruction.
If the object is copied to multiple variables (or cell or struct array elements) or
returned from a function, function will be executed after clearing the last copy of
the object. Note that if multiple local onCleanup variables are created, the order in
which they are called is unspecified. For similar functionality See Section 10.8 [The
unwind protect Statement], page 172.

12.2 Handling Warnings
Like an error, a warning is issued when something unexpected happens. Unlike an error,
a warning doesn’t abort the currently running program. A simple example of a warning is
when a number is divided by zero. In this case Octave will issue a warning and assign the
value Inf to the result.
a = 1/0
a warning: division by zero
⇒ a = Inf

224

GNU Octave

12.2.1 Issuing Warnings
It is possible to issue warnings from any code using the warning function. In its most simple
form, the warning function takes a string describing the warning as its input argument. As
an example, the following code controls if the variable ‘a’ is non-negative, and if not issues
a warning and sets ‘a’ to zero.
a = -1;
if (a < 0)
warning ("’a’ must be non-negative. Setting ’a’ to zero.");
a = 0;
endif
a ’a’ must be non-negative. Setting ’a’ to zero.
Since warnings aren’t fatal to a running program, it is not possible to catch a warning
using the try statement or something similar. It is however possible to access the last
warning as a string using the lastwarn function.
It is also possible to assign an identification string to a warning. If a warning has such an
ID the user can enable and disable this warning as will be described in the next section. To
assign an ID to a warning, simply call warning with two string arguments, where the first
is the identification string, and the second is the actual warning. Note that warning IDs are
in the format "NAMESPACE:WARNING-NAME". The namespace "Octave" is used for Octave’s
own warnings. Any other string is available as a namespace for user’s own warnings.
(template, . . . )
(id, template, . . . )
("on", id)
("off", id)
("query", id)
("error", id)
(state, "backtrace")
(state, id, "local")
Display a warning message or control the behavior of Octave’s warning system.
Format the optional arguments under the control of the template string template
using the same rules as the printf family of functions (see Section 14.2.4 [Formatted
Output], page 270) and print the resulting message on the stderr stream. The
message is prefixed by the character string ‘warning: ’. You should use this function
when you want to notify the user of an unusual condition, but only when it makes
sense for your program to go on.
The optional message identifier allows users to enable or disable warnings tagged by
id. A message identifier is of the form "NAMESPACE:WARNING-NAME". Octave’s
own warnings use the "Octave" namespace (see [warning ids], page 225). The special
identifier "all" may be used to set the state of all warnings.
If the first argument is "on" or "off", set the state of a particular warning using
the identifier id. If the first argument is "query", query the state of this warning
instead. If the identifier is omitted, a value of "all" is assumed. If you set the state
of a warning to "error", the warning named by id is handled as if it were an error
instead. So, for example, the following handles all warnings as errors:
warning ("error");

warning
warning
warning
warning
warning
warning
warning
warning

Chapter 12: Errors and Warnings

225

If the state is "on" or "off" and the third argument is "backtrace", then a stack
trace is printed along with the warning message when warnings occur inside function
calls. This option is enabled by default.
If the state is "on", "off", or "error" and the third argument is "local", then the
warning state will be set temporarily, until the end of the current function. Changes to
warning states that are set locally affect the current function and all functions called
from the current scope. The previous warning state is restored on return from the
current function. The "local" option is ignored if used in the top-level workspace.
Implementation Note: For compatibility with matlab, escape sequences in template
(e.g., "\n" => newline) are processed regardless of whether template has been defined
with single quotes, as long as there are two or more input arguments. To disable escape
sequence expansion use a second backslash before the sequence (e.g., "\\n") or use
the regexptranslate function.
See also: [warning ids], page 225, [lastwarn], page 225, [error], page 217.

[msg, msgid] = lastwarn ()
lastwarn (msg)
lastwarn (msg, msgid)
Query or set the last warning message.
When called without input arguments, return the last warning message and message
identifier.
With one argument, set the last warning message to msg.
With two arguments, also set the last message identifier.
See also: [warning], page 224, [lasterror], page 220, [lasterr], page 221.
The functions distributed with Octave can issue one of the following warnings.
Octave:abbreviated-property-match
By default, the Octave:abbreviated-property-match warning is enabled.
Octave:array-as-logical
If the Octave:array-as-logical warning is enabled, Octave will warn when
an array of size greater than 1x1 is used as a truth value in an if, while or until
statement. By default, the Octave:array-as-logical warning is disabled.
Octave:array-to-scalar
If the Octave:array-to-scalar warning is enabled, Octave will warn when an
implicit conversion from an array to a scalar value is attempted. By default,
the Octave:array-to-scalar warning is disabled.
Octave:array-to-vector
If the Octave:array-to-vector warning is enabled, Octave will warn when an
implicit conversion from an array to a vector value is attempted. By default,
the Octave:array-to-vector warning is disabled.
Octave:assign-as-truth-value
If the Octave:assign-as-truth-value warning is enabled, a warning is issued
for statements like
if (s = t)
...

226

GNU Octave

since such statements are not common, and it is likely that the intent was to
write
if (s == t)
...
instead.
There are times when it is useful to write code that contains assignments within
the condition of a while or if statement. For example, statements like
while (c = getc ())
...
are common in C programming.
It is possible to avoid all warnings about such statements by disabling the
Octave:assign-as-truth-value warning, but that may also let real errors
like
if (x = 1)
...

# intended to test (x == 1)!

slip by.
In such cases, it is possible suppress errors for specific statements by writing
them with an extra set of parentheses. For example, writing the previous example as
while ((c = getc ()))
...
will prevent the warning from being printed for this statement, while allowing
Octave to warn about other assignments used in conditional contexts.
By default, the Octave:assign-as-truth-value warning is enabled.
Octave:associativity-change
If the Octave:associativity-change warning is enabled, Octave will warn
about possible changes in the meaning of some code due to changes in associativity for some operators. Associativity changes have typically been made for
matlab compatibility. By default, the Octave:associativity-change warning is enabled.
Octave:autoload-relative-file-name
If the Octave:autoload-relative-file-name is enabled, Octave will warn
when parsing autoload() function calls with relative paths to function files.
This usually happens when using autoload() calls in PKG ADD files, when the
PKG ADD file is not in the same directory as the .oct file referred to by the
autoload() command. By default, the Octave:autoload-relative-file-name
warning is enabled.
Octave:built-in-variable-assignment
By default, the Octave:built-in-variable-assignment warning is enabled.
Octave:deprecated-function
If the Octave:deprecated-function warning is enabled, a warning is issued
when Octave encounters a function that is obsolete and scheduled for removal

Chapter 12: Errors and Warnings

227

from Octave. By default, the Octave:deprecated-function warning is enabled.
Octave:deprecated-keyword
If the Octave:deprecated-keyword warning is enabled, a warning is issued
when Octave encounters a keyword that is obsolete and scheduled for removal
from Octave. By default, the Octave:deprecated-keyword warning is enabled.
Octave:deprecated-property
If the Octave:deprecated-property warning is enabled, a warning is issued
when Octave encounters a graphics property that is obsolete and scheduled for
removal from Octave. By default, the Octave:deprecated-property warning
is enabled.
Octave:divide-by-zero
If the Octave:divide-by-zero warning is enabled, a warning is issued when
Octave encounters a division by zero. By default, the Octave:divide-by-zero
warning is enabled.
Octave:fopen-file-in-path
By default, the Octave:fopen-file-in-path warning is enabled.
Octave:function-name-clash
If the Octave:function-name-clash warning is enabled, a warning is issued
when Octave finds that the name of a function defined in a function file differs
from the name of the file. (If the names disagree, the name declared inside
the file is ignored.) By default, the Octave:function-name-clash warning is
enabled.
Octave:future-time-stamp
If the Octave:future-time-stamp warning is enabled, Octave will print a
warning if it finds a function file with a time stamp that is in the future. By
default, the Octave:future-time-stamp warning is enabled.
Octave:glyph-render
By default, the Octave:glyph-render warning is enabled.
Octave:imag-to-real
If the Octave:imag-to-real warning is enabled, a warning is printed for
implicit conversions of complex numbers to real numbers. By default, the
Octave:imag-to-real warning is disabled.
Octave:language-extension
Print warnings when using features that are unique to the Octave
language and that may still be missing in matlab.
By default, the
Octave:language-extension warning is disabled. The --traditional or
--braindead startup options for Octave may also be of use, see Section 2.1.1
[Command Line Options], page 15.
Octave:load-file-in-path
By default, the Octave:load-file-in-path warning is enabled.
Octave:logical-conversion
By default, the Octave:logical-conversion warning is enabled.

228

GNU Octave

Octave:missing-glyph
By default, the Octave:missing-glyph warning is enabled.
Octave:missing-semicolon
If the Octave:missing-semicolon warning is enabled, Octave will warn when
statements in function definitions don’t end in semicolons. By default the
Octave:missing-semicolon warning is disabled.
Octave:mixed-string-concat
If the Octave:mixed-string-concat warning is enabled, print a warning when
concatenating a mixture of double and single quoted strings. By default, the
Octave:mixed-string-concat warning is disabled.
Octave:neg-dim-as-zero
If the Octave:neg-dim-as-zero warning is enabled, print a warning for expressions like
eye (-1)
By default, the Octave:neg-dim-as-zero warning is disabled.
Octave:nested-functions-coerced
By default, the Octave:nested-functions-coerced warning is enabled.
Octave:noninteger-range-as-index
By default, the Octave:noninteger-range-as-index warning is enabled.
Octave:num-to-str
If the Octave:num-to-str warning is enable, a warning is printed for implicit
conversions of numbers to their ASCII character equivalents when strings are
constructed using a mixture of strings and numbers in matrix notation. For
example,
[ "f", 111, 111 ]
⇒ "foo"
elicits a warning if the Octave:num-to-str warning is enabled. By default, the
Octave:num-to-str warning is enabled.
Octave:possible-matlab-short-circuit-operator
If the Octave:possible-matlab-short-circuit-operator warning is
enabled, Octave will warn about using the not short circuiting operators &
and | inside if or while conditions. They normally never short circuit, but
matlab always short circuits if any logical operators are used in a condition.
You can turn on the option
do_braindead_shortcircuit_evaluation (1)
if you would like to enable this short-circuit evaluation in Octave. Note that
the && and || operators always short circuit in both Octave and matlab,
so it’s only necessary to enable matlab-style short-circuiting if it’s too arduous to modify existing code that relies on this behavior. By default, the
Octave:possible-matlab-short-circuit-operator warning is enabled.
Octave:precedence-change
If the Octave:precedence-change warning is enabled, Octave will warn about
possible changes in the meaning of some code due to changes in precedence

Chapter 12: Errors and Warnings

229

for some operators. Precedence changes have typically been made for matlab
compatibility. By default, the Octave:precedence-change warning is enabled.
Octave:recursive-path-search
By default, the Octave:recursive-path-search warning is enabled.
Octave:remove-init-dir
The path function changes the search path that Octave uses to find functions.
It is possible to set the path to a value which excludes Octave’s own built-in
functions. If the Octave:remove-init-dir warning is enabled then Octave will
warn when the path function has been used in a way that may render Octave
unworkable. By default, the Octave:remove-init-dir warning is enabled.
Octave:reload-forces-clear
If several functions have been loaded from the same file, Octave must
clear all the functions before any one of them can be reloaded. If the
Octave:reload-forces-clear warning is enabled, Octave will warn you
when this happens, and print a list of the additional functions that it is forced
to clear. By default, the Octave:reload-forces-clear warning is enabled.
Octave:resize-on-range-error
If the Octave:resize-on-range-error warning is enabled, print a warning
when a matrix is resized by an indexed assignment with indices outside the
current bounds. By default, the Octave:resize-on-range-error warning is
disabled.
Octave:separator-insert
Print warning if commas or semicolons might be inserted automatically in literal
matrices. By default, the Octave:separator-insert warning is disabled.
Octave:shadowed-function
By default, the Octave:shadowed-function warning is enabled.
Octave:single-quote-string
Print warning if a single quote character is used to introduce a string constant.
By default, the Octave:single-quote-string warning is disabled.
Octave:nearly-singular-matrix
Octave:singular-matrix
By default, the Octave:nearly-singular-matrix and Octave:singular-matrix
warnings are enabled.
Octave:sqrtm:SingularMatrix
By default, the Octave:sqrtm:SingularMatrix warning is enabled.
Octave:str-to-num
If the Octave:str-to-num warning is enabled, a warning is printed for implicit
conversions of strings to their numeric ASCII equivalents. For example,
"abc" + 0
⇒ 97 98 99
elicits a warning if the Octave:str-to-num warning is enabled. By default, the
Octave:str-to-num warning is disabled.

230

GNU Octave

Octave:undefined-return-values
If the Octave:undefined-return-values warning is disabled, print a warning
if a function does not define all the values in the return list which are expected.
By default, the Octave:undefined-return-values warning is enabled.
Octave:variable-switch-label
If the Octave:variable-switch-label warning is enabled, Octave will print a
warning if a switch label is not a constant or constant expression. By default,
the Octave:variable-switch-label warning is disabled.

12.2.2 Enabling and Disabling Warnings
The warning function also allows you to control which warnings are actually printed to
the screen. If the warning function is called with a string argument that is either "on" or
"off" all warnings will be enabled or disabled.
It is also possible to enable and disable individual warnings through their string identifications. The following code will issue a warning
warning ("example:non-negative-variable",
"’a’ must be non-negative. Setting ’a’ to zero.");
while the following won’t issue a warning
warning ("off", "example:non-negative-variable");
warning ("example:non-negative-variable",
"’a’ must be non-negative. Setting ’a’ to zero.");

231

13 Debugging
Octave includes a built-in debugger to aid in the development of scripts. This can be used
to interrupt the execution of an Octave script at a certain point, or when certain conditions
are met. Once execution has stopped, and debug mode is entered, the symbol table at the
point where execution has stopped can be examined and modified to check for errors.
The normal command-line editing and history functions are available in debug mode.

13.1 Entering Debug Mode
There are two basic means of interrupting the execution of an Octave script. These are
breakpoints (see Section 13.3 [Breakpoints], page 232), discussed in the next section, and
interruption based on some condition.
Octave supports three means to stop execution based on the values set in the functions
debug_on_interrupt, debug_on_warning, and debug_on_error.

val = debug_on_interrupt ()
old_val = debug_on_interrupt (new_val)
debug_on_interrupt (new_val, "local")
Query or set the internal variable that controls whether Octave will try to enter
debugging mode when it receives an interrupt signal (typically generated with C-c).
If a second interrupt signal is received before reaching the debugging mode, a normal
interrupt will occur.
When called from inside a function with the "local" option, the variable is changed
locally for the function and any subroutines it calls. The original variable value is
restored when exiting the function.
See also: [debug on error], page 231, [debug on warning], page 231.

val = debug_on_warning ()
old_val = debug_on_warning (new_val)
debug_on_warning (new_val, "local")
Query or set the internal variable that controls whether Octave will try to enter the
debugger when a warning is encountered.
When called from inside a function with the "local" option, the variable is changed
locally for the function and any subroutines it calls. The original variable value is
restored when exiting the function.
See also: [debug on error], page 231, [debug on interrupt], page 231.

val = debug_on_error ()
old_val = debug_on_error (new_val)
debug_on_error (new_val, "local")
Query or set the internal variable that controls whether Octave will try to enter the
debugger when an error is encountered.
This will also inhibit printing of the normal traceback message (you will only see the
top-level error message).

232

GNU Octave

When called from inside a function with the "local" option, the variable is changed
locally for the function and any subroutines it calls. The original variable value is
restored when exiting the function.
See also: [debug on warning], page 231, [debug on interrupt], page 231.

13.2 Leaving Debug Mode
Use either dbcont or return to leave the debug mode and continue the normal execution
of the script.

dbcont
Leave command-line debugging mode and continue code execution normally.
See also: [dbstep], page 237, [dbquit], page 232.
To quit debug mode and return directly to the prompt without executing any additional
code use dbquit.

dbquit
Quit debugging mode immediately without further code execution and return to the
Octave prompt.
See also: [dbcont], page 232, [dbstep], page 237.
Finally, typing exit or quit at the debug prompt will result in Octave terminating
normally.

13.3 Breakpoints
Breakpoints can be set in any m-file function by using the dbstop function.

dbstop func
dbstop func line
dbstop func line1 line2 . . .
dbstop line1 . . .
dbstop in func
dbstop in func at line
dbstop in func at line if "condition"
dbstop if event
dbstop if event ID
dbstop (bp_struct)
rline = dbstop . . .
Set breakpoints for the built-in debugger.
func is the name of a function on the current path. When already in debug mode the
func argument can be omitted and the current function will be used. Breakpoints
at subfunctions are set with the scope operator ‘>’. For example, If file.m has a
subfunction func2, then a breakpoint in func2 can be specified by file>func2.
line is the line number at which to break. If line is not specified, it defaults to the
first executable line in the file func.m. Multiple lines can be specified in a single

Chapter 13: Debugging

233

command; when function syntax is used, the lines may also be passed as a single
vector argument ([line1, line2, ...]).
condition is any Octave expression that can be evaluated in the code context that
exists at the breakpoint. When the breakpoint is encountered, condition will be evaluated, and execution will stop if condition is true. If condition cannot be evaluated,
for example because it refers to an undefined variable, an error will be thrown. Expressions with side effects (such as y++ > 1) will alter variables, and should generally
be avoided. Conditions containing quotes (‘"’, ‘’’) or comment characters (‘#’, ‘%’)
must be enclosed in quotes. (This does not apply to conditions entered from the
editor’s context menu.) For example:
dbstop in strread at 209 if ’any (format == "%f")’
The form specifying event does not cause a specific breakpoint at a given function and
line number. Instead it causes debug mode to be entered when certain unexpected
events are encountered. Possible values are
error

Stop when an error is reported. This is equivalent to specifying both
debug_on_error (true) and debug_on_interrupt (true).

caught error
Stop when an error is caught by a try-catch block (not yet implemented).
interrupt
Stop when an interrupt (Ctrl-C) occurs.
naninf

Stop when code returns a non-finite value (not yet implemented).

warning

Stop when a warning is reported. This is equivalent to specifying debug_
on_warning (true).

The events error, caught error, and warning can all be followed by a string specifying an error ID or warning ID. If that is done, only errors with the specified ID will
cause execution to stop. To stop on one of a set of IDs, multiple dbstop commands
must be issued.
Breakpoints and events can be removed using the dbclear command with the same
syntax.
It is possible to save all breakpoints and restore them at once by issuing the commands
bp_state = dbstatus; ...; dbstop (bp_state).
The optional output rline is the real line number where the breakpoint was set. This
can differ from the specified line if the line is not executable. For example, if a
breakpoint attempted on a blank line then Octave will set the real breakpoint at the
next executable line.
When a file is re-parsed, such as when it is modified outside the GUI, all breakpoints
within the file are cleared.
See also: [dbclear], page 234, [dbstatus], page 234, [dbstep], page 237,
[debug on error], page 231, [debug on warning], page 231, [debug on interrupt],
page 231.

234

GNU Octave

Breakpoints in class methods are also supported (e.g., dbstop ("@class/method")). However, breakpoints cannot be set in built-in functions (e.g., sin, etc.) or dynamically loaded
functions (i.e., oct-files).
To set a breakpoint immediately upon entering a function use line number 1, or omit
the line number entirely and just give the function name. When setting the breakpoint
Octave will ignore the leading comment block, and the breakpoint will be set on the first
executable statement in the function. For example:
dbstop ("asind", 1)
⇒ 29
Note that the return value of 29 means that the breakpoint was effectively set to line 29.
The status of breakpoints in a function can be queried with dbstatus.

dbstatus
dbstatus func
bp_list = dbstatus . . .
Report the location of active breakpoints.
When called with no input or output arguments, print the list of all functions with
breakpoints and the line numbers where those breakpoints are set.
If a function name func is specified then only report breakpoints for the named function and its subfunctions.
The optional return argument bp list is a struct array with the following fields.
name

The name of the function with a breakpoint. A subfunction, say func2
within an m-file, say file.m, is specified as file>func2.

file

The name of the m-file where the function code is located.

line

The line number with the breakpoint.

cond

The condition that must be satisfied for the breakpoint to be active, or
the empty string for unconditional breakpoints.

If dbstop if error is true but no explicit IDs are specified, the return value will have
an empty field called "errs". If IDs are specified, the errs field will have one row
per ID. If dbstop if error is false, there is no "errs" field. The "warn" field is set
similarly by dbstop if warning.
See also: [dbstop], page 232, [dbclear], page 234, [dbwhere], page 236, [dblist],
page 236, [dbstack], page 237.
Reusing the previous example, dbstatus ("asind") will return 29. The breakpoints listed
can then be cleared with the dbclear function.

dbclear
dbclear
dbclear
dbclear
dbclear
dbclear
dbclear

func
func line
func line1 line2 . . .
line . . .
all
in func
in func at line

Chapter 13: Debugging

dbclear
dbclear
dbclear
dbclear
dbclear
dbclear
dbclear

235

if event
("func")
("func", line)
("func", line1, line2, . . . )
("func", line1, . . . )
(line, . . . )
("all")

Delete a breakpoint at line number line in the function func.
Arguments are
func

Function name as a string variable. When already in debug mode this
argument can be omitted and the current function will be used.

line

Line number from which to remove a breakpoint. Multiple lines may be
given as separate arguments or as a vector.

event

An event such as error, interrupt, or warning (see [dbstop], page 232,
for details).

When called without a line number specification all breakpoints in the named function
are cleared.
If the requested line is not a breakpoint no action is performed.
The special keyword "all" will clear all breakpoints from all files.
See also: [dbstop], page 232, [dbstatus], page 234, [dbwhere], page 236.
A breakpoint may also be set in a subfunction. For example, if a file contains the
functions
function y = func1 (x)
y = func2 (x);
endfunction
function y = func2 (x)
y = x + 1;
endfunction
then a breakpoint can be set at the start of the subfunction directly with
dbstop (["func1", filemarker(), "func2"])
⇒ 5

Note that filemarker returns the character that marks subfunctions from the file containing them. Unless the default has been changed this character is ‘>’. Thus, a quicker
and more normal way to set the breakpoint would be
dbstop func1>func2
Another simple way of setting a breakpoint in an Octave script is the use of the keyboard
function.

keyboard ()
keyboard ("prompt")
Stop m-file execution and enter debug mode.

236

GNU Octave

When the keyboard function is executed, Octave prints a prompt and waits for user
input. The input strings are then evaluated and the results are printed. This makes it
possible to examine the values of variables within a function, and to assign new values
if necessary. To leave the prompt and return to normal execution type ‘return’ or
‘dbcont’. The keyboard function does not return an exit status.
If keyboard is invoked without arguments, a default prompt of ‘debug> ’ is used.
See also: [dbstop], page 232, [dbcont], page 232, [dbquit], page 232.
The keyboard function is placed in a script at the point where the user desires that the
execution be stopped. It automatically sets the running script into the debug mode.

13.4 Debug Mode
There are three additional support functions that allow the user to find out where in the
execution of a script Octave entered the debug mode, and to print the code in the script
surrounding the point where Octave entered debug mode.

dbwhere
In debugging mode, report the current file and line number where execution is
stopped.
See also: [dbstack], page 237, [dblist], page 236, [dbstatus], page 234, [dbcont],
page 232, [dbstep], page 237, [dbup], page 238, [dbdown], page 238.

dbtype
dbtype
dbtype
dbtype
dbtype
dbtype
dbtype
dbtype

lineno
startl:endl
startl:end
func
func lineno
func startl:endl
func startl:end

Display a script file with line numbers.
When called with no arguments in debugging mode, display the script file currently
being debugged.
An optional range specification can be used to list only a portion of the file. The
special keyword "end" is a valid line number specification for the last line of the file.
When called with the name of a function, list that script file with line numbers.
See also: [dblist], page 236, [dbwhere], page 236, [dbstatus], page 234, [dbstop],
page 232.

dblist
dblist n
In debugging mode, list n lines of the function being debugged centered around the
current line to be executed.
If unspecified n defaults to 10 (+/- 5 lines)
See also: [dbwhere], page 236, [dbtype], page 236, [dbstack], page 237.

Chapter 13: Debugging

237

You may also use isdebugmode to determine whether the debugger is currently active.

isdebugmode ()
Return true if in debugging mode, otherwise false.
See also: [dbwhere], page 236, [dbstack], page 237, [dbstatus], page 234.
Debug mode also allows single line stepping through a function using the command
dbstep.

dbstep
dbstep
dbstep
dbstep
dbnext

n
in
out
...

In debugging mode, execute the next n lines of code.
If n is omitted, execute the next single line of code. If the next line of code is itself
defined in terms of an m-file remain in the existing function.
Using dbstep in will cause execution of the next line to step into any m-files defined
on the next line.
Using dbstep out will cause execution to continue until the current function returns.
dbnext is an alias for dbstep.
See also: [dbcont], page 232, [dbquit], page 232.
When in debug mode the RETURN key will execute the last entered command. This is
useful, for example, after hitting a breakpoint and entering dbstep once. After that, one
can advance line by line through the code with only a single key stroke.

13.5 Call Stack
The function being debugged may be the leaf node of a series of function calls. After
examining values in the current subroutine it may turn out that the problem occurred in
earlier pieces of code. Use dbup and dbdown to move up and down through the series of
function calls to locate where variables first took on the wrong values. dbstack shows the
entire series of function calls and at what level debugging is currently taking place.

dbstack
dbstack n
dbstack -completenames
[stack, idx] = dbstack ( . . . )
Display or return current debugging function stack information.
With optional argument n, omit the n innermost stack frames.
Although accepted, the argument -completenames is silently ignored. Octave always
returns absolute filenames.
The arguments n and -completenames can be both specified in any order.
The optional return argument stack is a struct array with the following fields:
file

The name of the m-file where the function code is located.

238

GNU Octave

name

The name of the function with a breakpoint.

line

The line number of an active breakpoint.

column

The column number of the line where the breakpoint begins.

scope

Undocumented.

context

Undocumented.

The return argument idx specifies which element of the stack struct array is currently
active.
See also: [dbup], page 238, [dbdown], page 238, [dbwhere], page 236, [dblist], page 236,
[dbstatus], page 234.

dbup
dbup n
In debugging mode, move up the execution stack n frames.
If n is omitted, move up one frame.
See also: [dbstack], page 237, [dbdown], page 238.

dbdown
dbdown n
In debugging mode, move down the execution stack n frames.
If n is omitted, move down one frame.
See also: [dbstack], page 237, [dbup], page 238.

13.6 Profiling
Octave supports profiling of code execution on a per-function level. If profiling is enabled,
each call to a function (supporting built-ins, operators, functions in oct- and mex-files, userdefined functions in Octave code and anonymous functions) is recorded while running Octave
code. After that, this data can aid in analyzing the code behavior, and is in particular helpful
for finding “hot spots” in the code which use up a lot of computation time and are the best
targets to spend optimization efforts on.
The main command for profiling is profile, which can be used to start or stop the
profiler and also to query collected data afterwards. The data is returned in an Octave data
structure which can then be examined or further processed by other routines or tools.

profile on
profile off
profile resume
profile clear
S = profile ("status")
T = profile ("info")
Control the built-in profiler.
profile on
Start the profiler, clearing all previously collected data if there is any.

Chapter 13: Debugging

239

profile off
Stop profiling. The collected data can later be retrieved and examined
with T = profile ("info").
profile clear
Clear all collected profiler data.
profile resume
Restart profiling without clearing the old data. All newly collected statistics are added to the existing ones.
S = profile ("status")
Return a structure with information about the current status of the profiler. At the moment, the only field is ProfilerStatus which is either
"on" or "off".
T = profile ("info")
Return the collected profiling statistics in the structure T. The flat profile
is returned in the field FunctionTable which is an array of structures,
each entry corresponding to a function which was called and for which profiling statistics are present. In addition, the field Hierarchical contains
the hierarchical call tree. Each node has an index into the FunctionTable
identifying the function it corresponds to as well as data fields for number
of calls and time spent at this level in the call tree.
See also: [profshow], page 239, [profexplore], page 240.
An easy way to get an overview over the collected data is profshow. This function takes
the profiler data returned by profile as input and prints a flat profile, for instance:
Function Attr
Time (s)
Calls
--------------------------------------->myfib
R
2.195
13529
binary <=
0.061
13529
binary 0.050
13528
binary +
0.026
6764
This shows that most of the run time was spent executing the function ‘myfib’, and
some minor proportion evaluating the listed binary operators. Furthermore, it is shown
how often the function was called and the profiler also records that it is recursive.
(data)
(data, n)
()
(n)
Display flat per-function profiler results.

profshow
profshow
profshow
profshow

Print out profiler data (execution time, number of calls) for the most critical n functions. The results are sorted in descending order by the total time spent in each
function. If n is unspecified it defaults to 20.
The input data is the structure returned by profile ("info").
profshow will use the current profile dataset.

If unspecified,

240

GNU Octave

The attribute column displays ‘R’ for recursive functions, and is blank for all other
function types.
See also: [profexplore], page 240, [profile], page 238.
(dir)
(dir, data)
(dir, name)
(dir, name, data)
Export profiler data as HTML.
Export the profiling data in data into a series of HTML files in the folder dir. The
initial file will be data/index.html.
If name is specified, it must be a string that contains a “name” for the profile being
exported. This name is included in the HTML.
The input data is the structure returned by profile ("info"). If unspecified,
profexport will use the current profile dataset.

profexport
profexport
profexport
profexport

See also: [profshow], page 239, [profexplore], page 240, [profile], page 238.

profexplore ()
profexplore (data)
Interactively explore hierarchical profiler output.
Assuming data is the structure with profile data returned by profile ("info"),
this command opens an interactive prompt that can be used to explore the call-tree.
Type help to get a list of possible commands. If data is omitted, profile ("info")
is called and used in its place.
See also: [profile], page 238, [profshow], page 239.

13.7 Profiler Example
Below, we will give a short example of a profiler session. See Section 13.6 [Profiling],
page 238, for the documentation of the profiler functions in detail. Consider the code:
global N A;
N = 300;
A = rand (N, N);
function xt = timesteps (steps, x0, expM)
global N;
if (steps == 0)
xt = NA (N, 0);
else
xt = NA (N, steps);
x1 = expM * x0;
xt(:, 1) = x1;
xt(:, 2 : end) = timesteps (steps - 1, x1, expM);
endif

Chapter 13: Debugging

241

endfunction
function foo ()
global N A;
initial = @(x) sin (x);
x0 = (initial (linspace (0, 2 * pi, N)))’;
expA = expm (A);
xt = timesteps (100, x0, expA);
endfunction
function fib = bar (N)
if (N <= 2)
fib = 1;
else
fib = bar (N - 1) + bar (N - 2);
endif
endfunction

If we execute the two main functions, we get:

tic; foo; toc;
⇒ Elapsed time is 2.37338 seconds.
tic; bar (20); toc;
⇒ Elapsed time is 2.04952 seconds.
But this does not give much information about where this time is spent; for instance,
whether the single call to expm is more expensive or the recursive time-stepping itself. To
get a more detailed picture, we can use the profiler.

profile on;
foo;
profile off;
data = profile ("info");
profshow (data, 10);

This prints a table like:

242

GNU Octave

# Function Attr
Time (s)
Calls
--------------------------------------------7
expm
1.034
1
3 binary *
0.823
117
41 binary \
0.188
1
38 binary ^
0.126
2
43 timesteps
R
0.111
101
44
NA
0.029
101
39 binary +
0.024
8
34
norm
0.011
1
40 binary 0.004
101
33
balance
0.003
1
The entries are the individual functions which have been executed (only the 10 most
important ones), together with some information for each of them. The entries like ‘binary
*’ denote operators, while other entries are ordinary functions. They include both builtins like expm and our own routines (for instance timesteps). From this profile, we can
immediately deduce that expm uses up the largest proportion of the processing time, even
though it is only called once. The second expensive operation is the matrix-vector product
in the routine timesteps.1
Timing, however, is not the only information available from the profile. The attribute
column shows us that timesteps calls itself recursively. This may not be that remarkable
in this example (since it’s clear anyway), but could be helpful in a more complex setting.
As to the question of why is there a ‘binary \’ in the output, we can easily shed some
light on that too. Note that data is a structure array (Section 6.1.2 [Structure Arrays],
page 105) which contains the field FunctionTable. This stores the raw data for the profile
shown. The number in the first column of the table gives the index under which the shown
function can be found there. Looking up data.FunctionTable(41) gives:
scalar structure containing the fields:
FunctionName = binary \
TotalTime = 0.18765
NumCalls = 1
IsRecursive = 0
Parents = 7
Children = [](1x0)
Here we see the information from the table again, but have additional fields Parents
and Children. Those are both arrays, which contain the indices of functions which have
directly called the function in question (which is entry 7, expm, in this case) or been called
by it (no functions). Hence, the backslash operator has been used internally by expm.
Now let’s take a look at bar. For this, we start a fresh profiling session (profile on
does this; the old data is removed before the profiler is restarted):
1

We only know it is the binary multiplication operator, but fortunately this operator appears only at one
place in the code and thus we know which occurrence takes so much time. If there were multiple places,
we would have to use the hierarchical profile to find out the exact place which uses up the time which is
not covered in this example.

243

profile on;
bar (20);
profile off;
profshow (profile ("info"));
This gives:
#
Function Attr
Time (s)
Calls
------------------------------------------------------1
bar
R
2.091
13529
2
binary <=
0.062
13529
3
binary 0.042
13528
4
binary +
0.023
6764
5
profile
0.000
1
8
false
0.000
1
6
nargin
0.000
1
7
binary !=
0.000
1
9 __profiler_enable__
0.000
1
Unsurprisingly, bar is also recursive. It has been called 13,529 times in the course of
recursively calculating the Fibonacci number in a suboptimal way, and most of the time
was spent in bar itself.
Finally, let’s say we want to profile the execution of both foo and bar together. Since
we already have the run-time data collected for bar, we can restart the profiler without
clearing the existing data and collect the missing statistics about foo. This is done by:
profile resume;
foo;
profile off;
profshow (profile ("info"), 10);
As you can see in the table below, now we have both profiles mixed together.
# Function Attr
Time (s)
Calls
--------------------------------------------1
bar
R
2.091
13529
16
expm
1.122
1
12 binary *
0.798
117
46 binary \
0.185
1
45 binary ^
0.124
2
48 timesteps
R
0.115
101
2 binary <=
0.062
13529
3 binary 0.045
13629
4 binary +
0.041
6772
49
NA
0.036
101

245

14 Input and Output
Octave supports several ways of reading and writing data to or from the prompt or a file.
The simplest functions for data Input and Output (I/O) are easy to use, but only provide
limited control of how data is processed. For more control, a set of functions modeled after
the C standard library are also provided by Octave.

14.1 Basic Input and Output
14.1.1 Terminal Output
Since Octave normally prints the value of an expression as soon as it has been evaluated, the
simplest of all I/O functions is a simple expression. For example, the following expression
will display the value of ‘pi’
pi
a pi = 3.1416

This works well as long as it is acceptable to have the name of the variable (or ‘ans’)
printed along with the value. To print the value of a variable without printing its name,
use the function disp.
The format command offers some control over the way Octave prints values with disp
and through the normal echoing mechanism.

disp (x)
str = disp (x)
Display the value of x.
For example:
disp ("The value of pi is:"), disp (pi)
a the value of pi is:
a 3.1416

Note that the output from disp always ends with a newline.
If an output value is requested, disp prints nothing and returns the formatted output
in a string.
See also: [fdisp], page 256.

list_in_columns (arg, width, prefix)
Return a string containing the elements of arg listed in columns with an overall
maximum width of width and optional prefix prefix.
The argument arg must be a cell array of character strings or a character array.
If width is not specified or is an empty matrix, or less than or equal to zero, the width
of the terminal screen is used. Newline characters are used to break the lines in the
output string. For example:

246

GNU Octave

list_in_columns
⇒ abc
def
ghijkl

({"abc", "def", "ghijkl", "mnop", "qrs", "tuv"}, 20)
mnop
qrs
tuv

whos ans
⇒
Variables in the current scope:
Attr Name
==== ====
ans

Size
====
1x37

Bytes
=====
37

Class
=====
char

Total is 37 elements using 37 bytes

See also: [terminal size], page 246.

terminal_size ()
Return a two-element row vector containing the current size of the terminal window
in characters (rows and columns).
See also: [list in columns], page 245.

format
format options
Reset or specify the format of the output produced by disp and Octave’s normal
echoing mechanism.
This command only affects the display of numbers but not how they are stored or
computed. To change the internal representation from the default double use one of
the conversion functions such as single, uint8, int64, etc.
By default, Octave displays 5 significant digits in a human readable form (option
‘short’ paired with ‘loose’ format for matrices). If format is invoked without any
options, this default format is restored.
Valid formats for floating point numbers are listed in the following table.
short

Fixed point format with 5 significant figures in a field that is a maximum
of 10 characters wide. (default).
If Octave is unable to format a matrix so that columns line up on the
decimal point and all numbers fit within the maximum field width then
it switches to an exponential ‘e’ format.

long

Fixed point format with 15 significant figures in a field that is a maximum
of 20 characters wide.
As with the ‘short’ format, Octave will switch to an exponential ‘e’
format if it is unable to format a matrix properly using the current format.

short e
long e

Exponential format. The number to be represented is split between a
mantissa and an exponent (power of 10). The mantissa has 5 significant
digits in the short format and 15 digits in the long format. For example,
with the ‘short e’ format, pi is displayed as 3.1416e+00.

Chapter 14: Input and Output

short E
long E

short g
long g

247

Identical to ‘short e’ or ‘long e’ but displays an uppercase ‘E’ to indicate
the exponent. For example, with the ‘long E’ format, pi is displayed as
3.14159265358979E+00.
Optimally choose between fixed point and exponential format based on
the magnitude of the number. For example, with the ‘short g’ format,
pi .^ [2; 4; 8; 16; 32] is displayed as
ans =
9.8696
97.409
9488.5
9.0032e+07
8.1058e+15

short eng
long eng Identical to ‘short e’ or ‘long e’ but displays the value using an engineering format, where the exponent is divisible by 3. For example, with
the ‘short eng’ format, 10 * pi is displayed as 31.4159e+00.
long G
short G
free
none

Identical to ‘short g’ or ‘long g’ but displays an uppercase ‘E’ to indicate
the exponent.
Print output in free format, without trying to line up columns of matrices
on the decimal point. This also causes complex numbers to be formatted as numeric pairs like this ‘(0.60419, 0.60709)’ instead of like this
‘0.60419 + 0.60709i’.

The following formats affect all numeric output (floating point and integer types).
"+"
"+" chars
plus
plus chars
Print a ‘+’ symbol for matrix elements greater than zero, a ‘-’ symbol for
elements less than zero and a space for zero matrix elements. This format
can be very useful for examining the structure of a large sparse matrix.
The optional argument chars specifies a list of 3 characters to use for
printing values greater than zero, less than zero and equal to zero. For
example, with the ‘"+" "+-."’ format, [1, 0, -1; -1, 0, 1] is displayed
as
ans =
+.-.+
bank

Print in a fixed format with two digits to the right of the decimal point.

248

GNU Octave

native-hex
Print the hexadecimal representation of numbers as they are stored in
memory. For example, on a workstation which stores 8 byte real values
in IEEE format with the least significant byte first, the value of pi when
printed in native-hex format is 400921fb54442d18.
hex

The same as native-hex, but always print the most significant byte first.

native-bit
Print the bit representation of numbers as stored in memory. For example,
the value of pi is
01000000000010010010000111111011
01010100010001000010110100011000
(shown here in two 32 bit sections for typesetting purposes) when printed
in native-bit format on a workstation which stores 8 byte real values in
IEEE format with the least significant byte first.
bit

The same as native-bit, but always print the most significant bits first.

rat

Print a rational approximation, i.e., values are approximated as the ratio
of small integers. For example, with the ‘rat’ format, pi is displayed as
355/113.

The following two options affect the display of all matrices.
compact

Remove blank lines around column number labels and between matrices
producing more compact output with more data per page.

loose

Insert blank lines above and below column number labels and between
matrices to produce a more readable output with less data per page.
(default).

See also: [fixed point format], page 51, [output max field width], page 49,
[output precision], page 50, [split long rows], page 50, [print empty dimensions],
page 52, [rats], page 500.

14.1.1.1 Paging Screen Output
When running interactively, Octave normally sends any output intended for your terminal
that is more than one screen long to a paging program, such as less or more. This avoids
the problem of having a large volume of output stream by before you can read it. With
less (and some versions of more) you can also scan forward and backward, and search for
specific items.
Normally, no output is displayed by the pager until just before Octave is ready to print
the top level prompt, or read from the standard input (for example, by using the fscanf or
scanf functions). This means that there may be some delay before any output appears on
your screen if you have asked Octave to perform a significant amount of work with a single
command statement. The function fflush may be used to force output to be sent to the
pager (or any other stream) immediately.
You can select the program to run as the pager using the PAGER function, and you can
turn paging off by using the function more.

Chapter 14: Input and Output

249

more
more on
more off
Turn output pagination on or off.
Without an argument, more toggles the current state.
The current state can be determined via page_screen_output.
See also: [page screen output], page 249, [page output immediately], page 250,
[PAGER], page 249, [PAGER FLAGS], page 249.

val = PAGER ()
old_val = PAGER (new_val)
PAGER (new_val, "local")
Query or set the internal variable that specifies the program to use to display terminal
output on your system.
The default value is normally "less", "more", or "pg", depending on what programs
are installed on your system. See Appendix E [Installation], page 947.
When called from inside a function with the "local" option, the variable is changed
locally for the function and any subroutines it calls. The original variable value is
restored when exiting the function.
See also: [PAGER FLAGS], page 249, [page output immediately], page 250, [more],
page 248, [page screen output], page 249.

val = PAGER_FLAGS ()
old_val = PAGER_FLAGS (new_val)
PAGER_FLAGS (new_val, "local")
Query or set the internal variable that specifies the options to pass to the pager.
When called from inside a function with the "local" option, the variable is changed
locally for the function and any subroutines it calls. The original variable value is
restored when exiting the function.
See also: [PAGER], page 249, [more], page 248, [page screen output], page 249,
[page output immediately], page 250.

val = page_screen_output ()
old_val = page_screen_output (new_val)
page_screen_output (new_val, "local")
Query or set the internal variable that controls whether output intended for the
terminal window that is longer than one page is sent through a pager.
This allows you to view one screenful at a time. Some pagers (such as less—see
Appendix E [Installation], page 947) are also capable of moving backward on the
output.
When called from inside a function with the "local" option, the variable is changed
locally for the function and any subroutines it calls. The original variable value is
restored when exiting the function.
See also: [more], page 248, [page output immediately], page 250, [PAGER], page 249,
[PAGER FLAGS], page 249.

250

GNU Octave

val = page_output_immediately ()
old_val = page_output_immediately (new_val)
page_output_immediately (new_val, "local")
Query or set the internal variable that controls whether Octave sends output to the
pager as soon as it is available.
Otherwise, Octave buffers its output and waits until just before the prompt is printed
to flush it to the pager.
When called from inside a function with the "local" option, the variable is changed
locally for the function and any subroutines it calls. The original variable value is
restored when exiting the function.
See also: [page screen output], page 249, [more], page 248, [PAGER], page 249,
[PAGER FLAGS], page 249.

fflush (fid)
Flush output to file descriptor fid.
fflush returns 0 on success and an OS dependent error value (−1 on Unix) on error.
Programming Note: Flushing is useful for ensuring that all pending output makes it
to the screen before some other event occurs. For example, it is always a good idea
to flush the standard output stream before calling input.
See also: [fopen], page 267, [fclose], page 268.

14.1.2 Terminal Input
Octave has three functions that make it easy to prompt users for input. The input and
menu functions are normally used for managing an interactive dialog with a user, and the
keyboard function is normally used for doing simple debugging.

ans = input (prompt)
ans = input (prompt, "s")
Print prompt and wait for user input.
For example,
input ("Pick a number, any number! ")
prints the prompt
Pick a number, any number!
and waits for the user to enter a value. The string entered by the user is evaluated
as an expression, so it may be a literal constant, a variable name, or any other valid
Octave code.
The number of return arguments, their size, and their class depend on the expression
entered.
If you are only interested in getting a literal string value, you can call input with the
character string "s" as the second argument. This tells Octave to return the string
entered by the user directly, without evaluating it first.
Because there may be output waiting to be displayed by the pager, it is a good idea to
always call fflush (stdout) before calling input. This will ensure that all pending
output is written to the screen before your prompt.

Chapter 14: Input and Output

251

See also: [yes or no], page 251, [kbhit], page 251, [pause], page 825, [menu], page 251,
[listdlg], page 804.

choice = menu (title, opt1, . . . )
choice = menu (title, {opt1, . . . })

Display a menu with heading title and options opt1, . . . , and wait for user input.

If the GUI is running, the menu is displayed graphically using listdlg. Otherwise,
the title and menu options are printed on the console.
title is a string and the options may be input as individual strings or as a cell array
of strings.
The return value choice is the number of the option selected by the user counting
from 1. If the user aborts the dialog or makes an invalid selection then 0 is returned.
This function is useful for interactive programs. There is no limit to the number of
options that may be passed in, but it may be confusing to present more than will fit
easily on one screen.
See also: [input], page 250, [listdlg], page 804.

ans = yes_or_no ("prompt")
Ask the user a yes-or-no question.
Return logical true if the answer is yes or false if the answer is no.
Takes one argument, prompt, which is the string to display when asking the question.
prompt should end in a space; yes-or-no adds the string ‘(yes or no) ’ to it. The
user must confirm the answer with RET and can edit it until it has been confirmed.
See also: [input], page 250.
For input, the normal command line history and editing functions are available at the
prompt.
Octave also has a function that makes it possible to get a single character from the
keyboard without requiring the user to type a carriage return.

kbhit ()
kbhit (1)
Read a single keystroke from the keyboard.
If called with an argument, don’t wait for a keypress.
For example,
x = kbhit ();
will set x to the next character typed at the keyboard as soon as it is typed.
x = kbhit (1);
is identical to the above example, but doesn’t wait for a keypress, returning the empty
string if no key is available.
See also: [input], page 250, [pause], page 825.

252

GNU Octave

14.1.3 Simple File I/O
The save and load commands allow data to be written to and read from disk files in various
formats. The default format of files written by the save command can be controlled using
the functions save_default_options and save_precision.
As an example the following code creates a 3-by-3 matrix and saves it to the file
‘myfile.mat’.
A = [ 1:3; 4:6; 7:9 ];
save myfile.mat A
Once one or more variables have been saved to a file, they can be read into memory
using the load command.
load myfile.mat
A
a A =
a
1
2
a
4
5
a
7
8
a

3
6
9

save file
save options file
save options file v1 v2 . . .
save options file -struct STRUCT f1 f2 . . .
save - v1 v2 . . .
str = save ("-", "v1", "v2", . . . )
Save the named variables v1, v2, . . . , in the file file.
The special filename ‘-’ may be used to return the content of the variables as a string.
If no variable names are listed, Octave saves all the variables in the current scope.
Otherwise, full variable names or pattern syntax can be used to specify the variables to
save. If the -struct modifier is used, fields f1 f2 . . . of the scalar structure STRUCT
are saved as if they were variables with corresponding names. Valid options for the
save command are listed in the following table. Options that modify the output
format override the format specified by save_default_options.
If save is invoked using the functional form
save ("-option1", ..., "file", "v1", ...)
then the options, file, and variable name arguments (v1, . . . ) must be specified as
character strings.
If called with a filename of "-", write the output to stdout if nargout is 0, otherwise
return the output in a character string.
-append

Append to the destination instead of overwriting.

-ascii

Save a single matrix in a text file without header or any other information.

-binary

Save the data in Octave’s binary data format.

Chapter 14: Input and Output

253

-float-binary
Save the data in Octave’s binary data format but only using single precision. Only use this format if you know that all the values to be saved
can be represented in single precision.
-hdf5

Save the data in hdf5 format. (HDF5 is a free, portable binary format
developed by the National Center for Supercomputing Applications at
the University of Illinois.) This format is only available if Octave was
built with a link to the hdf5 libraries.

-float-hdf5
Save the data in hdf5 format but only using single precision. Only use
this format if you know that all the values to be saved can be represented
in single precision.
-V7
-v7
-7
-mat7-binary
Save the data in matlab’s v7 binary data format.
-V6
-v6
-6
-mat
-mat-binary
Save the data in matlab’s v6 binary data format.
-V4
-v4
-4
-mat4-binary
Save the data in the binary format written by matlab version 4.
-text
-zip
-z

Save the data in Octave’s text data format. (default).
Use the gzip algorithm to compress the file. This works equally on files
that are compressed with gzip outside of octave, and gzip can equally be
used to convert the files for backward compatibility. This option is only
available if Octave was built with a link to the zlib libraries.

The list of variables to save may use wildcard patterns containing the following special
characters:
?

Match any single character.

*

Match zero or more characters.

[ list ]

Match the list of characters specified by list. If the first character is ! or
^, match all characters except those specified by list. For example, the
pattern [a-zA-Z] will match all lower and uppercase alphabetic characters.

254

GNU Octave

Wildcards may also be used in the field name specifications when using
the -struct modifier (but not in the struct name itself).
Except when using the matlab binary data file format or the ‘-ascii’ format, saving
global variables also saves the global status of the variable. If the variable is restored
at a later time using ‘load’, it will be restored as a global variable.
The command
save -binary data a b*
saves the variable ‘a’ and all variables beginning with ‘b’ to the file data in Octave’s
binary format.
See also: [load], page 255, [save default options], page 254, [save header format string],
page 254, [dlmread], page 258, [csvread], page 258, [fread], page 279.
There are three functions that modify the behavior of save.

val = save_default_options ()
old_val = save_default_options (new_val)
save_default_options (new_val, "local")
Query or set the internal variable that specifies the default options for the save
command, and defines the default format.
Typical values include "-ascii", "-text -zip". The default value is -text.
When called from inside a function with the "local" option, the variable is changed
locally for the function and any subroutines it calls. The original variable value is
restored when exiting the function.
See also: [save], page 252.

val = save_precision ()
old_val = save_precision (new_val)
save_precision (new_val, "local")
Query or set the internal variable that specifies the number of digits to keep when
saving data in text format.
When called from inside a function with the "local" option, the variable is changed
locally for the function and any subroutines it calls. The original variable value is
restored when exiting the function.

val = save_header_format_string ()
old_val = save_header_format_string (new_val)
save_header_format_string (new_val, "local")
Query or set the internal variable that specifies the format string used for the comment
line written at the beginning of text-format data files saved by Octave.
The format string is passed to strftime and should begin with the character ‘#’ and
contain no newline characters. If the value of save_header_format_string is the
empty string, the header comment is omitted from text-format data files. The default
value is
"# Created by Octave VERSION, %a %b %d %H:%M:%S %Y %Z "

Chapter 14: Input and Output

255

When called from inside a function with the "local" option, the variable is changed
locally for the function and any subroutines it calls. The original variable value is
restored when exiting the function.
See also: [strftime], page 821, [save], page 252.

load file
load options file
load options file v1 v2 . . .
S = load ("options", "file", "v1", "v2", . . . )
load file options
load file options v1 v2 . . .
S = load ("file", "options", "v1", "v2", . . . )
Load the named variables v1, v2, . . . , from the file file.
If no variables are specified then all variables found in the file will be loaded. As with
save, the list of variables to extract can be full names or use a pattern syntax. The
format of the file is automatically detected but may be overridden by supplying the
appropriate option.
If load is invoked using the functional form
load ("-option1", ..., "file", "v1", ...)
then the options, file, and variable name arguments (v1, . . . ) must be specified as
character strings.
If a variable that is not marked as global is loaded from a file when a global symbol
with the same name already exists, it is loaded in the global symbol table. Also, if
a variable is marked as global in a file and a local symbol exists, the local symbol is
moved to the global symbol table and given the value from the file.
If invoked with a single output argument, Octave returns data instead of inserting variables in the symbol table. If the data file contains only numbers (TAB- or
space-delimited columns), a matrix of values is returned. Otherwise, load returns a
structure with members corresponding to the names of the variables in the file.
The load command can read data stored in Octave’s text and binary formats,
and matlab’s binary format. If compiled with zlib support, it can also load
gzip-compressed files. It will automatically detect the type of file and do conversion
from different floating point formats (currently only IEEE big and little endian,
though other formats may be added in the future).
Valid options for load are listed in the following table.
-force

This option is accepted for backward compatibility but is ignored. Octave
now overwrites variables currently in memory with those of the same name
found in the file.

-ascii

Force Octave to assume the file contains columns of numbers in text
format without any header or other information. Data in the file will be
loaded as a single numeric matrix with the name of the variable derived
from the name of the file.

-binary

Force Octave to assume the file is in Octave’s binary format.

256

GNU Octave

-hdf5

Force Octave to assume the file is in hdf5 format. (hdf5 is a free,
portable binary format developed by the National Center for Supercomputing Applications at the University of Illinois.) Note that Octave can
read hdf5 files not created by itself, but may skip some datasets in formats that it cannot support. This format is only available if Octave was
built with a link to the hdf5 libraries.

-import

This option is accepted for backward compatibility but is ignored. Octave
can now support multi-dimensional HDF data and automatically modifies
variable names if they are invalid Octave identifiers.

-mat
-mat-binary
-6
-v6
-7
-v7
Force Octave to assume the file is in matlab’s version 6 or 7 binary
format.
-mat4-binary
-4
-v4
-V4
Force Octave to assume the file is in the binary format written by matlab
version 4.
-text

Force Octave to assume the file is in Octave’s text format.

See also: [save], page 252, [dlmwrite], page 257, [csvwrite], page 258, [fwrite],
page 282.

str = fileread (filename)
Read the contents of filename and return it as a string.
See also: [fread], page 279, [textread], page 258, [sscanf], page 277.

native_float_format ()
Return the native floating point format as a string.
It is possible to write data to a file in a similar way to the disp function for writing data
to the screen. The fdisp works just like disp except its first argument is a file pointer as
created by fopen. As an example, the following code writes to data ‘myfile.txt’.
fid = fopen ("myfile.txt", "w");
fdisp (fid, "3/8 is ");
fdisp (fid, 3/8);
fclose (fid);
See Section 14.2.1 [Opening and Closing Files], page 267, for details on how to use fopen
and fclose.

fdisp (fid, x)
Display the value of x on the stream fid.

Chapter 14: Input and Output

257

For example:
fdisp (stdout, "The value of pi is:"), fdisp (stdout, pi)
a the value of pi is:
a 3.1416
Note that the output from fdisp always ends with a newline.
See also: [disp], page 245.
Octave can also read and write matrices text files such as comma separated lists.
(file, M)
(file, M, delim, r, c)
(file, M, key, val . . . )
(file, M, "-append", . . . )
(fid, . . . )
Write the numeric matrix M to the text file file using a delimiter.
file should be a filename or a writable file ID given by fopen.
The parameter delim specifies the delimiter to use to separate values on a row. If no
delimiter is specified the comma character ‘,’ is used.
The value of r specifies the number of delimiter-only lines to add to the start of the
file.
The value of c specifies the number of delimiters to prepend to each line of data.
If the argument "-append" is given, append to the end of file.
In addition, the following keyword value pairs may appear at the end of the argument
list:

dlmwrite
dlmwrite
dlmwrite
dlmwrite
dlmwrite

"append"

Either "on" or "off". See "-append" above.

"delimiter"
See delim above.
"newline"
The character(s) to separate each row. Three special cases exist for this
option. "unix" is changed into "\n", "pc" is changed into "\r\n", and
"mac" is changed into "\r". Any other value is used directly as the
newline separator.
"roffset"
See r above.
"coffset"
See c above.
"precision"
The precision to use when writing the file. It can either be a format string
(as used by fprintf) or a number of significant digits.
dlmwrite ("file.csv", reshape (1:16, 4, 4));
dlmwrite ("file.tex", a, "delimiter", "&", "newline", "\n")
See also: [dlmread], page 258, [csvread], page 258, [csvwrite], page 258.

258

data
data
data
data
data

GNU Octave

(file)
(file, sep)
(file, sep, r0, c0)
(file, sep, range)
( . . . , "emptyvalue", EMPTYVAL)
Read numeric data from the text file file which uses the delimiter sep between data
values.
If sep is not defined the separator between fields is determined from the file itself.
The optional scalar arguments r0 and c0 define the starting row and column of the
data to be read. These values are indexed from zero, i.e., the first data row corresponds
to an index of zero.
The range parameter specifies exactly which data elements are read. The first form of
the parameter is a 4-element vector containing the upper left and lower right corners
[R0,C0,R1,C1] where the indices are zero-based. Alternatively, a spreadsheet style
form such as "A2..Q15" or "T1:AA5" can be used. The lowest alphabetical index ’A’
refers to the first column. The lowest row index is 1.
file should be a filename or a file id given by fopen. In the latter case, the file is read
until end of file is reached.
The "emptyvalue" option may be used to specify the value used to fill empty fields.
The default is zero. Note that any non-numeric values, such as text, are also replaced
by the "emptyvalue".

=
=
=
=
=

dlmread
dlmread
dlmread
dlmread
dlmread

See also: [csvread], page 258, [textscan], page 260, [textread], page 258, [dlmwrite],
page 257.

csvwrite (filename, x)
csvwrite (filename, x, dlm_opt1, . . . )
Write the numeric matrix x to the file filename in comma-separated-value (CSV)
format.
This function is equivalent to
dlmwrite (filename, x, ",", dlm_opt1, ...)
Any optional arguments are passed directly to dlmwrite (see [dlmwrite], page 257).
See also: [csvread], page 258, [dlmwrite], page 257, [dlmread], page 258.

x = csvread (filename)
x = csvread (filename, dlm_opt1, . . . )
Read the comma-separated-value (CSV) file filename into the matrix x.
Note: only CSV files containing numeric data can be read.
This function is equivalent to
x = dlmread (filename, "," , dlm_opt1, ...)
Any optional arguments are passed directly to dlmread (see [dlmread], page 258).
See also: [dlmread], page 258, [textread], page 258, [textscan], page 260, [csvwrite],
page 258, [dlmwrite], page 257.
Formatted data from can be read from, or written to, text files as well.

Chapter 14: Input and Output

[a,
[a,
[a,
[a,
[a,

259

(filename)
(filename, format)
(filename, format, n)
(filename, format, prop1, value1, . . . )
(filename, format, n, prop1, value1, . . . )
Read data from a text file.

...]
...]
...]
...]
...]

=
=
=
=
=

textread
textread
textread
textread
textread

The file filename is read and parsed according to format. The function behaves like
strread except it works by parsing a file instead of a string. See the documentation
of strread for details.
In addition to the options supported by strread, this function supports two more:
• "headerlines": The first value number of lines of filename are skipped.
• "endofline": Specify a single character or "\r\n". If no value is given, it will be
inferred from the file. If set to "" (empty string) EOLs are ignored as delimiters.
The optional input n (format repeat count) specifies the number of times the format
string is to be used or the number of lines to be read, whichever happens first while
reading. The former is equivalent to requesting that the data output vectors should
be of length N. Note that when reading files with format strings referring to multiple
lines, n should rather be the number of lines to be read than the number of format
string uses.
If the format string is empty (not just omitted) and the file contains only numeric data
(excluding headerlines), textread will return a rectangular matrix with the number
of columns matching the number of numeric fields on the first data line of the file.
Empty fields are returned as zero values.
Examples:
Assume a data file like:
1 a 2 b
3 c 4 d
5 e
[a, b] = textread (f, "%f %s")
returns two columns of data, one with doubles, the other a
cellstr array:
a = [1; 2; 3; 4; 5]
b = {"a"; "b"; "c"; "d"; "e"}
[a, b] = textread (f, "%f %s", 3)
(read data into two culumns, try to use the format string
three times)
returns
a = [1; 2; 3]
b = {"a"; "b"; "c"}

260

GNU Octave

With a data file like:
1
a
2
b
[a, b] = textread (f,
returns a = 1 and b =
only once because the
data file. To obtain
(number of data lines
than 2.

"%f %s", 2)
{"a"}; i.e., the format string is used
format string refers to 2 lines of the
2x1 data output columns, specify N = 4
containing all requested data) rather

See also: [strread], page 84, [load], page 255, [dlmread], page 258, [fscanf], page 276,
[textscan], page 260.

C =
C =
C =
C =
C =
[C,

textscan (fid, format)
textscan (fid, format, repeat)
textscan (fid, format, param, value, . . . )
textscan (fid, format, repeat, param, value, . . . )
textscan (str, . . . )
position, errmsg] = textscan ( . . . )
Read data from a text file or string.
The string str or file associated with fid is read from and parsed according to format.
The function is an extension of strread and textread. Differences include: the
ability to read from either a file or a string, additional options, and additional format
specifiers.
The input is interpreted as a sequence of words, delimiters (such as whitespace), and
literals. The characters that form delimiters and whitespace are determined by the
options. The format consists of format specifiers interspersed between literals. In the
format, whitespace forms a delimiter between consecutive literals, but is otherwise
ignored.
The output C is a cell array where the number of columns is determined by the
number of format specifiers.
The first word of the input is matched to the first specifier of the format and placed
in the first column of the output; the second is matched to the second specifier and
placed in the second column and so forth. If there are more words than specifiers
then the process is repeated until all words have been processed or the limit imposed
by repeat has been met (see below).
The string format describes how the words in str should be parsed. As in fscanf, any
(non-whitespace) text in the format that is not one of these specifiers is considered a
literal. If there is a literal between two format specifiers then that same literal must
appear in the input stream between the matching words.
The following specifiers are valid:
%f
%f64
%n

The word is parsed as a number and converted to double.

Chapter 14: Input and Output

%f32
%d
%d8
%d16
%d32
%d64
%u
%u8
%u16
%u32
%u64

261

The word is parsed as a number and converted to single (float).

The word is parsed as a number and converted to int8, int16, int32, or
int64. If no size is specified then int32 is used.

The word is parsed as a number and converted to uint8, uint16, uint32,
or uint64. If no size is specified then uint32 is used.

%s

The word is parsed as a string ending at the last character before whitespace, an end-of-line, or a delimiter specified in the options.

%q

The word is parsed as a "quoted string". If the first character of the
string is a double quote (") then the string includes everything until a
matching double quote—including whitespace, delimiters, and end-of-line
characters. If a pair of consecutive double quotes appears in the input,
it is replaced in the output by a single double quote. For examples, the
input "He said ""Hello""" would return the value ’He said "Hello"’.

%c

The next character of the input is read. This includes delimiters, whitespace, and end-of-line characters.

%[...]
%[^...]

In the first form, the word consists of the longest run consisting of only
characters between the brackets. Ranges of characters can be specified
by a hyphen; for example, %[0-9a-zA-Z] matches all alphanumeric characters (if the underlying character set is ASCII). Since matlab treats
hyphens literally, this expansion only applies to alphanumeric characters.
To include ’-’ in the set, it should appear first or last in the brackets; to
include ’]’, it should be the first character. If the first character is ’^’ then
the word consists of characters not listed.

%N...

For %s, %c %d, %f, %n, %u, an optional width can be specified as %Ns,
etc. where N is an integer > 1. For %c, this causes exactly N characters
to be read instead of a single character. For the other specifiers, it is
an upper bound on the number of characters read; normal delimiters can
cause fewer characters to be read. For complex numbers, this limit applies
to the real and imaginary components individually. For %f and %n,
format specifiers like %N.Mf are allowed, where M is an upper bound on
number of characters after the decimal point to be considered; subsequent
digits are skipped. For example, the specifier %8.2f would read 12.345e6
as 1.234e7.

%*...

The word specified by the remainder of the conversion specifier is skipped.

262

GNU Octave

literals

In addition the format may contain literal character strings; these will be
skipped during reading. If the input string does not match this literal,
the processing terminates.

Parsed words corresponding to the first specifier are returned in the first output
argument and likewise for the rest of the specifiers.
By default, if there is only one input argument, format is "%f". This means that
numbers are read from the input into a single column vector. If format is explicitly
empty ("") then textscan will return data in a number of columns matching the
number of fields on the first data line of the input. Either of these is suitable only
when the input is exclusively numeric.
For example, the string
str = "\
Bunny Bugs
5.5\n\
Duck Daffy -7.5e-5\n\
Penguin Tux
6"

can be read using
a = textscan (str, "%s %s %f");
The optional numeric argument repeat can be used for limiting the number of items
read:
-1

Read all of the string or file until the end (default).

N

Read until the first of two conditions occurs: 1) the format has been
processed N times, or 2) N lines of the input have been processed. Zero
(0) is an acceptable value for repeat. Currently, end-of-line characters
inside %q, %c, and %[. . . ]$ conversions do not contribute to the line
count. This is incompatible with matlab and may change in future.

The behavior of textscan can be changed via property/value pairs. The following
properties are recognized:
"BufSize"
This specifies the number of bytes to use for the internal buffer. A modest
speed improvement may be obtained by setting this to a large value when
reading a large file, especially if the input contains long strings. The
default is 4096, or a value dependent on n if that is specified.
"CollectOutput"
A value of 1 or true instructs textscan to concatenate consecutive
columns of the same class in the output cell array. A value of 0 or false
(default) leaves output in distinct columns.
"CommentStyle"
Specify parts of the input which are considered comments and will be
skipped. value is the comment style and can be either (1) A string or
1x1 cell string, to skip everything to the right of it; (2) A cell array
of two strings, to skip everything between the first and second strings.
Comments are only parsed where whitespace is accepted and do not act
as delimiters.

Chapter 14: Input and Output

263

"Delimiter"
If value is a string, any character in value will be used to split the input
into words. If value is a cell array of strings, any string in the array will
be used to split the input into words. (default value = any whitespace.)
"EmptyValue"
Value to return for empty numeric values in non-whitespace delimited
data. The default is NaN. When the data type does not support NaN
(int32 for example), then the default is zero.
"EndOfLine"
value can be either an emtpy or one character specifying the end-of-line
character, or the pair "\r\n" (CRLF). In the latter case, any of "\r",
"\n" or "\r\n" is counted as a (single) newline. If no value is given,
"\r\n" is used.
"HeaderLines"
The first value number of lines of fid are skipped. Note that this does
not refer to the first non-comment lines, but the first lines of any type.
"MultipleDelimsAsOne"
If value is nonzero, treat a series of consecutive delimiters, without whitespace in between, as a single delimiter. Consecutive delimiter series need
not be vertically aligned. Without this option, a single delimiter before
the end of the line does not cause the line to be considered to end with
an empty value, but a single delimiter at the start of a line causes the
line to be considered to start with an empty value.
"TreatAsEmpty"
Treat single occurrences (surrounded by delimiters or whitespace) of the
string(s) in value as missing values.
"ReturnOnError"
If set to numerical 1 or true, return normally as soon as an error is
encountered, such as trying to read a string using %f. If set to 0 or false,
return an error and no data.
"Whitespace"
Any character in value will be interpreted as whitespace and trimmed;
The default value for whitespace is " \b\r\n\t" (note the space). Unless
whitespace is set to "" (empty) AND at least one "%s" format conversion
specifier is supplied, a space is always part of whitespace.
When the number of words in str or fid doesn’t match an exact multiple of the
number of format conversion specifiers, textscan’s behavior depends on whether the
last character of the string or file is an end-of-line as specified by the EndOfLine
option:
last character = end-of-line
Data columns are padded with empty fields, NaN or 0 (for integer fields)
so that all columns have equal length

264

GNU Octave

last character is not end-of-line
Data columns are not padded; textscan returns columns of unequal
length
The second output position provides the location, in characters from the beginning
of the file or string, where processing stopped.
See also: [dlmread], page 258, [fscanf], page 276, [load], page 255, [strread], page 84,
[textread], page 258.
The importdata function has the ability to work with a wide variety of data.

A =
A =
A =
[A,
[A,

(fname)
(fname, delimiter)
(fname, delimiter, header_rows)
= importdata ( . . . )
header_rows] = importdata ( . . . )
Import data from the file fname.
Input parameters:
• fname The name of the file containing data.
• delimiter The character separating columns of data. Use \t for tab. (Only valid
for ASCII files)
• header rows The number of header rows before the data begins. (Only valid for
ASCII files)

importdata
importdata
importdata
delimiter]
delimiter,

Different file types are supported:
• ASCII table
Import ASCII table using the specified number of header rows and the specified
delimiter.
• Image file
• matlab file
• Spreadsheet files (depending on external software)
• WAV file
See also: [textscan], page 260, [dlmread], page 258, [csvread], page 258, [load],
page 255.

14.1.3.1 Saving Data on Unexpected Exits
If Octave for some reason exits unexpectedly it will by default save the variables available in the workspace to a file in the current directory. By default this file is named
‘octave-workspace’ and can be loaded into memory with the load command. While the
default behavior most often is reasonable it can be changed through the following functions.

val = crash_dumps_octave_core ()
old_val = crash_dumps_octave_core (new_val)
crash_dumps_octave_core (new_val, "local")
Query or set the internal variable that controls whether Octave tries to save all current
variables to the file octave-workspace if it crashes or receives a hangup, terminate
or similar signal.

Chapter 14: Input and Output

265

When called from inside a function with the "local" option, the variable is changed
locally for the function and any subroutines it calls. The original variable value is
restored when exiting the function.
See also: [octave core file limit], page 265, [octave core file name], page 266,
[octave core file options], page 265.

val = sighup_dumps_octave_core ()
old_val = sighup_dumps_octave_core (new_val)
sighup_dumps_octave_core (new_val, "local")
Query or set the internal variable that controls whether Octave tries to save all current
variables to the file octave-workspace if it receives a hangup signal.
When called from inside a function with the "local" option, the variable is changed
locally for the function and any subroutines it calls. The original variable value is
restored when exiting the function.

val = sigterm_dumps_octave_core ()
old_val = sigterm_dumps_octave_core (new_val)
sigterm_dumps_octave_core (new_val, "local")
Query or set the internal variable that controls whether Octave tries to save all current
variables to the file octave-workspace if it receives a terminate signal.
When called from inside a function with the "local" option, the variable is changed
locally for the function and any subroutines it calls. The original variable value is
restored when exiting the function.

val = octave_core_file_options ()
old_val = octave_core_file_options (new_val)
octave_core_file_options (new_val, "local")
Query or set the internal variable that specifies the options used for saving the
workspace data if Octave aborts.
The value of octave_core_file_options should follow the same format as the options for the save function. The default value is Octave’s binary format.
When called from inside a function with the "local" option, the variable is changed
locally for the function and any subroutines it calls. The original variable value is
restored when exiting the function.
See also: [crash dumps octave core], page 264, [octave core file name], page 266,
[octave core file limit], page 265.

val = octave_core_file_limit ()
old_val = octave_core_file_limit (new_val)
octave_core_file_limit (new_val, "local")
Query or set the internal variable that specifies the maximum amount of memory (in
kilobytes) of the top-level workspace that Octave will attempt to save when writing
data to the crash dump file (the name of the file is specified by octave core file name).
If octave core file options flags specify a binary format, then octave core file limit
will be approximately the maximum size of the file. If a text file format is used, then
the file could be much larger than the limit. The default value is -1 (unlimited)

266

GNU Octave

When called from inside a function with the "local" option, the variable is changed
locally for the function and any subroutines it calls. The original variable value is
restored when exiting the function.
See also: [crash dumps octave core], page 264, [octave core file name], page 266,
[octave core file options], page 265.

val = octave_core_file_name ()
old_val = octave_core_file_name (new_val)
octave_core_file_name (new_val, "local")
Query or set the internal variable that specifies the name of the file used for saving
data from the top-level workspace if Octave aborts.
The default value is "octave-workspace"
When called from inside a function with the "local" option, the variable is changed
locally for the function and any subroutines it calls. The original variable value is
restored when exiting the function.
See also: [crash dumps octave core], page 264, [octave core file name], page 266,
[octave core file options], page 265.

14.2 C-Style I/O Functions
Octave’s C-style input and output functions provide most of the functionality of the C
programming language’s standard I/O library. The argument lists for some of the input
functions are slightly different, however, because Octave has no way of passing arguments
by reference.
In the following, file refers to a filename and fid refers to an integer file number, as
returned by fopen.
There are three files that are always available. Although these files can be accessed using
their corresponding numeric file ids, you should always use the symbolic names given in the
table below, since it will make your programs easier to understand.

stdin ()
Return the numeric value corresponding to the standard input stream.
When Octave is used interactively, stdin is filtered through the command line editing
functions.
See also: [stdout], page 266, [stderr], page 266.

stdout ()
Return the numeric value corresponding to the standard output stream.
Data written to the standard output is normally filtered through the pager.
See also: [stdin], page 266, [stderr], page 266.

stderr ()
Return the numeric value corresponding to the standard error stream.
Even if paging is turned on, the standard error is not sent to the pager. It is useful
for error messages and prompts.
See also: [stdin], page 266, [stdout], page 266.

Chapter 14: Input and Output

267

14.2.1 Opening and Closing Files
When reading data from a file it must be opened for reading first, and likewise when writing
to a file. The fopen function returns a pointer to an open file that is ready to be read or
written. Once all data has been read from or written to the opened file it should be closed.
The fclose function does this. The following code illustrates the basic pattern for writing
to a file, but a very similar pattern is used when reading a file.
filename = "myfile.txt";
fid = fopen (filename, "w");
# Do the actual I/O here...
fclose (fid);

fid = fopen (name)
fid = fopen (name, mode)
fid = fopen (name, mode, arch)
[fid, msg] = fopen ( . . . )
fid_list = fopen ("all")
[file, mode, arch] = fopen (fid)
Open a file for low-level I/O or query open files and file descriptors.
The first form of the fopen function opens the named file with the specified mode
(read-write, read-only, etc.) and architecture interpretation (IEEE big endian, IEEE
little endian, etc.), and returns an integer value that may be used to refer to the file
later. If an error occurs, fid is set to −1 and msg contains the corresponding system
error message. The mode is a one or two character string that specifies whether the
file is to be opened for reading, writing, or both.
The second form of the fopen function returns a vector of file ids corresponding to
all the currently open files, excluding the stdin, stdout, and stderr streams.
The third form of the fopen function returns information about the open file given
its file id.
For example,
myfile = fopen ("splat.dat", "r", "ieee-le");
opens the file splat.dat for reading. If necessary, binary numeric values will be read
assuming they are stored in IEEE format with the least significant bit first, and then
converted to the native representation.
Opening a file that is already open simply opens it again and returns a separate file
id. It is not an error to open a file several times, though writing to the same file
through several different file ids may produce unexpected results.
The possible values of mode are
‘r’ (default)
Open a file for reading.
‘w’

Open a file for writing. The previous contents are discarded.

‘a’

Open or create a file for writing at the end of the file.

‘r+’

Open an existing file for reading and writing.

‘w+’

Open a file for reading or writing. The previous contents are discarded.

268

GNU Octave

‘a+’

Open or create a file for reading or writing at the end of the file.

Append a "t" to the mode string to open the file in text mode or a "b" to open
in binary mode. On Windows systems, text mode reading and writing automatically
converts linefeeds to the appropriate line end character for the system (carriage-return
linefeed on Windows). The default when no mode is specified is binary.
Additionally, you may append a "z" to the mode string to open a gzipped file for
reading or writing. For this to be successful, you must also open the file in binary
mode.
The parameter arch is a string specifying the default data format for the file. Valid
values for arch are:
"native" or "n" (default)
The format of the current machine.
"ieee-be" or "b"
IEEE big endian format.
"ieee-le" or "l"
IEEE little endian format.
However, conversions are currently only supported for ‘native’, ‘ieee-be’, and
‘ieee-le’ formats.
When opening a new file that does not yet exist, permissions will be set to 0666 umask.
Compatibility Note: Octave opens files using buffered I/O. Small writes are accumulated until an internal buffer is filled, and then everything is written in a single
operation. This is very efficient and improves performance. matlab, however, opens
files using flushed I/O where every write operation is immediately performed. If the
write operation must be performed immediately after data has been written then the
write should be followed by a call to fflush to flush the internal buffer.
See also: [fclose], page 268, [fgets], page 270, [fgetl], page 269, [fscanf], page 276,
[fread], page 279, [fputs], page 269, [fdisp], page 256, [fprintf], page 271, [fwrite],
page 282, [fskipl], page 270, [fseek], page 285, [frewind], page 285, [ftell], page 285,
[feof], page 284, [ferror], page 284, [fclear], page 284, [fflush], page 250, [freport],
page 284, [umask], page 832.

fclose (fid)
fclose ("all")
status = fclose ("all")
Close the file specified by the file descriptor fid.
If successful, fclose returns 0, otherwise, it returns -1. The second form of the fclose
call closes all open files except stdin, stdout, stderr, and any FIDs associated with
gnuplot.
See also: [fopen], page 267, [fflush], page 250, [freport], page 284.

is_valid_file_id (fid)
Return true if fid refers to an open file.
See also: [freport], page 284, [fopen], page 267.

Chapter 14: Input and Output

269

14.2.2 Simple Output
Once a file has been opened for writing a string can be written to the file using the fputs
function. The following example shows how to write the string ‘Free Software is needed
for Free Science’ to the file ‘free.txt’.
filename = "free.txt";
fid = fopen (filename, "w");
fputs (fid, "Free Software is needed for Free Science");
fclose (fid);

fputs (fid, string)
status = fputs (fid, string)
Write the string string to the file with file descriptor fid.
The string is written to the file with no additional formatting. Use fdisp instead to
automatically append a newline character appropriate for the local machine.
Return a non-negative number on success or EOF on error.
See also: [fdisp], page 256, [fprintf], page 271, [fwrite], page 282, [fopen], page 267.
A function much similar to fputs is available for writing data to the screen. The puts
function works just like fputs except it doesn’t take a file pointer as its input.

puts (string)
status = puts (string)
Write a string to the standard output with no formatting.
The string is written verbatim to the standard output. Use disp to automatically
append a newline character appropriate for the local machine.
Return a non-negative number on success and EOF on error.
See also: [fputs], page 269, [disp], page 245.

14.2.3 Line-Oriented Input
To read from a file it must be opened for reading using fopen. Then a line can be read
from the file using fgetl as the following code illustrates
fid = fopen ("free.txt");
txt = fgetl (fid)
a Free Software is needed for Free Science
fclose (fid);
This of course assumes that the file ‘free.txt’ exists and contains the line ‘Free Software
is needed for Free Science’.

str = fgetl (fid)
str = fgetl (fid, len)
Read characters from a file, stopping after a newline, or EOF, or len characters have
been read.
The characters read, excluding the possible trailing newline, are returned as a string.
If len is omitted, fgetl reads until the next newline character.
If there are no more characters to read, fgetl returns −1.

270

GNU Octave

To read a line and return the terminating newline see fgets.
See also: [fgets], page 270, [fscanf], page 276, [fread], page 279, [fopen], page 267.

str = fgets (fid)
str = fgets (fid, len)
Read characters from a file, stopping after a newline, or EOF, or len characters have
been read.
The characters read, including the possible trailing newline, are returned as a string.
If len is omitted, fgets reads until the next newline character.
If there are no more characters to read, fgets returns −1.
To read a line and discard the terminating newline see fgetl.
See also: [fputs], page 269, [fgetl], page 269, [fscanf], page 276, [fread], page 279,
[fopen], page 267.

nlines = fskipl (fid)
nlines = fskipl (fid, count)
nlines = fskipl (fid, Inf)
Read and skip count lines from the file specified by the file descriptor fid.
fskipl discards characters until an end-of-line is encountered exactly count-times, or
until the end-of-file marker is found.
If count is omitted, it defaults to 1. count may also be Inf, in which case lines are
skipped until the end of the file. This form is suitable for counting the number of
lines in a file.
Returns the number of lines skipped (end-of-line sequences encountered).
See also: [fgetl], page 269, [fgets], page 270, [fscanf], page 276, [fopen], page 267.

14.2.4 Formatted Output
This section describes how to call printf and related functions.
The following functions are available for formatted output. They are modeled after the
C language functions of the same name, but they interpret the format template differently
in order to improve the performance of printing vector and matrix values.
Implementation Note: For compatibility with matlab, escape sequences in the template
string (e.g., "\n" => newline) are expanded even when the template string is defined with
single quotes.

printf (template, . . . )
Print optional arguments under the control of the template string template to the
stream stdout and return the number of characters printed.
See the Formatted Output section of the GNU Octave manual for a complete description of the syntax of the template string.
Implementation Note: For compatibility with matlab, escape sequences in the template string (e.g., "\n" => newline) are expanded even when the template string is
defined with single quotes.
See also: [fprintf], page 271, [sprintf], page 271, [scanf], page 277.

Chapter 14: Input and Output

271

fprintf (fid, template, . . . )
fprintf (template, . . . )
numbytes = fprintf ( . . . )
This function is equivalent to printf, except that the output is written to the file
descriptor fid instead of stdout.
If fid is omitted, the output is written to stdout making the function exactly equivalent to printf.
The optional output returns the number of bytes written to the file.
Implementation Note: For compatibility with matlab, escape sequences in the template string (e.g., "\n" => newline) are expanded even when the template string is
defined with single quotes.
See also: [fputs], page 269, [fdisp], page 256, [fwrite], page 282, [fscanf], page 276,
[printf], page 270, [sprintf], page 271, [fopen], page 267.

sprintf (template, . . . )
This is like printf, except that the output is returned as a string.
Unlike the C library function, which requires you to provide a suitably sized string
as an argument, Octave’s sprintf function returns the string, automatically sized to
hold all of the items converted.
Implementation Note: For compatibility with matlab, escape sequences in the template string (e.g., "\n" => newline) are expanded even when the template string is
defined with single quotes.
See also: [printf], page 270, [fprintf], page 271, [sscanf], page 277.
The printf function can be used to print any number of arguments. The template
string argument you supply in a call provides information not only about the number of
additional arguments, but also about their types and what style should be used for printing
them.
Ordinary characters in the template string are simply written to the output stream
as-is, while conversion specifications introduced by a ‘%’ character in the template cause
subsequent arguments to be formatted and written to the output stream. For example,
pct = 37;
filename = "foo.txt";
printf ("Processed %d%% of ’%s’.\nPlease be patient.\n",
pct, filename);
produces output like
Processed 37% of ’foo.txt’.
Please be patient.
This example shows the use of the ‘%d’ conversion to specify that a scalar argument
should be printed in decimal notation, the ‘%s’ conversion to specify printing of a string
argument, and the ‘%%’ conversion to print a literal ‘%’ character.
There are also conversions for printing an integer argument as an unsigned value in
octal, decimal, or hexadecimal radix (‘%o’, ‘%u’, or ‘%x’, respectively); or as a character
value (‘%c’).

272

GNU Octave

Floating-point numbers can be printed in normal, fixed-point notation using the ‘%f’
conversion or in exponential notation using the ‘%e’ conversion. The ‘%g’ conversion uses
either ‘%e’ or ‘%f’ format, depending on what is more appropriate for the magnitude of the
particular number.
You can control formatting more precisely by writing modifiers between the ‘%’ and
the character that indicates which conversion to apply. These slightly alter the ordinary
behavior of the conversion. For example, most conversion specifications permit you to
specify a minimum field width and a flag indicating whether you want the result left- or
right-justified within the field.
The specific flags and modifiers that are permitted and their interpretation vary depending on the particular conversion. They’re all described in more detail in the following
sections.

14.2.5 Output Conversion for Matrices
When given a matrix value, Octave’s formatted output functions cycle through the format
template until all the values in the matrix have been printed. For example:
printf ("%4.2f %10.2e %8.4g\n", hilb (3));
5.00e-01
0.3333
a 1.00
3.33e-01
0.25
a 0.50
2.50e-01
0.2
a 0.33
If more than one value is to be printed in a single call, the output functions do not
return to the beginning of the format template when moving on from one value to the next.
This can lead to confusing output if the number of elements in the matrices are not exact
multiples of the number of conversions in the format template. For example:
printf ("%4.2f %10.2e %8.4g\n", [1, 2], [3, 4]);
2.00e+00
3
a 1.00
a 4.00
If this is not what you want, use a series of calls instead of just one.

14.2.6 Output Conversion Syntax
This section provides details about the precise syntax of conversion specifications that can
appear in a printf template string.
Characters in the template string that are not part of a conversion specification are
printed as-is to the output stream.
The conversion specifications in a printf template string have the general form:
% flags width [ . precision ] type conversion
For example, in the conversion specifier ‘%-10.8ld’, the ‘-’ is a flag, ‘10’ specifies the field
width, the precision is ‘8’, the letter ‘l’ is a type modifier, and ‘d’ specifies the conversion
style. (This particular type specifier says to print a numeric argument in decimal notation,
with a minimum of 8 digits left-justified in a field at least 10 characters wide.)
In more detail, output conversion specifications consist of an initial ‘%’ character followed
in sequence by:

Chapter 14: Input and Output

273

• Zero or more flag characters that modify the normal behavior of the conversion specification.
• An optional decimal integer specifying the minimum field width. If the normal conversion produces fewer characters than this, the field is padded with spaces to the specified
width. This is a minimum value; if the normal conversion produces more characters
than this, the field is not truncated. Normally, the output is right-justified within the
field.
You can also specify a field width of ‘*’. This means that the next argument in the
argument list (before the actual value to be printed) is used as the field width. The
value is rounded to the nearest integer. If the value is negative, this means to set the
‘-’ flag (see below) and to use the absolute value as the field width.
• An optional precision to specify the number of digits to be written for the numeric
conversions. If the precision is specified, it consists of a period (‘.’) followed optionally
by a decimal integer (which defaults to zero if omitted).
You can also specify a precision of ‘*’. This means that the next argument in the
argument list (before the actual value to be printed) is used as the precision. The value
must be an integer, and is ignored if it is negative.
• An optional type modifier character. This character is ignored by Octave’s printf
function, but is recognized to provide compatibility with the C language printf.
• A character that specifies the conversion to be applied.
The exact options that are permitted and how they are interpreted vary between the
different conversion specifiers. See the descriptions of the individual conversions for information about the particular options that they use.

14.2.7 Table of Output Conversions
Here is a table summarizing what all the different conversions do:
‘%d’, ‘%i’

Print an integer as a signed decimal number. See Section 14.2.8 [Integer Conversions], page 274, for details. ‘%d’ and ‘%i’ are synonymous for output, but are
different when used with scanf for input (see Section 14.2.13 [Table of Input
Conversions], page 278).

‘%o’

Print an integer as an unsigned octal number. See Section 14.2.8 [Integer Conversions], page 274, for details.

‘%u’

Print an integer as an unsigned decimal number. See Section 14.2.8 [Integer
Conversions], page 274, for details.

‘%x’, ‘%X’

Print an integer as an unsigned hexadecimal number. ‘%x’ uses lowercase letters
and ‘%X’ uses uppercase. See Section 14.2.8 [Integer Conversions], page 274, for
details.

‘%f’

Print a floating-point number in normal (fixed-point) notation.
Section 14.2.9 [Floating-Point Conversions], page 275, for details.

‘%e’, ‘%E’

Print a floating-point number in exponential notation. ‘%e’ uses lowercase
letters and ‘%E’ uses uppercase. See Section 14.2.9 [Floating-Point Conversions],
page 275, for details.

See

274

GNU Octave

‘%g’, ‘%G’

Print a floating-point number in either normal (fixed-point) or exponential
notation, whichever is more appropriate for its magnitude. ‘%g’ uses lowercase
letters and ‘%G’ uses uppercase. See Section 14.2.9 [Floating-Point Conversions],
page 275, for details.

‘%c’

Print a single character.
page 275.

‘%s’

Print a string. See Section 14.2.10 [Other Output Conversions], page 275.

‘%%’

Print a literal ‘%’ character. See Section 14.2.10 [Other Output Conversions],
page 275.

See Section 14.2.10 [Other Output Conversions],

If the syntax of a conversion specification is invalid, unpredictable things will happen, so
don’t do this. In particular, matlab allows a bare percentage sign ‘%’ with no subsequent
conversion character. Octave will emit an error and stop if it sees such code. When the
string variable to be processed cannot be guaranteed to be free of potential format codes it
is better to use the two argument form of any of the printf functions and set the format
string to %s. Alternatively, for code which is not required to be backwards-compatible with
matlab the Octave function puts or disp can be used.
printf (strvar);
# Unsafe if strvar contains format codes
printf ("%s", strvar); # Safe
puts (strvar);
# Safe
If there aren’t enough function arguments provided to supply values for all the conversion
specifications in the template string, or if the arguments are not of the correct types, the
results are unpredictable. If you supply more arguments than conversion specifications, the
extra argument values are simply ignored; this is sometimes useful.

14.2.8 Integer Conversions
This section describes the options for the ‘%d’, ‘%i’, ‘%o’, ‘%u’, ‘%x’, and ‘%X’ conversion
specifications. These conversions print integers in various formats.
The ‘%d’ and ‘%i’ conversion specifications both print an numeric argument as a signed
decimal number; while ‘%o’, ‘%u’, and ‘%x’ print the argument as an unsigned octal, decimal,
or hexadecimal number (respectively). The ‘%X’ conversion specification is just like ‘%x’
except that it uses the characters ‘ABCDEF’ as digits instead of ‘abcdef’.
The following flags are meaningful:
‘-’

Left-justify the result in the field (instead of the normal right-justification).

‘+’

For the signed ‘%d’ and ‘%i’ conversions, print a plus sign if the value is positive.

‘’

For the signed ‘%d’ and ‘%i’ conversions, if the result doesn’t start with a plus
or minus sign, prefix it with a space character instead. Since the ‘+’ flag ensures
that the result includes a sign, this flag is ignored if you supply both of them.

‘#’

For the ‘%o’ conversion, this forces the leading digit to be ‘0’, as if by increasing
the precision. For ‘%x’ or ‘%X’, this prefixes a leading ‘0x’ or ‘0X’ (respectively) to
the result. This doesn’t do anything useful for the ‘%d’, ‘%i’, or ‘%u’ conversions.

‘0’

Pad the field with zeros instead of spaces. The zeros are placed after any
indication of sign or base. This flag is ignored if the ‘-’ flag is also specified, or
if a precision is specified.

Chapter 14: Input and Output

275

If a precision is supplied, it specifies the minimum number of digits to appear; leading
zeros are produced if necessary. If you don’t specify a precision, the number is printed with
as many digits as it needs. If you convert a value of zero with an explicit precision of zero,
then no characters at all are produced.

14.2.9 Floating-Point Conversions
This section discusses the conversion specifications for floating-point numbers: the ‘%f’,
‘%e’, ‘%E’, ‘%g’, and ‘%G’ conversions.
The ‘%f’ conversion prints its argument in fixed-point notation, producing output of the
form [-]ddd.ddd, where the number of digits following the decimal point is controlled by
the precision you specify.
The ‘%e’ conversion prints its argument in exponential notation, producing output of
the form [-]d.ddde[+|-]dd. Again, the number of digits following the decimal point is
controlled by the precision. The exponent always contains at least two digits. The ‘%E’
conversion is similar but the exponent is marked with the letter ‘E’ instead of ‘e’.
The ‘%g’ and ‘%G’ conversions print the argument in the style of ‘%e’ or ‘%E’ (respectively)
if the exponent would be less than -4 or greater than or equal to the precision; otherwise
they use the ‘%f’ style. Trailing zeros are removed from the fractional portion of the result
and a decimal-point character appears only if it is followed by a digit.
The following flags can be used to modify the behavior:
‘-’

Left-justify the result in the field. Normally the result is right-justified.

‘+’

Always include a plus or minus sign in the result.

‘’

If the result doesn’t start with a plus or minus sign, prefix it with a space
instead. Since the ‘+’ flag ensures that the result includes a sign, this flag is
ignored if you supply both of them.

‘#’

Specifies that the result should always include a decimal point, even if no digits
follow it. For the ‘%g’ and ‘%G’ conversions, this also forces trailing zeros after
the decimal point to be left in place where they would otherwise be removed.

‘0’

Pad the field with zeros instead of spaces; the zeros are placed after any sign.
This flag is ignored if the ‘-’ flag is also specified.

The precision specifies how many digits follow the decimal-point character for the ‘%f’,
‘%e’, and ‘%E’ conversions. For these conversions, the default precision is 6. If the precision
is explicitly 0, this suppresses the decimal point character entirely. For the ‘%g’ and ‘%G’
conversions, the precision specifies how many significant digits to print. Significant digits
are the first digit before the decimal point, and all the digits after it. If the precision is 0
or not specified for ‘%g’ or ‘%G’, it is treated like a value of 1. If the value being printed
cannot be expressed precisely in the specified number of digits, the value is rounded to the
nearest number that fits.

14.2.10 Other Output Conversions
This section describes miscellaneous conversions for printf.

276

GNU Octave

The ‘%c’ conversion prints a single character. The ‘-’ flag can be used to specify leftjustification in the field, but no other flags are defined, and no precision or type modifier
can be given. For example:
printf ("%c%c%c%c%c", "h", "e", "l", "l", "o");
prints ‘hello’.
The ‘%s’ conversion prints a string. The corresponding argument must be a string. A
precision can be specified to indicate the maximum number of characters to write; otherwise
characters in the string up to but not including the terminating null character are written
to the output stream. The ‘-’ flag can be used to specify left-justification in the field, but
no other flags or type modifiers are defined for this conversion. For example:
printf ("%3s%-6s", "no", "where");
prints ‘ nowhere ’ (note the leading and trailing spaces).

14.2.11 Formatted Input
Octave provides the scanf, fscanf, and sscanf functions to read formatted input. There
are two forms of each of these functions. One can be used to extract vectors of data from
a file, and the other is more ‘C-like’.

[val, count, errmsg] = fscanf (fid, template, size)
[v1, v2, ..., count, errmsg] = fscanf (fid, template, "C")
In the first form, read from fid according to template, returning the result in the
matrix val.
The optional argument size specifies the amount of data to read and may be one of
Inf

Read as much as possible, returning a column vector.

nr

Read up to nr elements, returning a column vector.

[nr, Inf] Read as much as possible, returning a matrix with nr rows. If the number
of elements read is not an exact multiple of nr, the last column is padded
with zeros.
[nr, nc]

Read up to nr * nc elements, returning a matrix with nr rows. If the
number of elements read is not an exact multiple of nr, the last column
is padded with zeros.

If size is omitted, a value of Inf is assumed.
A string is returned if template specifies only character conversions.
The number of items successfully read is returned in count.
If an error occurs, errmsg contains a system-dependent error message.
In the second form, read from fid according to template, with each conversion specifier
in template corresponding to a single scalar return value. This form is more “C-like”,
and also compatible with previous versions of Octave. The number of successful
conversions is returned in count
See the Formatted Input section of the GNU Octave manual for a complete description
of the syntax of the template string.
See also: [fgets], page 270, [fgetl], page 269, [fread], page 279, [scanf], page 277,
[sscanf], page 277, [fopen], page 267.

Chapter 14: Input and Output

277

[val, count, errmsg] = scanf (template, size)
[v1, v2, ..., count, errmsg] = scanf (template, "C")
This is equivalent to calling fscanf with fid = stdin.
It is currently not useful to call scanf in interactive programs.
See also: [fscanf], page 276, [sscanf], page 277, [printf], page 270.

[val, count, errmsg, pos] = sscanf (string, template, size)
[v1, v2, ..., count, errmsg] = sscanf (string, template, "C")
This is like fscanf, except that the characters are taken from the string string instead
of from a stream.
Reaching the end of the string is treated as an end-of-file condition. In addition to
the values returned by fscanf, the index of the next character to be read is returned
in pos.
See also: [fscanf], page 276, [scanf], page 277, [sprintf], page 271.
Calls to scanf are superficially similar to calls to printf in that arbitrary arguments are
read under the control of a template string. While the syntax of the conversion specifications
in the template is very similar to that for printf, the interpretation of the template is
oriented more towards free-format input and simple pattern matching, rather than fixedfield formatting. For example, most scanf conversions skip over any amount of “white
space” (including spaces, tabs, and newlines) in the input file, and there is no concept
of precision for the numeric input conversions as there is for the corresponding output
conversions. Ordinarily, non-whitespace characters in the template are expected to match
characters in the input stream exactly.
When a matching failure occurs, scanf returns immediately, leaving the first nonmatching character as the next character to be read from the stream, and scanf returns all
the items that were successfully converted.
The formatted input functions are not used as frequently as the formatted output functions. Partly, this is because it takes some care to use them properly. Another reason is
that it is difficult to recover from a matching error.

14.2.12 Input Conversion Syntax
A scanf template string is a string that contains ordinary multibyte characters interspersed
with conversion specifications that start with ‘%’.
Any whitespace character in the template causes any number of whitespace characters
in the input stream to be read and discarded. The whitespace characters that are matched
need not be exactly the same whitespace characters that appear in the template string. For
example, write ‘ , ’ in the template to recognize a comma with optional whitespace before
and after.
Other characters in the template string that are not part of conversion specifications
must match characters in the input stream exactly; if this is not the case, a matching
failure occurs.
The conversion specifications in a scanf template string have the general form:
% flags width type conversion

278

GNU Octave

In more detail, an input conversion specification consists of an initial ‘%’ character followed in sequence by:
• An optional flag character ‘*’, which says to ignore the text read for this specification.
When scanf finds a conversion specification that uses this flag, it reads input as directed
by the rest of the conversion specification, but it discards this input, does not return
any value, and does not increment the count of successful assignments.
• An optional decimal integer that specifies the maximum field width. Reading of characters from the input stream stops either when this maximum is reached or when a
non-matching character is found, whichever happens first. Most conversions discard
initial whitespace characters, and these discarded characters don’t count towards the
maximum field width. Conversions that do not discard initial whitespace are explicitly
documented.
• An optional type modifier character. This character is ignored by Octave’s scanf
function, but is recognized to provide compatibility with the C language scanf.
• A character that specifies the conversion to be applied.

The exact options that are permitted and how they are interpreted vary between the
different conversion specifiers. See the descriptions of the individual conversions for information about the particular options that they allow.

14.2.13 Table of Input Conversions
Here is a table that summarizes the various conversion specifications:
‘%d’

Matches an optionally signed integer written in decimal. See Section 14.2.14
[Numeric Input Conversions], page 279.

‘%i’

Matches an optionally signed integer in any of the formats that the C language
defines for specifying an integer constant. See Section 14.2.14 [Numeric Input
Conversions], page 279.

‘%o’

Matches an unsigned integer written in octal radix. See Section 14.2.14 [Numeric Input Conversions], page 279.

‘%u’

Matches an unsigned integer written in decimal radix. See Section 14.2.14
[Numeric Input Conversions], page 279.

‘%x’, ‘%X’

Matches an unsigned integer written in hexadecimal radix. See Section 14.2.14
[Numeric Input Conversions], page 279.

‘%e’, ‘%f’, ‘%g’, ‘%E’, ‘%G’
Matches an optionally signed floating-point number. See Section 14.2.14 [Numeric Input Conversions], page 279.
‘%s’

Matches a string containing only non-whitespace characters. See Section 14.2.15
[String Input Conversions], page 279.

‘%c’

Matches a string of one or more characters; the number of characters read is controlled by the maximum field width given for the conversion. See Section 14.2.15
[String Input Conversions], page 279.

‘%%’

This matches a literal ‘%’ character in the input stream. No corresponding
argument is used.

Chapter 14: Input and Output

279

If the syntax of a conversion specification is invalid, the behavior is undefined. If there
aren’t enough function arguments provided to supply addresses for all the conversion specifications in the template strings that perform assignments, or if the arguments are not of
the correct types, the behavior is also undefined. On the other hand, extra arguments are
simply ignored.

14.2.14 Numeric Input Conversions
This section describes the scanf conversions for reading numeric values.
The ‘%d’ conversion matches an optionally signed integer in decimal radix.
The ‘%i’ conversion matches an optionally signed integer in any of the formats that the
C language defines for specifying an integer constant.
For example, any of the strings ‘10’, ‘0xa’, or ‘012’ could be read in as integers under
the ‘%i’ conversion. Each of these specifies a number with decimal value 10.
The ‘%o’, ‘%u’, and ‘%x’ conversions match unsigned integers in octal, decimal, and hexadecimal radices, respectively.
The ‘%X’ conversion is identical to the ‘%x’ conversion. They both permit either uppercase
or lowercase letters to be used as digits.
Unlike the C language scanf, Octave ignores the ‘h’, ‘l’, and ‘L’ modifiers.

14.2.15 String Input Conversions
This section describes the scanf input conversions for reading string and character values:
‘%s’ and ‘%c’.
The ‘%c’ conversion is the simplest: it matches a fixed number of characters, always. The
maximum field with says how many characters to read; if you don’t specify the maximum,
the default is 1. This conversion does not skip over initial whitespace characters. It reads
precisely the next n characters, and fails if it cannot get that many.
The ‘%s’ conversion matches a string of non-whitespace characters. It skips and discards initial whitespace, but stops when it encounters more whitespace after having read
something.
For example, reading the input:
hello, world
with the conversion ‘%10c’ produces " hello, wo", but reading the same input with the
conversion ‘%10s’ produces "hello,".

14.2.16 Binary I/O
Octave can read and write binary data using the functions fread and fwrite, which are
patterned after the standard C functions with the same names. They are able to automatically swap the byte order of integer data and convert among the supported floating point
formats as the data are read.

val
val
val
val

=
=
=
=

fread
fread
fread
fread

(fid)
(fid, size)
(fid, size, precision)
(fid, size, precision, skip)

280

GNU Octave

val = fread (fid, size, precision, skip, arch)
[val, count] = fread ( . . . )
Read binary data from the file specified by the file descriptor fid.
The optional argument size specifies the amount of data to read and may be one of
Inf

Read as much as possible, returning a column vector.

nr

Read up to nr elements, returning a column vector.

[nr, Inf] Read as much as possible, returning a matrix with nr rows. If the number
of elements read is not an exact multiple of nr, the last column is padded
with zeros.
[nr, nc]

Read up to nr * nc elements, returning a matrix with nr rows. If the
number of elements read is not an exact multiple of nr, the last column
is padded with zeros.

If size is omitted, a value of Inf is assumed.
The optional argument precision is a string specifying the type of data to read and
may be one of
"schar"
"signed char"
Signed character.
"uchar"
"unsigned char"
Unsigned character.
"int8"
"integer*1"
8-bit signed integer.
"int16"
"integer*2"
16-bit signed integer.
"int32"
"integer*4"
32-bit signed integer.
"int64"
"integer*8"
64-bit signed integer.
"uint8"

8-bit unsigned integer.

"uint16"

16-bit unsigned integer.

"uint32"

32-bit unsigned integer.

"uint64"

64-bit unsigned integer.

"single"
"float32"
"real*4" 32-bit floating point number.

Chapter 14: Input and Output

281

"double"
"float64"
"real*8" 64-bit floating point number.
"char"
"char*1"

Single character.

"short"

Short integer (size is platform dependent).

"int"

Integer (size is platform dependent).

"long"

Long integer (size is platform dependent).

"ushort"
"unsigned short"
Unsigned short integer (size is platform dependent).
"uint"
"unsigned int"
Unsigned integer (size is platform dependent).
"ulong"
"unsigned long"
Unsigned long integer (size is platform dependent).
"float"

Single precision floating point number (size is platform dependent).

The default precision is "uchar".
The precision argument may also specify an optional repeat count. For example,
‘32*single’ causes fread to read a block of 32 single precision floating point numbers.
Reading in blocks is useful in combination with the skip argument.
The precision argument may also specify a type conversion.
For example,
‘int16=>int32’ causes fread to read 16-bit integer values and return an array of
32-bit integer values. By default, fread returns a double precision array. The special
form ‘*TYPE’ is shorthand for ‘TYPE=>TYPE’.
The conversion and repeat counts may be combined. For example, the specification
‘32*single=>single’ causes fread to read blocks of single precision floating point
values and return an array of single precision values instead of the default array of
double precision values.
The optional argument skip specifies the number of bytes to skip after each element
(or block of elements) is read. If it is not specified, a value of 0 is assumed. If the
final block read is not complete, the final skip is omitted. For example,
fread (f, 10, "3*single=>single", 8)
will omit the final 8-byte skip because the last read will not be a complete block of 3
values.
The optional argument arch is a string specifying the data format for the file. Valid
values are
"native" or "n"
The format of the current machine.

282

GNU Octave

"ieee-be" or "b"
IEEE big endian.
"ieee-le" or "l"
IEEE little endian.
If no arch is given the value used in the call to fopen which created the file descriptor is used. Otherwise, the value specified with fread overrides that of fopen and
determines the data format.
The output argument val contains the data read from the file.
The optional return value count contains the number of elements read.
See also: [fwrite], page 282, [fgets], page 270, [fgetl], page 269, [fscanf], page 276,
[fopen], page 267.

fwrite (fid, data)
fwrite (fid, data, precision)
fwrite (fid, data, precision, skip)
fwrite (fid, data, precision, skip, arch)
count = fwrite ( . . . )
Write data in binary form to the file specified by the file descriptor fid, returning the
number of values count successfully written to the file.
The argument data is a matrix of values that are to be written to the file. The values
are extracted in column-major order.
The remaining arguments precision, skip, and arch are optional, and are interpreted
as described for fread.
The behavior of fwrite is undefined if the values in data are too large to fit in the
specified precision.
See also: [fread], page 279, [fputs], page 269, [fprintf], page 271, [fopen], page 267.

14.2.17 Temporary Files
Sometimes one needs to write data to a file that is only temporary. This is most commonly
used when an external program launched from within Octave needs to access data. When
Octave exits all temporary files will be deleted, so this step need not be executed manually.

[fid, name, msg] = mkstemp ("template")
[fid, name, msg] = mkstemp ("template", delete)
Return the file descriptor fid corresponding to a new temporary file with a unique
name created from template.
The last six characters of template must be "XXXXXX" and these are replaced with a
string that makes the filename unique. The file is then created with mode read/write
and permissions that are system dependent (on GNU/Linux systems, the permissions
will be 0600 for versions of glibc 2.0.7 and later). The file is opened in binary mode
and with the O_EXCL flag.
If the optional argument delete is supplied and is true, the file will be deleted automatically when Octave exits.

Chapter 14: Input and Output

283

If successful, fid is a valid file ID, name is the name of the file, and msg is an empty
string. Otherwise, fid is -1, name is empty, and msg contains a system-dependent
error message.
See also: [tempname], page 283, [tempdir], page 283, [P tmpdir], page 283, [tmpfile],
page 283, [fopen], page 267.

[fid, msg] = tmpfile ()
Return the file ID corresponding to a new temporary file with a unique name.
The file is opened in binary read/write ("w+b") mode and will be deleted automatically
when it is closed or when Octave exits.
If successful, fid is a valid file ID and msg is an empty string. Otherwise, fid is -1
and msg contains a system-dependent error message.
See also: [tempname], page 283, [mkstemp], page 282, [tempdir], page 283,
[P tmpdir], page 283.

fname = tempname ()
fname = tempname (dir)
fname = tempname (dir, prefix)
Return a unique temporary filename as a string.
If prefix is omitted, a value of "oct-" is used.
If dir is also omitted, the default directory for temporary files (P_tmpdir) is used. If
dir is provided, it must exist, otherwise the default directory for temporary files is
used.
Programming Note: Because the named file is not opened by tempname, it is possible,
though relatively unlikely, that it will not be available by the time your program
attempts to open it. If this is a concern, see tmpfile.
See also: [mkstemp], page 282, [tempdir], page 283, [P tmpdir], page 283, [tmpfile],
page 283.

dir = tempdir ()
Return the name of the host system’s directory for temporary files.
The directory name is taken first from the environment variable TMPDIR. If that does
not exist the system default returned by P_tmpdir is used.
See also: [P tmpdir], page 283, [tempname], page 283, [mkstemp], page 282, [tmpfile],
page 283.

P_tmpdir ()
Return the name of the host system’s default directory for temporary files.
Programming Note: The value returned by P_tmpdir is always the default location.
This value may not agree with that returned from tempdir if the user has overridden
the default with the TMPDIR environment variable.
See also: [tempdir], page 283, [tempname], page 283, [mkstemp], page 282, [tmpfile],
page 283.

284

GNU Octave

14.2.18 End of File and Errors
Once a file has been opened its status can be acquired. As an example the feof functions
determines if the end of the file has been reached. This can be very useful when reading
small parts of a file at a time. The following example shows how to read one line at a time
from a file until the end has been reached.
filename = "myfile.txt";
fid = fopen (filename, "r");
while (! feof (fid) )
text_line = fgetl (fid);
endwhile
fclose (fid);
Note that in some situations it is more efficient to read the entire contents of a file and then
process it, than it is to read it line by line. This has the potential advantage of removing
the loop in the above code.

status = feof (fid)
Return 1 if an end-of-file condition has been encountered for the file specified by file
descriptor fid and 0 otherwise.
Note that feof will only return 1 if the end of the file has already been encountered,
not if the next read operation will result in an end-of-file condition.
See also: [fread], page 279, [frewind], page 285, [fseek], page 285, [fclear], page 284,
[fopen], page 267.

msg = ferror (fid)
[msg, err] = ferror (fid)
[...] = ferror (fid, "clear")
Query the error status of the stream specified by file descriptor fid
If an error condition exists then return a string msg describing the error. Otherwise,
return an empty string "".
The second input "clear" is optional. If supplied, the error state on the stream will
be cleared.
The optional second output is a numeric indication of the error status. err is 1 if an
error condition has been encountered and 0 otherwise.
Note that ferror indicates if an error has already occurred, not whether the next
operation will result in an error condition.
See also: [fclear], page 284, [fopen], page 267.

fclear (fid)
Clear the stream state for the file specified by the file descriptor fid.
See also: [ferror], page 284, [fopen], page 267.

freport ()
Print a list of which files have been opened, and whether they are open for reading,
writing, or both.

Chapter 14: Input and Output

285

For example:
freport ()
a
a
a
a
a
a

number
-----0
1
2
3

mode
---r
w
w
r

arch
---ieee-le
ieee-le
ieee-le
ieee-le

name
---stdin
stdout
stderr
myfile

See also: [fopen], page 267, [fclose], page 268, [is valid file id], page 268.

14.2.19 File Positioning
Three functions are available for setting and determining the position of the file pointer for
a given file.

pos = ftell (fid)
Return the position of the file pointer as the number of characters from the beginning
of the file specified by file descriptor fid.
See also: [fseek], page 285, [frewind], page 285, [feof], page 284, [fopen], page 267.

fseek (fid, offset)
fseek (fid, offset, origin)
status = fseek ( . . . )
Set the file pointer to the location offset within the file fid.
The pointer is positioned offset characters from the origin, which may be one of the
predefined variables SEEK_CUR (current position), SEEK_SET (beginning), or SEEK_END
(end of file) or strings "cof", "bof" or "eof". If origin is omitted, SEEK_SET is
assumed. offset may be positive, negative, or zero but not all combinations of origin
and offset can be realized.
fseek returns 0 on success and -1 on error.
See also: [fskipl], page 270, [frewind], page 285, [ftell], page 285, [fopen], page 267.

SEEK_SET ()
SEEK_CUR ()
SEEK_END ()
Return the numerical value to pass to fseek to perform one of the following actions:
SEEK_SET

Position file relative to the beginning.

SEEK_CUR

Position file relative to the current position.

SEEK_END

Position file relative to the end.

See also: [fseek], page 285.

frewind (fid)
status = frewind (fid)
Move the file pointer to the beginning of the file specified by file descriptor fid.

286

GNU Octave

frewind returns 0 for success, and -1 if an error is encountered. It is equivalent to
fseek (fid, 0, SEEK_SET).
See also: [fseek], page 285, [ftell], page 285, [fopen], page 267.
The following example stores the current file position in the variable marker, moves the
pointer to the beginning of the file, reads four characters, and then returns to the original
position.
marker = ftell (myfile);
frewind (myfile);
fourch = fgets (myfile, 4);
fseek (myfile, marker, SEEK_SET);

287

15 Plotting

15.1 Introduction to Plotting
Earlier versions of Octave provided plotting through the use of gnuplot. This capability is
still available. But, a newer plotting capability is provided by access to OpenGL. Which
plotting system is used is controlled by the graphics_toolkit function. See Section 15.4.7
[Graphics Toolkits], page 441.
The function call graphics_toolkit ("qt") selects the Qt/OpenGL system,
graphics_toolkit ("fltk") selects the FLTK/OpenGL system, and graphics_toolkit
("gnuplot") selects the gnuplot system. The three systems may be used selectively
through the use of the graphics_toolkit property of the graphics handle for each figure.
This is explained in Section 15.3 [Graphics Data Structures], page 376. Caution: The
OpenGL-based toolkits use single precision variables internally which limits the maximum
value that can be displayed to approximately 1038 . If your data contains larger values you
must use the gnuplot toolkit which supports values up to 10308 .

15.2 High-Level Plotting
Octave provides simple means to create many different types of two- and three-dimensional
plots using high-level functions.
If you need more detailed control, see Section 15.3 [Graphics Data Structures], page 376,
and Section 15.4 [Advanced Plotting], page 425.

15.2.1 Two-Dimensional Plots
The plot function allows you to create simple x-y plots with linear axes. For example,
x = -10:0.1:10;
plot (x, sin (x));
xlabel ("x");
ylabel ("sin (x)");
title ("Simple 2-D Plot");
displays a sine wave shown in Figure 15.1. On most systems, this command will open a
separate plot window to display the graph.

288

GNU Octave

Simple 2-D Plot
1

sin (x)

0.5

0

-0.5

-1
-10

-5

0

5

10

x

Figure 15.1: Simple Two-Dimensional Plot.

plot (y)
plot (x, y)
plot (x, y, fmt)
plot ( . . . , property, value, . . . )
plot (x1, y1, . . . , xn, yn)
plot (hax, . . . )
h = plot ( . . . )
Produce 2-D plots.
Many different combinations of arguments are possible. The simplest form is
plot (y)
where the argument is taken as the set of y coordinates and the x coordinates are
taken to be the range 1:numel (y).
If more than one argument is given, they are interpreted as
plot (y, property, value, ...)
or
plot (x, y, property, value, ...)
or
plot (x, y, fmt, ...)
and so on. Any number of argument sets may appear. The x and y values are
interpreted as follows:
• If a single data argument is supplied, it is taken as the set of y coordinates and
the x coordinates are taken to be the indices of the elements, starting with 1.
• If x and y are scalars, a single point is plotted.
• squeeze() is applied to arguments with more than two dimensions, but no more
than two singleton dimensions.

Chapter 15: Plotting

289

• If both arguments are vectors, the elements of y are plotted versus the elements
of x.
• If x is a vector and y is a matrix, then the columns (or rows) of y are plotted
versus x. (using whichever combination matches, with columns tried first.)
• If the x is a matrix and y is a vector, y is plotted versus the columns (or rows)
of x. (using whichever combination matches, with columns tried first.)
• If both arguments are matrices, the columns of y are plotted versus the columns
of x. In this case, both matrices must have the same number of rows and columns
and no attempt is made to transpose the arguments to make the number of rows
match.
Multiple property-value pairs may be specified, but they must appear in pairs.
These arguments are applied to the line objects drawn by plot. Useful properties
to modify are "linestyle", "linewidth", "color", "marker", "markersize",
"markeredgecolor", "markerfacecolor". See Section 15.3.3.4 [Line Properties],
page 399.
The fmt format argument can also be used to control the plot style. It is a string composed of four optional parts: "<;displayname;>". When a
marker is specified, but no linestyle, only the markers are plotted. Similarly, if a
linestyle is specified, but no marker, then only lines are drawn. If both are specified then lines and markers will be plotted. If no fmt and no property/value pairs
are given, then the default plot style is solid lines with no markers and the color
determined by the "colororder" property of the current axes.
Format arguments:
linestyle
‘-’
‘--’
‘:’
‘-.’

Use
Use
Use
Use

solid lines (default).
dashed lines.
dotted lines.
dash-dotted lines.

‘+’
‘o’
‘*’
‘.’
‘x’
‘s’
‘d’
‘^’
‘v’
‘>’
‘<’
‘p’
‘h’

crosshair
circle
star
point
cross
square
diamond
upward-facing triangle
downward-facing triangle
right-facing triangle
left-facing triangle
pentagram
hexagram

marker

color

290

GNU Octave

‘k’
‘r’
‘g’
‘b’
‘y’
‘m’
‘c’
‘w’

blacK
Red
Green
Blue
Yellow
Magenta
Cyan
White

";displayname;"
Here "displayname" is the label to use for the plot legend.
The fmt argument may also be used to assign legend labels. To do so, include the
desired label between semicolons after the formatting sequence described above, e.g.,
"+b;Key Title;". Note that the last semicolon is required and Octave will generate
an error if it is left out.
Here are some plot examples:
plot (x, y, "or", x, y2, x, y3, "m", x, y4, "+")
This command will plot y with red circles, y2 with solid lines, y3 with solid magenta
lines, and y4 with points displayed as ‘+’.
plot (b, "*", "markersize", 10)
This command will plot the data in the variable b, with points displayed as ‘*’ and a
marker size of 10.
t = 0:0.1:6.3;
plot (t, cos(t), "-;cos(t);", t, sin(t), "-b;sin(t);");
This will plot the cosine and sine functions and label them accordingly in the legend.
If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.
The optional return value h is a vector of graphics handles to the created line objects.
To save a plot, in one of several image formats such as PostScript or PNG, use the
print command.
See also: [axis], page 315, [box], page 352, [grid], page 353, [hold], page 361, [legend],
page 350, [title], page 350, [xlabel], page 351, [ylabel], page 351, [xlim], page 317,
[ylim], page 317, [ezplot], page 319, [errorbar], page 305, [fplot], page 318, [line],
page 378, [plot3], page 339, [polar], page 309, [loglog], page 292, [semilogx], page 291,
[semilogy], page 291, [subplot], page 356.
The plotyy function may be used to create a plot with two independent y axes.

plotyy (x1, y1, x2, y2)
plotyy ( . . . , fun)
plotyy ( . . . , fun1, fun2)
plotyy (hax, . . . )
[ax, h1, h2] = plotyy ( . . . )
Plot two sets of data with independent y-axes and a common x-axis.

Chapter 15: Plotting

291

The arguments x1 and y1 define the arguments for the first plot and x1 and y2 for
the second.
By default the arguments are evaluated with feval (@plot, x, y). However the type
of plot can be modified with the fun argument, in which case the plots are generated
by feval (fun, x, y). fun can be a function handle, an inline function, or a string
of a function name.
The function to use for each of the plots can be independently defined with fun1 and
fun2.
If the first argument hax is an axes handle, then it defines the principal axes in which
to plot the x1 and y1 data.
The return value ax is a vector with the axes handles of the two y-axes. h1 and h2
are handles to the objects generated by the plot commands.
x = 0:0.1:2*pi;
y1 = sin (x);
y2 = exp (x - 1);
ax = plotyy (x, y1, x - 1, y2, @plot, @semilogy);
xlabel ("X");
ylabel (ax(1), "Axis 1");
ylabel (ax(2), "Axis 2");
See also: [plot], page 288.
The functions semilogx, semilogy, and loglog are similar to the plot function, but
produce plots in which one or both of the axes use log scales.

semilogx (y)
semilogx (x, y)
semilogx (x, y, property, value, . . . )
semilogx (x, y, fmt)
semilogx (hax, . . . )
h = semilogx ( . . . )
Produce a 2-D plot using a logarithmic scale for the x-axis.
See the documentation of plot for a description of the arguments that semilogx will
accept.
If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.
The optional return value h is a graphics handle to the created plot.
See also: [plot], page 288, [semilogy], page 291, [loglog], page 292.

semilogy (y)
semilogy (x, y)
semilogy (x, y, property, value, . . . )
semilogy (x, y, fmt)
semilogy (h, . . . )
h = semilogy ( . . . )
Produce a 2-D plot using a logarithmic scale for the y-axis.

292

GNU Octave

See the documentation of plot for a description of the arguments that semilogy will
accept.
If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.
The optional return value h is a graphics handle to the created plot.
See also: [plot], page 288, [semilogx], page 291, [loglog], page 292.

loglog (y)
loglog (x, y)
loglog (x, y, prop, value, . . . )
loglog (x, y, fmt)
loglog (hax, . . . )
h = loglog ( . . . )
Produce a 2-D plot using logarithmic scales for both axes.
See the documentation of plot for a description of the arguments that loglog will
accept.
If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.
The optional return value h is a graphics handle to the created plot.
See also: [plot], page 288, [semilogx], page 291, [semilogy], page 291.

The functions bar, barh, stairs, and stem are useful for displaying discrete data. For
example,
randn ("state", 1);
hist (randn (10000, 1), 30);
xlabel ("Value");
ylabel ("Count");
title ("Histogram of 10,000 normally distributed random numbers");
produces the histogram of 10,000 normally distributed random numbers shown in
Figure 15.2. Note that, randn ("state", 1);, initializes the random number generator
for randn to a known value so that the returned values are reproducible; This guarantees
that the figure produced is identical to the one in this manual.

Chapter 15: Plotting

293

Histogram of 10,000 normally distributed random numbers
1000

Count

800

600

400

200

0
-4

-2

0

2

4

Value

Figure 15.2: Histogram.

bar
bar
bar
bar
bar
bar
h =

(y)
(x, y)
( . . . , w)
( . . . , style)
( . . . , prop, val, . . . )
(hax, . . . )
bar ( . . . , prop, val, . . . )
Produce a bar graph from two vectors of X-Y data.
If only one argument is given, y, it is taken as a vector of Y values and the X
coordinates are the range 1:numel (y).
The optional input w controls the width of the bars. A value of 1.0 will cause each
bar to exactly touch any adjacent bars. The default width is 0.8.
If y is a matrix, then each column of y is taken to be a separate bar graph plotted
on the same graph. By default the columns are plotted side-by-side. This behavior
can be changed by the style argument which can take the following values:
"grouped" (default)
Side-by-side bars with a gap between bars and centered over the Xcoordinate.
"stacked"
Bars are stacked so that each X value has a single bar composed of multiple segments.
"hist"

Side-by-side bars with no gap between bars and centered over the Xcoordinate.

"histc"

Side-by-side bars with no gap between bars and left-aligned to the Xcoordinate.

294

GNU Octave

Optional property/value pairs are passed directly to the underlying patch objects.
If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.
The optional return value h is a vector of handles to the created "bar series" hggroups
with one handle per column of the variable y. This series makes it possible to change
a common element in one bar series object and have the change reflected in the other
"bar series". For example,
h = bar (rand (5, 10));
set (h(1), "basevalue", 0.5);
changes the position on the base of all of the bar series.
The following example modifies the face and edge colors using property/value pairs.
bar (randn (1, 100), "facecolor", "r", "edgecolor", "b");
The color of the bars is taken from the figure’s colormap, such that
bar (rand (10, 3));
colormap (summer (64));
will change the colors used for the bars. The color of bars can also be set manually
using the "facecolor" property as shown below.
h =
set
set
set

bar (rand (10, 3));
(h(1), "facecolor", "r")
(h(2), "facecolor", "g")
(h(3), "facecolor", "b")

See also: [barh], page 294, [hist], page 295, [pie], page 309, [plot], page 288, [patch],
page 379.

barh (y)
barh (x, y)
barh ( . . . , w)
barh ( . . . , style)
barh ( . . . , prop, val, . . . )
barh (hax, . . . )
h = barh ( . . . , prop, val, . . . )
Produce a horizontal bar graph from two vectors of X-Y data.
If only one argument is given, it is taken as a vector of Y values and the X coordinates
are the range 1:numel (y).
The optional input w controls the width of the bars. A value of 1.0 will cause each
bar to exactly touch any adjacent bars. The default width is 0.8.
If y is a matrix, then each column of y is taken to be a separate bar graph plotted
on the same graph. By default the columns are plotted side-by-side. This behavior
can be changed by the style argument which can take the following values:
"grouped" (default)
Side-by-side bars with a gap between bars and centered over the Ycoordinate.

Chapter 15: Plotting

295

"stacked"
Bars are stacked so that each Y value has a single bar composed of multiple segments.
"hist"

Side-by-side bars with no gap between bars and centered over the Ycoordinate.

"histc"

Side-by-side bars with no gap between bars and left-aligned to the Ycoordinate.

Optional property/value pairs are passed directly to the underlying patch objects.
If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.
The optional return value h is a graphics handle to the created bar series hggroup.
For a description of the use of the bar series, see [bar], page 293.
See also: [bar], page 293, [hist], page 295, [pie], page 309, [plot], page 288, [patch],
page 379.

hist
hist
hist
hist
hist
hist
[nn,

(y)
(y, x)
(y, nbins)
(y, x, norm)
( . . . , prop, val, . . . )
(hax, . . . )
xx] = hist ( . . . )
Produce histogram counts or plots.
With one vector input argument, y, plot a histogram of the values with 10 bins. The
range of the histogram bins is determined by the range of the data. With one matrix
input argument, y, plot a histogram where each bin contains a bar per input column.
Given a second vector argument, x, use that as the centers of the bins, with the width
of the bins determined from the adjacent values in the vector.
If scalar, the second argument, nbins, defines the number of bins.
If a third argument is provided, the histogram is normalized such that the sum of the
bars is equal to norm.
Extreme values are lumped into the first and last bins.
The histogram’s appearance may be modified by specifying property/value pairs. For
example the face and edge color may be modified.
hist (randn (1, 100), 25, "facecolor", "r", "edgecolor", "b");
The histogram’s colors also depend upon the current colormap.
hist (rand (10, 3));
colormap (summer ());
If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.
With two output arguments, produce the values nn (numbers of elements) and xx
(bin centers) such that bar (xx, nn) will plot the histogram.
See also: [histc], page 656, [bar], page 293, [pie], page 309, [rose], page 301.

296

GNU Octave

stemleaf (x, caption)
stemleaf (x, caption, stem_sz)
plotstr = stemleaf ( . . . )
Compute and display a stem and leaf plot of the vector x.
The input x should be a vector of integers. Any non-integer values will be converted
to integer by x = fix (x). By default each element of x will be plotted with the last
digit of the element as a leaf value and the remaining digits as the stem. For example,
123 will be plotted with the stem ‘12’ and the leaf ‘3’. The second argument, caption,
should be a character array which provides a description of the data. It is included
as a heading for the output.
The optional input stem sz sets the width of each stem. The stem width is determined
by 10^(stem_sz + 1). The default stem width is 10.
The output of stemleaf is composed of two parts: a "Fenced Letter Display," followed
by the stem-and-leaf plot itself. The Fenced Letter Display is described in Exploratory
Data Analysis. Briefly, the entries are as shown:
Fenced Letter Display
#% nx|___________________
M% mi|
md
|
H% hi|hl
hu| hs
1
|x(1)
x(nx)|
_______
______|step |_______
f|ifl
ifh|
|nfl
nfh|
F|ofl
ofh|
|nFl
nFh|

nx = numel (x)
mi median index, md median
hi lower hinge index, hl,hu hinges,
hs h_spreadx(1), x(nx) first
and last data value.
step 1.5*h_spread
inner fence, lower and higher
no.\ of data points within fences
outer fence, lower and higher
no.\ of data points outside outer
fences

The stem-and-leaf plot shows on each line the stem value followed by the string made
up of the leaf digits. If the stem sz is not 1 the successive leaf values are separated
by ",".
With no return argument, the plot is immediately displayed. If an output argument
is provided, the plot is returned as an array of strings.
The leaf digits are not sorted. If sorted leaf values are desired, use xs = sort (x)
before calling stemleaf (xs).
The stem and leaf plot and associated displays are described in: Chapter 3, Exploratory Data Analysis by J. W. Tukey, Addison-Wesley, 1977.
See also: [hist], page 295, [printd], page 296.

printd (obj, filename)
out_file = printd ( . . . )
Convert any object acceptable to disp into the format selected by the suffix of
filename.
If the return argument out file is given, the name of the created file is returned.

Chapter 15: Plotting

297

This function is intended to facilitate manipulation of the output of functions such
as stemleaf.
See also: [stemleaf], page 296.

stairs (y)
stairs (x, y)
stairs ( . . . , style)
stairs ( . . . , prop, val, . . . )
stairs (hax, . . . )
h = stairs ( . . . )
[xstep, ystep] = stairs ( . . . )
Produce a stairstep plot.
The arguments x and y may be vectors or matrices. If only one argument is given, it
is taken as a vector of Y values and the X coordinates are taken to be the indices of
the elements (x = 1:numel (y)).
The style to use for the plot can be defined with a line style style of the same format
as the plot command.
Multiple property/value pairs may be specified, but they must appear in pairs.
If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.
If one output argument is requested, return a graphics handle to the created plot.
If two output arguments are specified, the data are generated but not plotted. For
example,
stairs (x, y);
and
[xs, ys] = stairs (x, y);
plot (xs, ys);
are equivalent.
See also: [bar], page 293, [hist], page 295, [plot], page 288, [stem], page 297.

stem (y)
stem (x, y)
stem ( . . . , linespec)
stem ( . . . , "filled")
stem ( . . . , prop, val, . . . )
stem (hax, . . . )
h = stem ( . . . )
Plot a 2-D stem graph.
If only one argument is given, it is taken as the y-values and the x-coordinates are
taken from the indices of the elements.
If y is a matrix, then each column of the matrix is plotted as a separate stem graph.
In this case x can either be a vector, the same length as the number of rows in y, or
it can be a matrix of the same size as y.

298

GNU Octave

The default color is "b" (blue), the default line style is "-", and the default marker
is "o". The line style can be altered by the linespec argument in the same manner
as the plot command. If the "filled" argument is present the markers at the top
of the stems will be filled in. For example,
x = 1:10;
y = 2*x;
stem (x, y, "r");
plots 10 stems with heights from 2 to 20 in red;
Optional property/value pairs may be specified to control the appearance of the plot.
If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.
The optional return value h is a handle to a "stem series" hggroup. The single
hggroup handle has all of the graphical elements comprising the plot as its children;
This allows the properties of multiple graphics objects to be changed by modifying
just a single property of the "stem series" hggroup.
For example,
x =
y =
h =
set
set

[0:10]’;
[sin(x), cos(x)]
stem (x, y);
(h(2), "color", "g");
(h(1), "basevalue", -1)

changes the color of the second "stem series" and moves the base line of the first.
Stem Series Properties
linestyle

The linestyle of the stem. (Default: "-")

linewidth

The width of the stem. (Default: 0.5)

color

The color of the stem, and if not separately specified, the marker. (Default: "b" [blue])

marker

The marker symbol to use at the top of each stem. (Default: "o")

markeredgecolor
The edge color of the marker. (Default: "color" property)
markerfacecolor
The color to use for "filling" the marker. (Default: "none" [unfilled])
markersize
The size of the marker. (Default: 6)
baseline

The handle of the line object which implements the baseline. Use set
with the returned handle to change graphic properties of the baseline.

basevalue

The y-value where the baseline is drawn. (Default: 0)

See also: [stem3], page 299, [bar], page 293, [hist], page 295, [plot], page 288, [stairs],
page 297.

Chapter 15: Plotting

299

stem3 (x, y, z)
stem3 ( . . . , linespec)
stem3 ( . . . , "filled")
stem3 ( . . . , prop, val, . . . )
stem3 (hax, . . . )
h = stem3 ( . . . )
Plot a 3-D stem graph.
Stems are drawn from the height z to the location in the x-y plane determined by x
and y. The default color is "b" (blue), the default line style is "-", and the default
marker is "o".
The line style can be altered by the linespec argument in the same manner as the
plot command. If the "filled" argument is present the markers at the top of the
stems will be filled in.
Optional property/value pairs may be specified to control the appearance of the plot.
If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.
The optional return value h is a handle to the "stem series" hggroup containing the
line and marker objects used for the plot. See [stem], page 297, for a description of
the "stem series" object.
Example:
theta = 0:0.2:6;
stem3 (cos (theta), sin (theta), theta);
plots 31 stems with heights from 0 to 6 lying on a circle.
Implementation Note: Color definitions with RGB-triples are not valid.
See also: [stem], page 297, [bar], page 293, [hist], page 295, [plot], page 288.

scatter (x, y)
scatter (x, y, s)
scatter (x, y, s, c)
scatter ( . . . , style)
scatter ( . . . , "filled")
scatter ( . . . , prop, val, . . . )
scatter (hax, . . . )
h = scatter ( . . . )
Draw a 2-D scatter plot.
A marker is plotted at each point defined by the coordinates in the vectors x and y.
The size of the markers is determined by s, which can be a scalar or a vector of the
same length as x and y. If s is not given, or is an empty matrix, then a default value
of 36 square points is used (The marker size itself is sqrt (s)).
The color of the markers is determined by c, which can be a string defining a fixed
color; a 3-element vector giving the red, green, and blue components of the color; a
vector of the same length as x that gives a scaled index into the current colormap; or
an Nx3 matrix defining the RGB color of each marker individually.

300

GNU Octave

The marker to use can be changed with the style argument; it is a string defining a
marker in the same manner as the plot command. If no marker is specified it defaults
to "o" or circles. If the argument "filled" is given then the markers are filled.
Additional property/value pairs are passed directly to the underlying patch object.
If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.
The optional return value h is a graphics handle to the created scatter object.
Example:
x = randn (100, 1);
y = randn (100, 1);
scatter (x, y, [], sqrt (x.^2 + y.^2));
See also: [scatter3], page 342, [patch], page 379, [plot], page 288.

plotmatrix (x, y)
plotmatrix (x)
plotmatrix ( . . . , style)
plotmatrix (hax, . . . )
[h, ax, bigax, p, pax] = plotmatrix ( . . . )
Scatter plot of the columns of one matrix against another.
Given the arguments x and y that have a matching number of rows, plotmatrix
plots a set of axes corresponding to
plot (x(:, i), y(:, j))
When called with a single argument x this is equivalent to
plotmatrix (x, x)
except that the diagonal of the set of axes will be replaced with the histogram hist
(x(:, i)).
The marker to use can be changed with the style argument, that is a string defining
a marker in the same manner as the plot command.
If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.
The optional return value h provides handles to the individual graphics objects in the
scatter plots, whereas ax returns the handles to the scatter plot axes objects.
bigax is a hidden axes object that surrounds the other axes, such that the commands
xlabel, title, etc., will be associated with this hidden axes.
Finally, p returns the graphics objects associated with the histogram and pax the
corresponding axes objects.
Example:
plotmatrix (randn (100, 3), "g+")
See also: [scatter], page 299, [plot], page 288.

pareto (y)
pareto (y, x)
pareto (hax, . . . )

Chapter 15: Plotting

301

h = pareto ( . . . )
Draw a Pareto chart.
A Pareto chart is a bar graph that arranges information in such a way that priorities
for process improvement can be established; It organizes and displays information to
show the relative importance of data. The chart is similar to the histogram or bar
chart, except that the bars are arranged in decreasing magnitude from left to right
along the x-axis.
The fundamental idea (Pareto principle) behind the use of Pareto diagrams is that the
majority of an effect is due to a small subset of the causes. For quality improvement,
the first few contributing causes (leftmost bars as presented on the diagram) to a
problem usually account for the majority of the result. Thus, targeting these "major
causes" for elimination results in the most cost-effective improvement scheme.
Typically only the magnitude data y is present in which case x is taken to be the
range 1 : length (y). If x is given it may be a string array, a cell array of strings,
or a numerical vector.
If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.
The optional return value h is a 2-element vector with a graphics handle for the
created bar plot and a second handle for the created line plot.
An example of the use of pareto is
Cheese = {"Cheddar", "Swiss", "Camembert", ...
"Munster", "Stilton", "Blue"};
Sold = [105, 30, 70, 10, 15, 20];
pareto (Sold, Cheese);
See also: [bar], page 293, [barh], page 294, [hist], page 295, [pie], page 309, [plot],
page 288.

rose (th)
rose (th, nbins)
rose (th, bins)
rose (hax, . . . )
h = rose ( . . . )
[thout rout] = rose ( . . . )
Plot an angular histogram.
With one vector argument, th, plot the histogram with 20 angular bins. If th is a
matrix then each column of th produces a separate histogram.
If nbins is given and is a scalar, then the histogram is produced with nbin bins. If
bins is a vector, then the center of each bin is defined by the values in bins and the
number of bins is given by the number of elements in bins.
If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.
The optional return value h is a vector of graphics handles to the line objects representing each histogram.
If two output arguments are requested then no plot is made and the polar vectors
necessary to plot the histogram are returned instead.

302

GNU Octave

Example
[th, r] = rose ([2*randn(1e5,1), pi + 2*randn(1e5,1)]);
polar (th, r);
Programming Note: When specifying bin centers with the bins input, the edges for
bins 2 to N-1 are spaced so that bins(i) is centered between the edges. The final
edge is drawn halfway between bin N and bin 1. This guarantees that all input th
will be placed into one of the bins, but also means that for some combinations bin 1
and bin N may not be centered on the user’s given values.
See also: [hist], page 295, [polar], page 309.
The contour, contourf and contourc functions produce two-dimensional contour plots
from three-dimensional data.

contour (z)
contour (z, vn)
contour (x, y, z)
contour (x, y, z, vn)
contour ( . . . , style)
contour (hax, . . . )
[c, h] = contour ( . . . )
Create a 2-D contour plot.
Plot level curves (contour lines) of the matrix z, using the contour matrix c computed
by contourc from the same arguments; see the latter for their interpretation.
The appearance of contour lines can be defined with a line style style in the same
manner as plot. Only line style and color are used; Any markers defined by style are
ignored.
If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.
The optional output c contains the contour levels in contourc format.
The optional return value h is a graphics handle to the hggroup comprising the contour
lines.
Example:
x = 0:2;
y = x;
z = x’ * y;
contour (x, y, z, 2:3)
See also: [ezcontour], page 320, [contourc], page 303, [contourf], page 302, [contour3],
page 304, [clabel], page 352, [meshc], page 324, [surfc], page 326, [caxis], page 317,
[colormap], page 757, [plot], page 288.

contourf
contourf
contourf
contourf
contourf

(z)
(z, vn)
(x, y, z)
(x, y, z, vn)
( . . . , style)

Chapter 15: Plotting

303

contourf (hax, . . . )
[c, h] = contourf ( . . . )
Create a 2-D contour plot with filled intervals.
Plot level curves (contour lines) of the matrix z and fill the region between lines with
colors from the current colormap.
The level curves are taken from the contour matrix c computed by contourc for the
same arguments; see the latter for their interpretation.
The appearance of contour lines can be defined with a line style style in the same
manner as plot. Only line style and color are used; Any markers defined by style are
ignored.
If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.
The optional output c contains the contour levels in contourc format.
The optional return value h is a graphics handle to the hggroup comprising the contour
lines.
The following example plots filled contours of the peaks function.
[x, y, z] = peaks (50);
contourf (x, y, z, -7:9)
See also: [ezcontourf], page 320, [contour], page 302, [contourc], page 303, [contour3],
page 304, [clabel], page 352, [meshc], page 324, [surfc], page 326, [caxis], page 317,
[colormap], page 757, [plot], page 288.

[c,
[c,
[c,
[c,

(z)
(z, vn)
(x, y, z)
(x, y, z, vn)
Compute contour lines (isolines of constant Z value).

lev]
lev]
lev]
lev]

=
=
=
=

contourc
contourc
contourc
contourc

The matrix z contains height values above the rectangular grid determined by x and
y. If only a single input z is provided then x is taken to be 1:columns (z) and y is
taken to be 1:rows (z).
The optional input vn is either a scalar denoting the number of contour lines to
compute or a vector containing the Z values where lines will be computed. When vn
is a vector the number of contour lines is numel (vn). However, to compute a single
contour line at a given value use vn = [val, val]. If vn is omitted it defaults to 10.
The return value c is a 2xn matrix containing the contour lines in the following format
c = [lev1, x1, x2, ..., levn, x1, x2, ...
len1, y1, y2, ..., lenn, y1, y2, ...]
in which contour line n has a level (height) of levn and length of lenn.
The optional return value lev is a vector with the Z values of the contour levels.
Example:

304

GNU Octave

x = 0:2;
y = x;
z = x’ * y;
contourc (x, y, z, 2:3)
⇒
2.0000
2.0000
2.0000
1.0000

1.0000
2.0000

3.0000
2.0000

1.5000
2.0000

2.0000
1.5000

See also: [contour], page 302, [contourf], page 302, [contour3], page 304, [clabel],
page 352.
(z)
(z, vn)
(x, y, z)
(x, y, z, vn)
( . . . , style)
(hax, . . . )
contour3 ( . . . )
Create a 3-D contour plot.

contour3
contour3
contour3
contour3
contour3
contour3
[c, h] =

contour3 plots level curves (contour lines) of the matrix z at a Z level corresponding
to each contour. This is in contrast to contour which plots all of the contour lines at
the same Z level and produces a 2-D plot.
The level curves are taken from the contour matrix c computed by contourc for the
same arguments; see the latter for their interpretation.
The appearance of contour lines can be defined with a line style style in the same
manner as plot. Only line style and color are used; Any markers defined by style are
ignored.
If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.
The optional output c are the contour levels in contourc format.
The optional return value h is a graphics handle to the hggroup comprising the contour
lines.
Example:
contour3 (peaks (19));
colormap cool;
hold on;
surf (peaks (19), "facecolor", "none", "edgecolor", "black");
See also: [contour], page 302, [contourc], page 303, [contourf], page 302, [clabel],
page 352, [meshc], page 324, [surfc], page 326, [caxis], page 317, [colormap], page 757,
[plot], page 288.
The errorbar, semilogxerr, semilogyerr, and loglogerr functions produce plots
with error bar markers. For example,

Chapter 15: Plotting

305

rand ("state", 2);
x = 0:0.1:10;
y = sin (x);
lerr = 0.1 .* rand (size (x));
uerr = 0.1 .* rand (size (x));
errorbar (x, y, lerr, uerr);
axis ([0, 10, -1.1, 1.1]);
xlabel ("x");
ylabel ("sin (x)");
title ("Errorbar plot of sin (x)");
produces the figure shown in Figure 15.3.
Errorbar plot of sin (x)
1

sin (x)

0.5

0

-0.5

-1
0

2

4

6

8

10

x

Figure 15.3: Errorbar plot.

errorbar (y, ey)
errorbar (y, . . . , fmt)
errorbar (x, y, ey)
errorbar (x, y, err, fmt)
errorbar (x, y, lerr, uerr, fmt)
errorbar (x, y, ex, ey, fmt)
errorbar (x, y, lx, ux, ly, uy, fmt)
errorbar (x1, y1, . . . , fmt, xn, yn, . . . )
errorbar (hax, . . . )
h = errorbar ( . . . )
Create a 2-D plot with errorbars.
Many different combinations of arguments are possible. The simplest form is
errorbar (y, ey)

306

GNU Octave

where the first argument is taken as the set of y coordinates, the second argument ey
are the errors around the y values, and the x coordinates are taken to be the indices
of the elements (1:numel (y)).
The general form of the function is
errorbar (x, y, err1, ..., fmt, ...)
After the x and y arguments there can be 1, 2, or 4 parameters specifying the error
values depending on the nature of the error values and the plot format fmt.
err (scalar)
When the error is a scalar all points share the same error value. The
errorbars are symmetric and are drawn from data-err to data+err. The
fmt argument determines whether err is in the x-direction, y-direction
(default), or both.
err (vector or matrix)
Each data point has a particular error value. The errorbars are symmetric
and are drawn from data(n)-err(n) to data(n)+err(n).
lerr, uerr (scalar)
The errors have a single low-side value and a single upper-side value. The
errorbars are not symmetric and are drawn from data-lerr to data+uerr.
lerr, uerr (vector or matrix)
Each data point has a low-side error and an upper-side error. The
errorbars are not symmetric and are drawn from data(n)-lerr(n) to
data(n)+uerr(n).
Any number of data sets (x1,y1, x2,y2, . . . ) may appear as long as they are separated
by a format string fmt.
If y is a matrix, x and the error parameters must also be matrices having the same
dimensions. The columns of y are plotted versus the corresponding columns of x and
errorbars are taken from the corresponding columns of the error parameters.
If fmt is missing, the yerrorbars ("~") plot style is assumed.
If the fmt argument is supplied then it is interpreted, as in normal plots, to specify
the line style, marker, and color. In addition, fmt may include an errorbar style which
must precede the ordinary format codes. The following errorbar styles are supported:
‘~’

Set yerrorbars plot style (default).

‘>’

Set xerrorbars plot style.

‘~>’

Set xyerrorbars plot style.

‘#~’

Set yboxes plot style.

‘#’

Set xboxes plot style.

‘#~>’

Set xyboxes plot style.

If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.

Chapter 15: Plotting

307

The optional return value h is a handle to the hggroup object representing the data
plot and errorbars.
Note: For compatibility with matlab a line is drawn through all data points. However, most scientific errorbar plots are a scatter plot of points with errorbars. To
accomplish this, add a marker style to the fmt argument such as ".". Alternatively, remove the line by modifying the returned graphic handle with set (h, "linestyle",
"none").
Examples:
errorbar (x, y, ex, ">.r")
produces an xerrorbar plot of y versus x with x errorbars drawn from x-ex to x+ex.
The marker "." is used so no connecting line is drawn and the errorbars appear in
red.
errorbar (x, y1, ey, "~",
x, y2, ly, uy)
produces yerrorbar plots with y1 and y2 versus x. Errorbars for y1 are drawn from
y1-ey to y1+ey, errorbars for y2 from y2-ly to y2+uy.
errorbar (x, y, lx, ux,
ly, uy, "~>")
produces an xyerrorbar plot of y versus x in which x errorbars are drawn from x-lx
to x+ux and y errorbars from y-ly to y+uy.
See also: [semilogxerr], page 307, [semilogyerr], page 308, [loglogerr], page 308, [plot],
page 288.

semilogxerr (y, ey)
semilogxerr (y, . . . , fmt)
semilogxerr (x, y, ey)
semilogxerr (x, y, err, fmt)
semilogxerr (x, y, lerr, uerr, fmt)
semilogxerr (x, y, ex, ey, fmt)
semilogxerr (x, y, lx, ux, ly, uy, fmt)
semilogxerr (x1, y1, . . . , fmt, xn, yn, . . . )
semilogxerr (hax, . . . )
h = semilogxerr ( . . . )
Produce 2-D plots using a logarithmic scale for the x-axis and errorbars at each data
point.
Many different combinations of arguments are possible. The most common form is
semilogxerr (x, y, ey, fmt)
which produces a semi-logarithmic plot of y versus x with errors in the y-scale defined
by ey and the plot format defined by fmt. See [errorbar], page 305, for available
formats and additional information.
If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.
See also: [errorbar], page 305, [semilogyerr], page 308, [loglogerr], page 308.

308

GNU Octave

semilogyerr (y, ey)
semilogyerr (y, . . . , fmt)
semilogyerr (x, y, ey)
semilogyerr (x, y, err, fmt)
semilogyerr (x, y, lerr, uerr, fmt)
semilogyerr (x, y, ex, ey, fmt)
semilogyerr (x, y, lx, ux, ly, uy, fmt)
semilogyerr (x1, y1, . . . , fmt, xn, yn, . . . )
semilogyerr (hax, . . . )
h = semilogyerr ( . . . )
Produce 2-D plots using a logarithmic scale for the y-axis and errorbars at each data
point.
Many different combinations of arguments are possible. The most common form is
semilogyerr (x, y, ey, fmt)
which produces a semi-logarithmic plot of y versus x with errors in the y-scale defined
by ey and the plot format defined by fmt. See [errorbar], page 305, for available
formats and additional information.
If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.
See also: [errorbar], page 305, [semilogxerr], page 307, [loglogerr], page 308.

loglogerr (y, ey)
loglogerr (y, . . . , fmt)
loglogerr (x, y, ey)
loglogerr (x, y, err, fmt)
loglogerr (x, y, lerr, uerr, fmt)
loglogerr (x, y, ex, ey, fmt)
loglogerr (x, y, lx, ux, ly, uy, fmt)
loglogerr (x1, y1, . . . , fmt, xn, yn, . . . )
loglogerr (hax, . . . )
h = loglogerr ( . . . )
Produce 2-D plots on a double logarithm axis with errorbars.
Many different combinations of arguments are possible. The most common form is
loglogerr (x, y, ey, fmt)
which produces a double logarithm plot of y versus x with errors in the y-scale defined
by ey and the plot format defined by fmt. See [errorbar], page 305, for available
formats and additional information.
If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.
See also: [errorbar], page 305, [semilogxerr], page 307, [semilogyerr], page 308.
Finally, the polar function allows you to easily plot data in polar coordinates. However,
the display coordinates remain rectangular and linear. For example,
polar (0:0.1:10*pi, 0:0.1:10*pi);
title ("Example polar plot from 0 to 10*pi");

Chapter 15: Plotting

309

produces the spiral plot shown in Figure 15.4.
Example polar plot from 0 to 10*pi
90

40

120

60

30
150

30

20
10

180

0

210

330

240

300
270

Figure 15.4: Polar plot.

polar (theta, rho)
polar (theta, rho, fmt)
polar (cplx)
polar (cplx, fmt)
polar (hax, . . . )
h = polar ( . . . )
Create a 2-D plot from polar coordinates theta and rho.
If a single complex input cplx is given then the real part is used for theta and the
imaginary part is used for rho.
The optional argument fmt specifies the line format in the same way as plot.
If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.
The optional return value h is a graphics handle to the created plot.
Implementation Note: The polar axis is drawn using line and text objects encapsulated in an hggroup. The hggroup properties are linked to the original axes object
such that altering an appearance property, for example fontname, will update the
polar axis. Two new properties are added to the original axes–rtick, ttick–which
replace xtick, ytick. The first is a list of tick locations in the radial (rho) direction; The second is a list of tick locations in the angular (theta) direction specified in
degrees, i.e., in the range 0–359.
See also: [rose], page 301, [compass], page 312, [plot], page 288.

pie (x)
pie ( . . . , explode)
pie ( . . . , labels)

310

GNU Octave

pie (hax, . . . );
h = pie ( . . . );
Plot a 2-D pie chart.
When called with a single vector argument, produce a pie chart of the elements in x.
The size of the ith slice is the percentage that the element xi represents of the total
sum of x: pct = x(i) / sum (x).
The optional input explode is a vector of the same length as x that, if nonzero,
"explodes" the slice from the pie chart.
The optional input labels is a cell array of strings of the same length as x specifying
the label for each slice.
If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.
The optional return value h is a list of handles to the patch and text objects generating
the plot.
Note: If sum (x) ≤ 1 then the elements of x are interpreted as percentages directly
and are not normalized by sum (x). Furthermore, if the sum is less than 1 then there
will be a missing slice in the pie plot to represent the missing, unspecified percentage.
See also: [pie3], page 310, [bar], page 293, [hist], page 295, [rose], page 301.

pie3 (x)
pie3 ( . . . , explode)
pie3 ( . . . , labels)
pie3 (hax, . . . );
h = pie3 ( . . . );
Plot a 3-D pie chart.
Called with a single vector argument, produces a 3-D pie chart of the elements in x.
The size of the ith slice is the percentage that the element xi represents of the total
sum of x: pct = x(i) / sum (x).
The optional input explode is a vector of the same length as x that, if nonzero,
"explodes" the slice from the pie chart.
The optional input labels is a cell array of strings of the same length as x specifying
the label for each slice.
If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.
The optional return value h is a list of graphics handles to the patch, surface, and
text objects generating the plot.
Note: If sum (x) ≤ 1 then the elements of x are interpreted as percentages directly
and are not normalized by sum (x). Furthermore, if the sum is less than 1 then there
will be a missing slice in the pie plot to represent the missing, unspecified percentage.
See also: [pie], page 309, [bar], page 293, [hist], page 295, [rose], page 301.

quiver (u, v)
quiver (x, y, u, v)
quiver ( . . . , s)

Chapter 15: Plotting

311

quiver ( . . . , style)
quiver ( . . . , "filled")
quiver (hax, . . . )
h = quiver ( . . . )
Plot a 2-D vector field with arrows.
Plot the (u, v) components of a vector field at the grid points defined by (x, y). If
the grid is uniform then x and y can be specified as vectors and meshgrid is used to
create the 2-D grid.
If x and y are not given they are assumed to be (1:m, 1:n) where [m, n] = size
(u).
The optional input s is a scalar defining a scaling factor to use for the arrows of the
field relative to the mesh spacing. A value of 1.0 will result in the longest vector
exactly filling one grid square. A value of 0 disables all scaling. The default value is
0.9.
The style to use for the plot can be defined with a line style style of the same format
as the plot command. If a marker is specified then the markers are drawn at the
origin of the vectors (which are the grid points defined by x and y). When a marker
is specified, the arrowhead is not drawn. If the argument "filled" is given then the
markers are filled.
If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.
The optional return value h is a graphics handle to a quiver object. A quiver object
regroups the components of the quiver plot (body, arrow, and marker), and allows
them to be changed together.
Example:
[x, y] = meshgrid (1:2:20);
h = quiver (x, y, sin (2*pi*x/10), sin (2*pi*y/10));
set (h, "maxheadsize", 0.33);
See also: [quiver3], page 311, [compass], page 312, [feather], page 312, [plot],
page 288.

quiver3 (u, v, w)
quiver3 (x, y, z, u, v, w)
quiver3 ( . . . , s)
quiver3 ( . . . , style)
quiver3 ( . . . , "filled")
quiver3 (hax, . . . )
h = quiver3 ( . . . )
Plot a 3-D vector field with arrows.
Plot the (u, v, w) components of a vector field at the grid points defined by (x, y,
z). If the grid is uniform then x, y, and z can be specified as vectors and meshgrid
is used to create the 3-D grid.
If x, y, and z are not given they are assumed to be (1:m, 1:n, 1:p) where [m, n] =
size (u) and p = max (size (w)).

312

GNU Octave

The optional input s is a scalar defining a scaling factor to use for the arrows of the
field relative to the mesh spacing. A value of 1.0 will result in the longest vector
exactly filling one grid cube. A value of 0 disables all scaling. The default value is
0.9.
The style to use for the plot can be defined with a line style style of the same format
as the plot command. If a marker is specified then the markers are drawn at the
origin of the vectors (which are the grid points defined by x, y, z). When a marker
is specified, the arrowhead is not drawn. If the argument "filled" is given then the
markers are filled.
If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.
The optional return value h is a graphics handle to a quiver object. A quiver object
regroups the components of the quiver plot (body, arrow, and marker), and allows
them to be changed together.
[x, y, z] = peaks (25);
surf (x, y, z);
hold on;
[u, v, w] = surfnorm (x, y, z / 10);
h = quiver3 (x, y, z, u, v, w);
set (h, "maxheadsize", 0.33);
See also: [quiver], page 310, [compass], page 312, [feather], page 312, [plot], page 288.

compass (u, v)
compass (z)
compass ( . . . , style)
compass (hax, . . . )
h = compass ( . . . )
Plot the (u, v) components of a vector field emanating from the origin of a polar
plot.
The arrow representing each vector has one end at the origin and the tip at [u(i),
v(i)]. If a single complex argument z is given, then u = real (z) and v = imag (z).
The style to use for the plot can be defined with a line style style of the same format
as the plot command.
If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.
The optional return value h is a vector of graphics handles to the line objects representing the drawn vectors.
a = toeplitz ([1;randn(9,1)], [1,randn(1,9)]);
compass (eig (a));
See also: [polar], page 309, [feather], page 312, [quiver], page 310, [rose], page 301,
[plot], page 288.

feather (u, v)
feather (z)
feather ( . . . , style)

Chapter 15: Plotting

313

feather (hax, . . . )
h = feather ( . . . )
Plot the (u, v) components of a vector field emanating from equidistant points on
the x-axis.
If a single complex argument z is given, then u = real (z) and v = imag (z).
The style to use for the plot can be defined with a line style style of the same format
as the plot command.
If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.
The optional return value h is a vector of graphics handles to the line objects representing the drawn vectors.
phi = [0 : 15 : 360] * pi/180;
feather (sin (phi), cos (phi));
See also: [plot], page 288, [quiver], page 310, [compass], page 312.

pcolor (x, y, c)
pcolor (c)
pcolor (hax, . . . )
h = pcolor ( . . . )
Produce a 2-D density plot.
A pcolor plot draws rectangles with colors from the matrix c over the two-dimensional
region represented by the matrices x and y. x and y are the coordinates of the mesh’s
vertices and are typically the output of meshgrid. If x and y are vectors, then a typical
vertex is (x(j), y(i), c(i,j)). Thus, columns of c correspond to different x values and
rows of c correspond to different y values.
The values in c are scaled to span the range of the current colormap. Limits may be
placed on the color axis by the command caxis, or by setting the clim property of
the parent axis.
The face color of each cell of the mesh is determined by interpolating the values of c
for each of the cell’s vertices; Contrast this with imagesc which renders one cell for
each element of c.
shading modifies an attribute determining the manner by which the face color of
each cell is interpolated from the values of c, and the visibility of the cells’ edges. By
default the attribute is "faceted", which renders a single color for each cell’s face
with the edge visible.
If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.
The optional return value h is a graphics handle to the created surface object.
See also: [caxis], page 317, [shading], page 342, [meshgrid], page 339, [contour],
page 302, [imagesc], page 754.

area (y)
area (x, y)
area ( . . . , lvl)

314

GNU Octave

area ( . . . , prop, val, . . . )
area (hax, . . . )
h = area ( . . . )
Area plot of the columns of y.
This plot shows the contributions of each column value to the row sum. It is functionally similar to plot (x, cumsum (y, 2)), except that the area under the curve is
shaded.
If the x argument is omitted it defaults to 1:rows (y). A value lvl can be defined
that determines where the base level of the shading under the curve should be defined.
The default level is 0.
Additional property/value pairs are passed directly to the underlying patch object.
If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.
The optional return value h is a graphics handle to the hggroup object comprising
the area patch objects. The "BaseValue" property of the hggroup can be used to
adjust the level where shading begins.
Example: Verify identity sin^2 + cos^2 = 1
t = linspace (0, 2*pi, 100)’;
y = [sin(t).^2, cos(t).^2];
area (t, y);
legend ("sin^2", "cos^2", "location", "NorthEastOutside");
See also: [plot], page 288, [patch], page 379.

fill (x, y, c)
fill (x1, y1, c1, x2, y2, c2)
fill ( . . . , prop, val)
fill (hax, . . . )
h = fill ( . . . )
Create one or more filled 2-D polygons.
The inputs x and y are the coordinates of the polygon vertices. If the inputs are matrices then the rows represent different vertices and each column produces a different
polygon. fill will close any open polygons before plotting.
The input c determines the color of the polygon. The simplest form is a single color
specification such as a plot format or an RGB-triple. In this case the polygon(s) will
have one unique color. If c is a vector or matrix then the color data is first scaled
using caxis and then indexed into the current colormap. A row vector will color each
polygon (a column from matrices x and y) with a single computed color. A matrix c
of the same size as x and y will compute the color of each vertex and then interpolate
the face color between the vertices.
Multiple property/value pairs for the underlying patch object may be specified, but
they must appear in pairs.
If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.
The optional return value h is a vector of graphics handles to the created patch
objects.

Chapter 15: Plotting

315

Example: red square
vertices = [0 0
1 0
1 1
0 1];
fill (vertices(:,1), vertices(:,2), "r");
axis ([-0.5 1.5, -0.5 1.5])
axis equal
See also: [patch], page 379, [caxis], page 317, [colormap], page 757.
(y)
(x, y)
(x, y, p)
(hax, . . . )
Produce a simple comet style animation along the trajectory provided by the input
coordinate vectors (x, y).
If x is not specified it defaults to the indices of y.
The speed of the comet may be controlled by p, which represents the time each point
is displayed before moving to the next one. The default for p is 0.1 seconds.
If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.

comet
comet
comet
comet

See also: [comet3], page 315.
(z)
(x, y, z)
(x, y, z, p)
(hax, . . . )
Produce a simple comet style animation along the trajectory provided by the input
coordinate vectors (x, y, z).
If only z is specified then x, y default to the indices of z.
The speed of the comet may be controlled by p, which represents the time each point
is displayed before moving to the next one. The default for p is 0.1 seconds.
If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.

comet3
comet3
comet3
comet3

See also: [comet], page 315.

15.2.1.1 Axis Configuration
The axis function may be used to change the axis limits of an existing plot and various
other axis properties, such as the aspect ratio and the appearance of tic marks.

axis
axis
axis
axis
axis

()
([x_lo
([x_lo
([x_lo
([x_lo

x_hi])
x_hi y_lo y_hi])
x_hi y_lo y_hi z_lo z_hi])
x_hi y_lo y_hi z_lo z_hi c_lo c_hi])

316

GNU Octave

axis (option)
axis (option1, option2, . . . )
axis (hax, . . . )
limits = axis ()
Set axis limits and appearance.
The argument limits should be a 2-, 4-, 6-, or 8-element vector. The first and second
elements specify the lower and upper limits for the x-axis. The third and fourth
specify the limits for the y-axis, the fifth and sixth specify the limits for the z-axis,
and the seventh and eighth specify the limits for the color axis. The special values
-Inf and Inf may be used to indicate that the limit should be automatically computed
based on the data in the axes.
Without any arguments, axis turns autoscaling on.
With one output argument, limits = axis returns the current axis limits.
The vector argument specifying limits is optional, and additional string arguments
may be used to specify various axis properties.
The following options control the aspect ratio of the axes.
"square"

Force a square axis aspect ratio.

"equal"

Force x-axis unit distance to equal y-axis (and z-axis) unit distance.

"normal"

Restore default aspect ratio.

The following options control the way axis limits are interpreted.
"auto"
"auto[xyz]"
Set the specified axes to have nice limits around the data or all if no axes
are specified.
"manual"

Fix the current axes limits.

"tight"

Fix axes to the limits of the data.

"image"

Equivalent to "tight" and "equal".

The following options affect the appearance of tick marks.
"tic[xyz]"
Turn tick marks on for all axes, or turn them on for the specified axes
and off for the remainder.
"label[xyz]"
Turn tick labels on for all axes, or turn them on for the specified axes
and off for the remainder.
"nolabel"
Turn tick labels off for all axes.
Note: If there are no tick marks for an axes then there can be no labels.
The following options affect the direction of increasing values on the axes.
"xy"

Default y-axis, larger values are near the top.

Chapter 15: Plotting

"ij"

317

Reverse y-axis, smaller values are near the top.

The following options affects the visibility of the axes.
"on"

Make the axes visible.

"off"

Hide the axes.

If the first argument hax is an axes handle, then operate on this axes rather than the
current axes returned by gca.
Example 1: set X/Y limits and force a square aspect ratio
axis ([1, 2, 3, 4], "square");
Example 2: enable tick marks on all axes, enable tick mark labels only on the y-axis
axis ("tic", "labely");
See also: [xlim], page 317, [ylim], page 317, [zlim], page 317, [caxis], page 317,
[daspect], page 344, [pbaspect], page 344, [box], page 352, [grid], page 353.
Similarly the axis limits of the colormap can be changed with the caxis function.

caxis ([cmin cmax])
caxis ("auto")
caxis ("manual")
caxis (hax, . . . )
limits = caxis ()
Query or set color axis limits for plots.
The limits argument should be a 2-element vector specifying the lower and upper
limits to assign to the first and last value in the colormap. Data values outside this
range are clamped to the first and last colormap entries.
If the "auto" option is given then automatic colormap limits are applied. The automatic algorithm sets cmin to the minimum data value and cmax to the maximum
data value. If "manual" is specified then the "climmode" property is set to "manual"
and the numeric values in the "clim" property are used for limits.
If the first argument hax is an axes handle, then operate on this axes rather than the
current axes returned by gca.
Called without arguments the current color axis limits are returned.
Programming Note: The color axis affects the display of image, patch, and surface graphics objects, but only if the "cdata" property has indexed data and the
"cdatamapping" property is set to "scaled". Graphic objects with true color cdata,
or "direct" cdatamapping are not affected.
See also: [colormap], page 757, [axis], page 315.
The xlim, ylim, and zlim functions may be used to get or set individual axis limits.
Each has the same form.

xlimits = xlim ()
xmode = xlim ("mode")
xlim ([x_lo x_hi])
xlim ("auto")

318

GNU Octave

xlim ("manual")
xlim (hax, . . . )
Query or set the limits of the x-axis for the current plot.
Called without arguments xlim returns the x-axis limits of the current plot.
With the input query "mode", return the current x-limit calculation mode which is
either "auto" or "manual".
If passed a 2-element vector [x lo x hi], the limits of the x-axis are set to these values
and the mode is set to "manual". The special values -Inf and Inf can be used to
indicate that either the lower axis limit or upper axis limit should be automatically
calculated.
The current plotting mode can be changed by using either "auto" or "manual" as
the argument.
If the first argument hax is an axes handle, then operate on this axes rather than the
current axes returned by gca.
Programming Note: The xlim function operates by modifying the "xlim" and
"xlimmode" properties of an axes object. These properties can be be directly
inspected and altered with get/set.
See also: [ylim], page 317, [zlim], page 317, [axis], page 315, [set], page 383, [get],
page 383, [gca], page 382.

15.2.1.2 Two-dimensional Function Plotting
Octave can plot a function from a function handle, inline function, or string defining the
function without the user needing to explicitly create the data to be plotted. The function
fplot also generates two-dimensional plots with linear axes using a function name and
limits for the range of the x-coordinate instead of the x and y data. For example,
fplot (@sin, [-10, 10], 201);
produces a plot that is equivalent to the one above, but also includes a legend displaying
the name of the plotted function.

fplot (fn, limits)
fplot ( . . . , tol)
fplot ( . . . , n)
fplot ( . . . , fmt)
[x, y] = fplot ( . . . )
Plot a function fn within the range defined by limits.
fn is a function handle, inline function, or string containing the name of the function
to evaluate.
The limits of the plot are of the form [xlo, xhi] or [xlo, xhi, ylo, yhi].
The next three arguments are all optional and any number of them may be given in
any order.
tol is the relative tolerance to use for the plot and defaults to 2e-3 (.2%).
n is the minimum number of points to use. When n is specified, the maximum stepsize
will be (xhi - xlo) / n. More than n points may still be used in order to meet the
relative tolerance requirement.

Chapter 15: Plotting

319

The fmt argument specifies the linestyle to be used by the plot command.
If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.
With no output arguments the results are immediately plotted. With two output
arguments the 2-D plot data is returned. The data can subsequently be plotted
manually with plot (x, y).
Example:
fplot (@cos, [0, 2*pi])
fplot ("[cos(x), sin(x)]", [0, 2*pi])
Programming Notes:
fplot works best with continuous functions. Functions with discontinuities are unlikely to plot well. This restriction may be removed in the future.
fplot requires that the function accept and return a vector argument. Consider this
when writing user-defined functions and use .*, ./, etc. See the function vectorize
for potentially converting inline or anonymous functions to vectorized versions.
See also: [ezplot], page 319, [plot], page 288, [vectorize], page 534.
Other functions that can create two-dimensional plots directly from a function include
ezplot, ezcontour, ezcontourf and ezpolar.

ezplot (f)
ezplot (f2v)
ezplot (fx, fy)
ezplot ( . . . , dom)
ezplot ( . . . , n)
ezplot (hax, . . . )
h = ezplot ( . . . )
Plot the 2-D curve defined by the function f.
The function f may be a string, inline function, or function handle and can have
either one or two variables. If f has one variable, then the function is plotted over
the domain -2*pi < x < 2*pi with 500 points.
If f2v is a function of two variables then the implicit function f(x,y) = 0 is calculated
over the meshed domain -2*pi <= x | y <= 2*pi with 60 points in each dimension.
For example:
ezplot (@(x, y) x.^2 - y.^2 - 1)
If two functions are passed as inputs then the parametric function
x = fx (t)
y = fy (t)
is plotted over the domain -2*pi <= t <= 2*pi with 500 points.
If dom is a two element vector, it represents the minimum and maximum values of
both x and y, or t for a parametric plot. If dom is a four element vector, then the
minimum and maximum values are [xmin xmax ymin ymax].
n is a scalar defining the number of points to use in plotting the function.

320

GNU Octave

If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.
The optional return value h is a vector of graphics handles to the created line objects.
See also: [plot], page 288, [ezplot3], page 345, [ezpolar], page 321, [ezcontour],
page 320, [ezcontourf], page 320, [ezmesh], page 345, [ezmeshc], page 346, [ezsurf],
page 347, [ezsurfc], page 347.

ezcontour (f)
ezcontour ( . . . , dom)
ezcontour ( . . . , n)
ezcontour (hax, . . . )
h = ezcontour ( . . . )
Plot the contour lines of a function.
f is a string, inline function, or function handle with two arguments defining the
function. By default the plot is over the meshed domain -2*pi <= x | y <= 2*pi
with 60 points in each dimension.
If dom is a two element vector, it represents the minimum and maximum values of
both x and y. If dom is a four element vector, then the minimum and maximum
values are [xmin xmax ymin ymax].
n is a scalar defining the number of points to use in each dimension.
If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.
The optional return value h is a graphics handle to the created plot.
Example:
f = @(x,y) sqrt (abs (x .* y)) ./ (1 + x.^2 + y.^2);
ezcontour (f, [-3, 3]);
See also: [contour], page 302, [ezcontourf], page 320, [ezplot], page 319, [ezmeshc],
page 346, [ezsurfc], page 347.

ezcontourf (f)
ezcontourf ( . . . , dom)
ezcontourf ( . . . , n)
ezcontourf (hax, . . . )
h = ezcontourf ( . . . )
Plot the filled contour lines of a function.
f is a string, inline function, or function handle with two arguments defining the
function. By default the plot is over the meshed domain -2*pi <= x | y <= 2*pi
with 60 points in each dimension.
If dom is a two element vector, it represents the minimum and maximum values of
both x and y. If dom is a four element vector, then the minimum and maximum
values are [xmin xmax ymin ymax].
n is a scalar defining the number of points to use in each dimension.
If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.

Chapter 15: Plotting

321

The optional return value h is a graphics handle to the created plot.
Example:
f = @(x,y) sqrt (abs (x .* y)) ./ (1 + x.^2 + y.^2);
ezcontourf (f, [-3, 3]);
See also: [contourf], page 302, [ezcontour], page 320, [ezplot], page 319, [ezmeshc],
page 346, [ezsurfc], page 347.

ezpolar (f)
ezpolar ( . . . , dom)
ezpolar ( . . . , n)
ezpolar (hax, . . . )
h = ezpolar ( . . . )
Plot a 2-D function in polar coordinates.
The function f is a string, inline function, or function handle with a single argument.
The expected form of the function is rho = f(theta). By default the plot is over the
domain 0 <= theta <= 2*pi with 500 points.
If dom is a two element vector, it represents the minimum and maximum values of
theta.
n is a scalar defining the number of points to use in plotting the function.
If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.
The optional return value h is a graphics handle to the created plot.
Example:
ezpolar (@(t) sin (5/4 * t), [0, 8*pi]);
See also: [polar], page 309, [ezplot], page 319.

15.2.1.3 Two-dimensional Geometric Shapes
rectangle ()
rectangle ( . . . , "Position", pos)
rectangle ( . . . , "Curvature", curv)
rectangle ( . . . , "EdgeColor", ec)
rectangle ( . . . , "FaceColor", fc)
rectangle (hax, . . . )
h = rectangle ( . . . )
Draw a rectangular patch defined by pos and curv.
The variable pos(1:2) defines the lower left-hand corner of the patch and pos(3:4)
defines its width and height. By default, the value of pos is [0, 0, 1, 1].
The variable curv defines the curvature of the sides of the rectangle and may be a
scalar or two-element vector with values between 0 and 1. A value of 0 represents
no curvature of the side, whereas a value of 1 means that the side is entirely curved
into the arc of a circle. If curv is a two-element vector, then the first element is the
curvature along the x-axis of the patch and the second along y-axis.

322

GNU Octave

If curv is a scalar, it represents the curvature of the shorter of the two sides of the
rectangle and the curvature of the other side is defined by
min (pos(1:2)) / max (pos(1:2)) * curv
Additional property/value pairs are passed to the underlying patch command.
If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.
The optional return value h is a graphics handle to the created rectangle object.
See also: [patch], page 379, [line], page 378, [cylinder], page 348, [ellipsoid], page 349,
[sphere], page 349.

15.2.2 Three-Dimensional Plots
The function mesh produces mesh surface plots. For example,
tx = ty = linspace (-8, 8, 41)’;
[xx, yy] = meshgrid (tx, ty);
r = sqrt (xx .^ 2 + yy .^ 2) + eps;
tz = sin (r) ./ r;
mesh (tx, ty, tz);
xlabel ("tx");
ylabel ("ty");
zlabel ("tz");
title ("3-D Sombrero plot");
produces the familiar “sombrero” plot shown in Figure 15.5. Note the use of the function
meshgrid to create matrices of X and Y coordinates to use for plotting the Z data. The
ndgrid function is similar to meshgrid, but works for N-dimensional matrices.
3-D Sombrero plot

1
0.8
0.6

tz

0.4
0.2
0
-0.2
-0.4
10
5

10
5

0

ty

0

-5
-10

-5
-10

tx

Figure 15.5: Mesh plot.
The meshc function is similar to mesh, but also produces a plot of contours for the
surface.

Chapter 15: Plotting

323

The plot3 function displays arbitrary three-dimensional data, without requiring it to
form a surface. For example,
t = 0:0.1:10*pi;
r = linspace (0, 1, numel (t));
z = linspace (0, 1, numel (t));
plot3 (r.*sin (t), r.*cos (t), z);
xlabel ("r.*sin (t)");
ylabel ("r.*cos (t)");
zlabel ("z");
title ("plot3 display of 3-D helix");
displays the spiral in three dimensions shown in Figure 15.6.
plot3 display of 3-D helix

1
0.8

z

0.6
0.4
0.2
0
1
0.5

1
0.5

0

r.*cos (t)

0

-0.5
-1

-0.5
-1

r.*sin (t)

Figure 15.6: Three-dimensional spiral.
Finally, the view function changes the viewpoint for three-dimensional plots.

mesh (x, y, z)
mesh (z)
mesh ( . . . , c)
mesh ( . . . , prop, val, . . . )
mesh (hax, . . . )
h = mesh ( . . . )
Plot a 3-D wireframe mesh.
The wireframe mesh is plotted using rectangles. The vertices of the rectangles [x,
y] are typically the output of meshgrid. over a 2-D rectangular region in the x-y
plane. z determines the height above the plane of each vertex. If only a single z
matrix is given, then it is plotted over the meshgrid x = 1:columns (z), y = 1:rows
(z). Thus, columns of z correspond to different x values and rows of z correspond to
different y values.

324

GNU Octave

The color of the mesh is computed by linearly scaling the z values to fit the range of the
current colormap. Use caxis and/or change the colormap to control the appearance.
Optionally, the color of the mesh can be specified independently of z by supplying a
color matrix, c.
Any property/value pairs are passed directly to the underlying surface object.
If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.
The optional return value h is a graphics handle to the created surface object.
See also: [ezmesh], page 345, [meshc], page 324, [meshz], page 324, [trimesh],
page 719, [contour], page 302, [surf], page 325, [surface], page 379, [meshgrid],
page 339, [hidden], page 325, [shading], page 342, [colormap], page 757, [caxis],
page 317.

meshc (x, y, z)
meshc (z)
meshc ( . . . , c)
meshc ( . . . , prop, val, . . . )
meshc (hax, . . . )
h = meshc ( . . . )
Plot a 3-D wireframe mesh with underlying contour lines.
The wireframe mesh is plotted using rectangles. The vertices of the rectangles [x,
y] are typically the output of meshgrid. over a 2-D rectangular region in the x-y
plane. z determines the height above the plane of each vertex. If only a single z
matrix is given, then it is plotted over the meshgrid x = 1:columns (z), y = 1:rows
(z). Thus, columns of z correspond to different x values and rows of z correspond to
different y values.
The color of the mesh is computed by linearly scaling the z values to fit the range of the
current colormap. Use caxis and/or change the colormap to control the appearance.
Optionally the color of the mesh can be specified independently of z by supplying a
color matrix, c.
Any property/value pairs are passed directly to the underlying surface object.
If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.
The optional return value h is a 2-element vector with a graphics handle to the created
surface object and to the created contour plot.
See also: [ezmeshc], page 346, [mesh], page 323, [meshz], page 324, [contour],
page 302, [surfc], page 326, [surface], page 379, [meshgrid], page 339, [hidden],
page 325, [shading], page 342, [colormap], page 757, [caxis], page 317.

meshz
meshz
meshz
meshz
meshz

(x, y, z)
(z)
( . . . , c)
( . . . , prop, val, . . . )
(hax, . . . )

Chapter 15: Plotting

325

h = meshz ( . . . )
Plot a 3-D wireframe mesh with a surrounding curtain.
The wireframe mesh is plotted using rectangles. The vertices of the rectangles [x,
y] are typically the output of meshgrid. over a 2-D rectangular region in the x-y
plane. z determines the height above the plane of each vertex. If only a single z
matrix is given, then it is plotted over the meshgrid x = 0:(columns (z) - 1), y =
0:(rows (z) - 1). Thus, columns of z correspond to different x values and rows of z
correspond to different y values.
The color of the mesh is computed by linearly scaling the z values to fit the range of the
current colormap. Use caxis and/or change the colormap to control the appearance.
Optionally the color of the mesh can be specified independently of z by supplying a
color matrix, c.
Any property/value pairs are passed directly to the underlying surface object.
If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.
The optional return value h is a graphics handle to the created surface object.
See also: [mesh], page 323, [meshc], page 324, [contour], page 302, [surf], page 325,
[surface], page 379, [waterfall], page 343, [meshgrid], page 339, [hidden], page 325,
[shading], page 342, [colormap], page 757, [caxis], page 317.

hidden
hidden on
hidden off
mode = hidden ( . . . )
Control mesh hidden line removal.
When called with no argument the hidden line removal state is toggled.
When called with one of the modes "on" or "off" the state is set accordingly.
The optional output argument mode is the current state.
Hidden Line Removal determines what graphic objects behind a mesh plot are visible.
The default is for the mesh to be opaque and lines behind the mesh are not visible. If
hidden line removal is turned off then objects behind the mesh can be seen through
the faces (openings) of the mesh, although the mesh grid lines are still opaque.
See also: [mesh], page 323, [meshc], page 324, [meshz], page 324, [ezmesh], page 345,
[ezmeshc], page 346, [trimesh], page 719, [waterfall], page 343.

surf (x, y, z)
surf (z)
surf ( . . . , c)
surf ( . . . , prop, val, . . . )
surf (hax, . . . )
h = surf ( . . . )
Plot a 3-D surface mesh.
The surface mesh is plotted using shaded rectangles. The vertices of the rectangles
[x, y] are typically the output of meshgrid. over a 2-D rectangular region in the x-y

326

GNU Octave

plane. z determines the height above the plane of each vertex. If only a single z
matrix is given, then it is plotted over the meshgrid x = 1:columns (z), y = 1:rows
(z). Thus, columns of z correspond to different x values and rows of z correspond to
different y values.
The color of the surface is computed by linearly scaling the z values to fit the range
of the current colormap. Use caxis and/or change the colormap to control the
appearance.
Optionally, the color of the surface can be specified independently of z by supplying
a color matrix, c.
Any property/value pairs are passed directly to the underlying surface object.
If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.
The optional return value h is a graphics handle to the created surface object.
Note: The exact appearance of the surface can be controlled with the shading command or by using set to control surface object properties.
See also: [ezsurf], page 347, [surfc], page 326, [surfl], page 327, [surfnorm], page 327,
[trisurf], page 720, [contour], page 302, [mesh], page 323, [surface], page 379,
[meshgrid], page 339, [hidden], page 325, [shading], page 342, [colormap], page 757,
[caxis], page 317.

surfc (x, y, z)
surfc (z)
surfc ( . . . , c)
surfc ( . . . , prop, val, . . . )
surfc (hax, . . . )
h = surfc ( . . . )
Plot a 3-D surface mesh with underlying contour lines.
The surface mesh is plotted using shaded rectangles. The vertices of the rectangles
[x, y] are typically the output of meshgrid. over a 2-D rectangular region in the x-y
plane. z determines the height above the plane of each vertex. If only a single z
matrix is given, then it is plotted over the meshgrid x = 1:columns (z), y = 1:rows
(z). Thus, columns of z correspond to different x values and rows of z correspond to
different y values.
The color of the surface is computed by linearly scaling the z values to fit the range
of the current colormap. Use caxis and/or change the colormap to control the
appearance.
Optionally, the color of the surface can be specified independently of z by supplying
a color matrix, c.
Any property/value pairs are passed directly to the underlying surface object.
If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.
The optional return value h is a graphics handle to the created surface object.
Note: The exact appearance of the surface can be controlled with the shading command or by using set to control surface object properties.

Chapter 15: Plotting

327

See also: [ezsurfc], page 347, [surf], page 325, [surfl], page 327, [surfnorm], page 327,
[trisurf], page 720, [contour], page 302, [mesh], page 323, [surface], page 379,
[meshgrid], page 339, [hidden], page 325, [shading], page 342, [colormap], page 757,
[caxis], page 317.

surfl (z)
surfl (x, y, z)
surfl ( . . . , lsrc)
surfl (x, y, z, lsrc, P)
surfl ( . . . , "cdata")
surfl ( . . . , "light")
surfl (hax, . . . )
h = surfl ( . . . )
Plot a 3-D surface using shading based on various lighting models.
The surface mesh is plotted using shaded rectangles. The vertices of the rectangles
[x, y] are typically the output of meshgrid. over a 2-D rectangular region in the x-y
plane. z determines the height above the plane of each vertex. If only a single z
matrix is given, then it is plotted over the meshgrid x = 1:columns (z), y = 1:rows
(z). Thus, columns of z correspond to different x values and rows of z correspond to
different y values.
The default lighting mode "cdata", changes the cdata property of the surface object
to give the impression of a lighted surface. Warning: The alternative mode "light"
mode which creates a light object to illuminate the surface is not implemented (yet).
The light source location can be specified using lsrc. It can be given as a 2-element
vector [azimuth, elevation] in degrees, or as a 3-element vector [lx, ly, lz]. The default
value is rotated 45 degrees counterclockwise to the current view.
The material properties of the surface can specified using a 4-element vector P =
[AM D SP exp] which defaults to p = [0.55 0.6 0.4 10].
"AM" strength of ambient light
"D" strength of diffuse reflection
"SP" strength of specular reflection
"EXP" specular exponent
If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.
The optional return value h is a graphics handle to the created surface object.
Example:
colormap (bone (64));
surfl (peaks);
shading interp;
See also: [diffuse], page 336, [specular], page 336, [surf], page 325, [shading], page 342,
[colormap], page 757, [caxis], page 317.

surfnorm (x, y, z)
surfnorm (z)
surfnorm ( . . . , prop, val, . . . )

328

GNU Octave

surfnorm (hax, . . . )
[nx, ny, nz] = surfnorm ( . . . )
Find the vectors normal to a meshgridded surface.
If x and y are vectors, then a typical vertex is (x(j), y(i), z(i,j)). Thus, columns of
z correspond to different x values and rows of z correspond to different y values. If
only a single input z is given then x is taken to be 1:columns (z) and y is 1:rows
(z).
If no return arguments are requested, a surface plot with the normal vectors to the
surface is plotted.
Any property/value input pairs are assigned to the surface object.
If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.
If output arguments are requested then the components of the normal vectors are
returned in nx, ny, and nz and no plot is made. The normal vectors are unnormalized
(magnitude != 1). To normalize, use
len = sqrt (nx.^2 + ny.^2 + nz.^2);
nx ./= len; ny ./= len; nz ./= len;
An example of the use of surfnorm is
surfnorm (peaks (25));
Algorithm: The normal vectors are calculated by taking the cross product of the
diagonals of each of the quadrilateral faces in the meshgrid to find the normal vectors
at the center of each face. Next, for each meshgrid point the four nearest normal
vectors are averaged to obtain the final normal to the surface at the meshgrid point.
For surface objects, the "VertexNormals" property contains equivalent information,
except possibly near the boundary of the surface where different interpolation schemes
may yield slightly different values.
See also: [isonormals], page 330, [quiver3], page 311, [surf], page 325, [meshgrid],
page 339.

fv = isosurface (v, isoval)
fv = isosurface (v)
fv = isosurface (x, y, z, v, isoval)
fv = isosurface (x, y, z, v)
fvc = isosurface ( . . . , col)
fv = isosurface ( . . . , "noshare")
fv = isosurface ( . . . , "verbose")
[f, v] = isosurface ( . . . )
[f, v, c] = isosurface ( . . . )
isosurface ( . . . )
Calculate isosurface of 3-D volume data.
An isosurface connects points with the same value and is analogous to a contour plot,
but in three dimensions.
The input argument v is a three-dimensional array that contains data sampled over
a volume.

Chapter 15: Plotting

329

The input isoval is a scalar that specifies the value for the isosurface. If isoval is
omitted or empty, a "good" value for an isosurface is determined from v.
When called with a single output argument isosurface returns a structure array
fv that contains the fields faces and vertices computed at the points [x, y, z] =
meshgrid (1:l, 1:m, 1:n) where [l, m, n] = size (v). The output fv can be used
directly as input to the patch function.
If called with additional input arguments x, y, and z that are three-dimensional arrays
with the same size as v or vectors with lengths corresponding to the dimensions of v,
then the volume data is taken at the specified points. If x, y, or z are empty, the grid
corresponds to the indices (1:n) in the respective direction (see [meshgrid], page 339).
The optional input argument col, which is a three-dimensional array of the same size
as v, specifies coloring of the isosurface. The color data is interpolated, as necessary,
to match isoval. The output structure array, in this case, has the additional field
facevertexcdata.
If given the string input argument "noshare", vertices may be returned multiple times
for different faces. The default behavior is to eliminate vertices shared by adjacent
faces with unique which may be time consuming.
The string input argument "verbose" is supported for matlab compatibility, but
has no effect.
Any string arguments must be passed after the other arguments.
If called with two or three output arguments, return the information about the faces
f, vertices v, and color data c as separate arrays instead of a single structure array.
If called with no output argument, the isosurface geometry is directly plotted with
the patch command and a light object is added to the axes if not yet present.
For example,
[x, y, z] = meshgrid (1:5, 1:5, 1:5);
v = rand (5, 5, 5);
isosurface (x, y, z, v, .5);
will directly draw a random isosurface geometry in a graphics window.
An example of an isosurface geometry with different additional coloring:
N = 15;
# Increase number of vertices in each direction
iso = .4; # Change isovalue to .1 to display a sphere
lin = linspace (0, 2, N);
[x, y, z] = meshgrid (lin, lin, lin);
v = abs ((x-.5).^2 + (y-.5).^2 + (z-.5).^2);
figure ();
subplot (2,2,1); view (-38, 20);
[f, vert] = isosurface (x, y, z, v, iso);
p = patch ("Faces", f, "Vertices", vert, "EdgeColor", "none");
pbaspect ([1 1 1]);
isonormals (x, y, z, v, p)
set (p, "FaceColor", "green", "FaceLighting", "gouraud");
light ("Position", [1 1 5]);
subplot (2,2,2); view (-38, 20);
p = patch ("Faces", f, "Vertices", vert, "EdgeColor", "blue");
pbaspect ([1 1 1]);

330

GNU Octave

isonormals (x, y, z, v, p)
set (p, "FaceColor", "none", "EdgeLighting", "gouraud");
light ("Position", [1 1 5]);
subplot (2,2,3); view (-38, 20);
[f, vert, c] = isosurface (x, y, z, v, iso, y);
p = patch ("Faces", f, "Vertices", vert, "FaceVertexCData", c, ...
"FaceColor", "interp", "EdgeColor", "none");
pbaspect ([1 1 1]);
isonormals (x, y, z, v, p)
set (p, "FaceLighting", "gouraud");
light ("Position", [1 1 5]);
subplot (2,2,4); view (-38, 20);
p = patch ("Faces", f, "Vertices", vert, "FaceVertexCData", c, ...
"FaceColor", "interp", "EdgeColor", "blue");
pbaspect ([1 1 1]);
isonormals (x, y, z, v, p)
set (p, "FaceLighting", "gouraud");
light ("Position", [1 1 5]);

See also: [isonormals], page 330, [isocolors], page 332, [isocaps], page 331, [smooth3],
page 333, [reducevolume], page 334, [reducepatch], page 334, [patch], page 379.

[vn] = isonormals (val, vert)
[vn] = isonormals (val, hp)
[vn] = isonormals (x, y, z, val, vert)
[vn] = isonormals (x, y, z, val, hp)
[vn] = isonormals ( . . . , "negate")
isonormals (val, hp)
isonormals (x, y, z, val, hp)
isonormals ( . . . , "negate")
Calculate normals to an isosurface.
The vertex normals vn are calculated from the gradient of the 3-dimensional array val
(size: lxmxn) with the data for an isosurface geometry. The normals point towards
lower values in val.
If called with one output argument vn and the second input argument vert holds
the vertices of an isosurface, the normals vn are calculated at the vertices vert on a
grid given by [x, y, z] = meshgrid (1:l, 1:m, 1:n). The output argument vn has
the same size as vert and can be used to set the "VertexNormals" property of the
corresponding patch.
If called with further input arguments x, y, and z which are 3-dimensional arrays
with the same size as val, the volume data is taken at these points. Instead of the
vertex data vert, a patch handle hp can be passed to this function.
If the last input argument is the string "negate", compute the reverse vector normals
of an isosurface geometry (i.e., pointed towards higher values in val).
If no output argument is given, the property "VertexNormals" of the patch associated
with the patch handle hp is changed directly.
See also: [isosurface], page 328, [isocolors], page 332, [smooth3], page 333.

Chapter 15: Plotting

331

fvc = isocaps (v, isoval)
fvc = isocaps (v)
fvc = isocaps (x, y, z, v, isoval)
fvc = isocaps (x, y, z, v)
fvc = isocaps ( . . . , which_caps)
fvc = isocaps ( . . . , which_plane)
fvc = isocaps ( . . . , "verbose")
[faces, vertices, fvcdata] = isocaps ( . . . )
isocaps ( . . . )
Create end-caps for isosurfaces of 3-D data.
This function places caps at the open ends of isosurfaces.
The input argument v is a three-dimensional array that contains data sampled over
a volume.
The input isoval is a scalar that specifies the value for the isosurface. If isoval is
omitted or empty, a "good" value for an isosurface is determined from v.
When called with a single output argument, isocaps returns a structure array fvc
with the fields: faces, vertices, and facevertexcdata. The results are computed
at the points [x, y, z] = meshgrid (1:l, 1:m, 1:n) where [l, m, n] = size (v).
The output fvc can be used directly as input to the patch function.
If called with additional input arguments x, y, and z that are three-dimensional arrays
with the same size as v or vectors with lengths corresponding to the dimensions of v,
then the volume data is taken at the specified points. If x, y, or z are empty, the grid
corresponds to the indices (1:n) in the respective direction (see [meshgrid], page 339).
The optional parameter which caps can have one of the following string values which
defines how the data will be enclosed:
"above", "a" (default)
for end-caps that enclose the data above isoval.
"below", "b"
for end-caps that enclose the data below isoval.
The optional parameter which plane can have one of the following string values to
define which end-cap should be drawn:
"all" (default)
for all of the end-caps.
"xmin"

for end-caps at the lower x-plane of the data.

"xmax"

for end-caps at the upper x-plane of the data.

"ymin"

for end-caps at the lower y-plane of the data.

"ymax"

for end-caps at the upper y-plane of the data.

"zmin"

for end-caps at the lower z-plane of the data.

"zmax"

for end-caps at the upper z-plane of the data.

332

GNU Octave

The string input argument "verbose" is supported for matlab compatibility, but
has no effect.
If called with two or three output arguments, the data for faces faces, vertices vertices,
and the color data facevertexcdata are returned in separate arrays instead of a single
structure.
If called with no output argument, the end-caps are drawn directly in the current
figure with the patch command.
See also: [isosurface], page 328, [isonormals], page 330, [patch], page 379.

[cd] = isocolors
[cd] = isocolors
[cd] = isocolors
[cd] = isocolors
[cd] = isocolors
isocolors ( . . . )

(c, v)
(x, y, z, c, v)
(x, y, z, r, g, b, v)
(r, g, b, v)
( . . . , p)

Compute isosurface colors.
If called with one output argument and the first input argument c is a
three-dimensional array that contains color values and the second input argument v
keeps the vertices of a geometry then return a matrix cd with color data information
for the geometry at computed points [x, y, z] = meshgrid (1:l, 1:m, 1:n). The
output argument cd can be taken to manually set FaceVertexCData of a patch.
If called with further input arguments x, y and z which are three–dimensional arrays
of the same size than c then the color data is taken at those given points. Instead
of the color data c this function can also be called with RGB values r, g, b. If input
argumnets x, y, z are not given then again meshgrid computed values are taken.
Optionally, the patch handle p can be given as the last input argument to all variations
of function calls instead of the vertices data v. Finally, if no output argument is given
then directly change the colors of a patch that is given by the patch handle p.
For example:
function isofinish (p)
set (gca, "PlotBoxAspectRatioMode", "manual", ...
"PlotBoxAspectRatio", [1 1 1]);
set (p, "FaceColor", "interp");
## set (p, "FaceLighting", "flat");
## light ("Position", [1 1 5]); # Available with JHandles
endfunction
N = 15;
# Increase number of vertices in each direction
iso = .4; # Change isovalue to .1 to display a sphere
lin = linspace (0, 2, N);
[x, y, z] = meshgrid (lin, lin, lin);
c = abs ((x-.5).^2 + (y-.5).^2 + (z-.5).^2);
figure (); # Open another figure window
subplot (2,2,1); view (-38, 20);

Chapter 15: Plotting

333

[f, v] = isosurface (x, y, z, c, iso);
p = patch ("Faces", f, "Vertices", v, "EdgeColor", "none");
cdat = rand (size (c));
# Compute random patch color data
isocolors (x, y, z, cdat, p); # Directly set colors of patch
isofinish (p);
# Call user function isofinish
subplot (2,2,2); view (-38, 20);
p = patch ("Faces", f, "Vertices", v, "EdgeColor", "none");
[r, g, b] = meshgrid (lin, 2-lin, 2-lin);
cdat = isocolors (x, y, z, c, v); # Compute color data vertices
set (p, "FaceVertexCData", cdat); # Set color data manually
isofinish (p);
subplot (2,2,3); view (-38, 20);
p = patch ("Faces", f, "Vertices", v, "EdgeColor", "none");
cdat = isocolors (r, g, b, c, p); # Compute color data patch
set (p, "FaceVertexCData", cdat); # Set color data manually
isofinish (p);
subplot (2,2,4); view (-38, 20);
p = patch ("Faces", f, "Vertices", v, "EdgeColor", "none");
r = g = b = repmat ([1:N] / N, [N, 1, N]); # Black to white
cdat = isocolors (x, y, z, r, g, b, v);
set (p, "FaceVertexCData", cdat);
isofinish (p);
See also: [isosurface], page 328, [isonormals], page 330.
(data)
(data, method)
(data, method, sz)
(data, method, sz, std_dev)
Smooth values of 3-dimensional matrix data.

smoothed_data
smoothed_data
smoothed_data
smoothed_data

=
=
=
=

smooth3
smooth3
smooth3
smooth3

This function can be used, for example, to reduce the impact of noise in data before
calculating isosurfaces.
data must be a non-singleton 3-dimensional matrix. The smoothed data from this
matrix is returned in smoothed data which is of the same size as data.
The option input method determines which convolution kernel is used for the smoothing process. Possible choices:
"box", "b" (default)
to use a convolution kernel with sharp edges.
"gaussian", "g"
to use a convolution kernel that is represented by a non-correlated trivariate normal distribution function.

334

GNU Octave

sz is either a vector of 3 elements representing the size of the convolution kernel in
x-, y- and z-direction or a scalar, in which case the same size is used in all three
dimensions. The default value is 3.
When method is "gaussian", std dev defines the standard deviation of the trivariate
normal distribution function. std dev is either a vector of 3 elements representing
the standard deviation of the Gaussian convolution kernel in x-, y- and z-directions
or a scalar, in which case the same value is used in all three dimensions. The default
value is 0.65.
See also: [isosurface], page 328, [isonormals], page 330, [patch], page 379.

[nx, ny, nz, nv] = reducevolume (v, r)
[nx, ny, nz, nv] = reducevolume (x, y, z, v, r)
nv = reducevolume ( . . . )
Reduce the volume of the dataset in v according to the values in r.
v is a matrix that is non-singleton in the first 3 dimensions.
r can either be a vector of 3 elements representing the reduction factors in the x-,
y-, and z-directions or a scalar, in which case the same reduction factor is used in all
three dimensions.
reducevolume reduces the number of elements of v by taking only every r-th element
in the respective dimension.
Optionally, x, y, and z can be supplied to represent the set of coordinates of v.
They can either be matrices of the same size as v or vectors with sizes according to
the dimensions of v, in which case they are expanded to matrices (see [meshgrid],
page 339).
If reducevolume is called with two arguments then x, y, and z are assumed to match
the respective indices of v.
The reduced matrix is returned in nv.
Optionally, the reduced set of coordinates are returned in nx, ny, and nz, respectively.
Examples:
v = reshape (1:6*8*4, [6 8 4]);
nv = reducevolume (v, [4 3 2]);
v = reshape (1:6*8*4, [6 8 4]);
x = 1:3:24; y = -14:5:11; z = linspace (16, 18, 4);
[nx, ny, nz, nv] = reducevolume (x, y, z, v, [4 3 2]);
See also: [isosurface], page 328, [isonormals], page 330.

reduced_fv = reducepatch (fv)
reduced_fv = reducepatch (faces, vertices)
reduced_fv = reducepatch (patch_handle)
reducepatch (patch_handle)
reduced_fv = reducepatch ( . . . , reduction_factor)
reduced_fv = reducepatch ( . . . , "fast")
reduced_fv = reducepatch ( . . . , "verbose")

Chapter 15: Plotting

335

[reduced_faces, reduces_vertices] = reducepatch ( . . . )
Reduce the number of faces and vertices in a patch object while retaining the overall
shape of the patch.
The input patch can be represented by a structure fv with the fields faces and
vertices, by two matrices faces and vertices (see, e.g., the result of isosurface), or
by a handle to a patch object patch handle (see [patch], page 379).
The number of faces and vertices in the patch is reduced by iteratively collapsing the
shortest edge of the patch to its midpoint (as discussed, e.g., here: http://libigl.
github.io/libigl/tutorial/tutorial.html#meshdecimation).
Currently, only patches consisting of triangles are supported. The resulting patch
also consists only of triangles.
If reducepatch is called with a handle to a valid patch patch handle, and without
any output arguments, then the given patch is updated immediately.
If the reduction factor is omitted, the resulting structure reduced fv includes approximately 50% of the faces of the original patch. If reduction factor is a fraction
between 0 (excluded) and 1 (excluded), a patch with approximately the corresponding
fraction of faces is determined. If reduction factor is an integer greater than or equal
to 1, the resulting patch has approximately reduction factor faces. Depending on the
geometry of the patch, the resulting number of faces can differ from the given value
of reduction factor. This is especially true when many shared vertices are detected.
For the reduction, it is necessary that vertices of touching faces are shared. Shared
vertices are detected automatically. This detection can be skipped by passing the
optional string argument "fast".
With the optional string arguments "verbose", additional status messages are printed
to the command window.
Any string input arguments must be passed after all other arguments.
If called with one output argument, the reduced faces and vertices are returned in a
structure reduced fv with the fields faces and vertices (see the one output option
of isosurface).
If called with two output arguments, the reduced faces and vertices are returned in
two separate matrices reduced faces and reduced vertices.
See also: [isosurface], page 328, [isonormals], page 330, [reducevolume], page 334,
[patch], page 379.

shrinkfaces (p, sf)
nfv = shrinkfaces (p, sf)
nfv = shrinkfaces (fv, sf)
nfv = shrinkfaces (f, v, sf)
[nf, nv] = shrinkfaces ( . . . )
Reduce the size of faces in a patch by the shrink factor sf.
The patch object can be specified by a graphics handle (p), a patch structure (fv)
with the fields "faces" and "vertices", or as two separate matrices (f, v) of faces
and vertices.
The shrink factor sf is a positive number specifying the percentage of the original
area the new face will occupy. If no factor is given the default is 0.3 (a reduction to

336

GNU Octave

30% of the original size). A factor greater than 1.0 will result in the expansion of
faces.
Given a patch handle as the first input argument and no output parameters, perform
the shrinking of the patch faces in place and redraw the patch.
If called with one output argument, return a structure with fields "faces",
"vertices", and "facevertexcdata" containing the data after shrinking. This
structure can be used directly as an input argument to the patch function.
Caution:: Performing the shrink operation on faces which are not convex can lead to
undesirable results.
Example: a triangulated 3/4 circle and the corresponding shrunken version.
[phi r] = meshgrid (linspace (0, 1.5*pi, 16), linspace (1, 2, 4));
tri = delaunay (phi(:), r(:));
v = [r(:).*sin(phi(:)) r(:).*cos(phi(:))];
clf ()
p = patch ("Faces", tri, "Vertices", v, "FaceColor", "none");
fv = shrinkfaces (p);
patch (fv)
axis equal
grid on
See also: [patch], page 379.

diffuse (sx, sy, sz, lv)
Calculate the diffuse reflection strength of a surface defined by the normal vector
elements sx, sy, sz.
The light source location vector lv can be given as a 2-element vector [azimuth,
elevation] in degrees or as a 3-element vector [x, y, z].
See also: [specular], page 336, [surfl], page 327.

specular (sx, sy, sz, lv, vv)
specular (sx, sy, sz, lv, vv, se)
Calculate the specular reflection strength of a surface defined by the normal vector
elements sx, sy, sz using Phong’s approximation.
The light source location and viewer location vectors are specified using parameters
lv and vv respectively. The location vectors can given as 2-element vectors [azimuth,
elevation] in degrees or as 3-element vectors [x, y, z].
An optional sixth argument specifies the specular exponent (spread) se. If not given,
se defaults to 10.
See also: [diffuse], page 336, [surfl], page 327.

lighting (type)
lighting (hax, type)
Set the lighting of patch or surface graphic objects.
Valid arguments for type are
"flat"

Draw objects with faceted lighting effects.

Chapter 15: Plotting

337

"gouraud"
Draw objects with linear interpolation of the lighting effects between the
vertices.
"none"

Draw objects without light and shadow effects.

If the first argument hax is an axes handle, then change the lighting effects of objects
in this axes, rather than the current axes returned by gca.
The lighting effects are only visible if at least one light object is present and visible
in the same axes.
See also: [light], page 380, [fill], page 314, [mesh], page 323, [patch], page 379, [pcolor],
page 313, [surf], page 325, [surface], page 379, [shading], page 342.

material shiny
material dull
material metal
material default
material ([as, ds, ss])
material ([as, ds, ss, se])
material ([as, ds, ss, se, scr])
material (hlist, . . . )
mtypes = material ()
refl_props = material (mtype_string)
Set reflectance properties for the lighting of surfaces and patches.
This function changes the ambient, diffuse, and specular strengths, as well as the
specular exponent and specular color reflectance, of all patch and surface objects
in the current axes. This can be used to simulate, to some extent, the reflectance
properties of certain materials when used with light.
When called with a string, the aforementioned properties are set according to the
values in the following table:
mtype

ambientstrength

diffusestrength

specularstrength

specularexponent

specularcolorreflectance
"shiny"
0.3
0.6
0.9
20
1.0
"dull"
0.3
0.8
0.0
10
1.0
"metal"
0.3
0.3
1.0
25
0.5
"default"
"default" "default" "default" "default" "default"
When called with a vector of three elements, the ambient, diffuse, and specular
strengths of all patch and surface objects in the current axes are updated. An
optional fourth vector element updates the specular exponent, and an optional fifth
vector element updates the specular color reflectance.
A list of graphic handles can also be passed as the first argument. In this case, the
properties of these handles and all child patch and surface objects will be updated.
Additionally, material can be called with a single output argument. If called without
input arguments, a column cell vector mtypes with the strings for all available materials is returned. If the one input argument mtype string is the name of a material, a

338

GNU Octave

1x5 cell vector refl props with the reflectance properties of that material is returned.
In both cases, no graphic properties are changed.
See also: [light], page 380, [fill], page 314, [mesh], page 323, [patch], page 379, [pcolor],
page 313, [surf], page 325, [surface], page 379.

camlight
camlight right
camlight left
camlight headlight
camlight (az, el)
camlight ( . . . , style)
camlight (hl, . . . )
h = camlight ( . . . )
Add a light object to a figure using a simple interface.
When called with no arguments, a light object is added to the current plot and is
placed slightly above and to the right of the camera’s current position: this is equivalent to camlight right. The commands camlight left and camlight headlight
behave similarly with the placement being either left of the camera position or centered on the camera position.
For more control, the light position can be specified by an azimuthal rotation az and
an elevation angle el, both in degrees, relative to the current properties of the camera.
The optional string style specifies whether the light is a local point source ("local",
the default) or placed at infinite distance ("infinite").
If the first argument hl is a handle to a light object, then act on this light object
rather than creating a new object.
The optional return value h is a graphics handle to the light object. This can be used
to move or further change properties of the light object.
Examples:
Add a light object to a plot
sphere (36);
camlight
Position the light source exactly
camlight (45, 30);
Here the light is first pitched upwards from the camera position by 30 degrees. It is
then yawed by 45 degrees to the right. Both rotations are centered around the camera
target.
Return a handle to further manipulate the light object
clf
sphere (36);
hl = camlight ("left");
set (hl, "color", "r");
See also: [light], page 380.

Chapter 15: Plotting

[xx,
[xx,
[xx,
[xx,

yy]
yy,
yy]
yy,

339

= meshgrid (x, y)
zz] = meshgrid (x, y, z)
= meshgrid (x)
zz] = meshgrid (x)

Given vectors of x and y coordinates, return matrices xx and yy corresponding to a
full 2-D grid.
The rows of xx are copies of x, and the columns of yy are copies of y. If y is omitted,
then it is assumed to be the same as x.
If the optional z input is given, or zz is requested, then the output will be a full 3-D
grid.
meshgrid is most frequently used to produce input for a 2-D or 3-D function that will
be plotted. The following example creates a surface plot of the “sombrero” function.
f = @(x,y) sin (sqrt (x.^2 + y.^2)) ./ sqrt (x.^2 + y.^2);
range = linspace (-8, 8, 41);
[X, Y] = meshgrid (range, range);
Z = f (X, Y);
surf (X, Y, Z);
Programming Note: meshgrid is restricted to 2-D or 3-D grid generation. The ndgrid
function will generate 1-D through N-D grids. However, the functions are not completely equivalent. If x is a vector of length M and y is a vector of length N, then
meshgrid will produce an output grid which is NxM. ndgrid will produce an output
which is MxN (transpose) for the same input. Some core functions expect meshgrid
input and others expect ndgrid input. Check the documentation for the function in
question to determine the proper input format.
See also: [ndgrid], page 339, [mesh], page 323, [contour], page 302, [surf], page 325.

[y1, y2, ..., yn] = ndgrid (x1, x2, . . . , xn)
[y1, y2, ..., yn] = ndgrid (x)
Given n vectors x1, . . . , xn, ndgrid returns n arrays of dimension n.
The elements of the i-th output argument contains the elements of the vector xi
repeated over all dimensions different from the i-th dimension. Calling ndgrid with
only one input argument x is equivalent to calling ndgrid with all n input arguments
equal to x:
[y1, y2, . . . , yn] = ndgrid (x, . . . , x)
Programming Note: ndgrid is very similar to the function meshgrid except that the
first two dimensions are transposed in comparison to meshgrid. Some core functions
expect meshgrid input and others expect ndgrid input. Check the documentation
for the function in question to determine the proper input format.
See also: [meshgrid], page 339.

plot3
plot3
plot3
plot3
plot3

(x, y, z)
(x, y, z, prop, value, . . . )
(x, y, z, fmt)
(x, cplx)
(cplx)

340

GNU Octave

plot3 (hax, . . . )
h = plot3 ( . . . )
Produce 3-D plots.
Many different combinations of arguments are possible. The simplest form is
plot3 (x, y, z)
in which the arguments are taken to be the vertices of the points to be plotted in three
dimensions. If all arguments are vectors of the same length, then a single continuous
line is drawn. If all arguments are matrices, then each column of is treated as a
separate line. No attempt is made to transpose the arguments to make the number
of rows match.
If only two arguments are given, as
plot3 (x, cplx)
the real and imaginary parts of the second argument are used as the y and z coordinates, respectively.
If only one argument is given, as
plot3 (cplx)
the real and imaginary parts of the argument are used as the y and z values, and
they are plotted versus their index.
Arguments may also be given in groups of three as
plot3 (x1, y1, z1, x2, y2, z2, ...)
in which each set of three arguments is treated as a separate line or set of lines in
three dimensions.
To plot multiple one- or two-argument groups, separate each group with an empty
format string, as
plot3 (x1, c1, "", c2, "", ...)
Multiple property-value pairs may be specified which will affect the line objects drawn
by plot3. If the fmt argument is supplied it will format the line objects in the same
manner as plot.
If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.
The optional return value h is a graphics handle to the created plot.
Example:
z = [0:0.05:5];
plot3 (cos (2*pi*z), sin (2*pi*z), z, ";helix;");
plot3 (z, exp (2i*pi*z), ";complex sinusoid;");
See also: [ezplot3], page 345, [plot], page 288.

view
view
view
view
view
view

(azimuth, elevation)
([azimuth elevation])
([x y z])
(2)
(3)
(hax, . . . )

Chapter 15: Plotting

341

[azimuth, elevation] = view ()
Query or set the viewpoint for the current axes.
The parameters azimuth and elevation can be given as two arguments or as 2-element
vector. The viewpoint can also be specified with Cartesian coordinates x, y, and z.
The call view (2) sets the viewpoint to azimuth = 0 and elevation = 90, which is the
default for 2-D graphs.
The call view (3) sets the viewpoint to azimuth = -37.5 and elevation = 30, which
is the default for 3-D graphs.
If the first argument hax is an axes handle, then operate on this axes rather than the
current axes returned by gca.
If no inputs are given, return the current azimuth and elevation.

slice (x, y, z, v, sx, sy, sz)
slice (x, y, z, v, xi, yi, zi)
slice (v, sx, sy, sz)
slice (v, xi, yi, zi)
slice ( . . . , method)
slice (hax, . . . )
h = slice ( . . . )
Plot slices of 3-D data/scalar fields.
Each element of the 3-dimensional array v represents a scalar value at a location given
by the parameters x, y, and z. The parameters x, x, and z are either 3-dimensional
arrays of the same size as the array v in the "meshgrid" format or vectors. The
parameters xi, etc. respect a similar format to x, etc., and they represent the points
at which the array vi is interpolated using interp3. The vectors sx, sy, and sz contain
points of orthogonal slices of the respective axes.
If x, y, z are omitted, they are assumed to be x = 1:size (v, 2), y = 1:size (v, 1)
and z = 1:size (v, 3).
method is one of:
"nearest"
Return the nearest neighbor.
"linear"

Linear interpolation from nearest neighbors.

"cubic"

Cubic interpolation from four nearest neighbors (not implemented yet).

"spline"

Cubic spline interpolation—smooth first and second derivatives throughout the curve.

The default method is "linear".
If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.
The optional return value h is a graphics handle to the created surface object.
Examples:

342

GNU Octave

[x, y, z] = meshgrid (linspace (-8, 8, 32));
v = sin (sqrt (x.^2 + y.^2 + z.^2)) ./ (sqrt (x.^2 + y.^2 + z.^2));
slice (x, y, z, v, [], 0, []);
[xi, yi] = meshgrid (linspace (-7, 7));
zi = xi + yi;
slice (x, y, z, v, xi, yi, zi);
See also: [interp3], page 712, [surface], page 379, [pcolor], page 313.

ribbon (y)
ribbon (x, y)
ribbon (x, y, width)
ribbon (hax, . . . )
h = ribbon ( . . . )
Draw a ribbon plot for the columns of y vs. x.
If x is omitted, a vector containing the row numbers is assumed (1:rows (Y)). Alternatively, x can also be a vector with same number of elements as rows of y in which
case the same x is used for each column of y.
The optional parameter width specifies the width of a single ribbon (default is 0.75).
If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.
The optional return value h is a vector of graphics handles to the surface objects
representing each ribbon.
See also: [surface], page 379, [waterfall], page 343.

shading (type)
shading (hax, type)
Set the shading of patch or surface graphic objects.
Valid arguments for type are
"flat"

Single colored patches with invisible edges.

"faceted"
Single colored patches with black edges.
"interp"

Colors between patch vertices are interpolated and the patch edges are
invisible.

If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.
See also: [fill], page 314, [mesh], page 323, [patch], page 379, [pcolor], page 313, [surf],
page 325, [surface], page 379, [hidden], page 325, [lighting], page 336.

scatter3
scatter3
scatter3
scatter3
scatter3

(x, y, z)
(x, y, z, s)
(x, y, z, s, c)
( . . . , style)
( . . . , "filled")

Chapter 15: Plotting

343

scatter3 ( . . . , prop, val)
scatter3 (hax, . . . )
h = scatter3 ( . . . )
Draw a 3-D scatter plot.
A marker is plotted at each point defined by the coordinates in the vectors x, y, and
z.
The size of the markers is determined by s, which can be a scalar or a vector of the
same length as x, y, and z. If s is not given, or is an empty matrix, then a default
value of 8 points is used.
The color of the markers is determined by c, which can be a string defining a fixed
color; a 3-element vector giving the red, green, and blue components of the color; a
vector of the same length as x that gives a scaled index into the current colormap; or
an Nx3 matrix defining the RGB color of each marker individually.
The marker to use can be changed with the style argument, that is a string defining a
marker in the same manner as the plot command. If no marker is specified it defaults
to "o" or circles. If the argument "filled" is given then the markers are filled.
Additional property/value pairs are passed directly to the underlying patch object.
If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.
The optional return value h is a graphics handle to the hggroup object representing
the points.
[x, y, z] = peaks (20);
scatter3 (x(:), y(:), z(:), [], z(:));
See also: [scatter], page 299, [patch], page 379, [plot], page 288.

waterfall (x, y, z)
waterfall (z)
waterfall ( . . . , c)
waterfall ( . . . , prop, val, . . . )
waterfall (hax, . . . )
h = waterfall ( . . . )
Plot a 3-D waterfall plot.
A waterfall plot is similar to a meshz plot except only mesh lines for the rows of z
(x-values) are shown.
The wireframe mesh is plotted using rectangles. The vertices of the rectangles [x,
y] are typically the output of meshgrid. over a 2-D rectangular region in the x-y
plane. z determines the height above the plane of each vertex. If only a single z
matrix is given, then it is plotted over the meshgrid x = 1:columns (z), y = 1:rows
(z). Thus, columns of z correspond to different x values and rows of z correspond to
different y values.
The color of the mesh is computed by linearly scaling the z values to fit the range of the
current colormap. Use caxis and/or change the colormap to control the appearance.
Optionally the color of the mesh can be specified independently of z by supplying a
color matrix, c.

344

GNU Octave

Any property/value pairs are passed directly to the underlying surface object.
If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.
The optional return value h is a graphics handle to the created surface object.
See also: [meshz], page 324, [mesh], page 323, [meshc], page 324, [contour], page 302,
[surf], page 325, [surface], page 379, [ribbon], page 342, [meshgrid], page 339, [hidden],
page 325, [shading], page 342, [colormap], page 757, [caxis], page 317.

15.2.2.1 Aspect Ratio
For three-dimensional plots the aspect ratio can be set for data with daspect and for the
plot box with pbaspect. See Section 15.2.1.1 [Axis Configuration], page 315, for controlling
the x-, y-, and z-limits for plotting.

data_aspect_ratio = daspect ()
daspect (data_aspect_ratio)
daspect (mode)
data_aspect_ratio_mode = daspect ("mode")
daspect (hax, . . . )
Query or set the data aspect ratio of the current axes.
The aspect ratio is a normalized 3-element vector representing the span of the x, y,
and z-axis limits.
daspect (mode)
Set the data aspect ratio mode of the current axes. mode is either "auto" or
"manual".
daspect ("mode")
Return the data aspect ratio mode of the current axes.
daspect (hax, ...)
Operate on the axes in handle hax instead of the current axes.
See also: [axis], page 315, [pbaspect], page 344, [xlim], page 317, [ylim], page 317,
[zlim], page 317.

plot_box_aspect_ratio = pbaspect ( )
pbaspect (plot_box_aspect_ratio)
pbaspect (mode)
plot_box_aspect_ratio_mode = pbaspect ("mode")
pbaspect (hax, . . . )
Query or set the plot box aspect ratio of the current axes.
The aspect ratio is a normalized 3-element vector representing the rendered lengths
of the x, y, and z axes.
pbaspect(mode)
Set the plot box aspect ratio mode of the current axes. mode is either "auto" or
"manual".
pbaspect ("mode")
Return the plot box aspect ratio mode of the current axes.

Chapter 15: Plotting

345

pbaspect (hax, ...)
Operate on the axes in handle hax instead of the current axes.
See also: [axis], page 315, [daspect], page 344, [xlim], page 317, [ylim], page 317,
[zlim], page 317.

15.2.2.2 Three-dimensional Function Plotting
ezplot3 (fx, fy, fz)
ezplot3 ( . . . , dom)
ezplot3 ( . . . , n)
ezplot3 ( . . . , "animate")
ezplot3 (hax, . . . )
h = ezplot3 ( . . . )
Plot a parametrically defined curve in three dimensions.
fx, fy, and fz are strings, inline functions, or function handles with one argument
defining the function. By default the plot is over the domain 0 <= t <= 2*pi with 500
points.
If dom is a two element vector, it represents the minimum and maximum values of t.
n is a scalar defining the number of points to use in plotting the function.
If the "animate" option is given then the plotting is animated in the style of comet3.
If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.
The optional return value h is a graphics handle to the created plot.
fx = @(t) cos (t);
fy = @(t) sin (t);
fz = @(t) t;
ezplot3 (fx, fy, fz, [0, 10*pi], 100);
See also: [plot3], page 339, [comet3], page 315, [ezplot], page 319, [ezmesh], page 345,
[ezsurf], page 347.

ezmesh (f)
ezmesh (fx, fy, fz)
ezmesh ( . . . , dom)
ezmesh ( . . . , n)
ezmesh ( . . . , "circ")
ezmesh (hax, . . . )
h = ezmesh ( . . . )
Plot the mesh defined by a function.
f is a string, inline function, or function handle with two arguments defining the
function. By default the plot is over the meshed domain -2*pi <= x | y <= 2*pi
with 60 points in each dimension.
If three functions are passed, then plot the parametrically defined function [fx(s,
t), fy(s, t), fz(s, t)].

346

GNU Octave

If dom is a two element vector, it represents the minimum and maximum values of
both x and y. If dom is a four element vector, then the minimum and maximum
values are [xmin xmax ymin ymax].
n is a scalar defining the number of points to use in each dimension.
If the argument "circ" is given, then the function is plotted over a disk centered on
the middle of the domain dom.
If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.
The optional return value h is a graphics handle to the created surface object.
Example 1: 2-argument function
f = @(x,y) sqrt (abs (x .* y)) ./ (1 + x.^2 + y.^2);
ezmesh (f, [-3, 3]);
Example 2: parametrically defined function
fx = @(s,t)
fy = @(s,t)
fz = @(s,t)
ezmesh (fx,

cos
sin
sin
fy,

(s) .* cos (t);
(s) .* cos (t);
(t);
fz, [-pi, pi, -pi/2, pi/2], 20);

See also: [mesh], page 323, [ezmeshc], page 346, [ezplot], page 319, [ezsurf], page 347,
[ezsurfc], page 347, [hidden], page 325.

ezmeshc (f)
ezmeshc (fx, fy, fz)
ezmeshc ( . . . , dom)
ezmeshc ( . . . , n)
ezmeshc ( . . . , "circ")
ezmeshc (hax, . . . )
h = ezmeshc ( . . . )
Plot the mesh and contour lines defined by a function.
f is a string, inline function, or function handle with two arguments defining the
function. By default the plot is over the meshed domain -2*pi <= x | y <= 2*pi
with 60 points in each dimension.
If three functions are passed, then plot the parametrically defined function [fx(s,
t), fy(s, t), fz(s, t)].
If dom is a two element vector, it represents the minimum and maximum values of
both x and y. If dom is a four element vector, then the minimum and maximum
values are [xmin xmax ymin ymax].
n is a scalar defining the number of points to use in each dimension.
If the argument "circ" is given, then the function is plotted over a disk centered on
the middle of the domain dom.
If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.
The optional return value h is a 2-element vector with a graphics handle for the
created mesh plot and a second handle for the created contour plot.

Chapter 15: Plotting

347

Example: 2-argument function
f = @(x,y) sqrt (abs (x .* y)) ./ (1 + x.^2 + y.^2);
ezmeshc (f, [-3, 3]);
See also: [meshc], page 324, [ezmesh], page 345, [ezplot], page 319, [ezsurf], page 347,
[ezsurfc], page 347, [hidden], page 325.

ezsurf (f)
ezsurf (fx, fy, fz)
ezsurf ( . . . , dom)
ezsurf ( . . . , n)
ezsurf ( . . . , "circ")
ezsurf (hax, . . . )
h = ezsurf ( . . . )
Plot the surface defined by a function.
f is a string, inline function, or function handle with two arguments defining the
function. By default the plot is over the meshed domain -2*pi <= x | y <= 2*pi
with 60 points in each dimension.
If three functions are passed, then plot the parametrically defined function [fx(s,
t), fy(s, t), fz(s, t)].
If dom is a two element vector, it represents the minimum and maximum values of
both x and y. If dom is a four element vector, then the minimum and maximum
values are [xmin xmax ymin ymax].
n is a scalar defining the number of points to use in each dimension.
If the argument "circ" is given, then the function is plotted over a disk centered on
the middle of the domain dom.
If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.
The optional return value h is a graphics handle to the created surface object.
Example 1: 2-argument function
f = @(x,y) sqrt (abs (x .* y)) ./ (1 + x.^2 + y.^2);
ezsurf (f, [-3, 3]);
Example 2: parametrically defined function
fx = @(s,t) cos (s) .* cos (t);
fy = @(s,t) sin (s) .* cos (t);
fz = @(s,t) sin (t);
ezsurf (fx, fy, fz, [-pi, pi, -pi/2, pi/2], 20);
See also: [surf], page 325, [ezsurfc], page 347, [ezplot], page 319, [ezmesh], page 345,
[ezmeshc], page 346, [shading], page 342.

ezsurfc
ezsurfc
ezsurfc
ezsurfc
ezsurfc

(f)
(fx, fy, fz)
( . . . , dom)
( . . . , n)
( . . . , "circ")

348

GNU Octave

ezsurfc (hax, . . . )
h = ezsurfc ( . . . )
Plot the surface and contour lines defined by a function.
f is a string, inline function, or function handle with two arguments defining the
function. By default the plot is over the meshed domain -2*pi <= x | y <= 2*pi
with 60 points in each dimension.
If three functions are passed, then plot the parametrically defined function [fx(s,
t), fy(s, t), fz(s, t)].
If dom is a two element vector, it represents the minimum and maximum values of
both x and y. If dom is a four element vector, then the minimum and maximum
values are [xmin xmax ymin ymax].
n is a scalar defining the number of points to use in each dimension.
If the argument "circ" is given, then the function is plotted over a disk centered on
the middle of the domain dom.
If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.
The optional return value h is a 2-element vector with a graphics handle for the
created surface plot and a second handle for the created contour plot.
Example:
f = @(x,y) sqrt (abs (x .* y)) ./ (1 + x.^2 + y.^2);
ezsurfc (f, [-3, 3]);
See also: [surfc], page 326, [ezsurf], page 347, [ezplot], page 319, [ezmesh], page 345,
[ezmeshc], page 346, [shading], page 342.

15.2.2.3 Three-dimensional Geometric Shapes
cylinder
cylinder (r)
cylinder (r, n)
cylinder (hax, . . . )
[x, y, z] = cylinder ( . . . )
Plot a 3-D unit cylinder.
The optional input r is a vector specifying the radius along the unit z-axis. The
default is [1 1] indicating radius 1 at Z == 0 and at Z == 1.
The optional input n determines the number of faces around the circumference of the
cylinder. The default value is 20.
If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.
If outputs are requested cylinder returns three matrices in meshgrid format, such
that surf (x, y, z) generates a unit cylinder.
Example:
[x, y, z] = cylinder (10:-1:0, 50);
surf (x, y, z);
title ("a cone");

Chapter 15: Plotting

349

See also: [ellipsoid], page 349, [rectangle], page 321, [sphere], page 349.

sphere
sphere
sphere
[x, y,

()
(n)
(hax, . . . )

z] = sphere ( . . . )
Plot a 3-D unit sphere.

The optional input n determines the number of faces around the circumference of the
sphere. The default value is 20.
If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.
If outputs are requested sphere returns three matrices in meshgrid format such that
surf (x, y, z) generates a unit sphere.
Example:
[x, y, z] = sphere (40);
surf (3*x, 3*y, 3*z);
axis equal;
title ("sphere of radius 3");
See also: [cylinder], page 348, [ellipsoid], page 349, [rectangle], page 321.
(xc, yc, zc, xr, yr, zr, n)
( . . . , n)
(hax, . . . )
= ellipsoid ( . . . )
Plot a 3-D ellipsoid.

ellipsoid
ellipsoid
ellipsoid
[x, y, z]

The inputs xc, yc, zc specify the center of the ellipsoid. The inputs xr, yr, zr specify
the semi-major axis lengths.
The optional input n determines the number of faces around the circumference of the
cylinder. The default value is 20.
If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.
If outputs are requested ellipsoid returns three matrices in meshgrid format, such
that surf (x, y, z) generates the ellipsoid.
See also: [cylinder], page 348, [rectangle], page 321, [sphere], page 349.

15.2.3 Plot Annotations
You can add titles, axis labels, legends, and arbitrary text to an existing plot. For example:
x = -10:0.1:10;
plot (x, sin (x));
title ("sin(x) for x = -10:0.1:10");
xlabel ("x");
ylabel ("sin (x)");
text (pi, 0.7, "arbitrary text");
legend ("sin (x)");

350

GNU Octave

The functions grid and box may also be used to add grid and border lines to the plot.
By default, the grid is off and the border lines are on.
Finally, arrows, text and rectangular or elliptic boxes can be added to highlight parts of
a plot using the annotation function. Those objects are drawn in an invisible axes, on top
of every other axes.

title (string)
title (string, prop, val, . . . )
title (hax, . . . )
h = title ( . . . )
Specify the string used as a title for the current axis.
An optional list of property/value pairs can be used to change the appearance of the
created title text object.
If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.
The optional return value h is a graphics handle to the created text object.
See also: [xlabel], page 351, [ylabel], page 351, [zlabel], page 351, [text], page 351.

legend
legend
legend
legend
legend
legend
legend
legend
legend
[hleg,

(str1, str2, . . . )
(matstr)
(cellstr)
( . . . , "location", pos)
( . . . , "orientation", orient)
(hax, . . . )
(hobjs, . . . )
(hax, hobjs, . . . )
("option")

hleg_obj, hplot, labels] = legend ( . . . )

Display a legend for the current axes using the specified strings as labels.
Legend entries may be specified as individual character string arguments, a character
array, or a cell array of character strings.
If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca. If the handles, hobjs, are not specified then the legend’s
strings will be associated with the axes’ descendants. legend works on line graphs,
bar graphs, etc. A plot must exist before legend is called.
The optional parameter pos specifies the location of the legend as follows:
pos
north
south
east
west
northeast
northwest
southeast

location of the legend
center top
center bottom
right center
left center
right top (default)
left top
right bottom

Chapter 15: Plotting

southwest
outside

351

left bottom
can be appended to any location string

The optional parameter orient determines if the key elements are placed vertically or
horizontally. The allowed values are "vertical" (default) or "horizontal".
The following customizations are available using option:
"show"

Show legend on the plot

"hide"

Hide legend on the plot

"toggle"

Toggles between "hide" and "show"

"boxon"

Show a box around legend (default)

"boxoff"

Hide the box around legend

"right"

Place label text to the right of the keys (default)

"left"

Place label text to the left of the keys

"off"

Delete the legend object

The optional output values are
hleg

The graphics handle of the legend object.

hleg obj

Graphics handles to the text and line objects which make up the legend.

hplot

Graphics handles to the plot objects which were used in making the legend.

labels

A cell array of strings of the labels in the legend.

The legend label text is either provided in the call to legend or is taken from the
DisplayName property of graphics objects. If no labels or DisplayNames are available,
then the label text is simply "data1", "data2", . . . , "dataN".
Implementation Note: A legend is implemented as an additional axes object of the
current figure with the "tag" set to "legend". Properties of the legend object may
be manipulated directly by using set.

text (x, y, string)
text (x, y, z, string)
text ( . . . , prop, val, . . . )
h = text ( . . . )
Create a text object with text string at position x, y, (z) on the current axes.
Multiple locations can be specified if x, y, (z) are vectors. Multiple strings can be
specified with a character matrix or a cell array of strings.
Optional property/value pairs may be used to control the appearance of the text.
The optional return value h is a vector of graphics handles to the created text objects.
See also: [gtext], page 374, [title], page 350, [xlabel], page 351, [ylabel], page 351,
[zlabel], page 351.
See Section 15.3.3.5 [Text Properties], page 401, for the properties that you can set.

352

GNU Octave

xlabel (string)
xlabel (string, property, val, . . . )
xlabel (hax, . . . )
h = xlabel ( . . . )
Specify the string used to label the x-axis of the current axis.
An optional list of property/value pairs can be used to change the properties of the
created text label.
If the first argument hax is an axes handle, then operate on this axes rather than the
current axes returned by gca.
The optional return value h is a graphics handle to the created text object.
See also: [ylabel], page 351, [zlabel], page 351, [datetick], page 829, [title], page 350,
[text], page 351.

clabel (c, h)
clabel (c, h, v)
clabel (c, h, "manual")
clabel (c)
clabel ( . . . , prop, val, . . . )
h = clabel ( . . . )
Add labels to the contours of a contour plot.
The contour levels are specified by the contour matrix c which is returned by contour,
contourc, contourf, and contour3. Contour labels are rotated to match the local
line orientation and centered on the line. The position of labels along the contour
line is chosen randomly.
If the argument h is a handle to a contour group object, then label this plot rather
than the one in the current axes returned by gca.
By default, all contours are labeled. However, the contours to label can be specified
by the vector v. If the "manual" argument is given then the contours to label can be
selected with the mouse.
Additional property/value pairs that are valid properties of text objects can be given
and are passed to the underlying text objects. Moreover, the contour group property "LabelSpacing" is available which determines the spacing between labels on a
contour to be specified. The default is 144 points, or 2 inches.
The optional return value h is a vector of graphics handles to the text objects representing each label. The "userdata" property of the text objects contains the numerical value of the contour label.
An example of the use of clabel is
[c, h] = contour (peaks (), -4 : 6);
clabel (c, h, -4:2:6, "fontsize", 12);
See also: [contour], page 302, [contourf], page 302, [contour3], page 304, [meshc],
page 324, [surfc], page 326, [text], page 351.

box
box on
box off

Chapter 15: Plotting

353

box (hax, . . . )
Control display of the axes border.
The argument may be either "on" or "off". If it is omitted, the current box state is
toggled.
If the first argument hax is an axes handle, then operate on this axes rather than the
current axes returned by gca.
See also: [axis], page 315, [grid], page 353.

grid
grid
grid
grid
grid
grid
grid

on
off
minor
minor on
minor off
(hax, . . . )
Control the display of plot grid lines.
The function state input may be either "on" or "off". If it is omitted, the current
grid state is toggled.
When the first argument is "minor" all subsequent commands modify the minor grid
rather than the major grid.
If the first argument hax is an axes handle, then operate on this axes rather than the
current axes returned by gca.
To control the grid lines for an individual axes use the set function. For example:
set (gca, "ygrid", "on");
See also: [axis], page 315, [box], page 352.

colorbar
colorbar (loc)
colorbar (delete_option)
colorbar (hcb, . . . )
colorbar (hax, . . . )
colorbar ( . . . , "peer", hax, . . . )
colorbar ( . . . , "location", loc, . . . )
colorbar ( . . . , prop, val, . . . )
h = colorbar ( . . . )
Add a colorbar to the current axes.
A colorbar displays the current colormap along with numerical rulings so that the
color scale can be interpreted.
The optional input loc determines the location of the colorbar. Valid values for loc
are
"EastOutside"
Place the colorbar outside the plot to the right. This is the default.
"East"

Place the colorbar inside the plot to the right.

354

GNU Octave

"WestOutside"
Place the colorbar outside the plot to the left.
"West"

Place the colorbar inside the plot to the left.

"NorthOutside"
Place the colorbar above the plot.
"North"

Place the colorbar at the top of the plot.

"SouthOutside"
Place the colorbar under the plot.
"South"

Place the colorbar at the bottom of the plot.

To remove a colorbar from a plot use any one of the following keywords for the
delete option: "delete", "hide", "off".
If the argument "peer" is given, then the following argument is treated as the axes
handle in which to add the colorbar. Alternatively, If the first argument hax is an
axes handle, then the colorbar is added to this axes, rather than the current axes
returned by gca.
If the first argument hcb is a handle to a colorbar object, then operate on this colorbar
directly.
Additional property/value pairs are passed directly to the underlying axes object.
The optional return value h is a graphics handle to the created colorbar object.
Implementation Note: A colorbar is created as an additional axes to the current figure
with the "tag" property set to "colorbar". The created axes object has the extra
property "location" which controls the positioning of the colorbar.
See also: [colormap], page 757.

annotation (type)
annotation ("line", x, y)
annotation ("arrow", x, y)
annotation ("doublearrow", x, y)
annotation ("textarrow", x, y)
annotation ("textbox", pos)
annotation ("rectangle", pos)
annotation ("ellipse", pos)
annotation ( . . . , prop, val)
annotation (hf, . . . )
h = annotation ( . . . )
Draw annotations to emphasize parts of a figure.
You may build a default annotation by specifying only the type of the annotation.
Otherwise you can select the type of annotation and then set its position using either
x and y coordinates for line-based annotations or a position vector pos for others. In
either case, coordinates are interpreted using the "units" property of the annotation
object. The default is "normalized", which means the lower left hand corner of the
figure has coordinates ‘[0 0]’ and the upper right hand corner ‘[1 1]’.

Chapter 15: Plotting

355

If the first argument hf is a figure handle, then plot into this figure, rather than the
current figure returned by gcf.
Further arguments can be provided in the form of prop/val pairs to customize the
annotation appearance.
The optional return value h is a graphics handle to the created annotation object.
This can be used with the set function to customize an existing annotation object.
All annotation objects share two properties:
• "units": the units in which coordinates are interpreted.
Its value may be one of "centimeters" | "characters" | "inches" |
"{normalized}" | "pixels" | "points".
• "position": a four-element vector [x0 y0 width height].
The vector specifies the coordinates (x0,y0) of the origin of the annotation object,
its width, and its height. The width and height may be negative, depending on
the orientation of the object.
Valid annotation types and their specific properties are described below:
"line"

Constructs a line. x and y must be two-element vectors specifying the x
and y coordinates of the two ends of the line.
The line can be customized using "linewidth", "linestyle", and
"color" properties the same way as for line objects.

"arrow"

Construct an arrow. The second point in vectors x and y specifies the
arrowhead coordinates.
Besides line properties, the arrowhead can be customized using
"headlength", "headwidth", and "headstyle" properties. Supported
values for "headstyle" property are: ["diamond" | "ellipse" |
"plain" | "rectangle" | "vback1" | "{vback2}" | "vback3"]

"doublearrow"
Construct a double arrow. Vectors x and y specify the arrowhead coordinates.
The line and the arrowhead can be customized as for arrow annotations, but some property names are duplicated:
"head1length"/"head2length", "head1width"/"head2width", etc.
The index 1 marks the properties of the arrowhead at the first point in
x and y coordinates.
"textarrow"
Construct an arrow with a text label at the opposite end from the arrowhead.
Use the "string" property to change the text string. The line and the
arrowhead can be customized as for arrow annotations, and the text
can be customized using the same properties as text graphics objects.
Note, however, that some text property names are prefixed with "text"
to distinguish them from arrow properties: "textbackgroundcolor",
"textcolor", "textedgecolor", "textlinewidth", "textmargin",
"textrotation".

356

GNU Octave

"textbox"
Construct a box with text inside. pos specifies the "position" property
of the annotation.
Use the "string" property to change the text string. You may use
"backgroundcolor", "edgecolor", "linestyle", and "linewidth"
properties to customize the box background color and edge appearance.
A limited set of text objects properties are also available; Besides
"font..." properties, you may also use "horizontalalignment" and
"verticalalignment" to position the text inside the box.
Finally, the "fitboxtotext" property controls the actual extent of the
box. If "on" (the default) the box limits are fitted to the text extent.
"rectangle"
Construct a rectangle. pos specifies the "position" property of the annotation.
You may use "facecolor", "color", "linestyle", and "linewidth"
properties to customize the rectangle background color and edge appearance.
"ellipse"
Construct an ellipse. pos specifies the "position" property of the annotation.
See "rectangle" annotations for customization.
See also: [xlabel], page 351, [ylabel], page 351, [zlabel], page 351, [title], page 350,
[text], page 351, [gtext], page 374, [legend], page 350, [colorbar], page 353.

15.2.4 Multiple Plots on One Page
Octave can display more than one plot in a single figure. The simplest way to do this is to
use the subplot function to divide the plot area into a series of subplot windows that are
indexed by an integer. For example,
subplot (2, 1, 1)
fplot (@sin, [-10, 10]);
subplot (2, 1, 2)
fplot (@cos, [-10, 10]);
creates a figure with two separate axes, one displaying a sine wave and the other a cosine
wave. The first call to subplot divides the figure into two plotting areas (two rows and one
column) and makes the first plot area active. The grid of plot areas created by subplot is
numbered in row-major order (left to right, top to bottom). After plotting a sine wave, the
next call to subplot activates the second subplot area, but does not re-partition the figure.

subplot
subplot
subplot
subplot
subplot
subplot

(rows, cols, index)
(rcn)
(hax)
( . . . , "align")
( . . . , "replace")
( . . . , "position", pos)

Chapter 15: Plotting

357

subplot ( . . . , prop, val, . . . )
hax = subplot ( . . . )
Set up a plot grid with rows by cols subwindows and set the current axes for plotting
(gca) to the location given by index.
If only one numeric argument is supplied, then it must be a three digit value specifying
the number of rows in digit 1, the number of columns in digit 2, and the plot index
in digit 3.
The plot index runs row-wise; First, all columns in a row are numbered and then the
next row is filled.
For example, a plot with 2x3 grid will have plot indices running as follows:
1

2

3

4

5

6

index may also be a vector. In this case, the new axes will enclose the grid locations
specified. The first demo illustrates this:
demo ("subplot", 1)
The index of the subplot to make active may also be specified by its axes handle, hax,
returned from a previous subplot command.
If the option "align" is given then the plot boxes of the subwindows will align, but
this may leave no room for axes tick marks or labels.
If the option "replace" is given then the subplot axes will be reset, rather than just
switching the current axes for plotting to the requested subplot.
The "position" property can be used to exactly position the subplot axes within
the current figure. The option pos is a 4-element vector [x, y, width, height] that
determines the location and size of the axes. The values in pos are normalized in the
range [0,1].
Any property/value pairs are passed directly to the underlying axes object.
If the output hax is requested, subplot returns the axes handle for the subplot. This
is useful for modifying the properties of a subplot using set.
See also: [axes], page 378, [plot], page 288, [gca], page 382, [set], page 383.

15.2.5 Multiple Plot Windows
You can open multiple plot windows using the figure function. For example,
figure (1);
fplot (@sin, [-10, 10]);
figure (2);
fplot (@cos, [-10, 10]);
creates two figures, with the first displaying a sine wave and the second a cosine wave.
Figure numbers must be positive integers.

358

GNU Octave

figure
figure n
figure (n)
figure ( . . . , "property", value, . . . )
h = figure ( . . . )
Create a new figure window for plotting.
If no arguments are specified, a new figure with the next available number is created.
If called with an integer n, and no such numbered figure exists, then a new figure with
the specified number is created. If the figure already exists then it is made visible
and becomes the current figure for plotting.
Multiple property-value pairs may be specified for the figure object, but they must
appear in pairs.
The optional return value h is a graphics handle to the created figure object.
See also: [axes], page 378, [gcf], page 381, [clf], page 362, [close], page 363.

15.2.6 Manipulation of Plot Objects
pan
pan
pan
pan
pan
pan

on
off
xon
yon
(hfig, option)
Control the interactive panning mode of a figure in the GUI.
Given the option "on" or "off", set the interactive pan mode on or off.
With no arguments, toggle the current pan mode on or off.
Given the option "xon" or "yon", enable pan mode for the x or y axis only.
If the first argument hfig is a figure, then operate on the given figure rather than the
current figure as returned by gcf.
See also: [rotate3d], page 358, [zoom], page 359.

rotate (h, dir, alpha)
rotate ( . . . , origin)
Rotate the plot object h through alpha degrees around the line with direction dir and
origin origin.
The default value of origin is the center of the axes object that is the parent of h.
If h is a vector of handles, they must all have the same parent axes object.
Graphics objects that may be rotated are lines, surfaces, patches, and images.

rotate3d
rotate3d on
rotate3d off
rotate3d (hfig, option)
Control the interactive 3-D rotation mode of a figure in the GUI.
Given the option "on" or "off", set the interactive rotate mode on or off.

Chapter 15: Plotting

359

With no arguments, toggle the current rotate mode on or off.
If the first argument hfig is a figure, then operate on the given figure rather than the
current figure as returned by gcf.
See also: [pan], page 358, [zoom], page 359.

zoom
zoom
zoom
zoom
zoom
zoom
zoom
zoom
zoom

(factor)

on
off
xon
yon
out
reset
(hfig, option)
Zoom the current axes object or control the interactive zoom mode of a figure in the
GUI.
Given a numeric argument greater than zero, zoom by the given factor. If the zoom
factor is greater than one, zoom in on the plot. If the factor is less than one, zoom
out. If the zoom factor is a two- or three-element vector, then the elements specify
the zoom factors for the x, y, and z axes respectively.
Given the option "on" or "off", set the interactive zoom mode on or off.
With no arguments, toggle the current zoom mode on or off.
Given the option "xon" or "yon", enable zoom mode for the x or y-axis only.
Given the option "out", zoom to the initial zoom setting.
Given the option "reset", store the current zoom setting so that zoom out will return
to this zoom level.
If the first argument hfig is a figure, then operate on the given figure rather than the
current figure as returned by gcf.
See also: [pan], page 358, [rotate3d], page 358.

15.2.7 Manipulation of Plot Windows
By default, Octave refreshes the plot window when a prompt is printed, or when waiting
for input. The drawnow function is used to cause a plot window to be updated.

drawnow ()
drawnow ("expose")
drawnow (term, file, debug_file)
Update figure windows and their children.
The event queue is flushed and any callbacks generated are executed.
With the optional argument "expose", only graphic objects are updated and no other
events or callbacks are processed.
The third calling form of drawnow is for debugging and is undocumented.
See also: [refresh], page 360.

360

GNU Octave

Only figures that are modified will be updated. The refresh function can also be used
to cause an update of the current figure, even if it is not modified.

refresh ()
refresh (h)
Refresh a figure, forcing it to be redrawn.
When called without an argument the current figure is redrawn. Otherwise, the figure
with graphic handle h is redrawn.
See also: [drawnow], page 359.
Normally, high-level plot functions like plot or mesh call newplot to initialize the state
of the current axes so that the next plot is drawn in a blank window with default property
settings. To have two plots superimposed over one another, use the hold function. For
example,
hold on;
x = -10:0.1:10;
plot (x, sin (x));
plot (x, cos (x));
hold off;
displays sine and cosine waves on the same axes. If the hold state is off, consecutive plotting
commands like this will only display the last plot.

newplot ()
newplot (hfig)
newplot (hax)
hax = newplot ( . . . )
Prepare graphics engine to produce a new plot.
This function is called at the beginning of all high-level plotting functions. It is not
normally required in user programs. newplot queries the "NextPlot" field of the
current figure and axes to determine what to do.
Figure NextPlot
"new"

Action
Create a new figure and make it the current figure.

"add" (default)

Add new graphic objects to the current figure.

"replacechildren"

Delete child objects whose HandleVisibility is set to "on". Set
NextPlot property to "add". This typically clears a figure, but
leaves in place hidden objects such as menubars. This is equivalent
to clf.

"replace"

Delete all child objects of the figure and reset all figure properties
to their defaults. However, the following four properties are not
reset: Position, Units, PaperPosition, PaperUnits. This is equivalent to clf reset.

Chapter 15: Plotting

361

Axes NextPlot
"add"

Action
Add new graphic objects to the current axes. This is equivalent
to hold on.

"replacechildren"

Delete child objects whose HandleVisibility is set to "on", but
leave axes properties unmodified. This typically clears a plot, but
preserves special settings such as log scaling for axes. This is
equivalent to cla.

"replace" (default)

Delete all child objects of the axes and reset all axes properties
to their defaults. However, the following properties are not reset:
Position, Units. This is equivalent to cla reset.
If the optional input hfig or hax is given then prepare the specified figure or axes
rather than the current figure and axes.
The optional return value hax is a graphics handle to the created axes object (not
figure).
Caution: Calling newplot may change the current figure and current axes.

hold
hold on
hold off
hold (hax, . . . )
Toggle or set the "hold" state of the plotting engine which determines whether new
graphic objects are added to the plot or replace the existing objects.
hold on

Retain plot data and settings so that subsequent plot commands are
displayed on a single graph. Line color and line style are advanced for
each new plot added.

hold all (deprecated)
Equivalent to hold on.
hold off

Restore default graphics settings which clear the graph and reset axes
properties before each new plot command. (default).

hold

Toggle the current hold state.

When given the additional argument hax, the hold state is modified for this axes
rather than the current axes returned by gca.
To query the current hold state use the ishold function.
See also: [ishold], page 361, [cla], page 362, [clf], page 362, [newplot], page 360.

ishold
ishold (hax)
ishold (hfig)
Return true if the next plot will be added to the current plot, or false if the plot
device will be cleared before drawing the next plot.
If the first argument is an axes handle hax or figure handle hfig then operate on this
plot rather than the current one.
See also: [hold], page 361, [newplot], page 360.

362

GNU Octave

To clear the current figure, call the clf function. To clear the current axis, call the cla
function. To bring the current figure to the top of the window stack, call the shg function.
To delete a graphics object, call delete on its index. To close the figure window, call the
close function.

clf
clf
clf
clf
h =

reset
(hfig)
(hfig, "reset")
clf ( . . . )
Clear the current figure window.
clf operates by deleting child graphics objects with visible handles (HandleVisibility
= "on").
If the optional argument "reset" is specified, delete all child objects including those
with hidden handles and reset all figure properties to their defaults. However, the
following properties are not reset: Position, Units, PaperPosition, PaperUnits.
If the first argument hfig is a figure handle, then operate on this figure rather than
the current figure returned by gcf.
The optional return value h is the graphics handle of the figure window that was
cleared.
See also: [cla], page 362, [close], page 363, [delete], page 362, [reset], page 424.

cla
cla reset
cla (hax)
cla (hax, "reset")
Clear the current axes.
cla operates by deleting child graphic objects with visible handles (HandleVisibility
= "on").
If the optional argument "reset" is specified, delete all child objects including those
with hidden handles and reset all axes properties to their defaults. However, the
following properties are not reset: Position, Units.
If the first argument hax is an axes handle, then operate on this axes rather than the
current axes returned by gca.
See also: [clf], page 362, [delete], page 362, [reset], page 424.

shg
Show the graph window.
Currently, this is the same as executing drawnow.
See also: [drawnow], page 359, [figure], page 357.

delete (file)
delete (file1, file2, . . . )
delete (handle)
Delete the named file or graphics handle.

Chapter 15: Plotting

363

file may contain globbing patterns such as ‘*’. Multiple files to be deleted may be
specified in the same function call.
handle may be a scalar or vector of graphic handles to delete.
Programming Note: Deleting graphics objects is the proper way to remove features
from a plot without clearing the entire figure.
See also: [clf], page 362, [cla], page 362, [unlink], page 831, [rmdir], page 832.

close
close
close
close
close
close
close

h
(h)
(h, "force")
all
all hidden
all force

Close figure window(s).
When called with no arguments, close the current figure. This is equivalent to close
(gcf). If the input h is a graphic handle, or vector of graphics handles, then close
each figure in h.
If the argument "all" is given then all figures with visible handles (HandleVisibility
= "on") are closed.
If the argument "all hidden" is given then all figures, including hidden ones, are
closed.
If the argument "force" is given then figures are closed even when
"closerequestfcn" has been altered to prevent closing the window.
Implementation Note: close operates by calling the function specified by the
"closerequestfcn" property for each figure. By default, the function closereq is
used. It is possible that the function invoked will delay or abort removing the figure.
To remove a figure without executing any callback functions use delete. When
writing a callback function to close a window do not use close to avoid recursion.
See also: [closereq], page 363, [delete], page 362.

closereq ()
Close the current figure and delete all graphics objects associated with it.
By default, the "closerequestfcn" property of a new plot figure points to this
function.
See also: [close], page 363, [delete], page 362.

15.2.8 Use of the interpreter Property
All text objects—such as titles, labels, legends, and text—include the property
"interpreter" that determines the manner in which special control sequences in the text
are rendered.
The interpreter property can take three values: "none", "tex", "latex". If the interpreter is set to "none" then no special rendering occurs—the displayed text is a verbatim
copy of the specified text. Currently, the "latex" interpreter is not implemented and is
equivalent to "none".

364

GNU Octave

The "tex" option implements a subset of TEX functionality when rendering text. This
allows the insertion of special glyphs such as Greek characters or mathematical symbols.
Special characters are inserted by using a backslash (\) character followed by a code, as
shown in Table 15.1.

Besides special glyphs, the formatting of the text can be changed within the string by
using the codes

\bf
\it
\sl
\rm

Bold font
Italic font
Oblique Font
Normal font

These codes may be used in conjunction with the { and } characters to limit the change
to a part of the string. For example,

xlabel (’{\bf H} = a {\bf V}’)

where the character ’a’ will not appear in bold font. Note that to avoid having Octave
interpret the backslash character in the strings, the strings themselves should be in single
quotes.

It is also possible to change the fontname and size within the text

\fontname{fontname}
\fontsize{size}

Specify the font to use
Specify the size of the font to use

The color of the text may also be changed inline using either a string (e.g., "red") or
numerically with a Red-Green-Blue (RGB) specification (.e.g., [1 0 0], also red).

\color{color}
\color[rgb]{R G B}

Specify the color as a string
Specify the color numerically

Finally, superscripting and subscripting can be controlled with the ’^’ and ’_’ characters. If the ’^’ or ’_’ is followed by a { character, then all of the block surrounded by the {
} pair is superscripted or subscripted. Without the { } pair, only the character immediately
following the ’^’ or ’_’ is changed.

Chapter 15: Plotting

Code
\alpha
\delta
\eta
\iota
\mu
\o
\rho
\tau
\chi

Greek Lowercase
Sym
α
δ
η
ι
µ
o
ρ
τ
χ

365

Letters
Code
\beta
\epsilon
\theta
\kappa
\nu
\pi
\sigma
\upsilon
\psi

Sym
β

θ
κ
ν
π
σ
υ
ψ

Code
\gamma
\zeta
\vartheta
\lambda
\xi
\varpi
\varsigma
\phi
\omega

Sym
γ
ζ
ϑ
λ
ξ
$
ς
φ
ω

Sym
Δ
Ξ
ϒ
Ω

Code
\Theta
\Pi
\Phi

Sym
Θ
Π
Φ

Sym
℘
∂
∇
∀
♣
♠

Code
\Re
\infty
\surd
\exists
\diamondsuit

Sym
<
∞
√

Code

Sym

Code

Sym

Code
\cdot
\circ
\cap
\wedge
\oslash

Sym
·
◦
∩
∧

Code
\times
\bullet
\cup
\oplus

Sym
×
•
∪
⊕

Greek Uppercase Letters
Code
\Gamma
\Lambda
\Sigma
\Psi

Sym
Γ
Λ
Σ
Ψ

Code
\Delta
\Xi
\Upsilon
\Omega

Misc Symbols Type Ord
Code
\aleph
\Im
\prime
\angle
\neg
\heartsuit

Sym
ℵ
=
0
6

¬
♥

Code
\wp
\partial
\nabla
\forall
\clubsuit
\spadesuit

∃
♦

“Large” Operators
Code
\int

Sym
R

Binary operators
Code
\pm
\ast
\div
\vee
\otimes

Sym
±
∗
÷
∨
⊗

Table 15.1: Available special characters in TEX mode

366

GNU Octave

Relations
Code
\leq
\in
\supseteq
\equiv
\cong

Sym
≤
∈
⊇
≡
∼
=

Code
\subset
\geq
\ni
\sim
\propto

Sym
⊂
≥
3
∼
∝

Code
\subseteq
\supset
\mid
\approx
\perp

Sym
⊆
⊃
|
≈
⊥

Sym
←
⇒
↓

Code
\Leftarrow
\leftrightarrow

Sym
⇐
↔

Code
\rightarrow
\uparrow

Sym
→
↑

Sym
h
i

Code
\lceil
\rceil

Sym
d
e

Sym

Code

Sym

Sym

Code
\copyright

Sym
c

Arrows
Code
\leftarrow
\Rightarrow
\downarrow

Openings and Closings
Code
\lfloor
\rfloor

Sym
b
c

Code
\langle
\rangle

Alternate Names
Sym
Code
6=

Code
\neq

Other (not in Appendix F Tables)
Code
\ldots
\deg

Sym
...
◦

Code
\0

Table 15.1: Available special characters in TEX mode (cont.)

15.2.8.1 Degree Symbol
Conformance to both TEX and matlab with respect to the \circ symbol is impossible. While TEX translates this symbol to Unicode 2218 (U+2218), matlab maps this to
Unicode 00B0 (U+00B0) instead. Octave has chosen to follow the TEX specification, but
has added the additional symbol \deg which maps to the degree symbol (U+00B0).

15.2.9 Printing and Saving Plots
The print command allows you to send plots to you printer and to save plots in a variety
of formats. For example,
print -dpsc
prints the current figure to a color PostScript printer. And,

Chapter 15: Plotting

367

print -deps foo.eps

saves the current figure to an encapsulated PostScript file called foo.eps.

The current graphic toolkits produce very similar graphic displays but differ in their
capability to display text and in print capabilities. In particular, the OpenGL based toolkits
such as fltk and qt do not support the "interpreter" property of text objects. This
means that when using OpenGL toolkits special symbols drawn with the "tex" interpreter
will appear correctly on-screen but will be rendered with interpreter "none" when printing
unless one of the standalone (see below) modes is used. These modes provide access to
the pdflatex processor and therefore allow full use of LATEX commands.

A complete example showing
-dpdflatexstandalone option is:

the

capabilities

of

text

printing

using

the

x = 0:0.01:3;
hf = figure ();
plot (x, erf (x));
hold on;
plot (x, x, "r");
axis ([0, 3, 0, 1]);
text (0.65, 0.6175, [’$\displaystyle\leftarrow x = {2\over\sqrt{\pi}}’...
’\int_{0}^{x}e^{-t^2} dt = 0.6175$’]);
xlabel ("x");
ylabel ("erf (x)");
title ("erf (x) with text annotation");
set (hf, "visible", "off");
print (hf, "plot15_7.pdf", "-dpdflatexstandalone");
set (hf, "visible", "on");
system ("pdflatex plot15_7");
open ("plot15_7.pdf");

The result of this example can be seen in Figure 15.7

368

GNU Octave

erf (x) with text annotation
1

erf (x)

0.8
2
←x= √
π

0.6

Z

x

2

e−t dt = 0.6175

0

0.4

0.2

0

0

0.5

1

1.5

x

2

2.5

3

Figure 15.7: Example of inclusion of text with use of -dpdflatexstandalone
()
(options)
(filename, options)
(h, filename, options)
Print a plot, or save it to a file.

print
print
print
print

Both output formatted for printing (PDF and PostScript), and many bitmapped and
vector image formats are supported.
filename defines the name of the output file. If the filename has no suffix, one is
inferred from the specified device and appended to the filename. If no filename is
specified, the output is sent to the printer.
h specifies the handle of the figure to print. If no handle is specified the current figure
is used.
For output to a printer, PostScript file, or PDF file, the paper size is specified by
the figure’s papersize property. The location and size of the image on the page
are specified by the figure’s paperposition property. The orientation of the page is
specified by the figure’s paperorientation property.
The width and height of images are specified by the figure’s paperposition(3:4)
property values.
The print command supports many options:
-fh

Specify the handle, h, of the figure to be printed. The default is the
current figure.

-Pprinter
Set the printer name to which the plot is sent if no filename is specified.

Chapter 15: Plotting

369

-Gghostscript_command
Specify the command for calling Ghostscript. For Unix and Windows the
defaults are "gs" and "gswin32c", respectively.
-color
-mono

Color or monochrome output.

-solid
-dashed

Force all lines to be solid or dashed, respectively.

-portrait
-landscape
Specify the orientation of the plot for printed output. For non-printed
output the aspect ratio of the output corresponds to the plot area defined
by the "paperposition" property in the orientation specified. This option is equivalent to changing the figure’s "paperorientation" property.
-TextAlphaBits=n
-GraphicsAlphaBits=n
Octave is able to produce output for various printers, bitmaps, and vector
formats by using Ghostscript. For bitmap and printer output anti-aliasing
is applied using Ghostscript’s TextAlphaBits and GraphicsAlphaBits options. The default number of bits are 4 and 1 respectively. Allowed values
for N are 1, 2, or 4.
-ddevice

The available output format is specified by the option device, and is one
of:
ps
ps2
psc
psc2
eps
eps2
epsc
epsc2

PostScript (level 1 and 2, mono and color). The OpenGLbased toolkits always generate PostScript level 3.0.

Encapsulated PostScript (level 1 and 2, mono and color). The
OpenGL-based toolkits always generate PostScript level 3.0.

pslatex
epslatex
pdflatex
pslatexstandalone
epslatexstandalone
pdflatexstandalone
Generate a LATEX file filename.tex for the text portions of
a plot and a file filename.(ps|eps|pdf) for the remaining
graphics. The graphics file suffix .ps|eps|pdf is determined
by the specified device type. The LATEX file produced by the
‘standalone’ option can be processed directly by LATEX. The

370

GNU Octave

file generated without the ‘standalone’ option is intended to
be included from another LATEX document. In either case, the
LATEX file contains an \includegraphics command so that
the generated graphics file is automatically included when
the LATEX file is processed. The text that is written to the
LATEX file contains the strings exactly as they were specified in the plot. If any special characters of the TEX mode
interpreter were used, the file must be edited before LATEX
processing. Specifically, the special characters must be enclosed with dollar signs ($ ... $), and other characters that
are recognized by LATEX may also need editing (.e.g., braces).
The ‘pdflatex’ device, and any of the ‘standalone’ formats,
are not available with the Gnuplot toolkit.
epscairo
pdfcairo
epscairolatex
pdfcairolatex
epscairolatexstandalone
pdfcairolatexstandalone
Generate Cairo based output when using the Gnuplot graphics toolkit. The ‘epscairo’ and ‘pdfcairo’ devices are synonymous with the ‘epsc’ device. The LATEX variants generate a LATEX file, filename.tex, for the text portions of a
plot, and an image file, filename.(eps|pdf), for the graph
portion of the plot. The ‘standalone’ variants behave as
described for ‘epslatexstandalone’ above.
ill
aifm
canvas

Adobe Illustrator (Obsolete for Gnuplot versions > 4.2)
Javascript-based drawing on HTML5 canvas viewable in a
web browser (only available for the Gnuplot graphics toolkit).

cdr
corel

CorelDraw

dxf

AutoCAD

emf
meta

Microsoft Enhanced Metafile

fig

XFig. For the Gnuplot graphics toolkit, the additional options -textspecial or -textnormal can be used to control
whether the special flag should be set for the text in the figure. (default is -textnormal)

gif

GIF image (only available for the Gnuplot graphics toolkit)

hpgl

HP plotter language

jpg
jpeg

JPEG image

Chapter 15: Plotting

svg

371

latex

LATEX picture environment (only available for the Gnuplot
graphics toolkit).

mf

Metafont

png

Portable network graphics

pbm

PBMplus

pdf

Portable document format

Scalable vector graphics

tikz
tikzstandalone
Generate a LATEX file using PGF/TikZ format. The OpenGL-based
toolkits create a PGF file while Gnuplot creates a TikZ file. The
‘tikzstandalone’ device produces a LATEX document which includes
the TikZ file (‘tikzstandalone’ and is only available for the Gnuplot
graphics toolkit).
If the device is omitted, it is inferred from the file extension, or if there
is no filename it is sent to the printer as PostScript.
-dghostscript_device
Additional devices are supported by Ghostscript. Some examples are;
pdfwrite

Produces pdf output from eps

ljet2p

HP LaserJet IIP

pcx24b

24-bit color PCX file format

ppm

Portable Pixel Map file format

For a complete list, type system ("gs -h") to see what formats and
devices are available.
When Ghostscript output is sent to a printer the size is determined by the
figure’s "papersize" property. When the output is sent to a file the size
is determined by the plot box defined by the figure’s "paperposition"
property.
-append

Append PostScript or PDF output to a pre-existing file of the same type.

-rNUM

Resolution of bitmaps in pixels per inch. For both metafiles and SVG the
default is the screen resolution; for other formats it is 150 dpi. To specify
screen resolution, use "-r0".

-loose
-tight

Force a tight or loose bounding box for eps files. The default is loose.

-preview

Add a preview to eps files. Supported formats are:
-interchange
Provide an interchange preview.
-metafile
Provide a metafile preview.

372

GNU Octave

-pict

Provide pict preview.

-tiff

Provide a tiff preview.

-Sxsize,ysize
Plot size in pixels for EMF, GIF, JPEG, PBM, PNG, and SVG. For
PS, EPS, PDF, and other vector formats the plot size is in points. This
option is equivalent to changing the size of the plot box associated with
the "paperposition" property. When using the command form of the
print function you must quote the xsize,ysize option. For example, by
writing "-S640,480".
-Ffontname
-Ffontname:size
-F:size
Use fontname and/or fontsize for all text. fontname is ignored for some
devices: dxf, fig, hpgl, etc.
The filename and options can be given in any order.
Example: Print to a file using the pdf device.
figure (1);
clf ();
surf (peaks);
print figure1.pdf
Example: Print to a file using jpg device.
clf ();
surf (peaks);
print -djpg figure2.jpg
Example: Print to printer named PS printer using ps format.
clf ();
surf (peaks);
print -dpswrite -PPS_printer
See also: [saveas], page 372, [hgsave], page 373, [orient], page 373, [figure], page 357.

saveas (h, filename)
saveas (h, filename, fmt)
Save graphic object h to the file filename in graphic format fmt.
fmt should be one of the following formats:
ps

PostScript

eps

Encapsulated PostScript

jpg

JPEG Image

png

PNG Image

emf

Enhanced Meta File

pdf

Portable Document Format

Chapter 15: Plotting

373

All device formats specified in print may also be used. If fmt is omitted it is extracted
from the extension of filename. The default format is "pdf".
clf ();
surf (peaks);
saveas (1, "figure1.png");
See also: [print], page 368, [hgsave], page 373, [orient], page 373.

orient (orientation)
orient (hfig, orientation)
orientation = orient ()
orientation = orient (hfig)
Query or set the print orientation for figure hfig.
Valid values for orientation are "portrait", "landscape", and "tall".
The "landscape" option changes the orientation so the plot width is larger than the
plot height. The "paperposition" is also modified so that the plot fills the page,
while leaving a 0.25 inch border.
The "tall" option sets the orientation to "portrait" and fills the page with the
plot, while leaving a 0.25 inch border.
The "portrait" option (default) changes the orientation so the plot height is larger
than the plot width. It also restores the default "paperposition" property.
When called with no arguments, return the current print orientation.
If the argument hfig is omitted, then operate on the current figure returned by gcf.
See also: [print], page 368, [saveas], page 372.
print and saveas are used when work on a plot has finished and the output must be in a
publication-ready format. During intermediate stages it is often better to save the graphics
object and all of its associated information so that changes—to colors, axis limits, marker
styles, etc.—can be made easily from within Octave. The hgsave/hgload commands can
be used to save and re-create a graphics object.

hgsave (filename)
hgsave (h, filename)
hgsave (h, filename, fmt)
Save the graphics handle h to the file filename in the format fmt.
If unspecified, h is the current figure as returned by gcf.
When filename does not have an extension the default filename extension .ofig will
be appended.
If present, fmt should be one of the following:
• -binary, -float-binary
• -hdf5, -float-hdf5
• -V7, -v7, -7, -mat7-binary
• -V6, -v6, -6, -mat6-binary
• -text
• -zip, -z

374

GNU Octave

When producing graphics for final publication use print or saveas. When it is important to be able to continue to edit a figure as an Octave object, use hgsave/hgload.
See also: [hgload], page 374, [hdl2struct], page 385, [saveas], page 372, [print],
page 368.

h = hgload (filename)
Load the graphics object in filename into the graphics handle h.
If filename has no extension, Octave will try to find the file with and without the
standard extension of .ofig.
See also: [hgsave], page 373, [struct2hdl], page 385.

15.2.10 Interacting with Plots
The user can select points on a plot with the ginput function or select the position at which
to place text on the plot with the gtext function using the mouse.

[x, y, buttons] = ginput (n)
[x, y, buttons] = ginput ()
Return the position and type of mouse button clicks and/or key strokes in the current
figure window.
If n is defined, then capture n events before returning. When n is not defined ginput
will loop until the return key RET is pressed.
The return values x, y are the coordinates where the mouse was clicked in the units
of the current axes. The return value button is 1, 2, or 3 for the left, middle, or right
button. If a key is pressed the ASCII value is returned in button.
Implementation Note: ginput is intenteded for 2-D plots. For 3-D plots see the
currentpoint property of the current axes which can be transformed with knowledge
of the current view into data units.
See also: [gtext], page 374, [waitforbuttonpress], page 374.

waitforbuttonpress ()
b = waitforbuttonpress ()
Wait for mouse click or key press over the current figure window.
The return value of b is 0 if a mouse button was pressed or 1 if a key was pressed.
See also: [waitfor], page 815, [ginput], page 374, [kbhit], page 251.

gtext (s)
gtext ({s1, s2, . . . })
gtext ({s1; s2; . . . })
gtext ( . . . , prop, val, . . . )
h = gtext ( . . . )
Place text on the current figure using the mouse.
The text is defined by the string s. If s is a cell string organized as a row vector then
each string of the cell array is written to a separate line. If s is organized as a column
vector then one string element of the cell array is placed for every mouse click.
Optional property/value pairs are passed directly to the underlying text objects.

Chapter 15: Plotting

375

The optional return value h is a graphics handle to the created text object(s).
See also: [ginput], page 374, [text], page 351.
More sophisticated user interaction mechanisms can be obtained using the ui* family of
functions, see Section 35.3 [UI Elements], page 808.

15.2.11 Test Plotting Functions
The functions sombrero and peaks provide a way to check that plotting is working. Typing
either sombrero or peaks at the Octave prompt should display a three-dimensional plot.

sombrero ()
sombrero (n)
z = sombrero ( . . . )
[x, y, z] = sombrero ( . . . )
Plot the familiar 3-D sombrero function.
The function plotted is
p
sin( (x2 + y2 ))
z= p 2
(x + y2 )
Called without a return argument, sombrero plots the surface of the above function
over the meshgrid [-8,8] using surf.
If n is a scalar the plot is made with n grid lines. The default value for n is 41.
When called with output arguments, return the data for the function evaluated over
the meshgrid. This can subsequently be plotted with surf (x, y, z).
See also: [peaks], page 375, [meshgrid], page 339, [mesh], page 323, [surf], page 325.

peaks ()
peaks (n)
peaks (x, y)
z = peaks ( . . . )
[x, y, z] = peaks ( . . . )
Plot a function with lots of local maxima and minima.
The function has the form




2
2
2
2
x
1
f (x, y) = 3(1 − x) e(−x −(y+1) ) − 10
− x3 − y 5 − e(−(x+1) −y )
5
3
Called without a return argument, peaks plots the surface of the above function using
surf.
If n is a scalar, peaks plots the value of the above function on an n-by-n mesh over
the range [-3,3]. The default value for n is 49.
If n is a vector, then it represents the grid values over which to calculate the function.
If x and y are specified then the function value is calculated over the specified grid
of vertices.
When called with output arguments, return the data for the function evaluated over
the meshgrid. This can subsequently be plotted with surf (x, y, z).

2

See also: [sombrero], page 375, [meshgrid], page 339, [mesh], page 323, [surf],
page 325.

376

GNU Octave

15.3 Graphics Data Structures
15.3.1 Introduction to Graphics Structures
The graphics functions use pointers, which are of class graphics handle, in order to address
the data structures which control visual display. A graphics handle may point to any one of
a number of different base object types and these objects are the graphics data structures
themselves. The primitive graphic object types are: figure, axes, line, text, patch,
surface, text, image, and light.
Each of these objects has a function by the same name, and, each of these functions
returns a graphics handle pointing to an object of the corresponding type. In addition
there are several functions which operate on properties of the graphics objects and which
also return handles: the functions plot and plot3 return a handle pointing to an object
of type line, the function subplot returns a handle pointing to an object of type axes, the
function fill returns a handle pointing to an object of type patch, the functions area,
bar, barh, contour, contourf, contour3, surf, mesh, surfc, meshc, errorbar, quiver,
quiver3, scatter, scatter3, stair, stem, stem3 each return a handle to a complex data
structure as documented in [Data Sources], page 433.
The graphics objects are arranged in a hierarchy:
1. The root is at 0. In other words, get (0) returns the properties of the root object.
2. Below the root are figure objects.
3. Below the figure objects are axes objects.
4. Below the axes objects are line, text, patch, surface, image, and light objects.
Graphics handles may be distinguished from function handles (see Section 11.11.1 [Function Handles], page 211) by means of the function ishandle. ishandle returns true if its
argument is a handle of a graphics object. In addition, a figure or axes object may be tested
using isfigure or isaxes respectively. The test functions return true only if the argument
is both a handle and of the correct type (figure or axes).
The whos function can be used to show the object type of each currently defined graphics
handle. (Note: this is not true today, but it is, I hope, considered an error in whos. It may
be better to have whos just show graphics handle as the class, and provide a new function
which, given a graphics handle, returns its object type. This could generalize the ishandle()
functions and, in fact, replace them.)
The get and set commands are used to obtain and set the values of properties of graphics
objects. In addition, the get command may be used to obtain property names.
For example, the property "type" of the graphics object pointed to by the graphics
handle h may be displayed by:
get (h, "type")
The properties and their current values are returned by get (h) where h is a handle
of a graphics object. If only the names of the allowed properties are wanted they may be
displayed by: get (h, "").
Thus, for example:
h = figure ();
get (h, "type")
ans = figure

Chapter 15: Plotting

377

get (h, "");
error: get: ambiguous figure property name ; possible matches:
__enhanced__
__graphics_toolkit__
__guidata__
__modified__
__myhandle__
__plot_stream__
alphamap
beingdeleted
busyaction
buttondownfcn
children
clipping
closerequestfcn
color
colormap
createfcn
currentaxes
currentcharacter
currentobject
currentpoint
deletefcn
dockcontrols
doublebuffer
filename
handlevisibility

hittest
integerhandle
interruptible
inverthardcopy
keypressfcn
keyreleasefcn
menubar
mincolormap
name
nextplot
numbertitle
outerposition
paperorientation
paperposition
paperpositionmode
papersize
papertype
paperunits
parent
pointer
pointershapecdata
pointershapehotspot
position
renderer
renderermode

resize
resizefcn
selected
selectionhighlight
selectiontype
tag
toolbar
type
uicontextmenu
units
userdata
visible
windowbuttondownfcn
windowbuttonmotionfcn
windowbuttonupfcn
windowkeypressfcn
windowkeyreleasefcn
windowscrollwheelfcn
windowstyle
wvisual
wvisualmode
xdisplay
xvisual
xvisualmode

The root figure has index 0. Its properties may be displayed by: get (0, "").
The uses of get and set are further explained in [get], page 383, [set], page 383.

res = isprop (obj, "prop")
Return true if prop is a property of the object obj.
obj may also be an array of objects in which case res will be a logical array indicating
whether each handle has the property prop.
For plotting, obj is a handle to a graphics object. Otherwise, obj should be an
instance of a class.
See also: [get], page 383, [set], page 383, [ismethod], page 777, [isobject], page 776.

15.3.2 Graphics Objects
The hierarchy of graphics objects was explained above. See Section 15.3.1 [Introduction to
Graphics Structures], page 376. Here the specific objects are described, and the properties
contained in these objects are discussed. Keep in mind that graphics objects are always
referenced by handle.
root figure the top level of the hierarchy and the parent of all figure objects. The handle
index of the root figure is 0.
figure

A figure window.

axes

A set of axes. This object is a child of a figure object and may be a parent of
line, text, image, patch, surface, or light objects.

line

A line in two or three dimensions.

378

GNU Octave

text

Text annotations.

image

A bitmap image.

patch

A filled polygon, currently limited to two dimensions.

surface

A three-dimensional surface.

light

A light object used for lighting effects on patches and surfaces.

15.3.2.1 Creating Graphics Objects
You can create any graphics object primitive by calling the function of the same name as
the object; In other words, figure, axes, line, text, image, patch, surface, and light
functions. These fundamental graphic objects automatically become children of the current
axes object as if hold on was in place. Separately, axes will automatically become children
of the current figure object and figures will become children of the root object 0.
If this auto-joining feature is not desired then it is important to call newplot first to
prepare a new figure and axes for plotting. Alternatively, the easier way is to call a highlevel graphics routine which will both create the plot and then populate it with low-level
graphics objects. Instead of calling line, use plot. Or use surf instead of surface. Or
use fill instead of patch.

axes ()
axes (property, value, . . . )
axes (hax)
h = axes ( . . . )
Create an axes object and return a handle to it, or set the current axes to hax.
Called without any arguments, or with property/value pairs, construct a new axes.
For accepted properties and corresponding values, see [set], page 383.
Called with a single axes handle argument hax, the function makes hax the current
axes. It also restacks the axes in the corresponding figure so that hax is the first
entry in the list of children. This causes hax to be displayed on top of any other axes
objects (Z-order stacking).
See also: [gca], page 382, [set], page 383, [get], page 383.

line ()
line (x, y)
line (x, y, property, value, . . . )
line (x, y, z)
line (x, y, z, property, value, . . . )
line (property, value, . . . )
line (hax, . . . )
h = line ( . . . )
Create line object from x and y (and possibly z) and insert in the current axes.
Multiple property-value pairs may be specified for the line object, but they must
appear in pairs.
If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.

Chapter 15: Plotting

379

The optional return value h is a graphics handle (or vector of handles) to the line
objects created.
See also: [image], page 754, [patch], page 379, [rectangle], page 321, [surface],
page 379, [text], page 351.

patch ()
patch (x, y, c)
patch (x, y, z, c)
patch ("Faces", faces, "Vertices", verts, . . . )
patch ( . . . , prop, val, . . . )
patch ( . . . , propstruct, . . . )
patch (hax, . . . )
h = patch ( . . . )
Create patch object in the current axes with vertices at locations (x, y) and of color
c.
If the vertices are matrices of size MxN then each polygon patch has M vertices and a
total of N polygons will be created. If some polygons do not have M vertices use NaN
to represent "no vertex". If the z input is present then 3-D patches will be created.
The color argument c can take many forms. To create polygons which all share a
single color use a string value (e.g., "r" for red), a scalar value which is scaled by
caxis and indexed into the current colormap, or a 3-element RGB vector with the
precise TrueColor.
If c is a vector of length N then the ith polygon will have a color determined by scaling
entry c(i) according to caxis and then indexing into the current colormap. More
complicated coloring situations require directly manipulating patch property/value
pairs.
Instead of specifying polygons by matrices x and y, it is possible to present a unique
list of vertices and then a list of polygon faces created from those vertices. In this case
the "Vertices" matrix will be an Nx2 (2-D patch) or Nx3 (3-D patch). The MxN
"Faces" matrix describes M polygons having N vertices—each row describes a single
polygon and each column entry is an index into the "Vertices" matrix to identify a
vertex. The patch object can be created by directly passing the property/value pairs
"Vertices"/verts, "Faces"/faces as inputs.
Instead of using property/value pairs, any property can be set by passing a structure
propstruct with the respective field names.
If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.
The optional return value h is a graphics handle to the created patch object.
Implementation Note: Patches are highly configurable objects. To truly customize
them requires setting patch properties directly. Useful patch properties are: "cdata",
"edgecolor", "facecolor", "faces", "facevertexcdata".
See also: [fill], page 314, [get], page 383, [set], page 383.

surface (x, y, z, c)
surface (x, y, z)

380

GNU Octave

surface (z, c)
surface (z)
surface ( . . . , prop, val, . . . )
surface (hax, . . . )
h = surface ( . . . )
Create a surface graphic object given matrices x and y from meshgrid and a matrix
of values z corresponding to the x and y coordinates of the surface.
If x and y are vectors, then a typical vertex is (x(j), y(i), z(i,j)). Thus, columns of
z correspond to different x values and rows of z correspond to different y values. If
only a single input z is given then x is taken to be 1:columns (z) and y is 1:rows
(z).
Any property/value input pairs are assigned to the surface object.
If the first argument hax is an axes handle, then plot into this axes, rather than the
current axes returned by gca.
The optional return value h is a graphics handle to the created surface object.
See also: [surf], page 325, [mesh], page 323, [patch], page 379, [line], page 378.

light ()
light ( . . . , "prop", val, . . . )
light (hax, . . . )
h = light ( . . . )
Create a light object in the current axes or for axes hax.
When a light object is present in an axes object, and the properties "EdgeLighting"
or "FaceLighting" of a patch or surface object are set to a value other than "none",
these objects are drawn with light and shadow effects. Supported values for Lighting
properties are "none" (no lighting effects), "flat" (faceted look of the objects), and
"gouraud" (linear interpolation of the lighting effects between the vertices). For
patch objects, the normals must be set manually (property "VertexNormals").
Up to eight light objects are supported per axes.
Lighting is only supported for OpenGL graphic toolkits (i.e., "fltk" and "qt").
A light object has the following properties which alter the appearance of the plot.
"Color": The color of the light can be passed as an
RGB-vector (e.g., [1 0 0] for red) or as a string (e.g., "r" for red). The
default color is white ([1 1 1]).
"Position": The direction from which the light emanates as a
1x3-vector. The default direction is [1 0 1].
"Style": This string defines whether the light emanates from a
light source at infinite distance ("infinite") or from a local point source
("local"). The default is "infinite".
If the first argument hax is an axes handle, then add the light object to this axes,
rather than the current axes returned by gca.
The optional return value h is a graphics handle to the created light object.
See also: [lighting], page 336, [material], page 337, [patch], page 379, [surface],
page 379.

Chapter 15: Plotting

381

15.3.2.2 Handle Functions
To determine whether a variable is a graphics object index, or an index to an axes or figure,
use the functions ishandle, isaxes, and isfigure.

ishandle (h)
Return true if h is a graphics handle and false otherwise.
h may also be a matrix of handles in which case a logical array is returned that is
true where the elements of h are graphics handles and false where they are not.
See also: [isaxes], page 381, [isfigure], page 381.

ishghandle (h)
ishghandle (h, type)
Return true if h is a graphics handle (of type type) and false otherwise.
When no type is specified the function is equivalent to ishandle.
See also: [ishandle], page 381.

isaxes (h)
Return true if h is an axes graphics handle and false otherwise.
If h is a matrix then return a logical array which is true where the elements of h are
axes graphics handles and false where they are not.
See also: [isaxes], page 381, [ishandle], page 381.

isfigure (h)
Return true if h is a figure graphics handle and false otherwise.
If h is a matrix then return a logical array which is true where the elements of h are
figure graphics handles and false where they are not.
See also: [isaxes], page 381, [ishandle], page 381.
The function gcf returns an index to the current figure object, or creates one if none
exists. Similarly, gca returns the current axes object, or creates one (and its parent figure
object) if none exists.

h = gcf ()
Return a handle to the current figure.
The current figure is the default target for graphics output. If multiple figures exist,
gcf returns the last created figure or the last figure that was clicked on with the
mouse.
If a current figure does not exist, create one and return its handle. The handle may
then be used to examine or set properties of the figure. For example,
fplot (@sin, [-10, 10]);
fig = gcf ();
set (fig, "numbertitle", "off", "name", "sin plot")
plots a sine wave, finds the handle of the current figure, and then renames the figure
window to describe the contents.

382

GNU Octave

Note: To find the current figure without creating a new one if it does not exist, query
the "CurrentFigure" property of the root graphics object.
get (0, "currentfigure");
See also: [gca], page 382, [gco], page 382, [gcbf], page 427, [gcbo], page 427, [get],
page 383, [set], page 383.

h = gca ()
Return a handle to the current axes object.
The current axes is the default target for graphics output. In the case of a figure with
multiple axes, gca returns the last created axes or the last axes that was clicked on
with the mouse.
If no current axes object exists, create one and return its handle. The handle may
then be used to examine or set properties of the axes. For example,
ax = gca ();
set (ax, "position", [0.5, 0.5, 0.5, 0.5]);
creates an empty axes object and then changes its location and size in the figure
window.
Note: To find the current axes without creating a new axes object if it does not exist,
query the "CurrentAxes" property of a figure.
get (gcf, "currentaxes");
See also: [gcf], page 381, [gco], page 382, [gcbf], page 427, [gcbo], page 427, [get],
page 383, [set], page 383.

h = gco ()
h = gco (fig)
Return a handle to the current object of the current figure, or a handle to the current
object of the figure with handle fig.
The current object of a figure is the object that was last clicked on. It is stored in
the "CurrentObject" property of the target figure.
If the last mouse click did not occur on any child object of the figure, then the current
object is the figure itself.
If no mouse click occurred in the target figure, this function returns an empty matrix.
Programming Note: The value returned by this function is not necessarily the same
as the one returned by gcbo during callback execution. An executing callback can be
interrupted by another callback and the current object may be changed.
See also: [gcbo], page 427, [gca], page 382, [gcf], page 381, [gcbf], page 427, [get],
page 383, [set], page 383.
The get and set functions may be used to examine and set properties for graphics
objects. For example,

Chapter 15: Plotting

383

get (0)
⇒ ans =
{
type = root
currentfigure = [](0x0)
children = [](0x0)
visible = on
...
}
returns a structure containing all the properties of the root figure. As with all functions
in Octave, the structure is returned by value, so modifying it will not modify the internal
root figure plot object. To do that, you must use the set function. Also, note that in this
case, the currentfigure property is empty, which indicates that there is no current figure
window.
The get function may also be used to find the value of a single property. For example,
get (gca (), "xlim")
⇒ [ 0 1 ]

returns the range of the x-axis for the current axes object in the current figure.
To set graphics object properties, use the set function. For example,
set (gca (), "xlim", [-10, 10]);
sets the range of the x-axis for the current axes object in the current figure to ‘[-10, 10]’.
Default property values can also be queried if the set function is called without a value
argument. When only one argument is given (a graphic handle) then a structure with
defaults for all properties of the given object type is returned. For example,
set (gca ())
returns a structure containing the default property values for axes objects. If set is called
with two arguments (a graphic handle and a property name) then only the defaults for the
requested property are returned.

val = get (h)
val = get (h, p)
Return the value of the named property p from the graphics handle h.
If p is omitted, return the complete property list for h.
If h is a vector, return a cell array including the property values or lists respectively.
See also: [set], page 383.

set (h, property, value, . . . )
set (h, properties, values)
set (h, pv)
value_list = set (h, property)
all_value_list = set (h)
Set named property values for the graphics handle (or vector of graphics handles) h.

384

GNU Octave

There are three ways to give the property names and values:
• as a comma separated list of property, value pairs

Here, each property is a string containing the property name, each value is a
value of the appropriate type for the property.

• as a cell array of strings properties containing property names and a cell array
values containing property values.
In this case, the number of columns of values must match the number of elements
in properties. The first column of values contains values for the first entry in
properties, etc. The number of rows of values must be 1 or match the number
of elements of h. In the first case, each handle in h will be assigned the same
values. In the latter case, the first handle in h will be assigned the values from
the first row of values and so on.
• as a structure array pv

Here, the field names of pv represent the property names, and the field values
give the property values. In contrast to the previous case, all elements of pv will
be set in all handles in h independent of the dimensions of pv.

set is also used to query the list of values a named property will take. clist = set
(h, "property") will return the list of possible values for "property" in the cell list
clist. If no output variable is used then the list is formatted and printed to the screen.
If no property is specified (slist = set (h)) then a structure slist is returned where
the fieldnames are the properties of the object h and the fields are the list of possible
values for each property. If no output variable is used then the list is formatted and
printed to the screen.
For example,
hf = figure ();
set (hf, "paperorientation")
⇒ paperorientation: [ landscape | {portrait} | rotated ]

shows the paperorientation property can take three values with the default being
"portrait".
See also: [get], page 383.

parent = ancestor (h, type)
parent = ancestor (h, type, "toplevel")
Return the first ancestor of handle object h whose type matches type, where type is
a character string.
If type is a cell array of strings, return the first parent whose type matches any of the
given type strings.
If the handle object h itself is of type type, return h.
If "toplevel" is given as a third argument, return the highest parent in the object
hierarchy that matches the condition, instead of the first (nearest) one.
See also: [findobj], page 422, [findall], page 423, [allchild], page 385.

Chapter 15: Plotting

385

h = allchild (handles)
Find all children, including hidden children, of a graphics object.
This function is similar to get (h, "children"), but also returns hidden objects
(HandleVisibility = "off").
If handles is a scalar, h will be a vector. Otherwise, h will be a cell matrix of the
same size as handles and each cell will contain a vector of handles.
See also: [findall], page 423, [findobj], page 422, [get], page 383, [set], page 383.

findfigs ()
Find all visible figures that are currently off the screen and move them onto the screen.
See also: [allchild], page 385, [figure], page 357, [get], page 383, [set], page 383.
Figures can be printed or saved in many graphics formats with print and saveas.
Occasionally, however, it may be useful to save the original Octave handle graphic directly
so that further modifications can be made such as modifying a title or legend.
This can be accomplished with the following functions by
fig_struct = hdl2struct (gcf);
save myplot.fig -struct fig_struct;
...
fig_struct = load ("myplot.fig");
struct2hdl (fig_struct);

s = hdl2struct (h)
Return a structure, s, whose fields describe the properties of the object, and its
children, associated with the handle, h.
The fields of the structure s are "type", "handle", "properties", "children", and
"special".
See also: [struct2hdl], page 385, [hgsave], page 373, [findobj], page 422.

h = struct2hdl (s)
h = struct2hdl (s, p)
h = struct2hdl (s, p, hilev)
Construct a graphics handle object h from the structure s.
The structure must contain the fields "handle", "type", "children", "properties",
and "special".
If the handle of an existing figure or axes is specified, p, the new object will be created
as a child of that object. If no parent handle is provided then a new figure and the
necessary children will be constructed using the default values from the root figure.
A third boolean argument hilev can be passed to specify whether the function should
preserve listeners/callbacks, e.g., for legends or hggroups. The default is false.
See also: [hdl2struct], page 385, [hgload], page 374, [findobj], page 422.

hnew = copyobj (horig)
hnew = copyobj (horig, hparent)
Construct a copy of the graphic objects associated with the handles horig and return
new handles hnew to the new objects.

386

GNU Octave

If a parent handle hparent (root, figure, axes, or hggroup) is specified, the copied
object will be created as a child of hparent.
If horig is a vector of handles, and hparent is a scalar, then each handle in the vector
hnew has its "Parent" property set to hparent. Conversely, if horig is a scalar and
hparent a vector, then each parent object will receive a copy of horig. If horig and
hparent are both vectors with the same number of elements then hnew(i) will have
parent hparent(i).
See also: [struct2hdl], page 385, [hdl2struct], page 385, [findobj], page 422.

15.3.3 Graphics Object Properties
In this Section the graphics object properties are discussed in detail, starting with the root
figure properties and continuing through the objects hierarchy. The documentation about a
specific graphics object can be displayed using doc function, e.g., doc ("axes properties")
will show Section 15.3.3.3 [Axes Properties], page 393.
The allowed values for radio (string) properties can be retrieved programmatically or
displayed using the one or two arguments call to set function. See [set], page 383.
In the following documentation, default values are enclosed in { }.

15.3.3.1 Root Figure Properties
The root figure properties are:
__modified__: "off" | {"on"}
beingdeleted: {"off"} | "on"
beingdeleted is unused.
busyaction: "cancel" | {"queue"}
busyaction is unused.
buttondownfcn: string | function handle, def. [](0x0)
buttondownfcn is unused.
callbackobject (read-only): graphics handle, def. [](0x0)
Graphics handle of the current object whose callback is executing.
children (read-only): vector of graphics handles, def. [](0x1)
Graphics handles of the root’s children.
clipping: "off" | {"on"}
clipping is unused.
commandwindowsize (read-only): def. [0 0]
createfcn: string | function handle, def. [](0x0)
createfcn is unused.
currentfigure: graphics handle, def. [](0x0)
Graphics handle of the current figure.
deletefcn: string | function handle, def. [](0x0)
deletefcn is unused.

Chapter 15: Plotting

387

fixedwidthfontname: string, def. "Courier"
handlevisibility: "callback" | "off" | {"on"}
handlevisibility is unused.
hittest: "off" | {"on"}
hittest is unused.
interruptible: "off" | {"on"}
interruptible is unused.
monitorpositions (read-only):
monitorpositions is unused.
parent: graphics handle, def. [](0x0)
Root figure has no parent graphics object. parent is always empty.
pointerlocation: two-element vector, def. [0 0]
pointerlocation is unused.
pointerwindow (read-only): graphics handle, def. 0
pointerwindow is unused.
screendepth (read-only): double
screenpixelsperinch (read-only): double
screensize (read-only): four-element vector
selected: {"off"} | "on"
selected is unused.
selectionhighlight: "off" | {"on"}
selectionhighlight is unused.
showhiddenhandles: {"off"} | "on"
If showhiddenhandles is "on", all graphics objects handles are visible in their
parents’ children list, regardless of the value of their handlevisibility property.
tag: string, def. ""
A user-defined string to label the graphics object.
type (read-only): string
Class name of the graphics object. type is always "root"
uicontextmenu: graphics handle, def. [](0x0)
uicontextmenu is unused.
units: "centimeters" | "inches" | "normalized" | {"pixels"} | "points"
userdata: Any Octave data, def. [](0x0)
User-defined data to associate with the graphics object.
visible: "off" | {"on"}
visible is unused.

388

GNU Octave

15.3.3.2 Figure Properties
The figure properties are:
__modified__: "off" | {"on"}
alphamap: def. 64-by-1 double
Transparency is not yet implemented for figure objects. alphamap is unused.
beingdeleted: {"off"} | "on"
busyaction: "cancel" | {"queue"}
buttondownfcn: string | function handle, def. [](0x0)
For information on how to write graphics listener functions see Section 15.4.4
[Callbacks section], page 426.
children (read-only): vector of graphics handles, def. [](0x1)
Graphics handles of the figure’s children.
clipping: "off" | {"on"}
clipping is unused.
closerequestfcn: string | function handle, def. "closereq"
Function that is executed when a figure is deleted. See [closereq function],
page 363.
For information on how to write graphics listener functions see Section 15.4.4
[Callbacks section], page 426.
color: colorspec, def. [1 1 1]
Color of the figure background. See Section 15.4.1 [colorspec], page 425.
colormap: N-by-3 matrix, def. 64-by-3 double
A matrix containing the RGB color map for the current axes.
createfcn: string | function handle, def. [](0x0)
Callback function executed immediately after figure has been created.
Function is set by using default property on root object, e.g., set (0,
"defaultfigurecreatefcn", ’disp ("figure created!")’).
For information on how to write graphics listener functions see Section 15.4.4
[Callbacks section], page 426.
currentaxes: graphics handle, def. [](0x0)
Handle to the graphics object of the current axes.
currentcharacter (read-only): def. ""
currentcharacter is unused.
currentobject (read-only): graphics handle, def. [](0x0)
currentpoint (read-only): two-element vector, def. [0; 0]
A 1-by-2 matrix which holds the coordinates of the point over which the mouse
pointer was when a mouse event occurred. The X and Y coordinates are in
units defined by the figure’s units property and their origin is the lower left
corner of the plotting area.
Events which set currentpoint are
A mouse button was pressed
always

Chapter 15: Plotting

389

A mouse button was released
only if the figure’s callback windowbuttonupfcn is defined
The pointer was moved while pressing the mouse button (drag)
only if the figure’s callback windowbuttonmotionfcn is defined
deletefcn: string | function handle, def. [](0x0)
Callback function executed immediately before figure is deleted.
For information on how to write graphics listener functions see Section 15.4.4
[Callbacks section], page 426.
dockcontrols: {"off"} | "on"
dockcontrols is unused.
filename: string, def. ""
The filename used when saving the plot figure.
graphicssmoothing: "off" | {"on"}
Use smoothing techniques to reduce the appearance of jagged lines.
handlevisibility: "callback" | "off" | {"on"}
If handlevisibility is "off", the figure’s handle is not visible in its parent’s
"children" property.
hittest: "off" | {"on"}
integerhandle: "off" | {"on"}
Assign the next lowest unused integer as the Figure number.
interruptible: "off" | {"on"}
inverthardcopy: "off" | {"on"}
Replace the figure and axes background color with white when printing.
keypressfcn: string | function handle, def. [](0x0)
Callback function executed when a keystroke event happens while the figure
has focus. The actual key that was pressed can be retrieved using the second
argument ’evt’ of the function. For information on how to write graphics listener
functions see Section 15.4.4 [Callbacks section], page 426.
keyreleasefcn: string | function handle, def. [](0x0)
With keypressfcn, the keyboard callback functions. These callback functions
are called when a key is pressed/released respectively. The functions are called
with two input arguments. The first argument holds the handle of the calling
figure. The second argument holds an event structure which has the following
members:
Character:
The ASCII value of the key
Key:

Lowercase value of the key

Modifier:
A cell array containing strings representing the modifiers pressed
with the key.

390

GNU Octave

For information on how to write graphics listener functions see Section 15.4.4
[Callbacks section], page 426.
menubar: {"figure"} | "none"
Control the display of the figure menu bar at the top of the figure.
name: string, def. ""
Name to be displayed in the figure title bar. The name is displayed to the right
of any title determined by the numbertitle property.
nextplot: {"add"} | "new" | "replace" | "replacechildren"
nextplot is used by high level plotting functions to decide what to do with
axes already present in the figure. See [newplot function], page 360.
numbertitle: "off" | {"on"}
Display "Figure" followed by the numerical figure handle value in the figure
title bar.
outerposition: four-element vector, def. [-1 -1 -1 -1]
Specify the position and size of the figure including the top menubar and the
bottom status bar. The four elements of the vector are the coordinates of
the lower left corner and width and height of the figure. See [units property],
page 392.
paperorientation: "landscape" | {"portrait"}
The value for the papersize, and paperposition properties depends upon
paperorientation. The horizontal and vertical values for papersize and
paperposition reverse order when paperorientation is switched between
"portrait" and "landscape".
paperposition: four-element vector, def. [0.25000 2.50000 8.00000 6.00000]
Vector [left bottom width height] defining the position and size of the figure
(in paperunits units) on the printed page. The position [left bottom] defines
the lower left corner of the figure on the page, and the size is defined by [width
height]. For output formats not implicitly rendered on paper, width and
height define the size of the image and the position information is ignored.
Setting paperposition also forces the paperpositionmode property to be set
to "manual".
paperpositionmode: "auto" | {"manual"}
If paperpositionmode is set to "auto", the paperposition property is automatically computed: the printed figure will have the same size as the on-screen
figure and will be centered on the output page. Setting the paperpositionmode
to "auto" does not modify the value of the paperposition property.
papersize: two-element vector, def. [8.5000 11.0000]
Vector [width height] defining the size of the paper for printing. Setting
the papersize property to a value, not associated with one of the defined
papertypes and consistent with the setting for paperorientation, forces
the papertype property to the value "". If papersize is set to
a value associated with a supported papertype and consistent with the
paperorientation, the papertype value is modified to the associated value.

Chapter 15: Plotting

391

papertype: "" | "a" | "a0" | "a1" | "a2" | "a3" | "a4" | "a5" | "arch-a" |
"arch-b" | "arch-c" | "arch-d" | "arch-e" | "b" | "b0" | "b1" | "b2" | "b3" | "b4"
| "b5" | "c" | "d" | "e" | "tabloid" | "uslegal" | {"usletter"}
Name of the paper used for printed output. Setting papertype also changes
papersize, while maintaining consistency with the paperorientation property.
paperunits: "centimeters" | {"inches"} | "normalized" | "points"
The unit used to compute the paperposition property. For paperunits set to
"pixels", the conversion between physical units (ex: "inches") and "pixels"
is dependent on the screenpixelsperinch property of the root object.
parent: graphics handle, def. 0
Handle of the parent graphics object.
pointer: {"arrow"} | "botl" | "botr" | "bottom" | "circle" | "cross" |
"crosshair" | "custom" | "fleur" | "fullcrosshair" | "hand" | "ibeam" | "left" |
"right" | "top" | "topl" | "topr" | "watch"
pointer is unused.
pointershapecdata: def. 16-by-16 double
pointershapecdata is unused.
pointershapehotspot: def. [0 0]
pointershapehotspot is unused.
position: four-element vector, def. [300 200 560 420]
Specify the position and size of the figure canvas. The four elements of the
vector are the coordinates of the lower left corner and width and height of the
figure. See [units property], page 392.
renderer: {"opengl"} | "painters"
renderer is unused.
renderermode: {"auto"} | "manual"
renderermode is unused.
resize: "off" | {"on"}
resize is unused.
resizefcn: string | function handle, def. [](0x0)
resizefcn is deprecated. Use sizechangedfcn instead.
selected: {"off"} | "on"
selectionhighlight: "off" | {"on"}
selectiontype: "alt" | "extend" | {"normal"} | "open"
sizechangedfcn: string | function handle, def. [](0x0)
Callback triggered when the figure window size is changed.
For information on how to write graphics listener functions see Section 15.4.4
[Callbacks section], page 426.
tag: string, def. ""
A user-defined string to label the graphics object.

392

GNU Octave

toolbar: {"auto"} | "figure" | "none"
Control the display of the toolbar (along the bottom of the menubar) and the
status bar. When set to "auto", the display is based on the value of the menubar
property.
type (read-only): string
Class name of the graphics object. type is always "figure"
uicontextmenu: graphics handle, def. [](0x0)
Graphics handle of the uicontextmenu object that is currently associated to
this figure object.
units: "centimeters" | "characters" | "inches" | "normalized" | {"pixels"} |
"points"
The unit used to compute the position and outerposition properties.
userdata: Any Octave data, def. [](0x0)
User-defined data to associate with the graphics object.
visible: "off" | {"on"}
If visible is "off", the figure is not rendered on screen.
windowbuttondownfcn: string | function handle, def. [](0x0)
See [windowbuttonupfcn property], page 392.
windowbuttonmotionfcn: string | function handle, def. [](0x0)
See [windowbuttonupfcn property], page 392.
windowbuttonupfcn: string | function handle, def. [](0x0)
With windowbuttondownfcn and windowbuttonmotionfcn, the mouse callback
functions. These callback functions are called when a mouse button is pressed,
dragged, or released respectively. When these callback functions are executed,
the currentpoint property holds the current coordinates of the cursor.
For information on how to write graphics listener functions see Section 15.4.4
[Callbacks section], page 426.
windowkeypressfcn: string | function handle, def. [](0x0)
Function that is executed when a key is pressed and the figure has focus.
For information on how to write graphics listener functions see Section 15.4.4
[Callbacks section], page 426.
windowkeyreleasefcn: string | function handle, def. [](0x0)
Function that is executed when a key is released and the figure has focus.
For information on how to write graphics listener functions see Section 15.4.4
[Callbacks section], page 426.
windowscrollwheelfcn: string | function handle, def. [](0x0)
windowscrollwheelfcn is unused.
windowstyle: "docked" | "modal" | {"normal"}
The window style of a figure. One of the following values:
normal

Set the window style as non modal.

Chapter 15: Plotting

393

modal

Set the window as modal so that it will stay on top of all normal
figures.

docked

Setting the window style as docked currently does not dock the
window.

Changing modes of a visible figure may cause the figure to close and reopen.

15.3.3.3 Axes Properties
The axes properties are:
__modified__: "off" | {"on"}
activepositionproperty: {"outerposition"} | "position"
Specify which of "position" or "outerposition" properties takes precedence
when axes annotations extent changes. See [position property], page 396, and
[outerposition property], page 396.
alim: def. [0 1]
Transparency is not yet implemented for axes objects. alim is unused.
alimmode: {"auto"} | "manual"
ambientlightcolor: def. [1 1 1]
ambientlightcolor is unused.
beingdeleted: {"off"} | "on"
box: {"off"} | "on"
Control whether the axes has a surrounding box.
boxstyle: {"back"} | "full"
For 3-D axes, control whether the "full" box is drawn or only the 3 "back"
axes
busyaction: "cancel" | {"queue"}
buttondownfcn: string | function handle, def. [](0x0)
For information on how to write graphics listener functions see Section 15.4.4
[Callbacks section], page 426.
cameraposition: three-element vector, def. [0.50000 0.50000 9.16025]
camerapositionmode: {"auto"} | "manual"
cameratarget: three-element vector, def. [0.50000 0.50000 0.50000]
cameratargetmode: {"auto"} | "manual"
cameraupvector: three-element vector, def. [-0 1 0]
cameraupvectormode: {"auto"} | "manual"
cameraviewangle: scalar, def. 6.6086
cameraviewanglemode: {"auto"} | "manual"
children (read-only): vector of graphics handles, def. [](0x1)
Graphics handles of the axes’s children.
clim: two-element vector, def. [0 1]
Define the limits for the color axis of image children. Setting clim also forces
the climmode property to be set to "manual". See [pcolor function], page 313.

394

GNU Octave

climmode: {"auto"} | "manual"
clipping: "off" | {"on"}
clipping is unused.
clippingstyle: {"3dbox"} | "rectangle"
clippingstyle is unused.
color: colorspec, def. [1 1 1]
Color of the axes background. See Section 15.4.1 [colorspec], page 425.
colororder: N-by-3 RGB matrix, def. 7-by-3 double
RGB values used by plot function for automatic line coloring.
colororderindex: def. 1
colororderindex is unused.
createfcn: string | function handle, def. [](0x0)
Callback function executed immediately after axes has been created.
Function is set by using default property on root object, e.g., set (0,
"defaultaxescreatefcn", ’disp ("axes created!")’).
For information on how to write graphics listener functions see Section 15.4.4
[Callbacks section], page 426.
currentpoint: 2-by-3 matrix, def. 2-by-3 double
Matrix [xf, yf, zf; xb, yb, zb] which holds the coordinates (in axes data
units) of the point over which the mouse pointer was when the mouse button
was pressed. If a mouse callback function is defined, currentpoint holds the
pointer coordinates at the time the mouse button was pressed. For 3-D plots,
the first row of the returned matrix specifies the point nearest to the current
camera position and the second row the furthest point. The two points forms
a line which is perpendicular to the screen.
dataaspectratio: three-element vector, def. [1 1 1]
Specify the relative height and width of the data displayed in the axes. Setting
dataaspectratio to [1, 2] causes the length of one unit as displayed on the xaxis to be the same as the length of 2 units on the y-axis. See [daspect function],
page 344. Setting dataaspectratio also forces the dataaspectratiomode
property to be set to "manual".
dataaspectratiomode: {"auto"} | "manual"
deletefcn: string | function handle, def. [](0x0)
Callback function executed immediately before axes is deleted.
For information on how to write graphics listener functions see Section 15.4.4
[Callbacks section], page 426.
drawmode: "fast" | {"normal"}
fontangle: "italic" | {"normal"}
Control whether the font is italic or normal.
fontname: string, def. "*"
Name of the font used for axes annotations.

Chapter 15: Plotting

395

fontsize: scalar, def. 10
Size of the font used for axes annotations. See [fontunits property], page 395.
fontsmoothing: "off" | {"on"}
fontsmoothing is unused.
fontunits: "centimeters" | "inches" | "normalized" | "pixels" | {"points"}
Unit used to interpret fontsize property.
fontweight: "bold" | {"normal"}
Control variant of base font used: bold, demi, light, normal.
gridalpha: def. 0.15000
Transparency is not yet implemented for axes objects. gridalpha is unused.
gridalphamode: {"auto"} | "manual"
gridalphamode is unused.
gridcolor: def. [0.15000 0.15000 0.15000]
gridcolor is unused.
gridcolormode: {"auto"} | "manual"
gridcolormode is unused.
gridlinestyle: {"-"} | "--" | "-." | ":" | "none"
handlevisibility: "callback" | "off" | {"on"}
If handlevisibility is "off", the axes’s handle is not visible in its parent’s
"children" property.
hittest: "off" | {"on"}
interruptible: "off" | {"on"}
labelfontsizemultiplier: def. 1.1000
Ratio between the x/y/zlabel fontsize and the tick label fontsize
layer: {"bottom"} | "top"
Control whether the axes is drawn below child graphics objects (ticks, labels,
etc. covered by plotted objects) or above.
linestyleorder: def. "-"
linestyleorder is unused.
linestyleorderindex: def. 1
linestyleorderindex is unused.
linewidth: def. 0.50000
Width of the main axes lines
minorgridalpha: def. 0.25000
Transparency is not yet implemented for axes objects. minorgridalpha is
unused.
minorgridalphamode: {"auto"} | "manual"
minorgridalphamode is unused.
minorgridcolor: def. [0.10000 0.10000 0.10000]
minorgridcolor is unused.

396

GNU Octave

minorgridcolormode: {"auto"} | "manual"
minorgridcolormode is unused.
minorgridlinestyle: "-" | "--" | "-." | {":"} | "none"
mousewheelzoom: scalar in the range (0, 1), def. 0.50000
Fraction of axes limits to zoom for each wheel movement.
nextplot: "add" | {"replace"} | "replacechildren"
nextplot is used by high level plotting functions to decide what to do with
graphics objects already present in the axes. See [newplot function], page 360.
The state of nextplot is typically controlled using the hold function. See
[hold function], page 361.
outerposition: four-element vector, def. [0 0 1 1]
Specify the position of the plot including titles, axes, and legend. The four
elements of the vector are the coordinates of the lower left corner and width
and height of the plot, in units normalized to the width and height of the plot
window. For example, [0.2, 0.3, 0.4, 0.5] sets the lower left corner of the
axes at (0.2, 0.3) and the width and height to be 0.4 and 0.5 respectively. See
[position property], page 396.
parent: graphics handle
Handle of the parent graphics object.
pickableparts: "all" | "none" | {"visible"}
pickableparts is unused.
plotboxaspectratio: def. [1 1 1]
See [pbaspect function], page 344. Setting plotboxaspectratio also forces the
plotboxaspectratiomode property to be set to "manual".
plotboxaspectratiomode: {"auto"} | "manual"
position: four-element vector, def. [0.13000 0.11000 0.77500 0.81500]
Specify the position of the plot excluding titles, axes, and legend. The four
elements of the vector are the coordinates of the lower left corner and width
and height of the plot, in units normalized to the width and height of the plot
window. For example, [0.2, 0.3, 0.4, 0.5] sets the lower left corner of the
axes at (0.2, 0.3) and the width and height to be 0.4 and 0.5 respectively. See
[outerposition property], page 396.
projection: {"orthographic"} | "perspective"
projection is unused.
selected: {"off"} | "on"
selectionhighlight: "off" | {"on"}
sortmethod: "childorder" | {"depth"}
sortmethod is unused.
tag: string, def. ""
A user-defined string to label the graphics object.
tickdir: {"in"} | "out"
Control whether axes tick marks project "in" to the plot box or "out". Setting
tickdir also forces the tickdirmode property to be set to "manual".

Chapter 15: Plotting

tickdirmode: {"auto"} | "manual"
ticklabelinterpreter: "latex" | "none" | {"tex"}
Control the way x/y/zticklabel properties
[Use of the interpreter property], page 363.

397

are

interpreted.

See

ticklength: two-element vector, def. [0.010000 0.025000]
Two-element vector [2Dlen 3Dlen] specifying the length of the tickmarks relative to the longest visible axis.
tightinset (read-only): four-element vector, def. [0.042857 0.038106 0.000000
0.023810]
Size of the [left bottom right top] margins around the axes that enclose
labels and title annotations.
title: graphics handle
Graphics handle of the title text object.
titlefontsizemultiplier: positive scalar, def. 1.1000
Ratio between the title fontsize and the tick label fontsize
titlefontweight: {"bold"} | "normal"
Control variant of base font used for the axes title.
type (read-only): string
Class name of the graphics object. type is always "axes"
uicontextmenu: graphics handle, def. [](0x0)
Graphics handle of the uicontextmenu object that is currently associated to
this axes object.
units: "centimeters" | "characters" | "inches" | {"normalized"} | "pixels" |
"points"
Units used to interpret the "position", "outerposition", and "tightinset"
properties.
userdata: Any Octave data, def. [](0x0)
User-defined data to associate with the graphics object.
view: two-element vector, def. [0 90]
Two-element vector [azimuth elevation] specifying the viewpoint for threedimensional plots
visible: "off" | {"on"}
If visible is "off", the axes is not rendered on screen.
xaxislocation: {"bottom"} | "origin" | "top" | "zero"
Control the x axis location.
xcolor: {colorspec} | "none", def. [0.15000 0.15000 0.15000]
Color of the x-axis. See Section 15.4.1 [colorspec], page 425. Setting xcolor
also forces the xcolormode property to be set to "manual".
xcolormode: {"auto"} | "manual"
xdir: {"normal"} | "reverse"
Direction of the x axis: "normal" is left to right.

398

GNU Octave

xgrid: {"off"} | "on"
Control whether major x grid lines are displayed.
xlabel: graphics handle
Graphics handle of the x label text object.
xlim: two-element vector, def. [0 1]
Two-element vector [xmin xmax] specifying the limits for the x-axis. Setting xlim also forces the xlimmode property to be set to "manual". See
[xlim function], page 317.
xlimmode: {"auto"} | "manual"
xminorgrid: {"off"} | "on"
Control whether minor x grid lines are displayed.
xminortick: {"off"} | "on"
xscale: {"linear"} | "log"
xtick: vector
Position of x tick marks. Setting xtick also forces the xtickmode property to
be set to "manual".
xticklabel: string | cell array of strings, def. 1-by-6 cell
Labels of x tick marks. Setting xticklabel also forces the xticklabelmode
property to be set to "manual".
xticklabelmode: {"auto"} | "manual"
xticklabelrotation: def. 0
xticklabelrotation is unused.
xtickmode: {"auto"} | "manual"
yaxislocation: {"left"} | "origin" | "right" | "zero"
Control the y-axis location.
ycolor: {colorspec} | "none", def. [0.15000 0.15000 0.15000]
Color of the y-axis. See Section 15.4.1 [colorspec], page 425.
ycolormode: {"auto"} | "manual"
ydir: {"normal"} | "reverse"
Direction of the y-axis: "normal" is bottom to top.
ygrid: {"off"} | "on"
Control whether major y grid lines are displayed.
ylabel: graphics handle
Graphics handle of the y label text object.
ylim: two-element vector, def. [0 1]
Two-element vector [ymin ymax] specifying the limits for the y-axis. Setting ylim also forces the ylimmode property to be set to "manual". See
[ylim function], page 317.
ylimmode: {"auto"} | "manual"
yminorgrid: {"off"} | "on"
Control whether minor y grid lines are displayed.

Chapter 15: Plotting

399

yminortick: {"off"} | "on"
yscale: {"linear"} | "log"
ytick: vector
Position of y tick marks. Setting ytick also forces the ytickmode property to
be set to "manual".
yticklabel: string | cell array of strings, def. 1-by-6 cell
Labels of y tick marks. Setting yticklabel also forces the yticklabelmode
property to be set to "manual".
yticklabelmode: {"auto"} | "manual"
yticklabelrotation: def. 0
yticklabelrotation is unused.
ytickmode: {"auto"} | "manual"
zcolor: {colorspec} | "none", def. [0.15000 0.15000 0.15000]
Color of the z-axis. See Section 15.4.1 [colorspec], page 425.
zcolormode: {"auto"} | "manual"
zdir: {"normal"} | "reverse"
zgrid: {"off"} | "on"
Control whether major z grid lines are displayed.
zlabel: graphics handle
Graphics handle of the z label text object.
zlim: two-element vector, def. [0 1]
Two-element vector [zmin zmaz] specifying the limits for the z-axis. Setting zlim also forces the zlimmode property to be set to "manual". See
[zlim function], page 317.
zlimmode: {"auto"} | "manual"
zminorgrid: {"off"} | "on"
Control whether minor z grid lines are displayed.
zminortick: {"off"} | "on"
zscale: {"linear"} | "log"
ztick: vector
Position of z tick marks. Setting ztick also forces the ztickmode property to
be set to "manual".
zticklabel: string | cell array of strings, def. 1-by-6 cell
Labels of z tick marks. Setting zticklabel also forces the zticklabelmode
property to be set to "manual".
zticklabelmode: {"auto"} | "manual"
zticklabelrotation: def. 0
zticklabelrotation is unused.
ztickmode: {"auto"} | "manual"

15.3.3.4 Line Properties
The line properties are:

400

GNU Octave

__modified__: "off" | {"on"}
beingdeleted: {"off"} | "on"
busyaction: "cancel" | {"queue"}
buttondownfcn: string | function handle, def. [](0x0)
For information on how to write graphics listener functions see Section 15.4.4
[Callbacks section], page 426.
children (read-only): vector of graphics handles, def. [](0x1)
children is unused.
clipping: "off" | {"on"}
If clipping is "on", the line is clipped in its parent axes limits.
color: colorspec, def. [0 0 0]
Color of the line object. See Section 15.4.1 [colorspec], page 425.
createfcn: string | function handle, def. [](0x0)
Callback function executed immediately after line has been created.
Function is set by using default property on root object, e.g., set (0,
"defaultlinecreatefcn", ’disp ("line created!")’).
For information on how to write graphics listener functions see Section 15.4.4
[Callbacks section], page 426.
deletefcn: string | function handle, def. [](0x0)
Callback function executed immediately before line is deleted.
For information on how to write graphics listener functions see Section 15.4.4
[Callbacks section], page 426.
displayname: string | cell array of strings, def. ""
Text for the legend entry corresponding to this line.
handlevisibility: "callback" | "off" | {"on"}
If handlevisibility is "off", the line’s handle is not visible in its parent’s
"children" property.
hittest: "off" | {"on"}
interpreter: "latex" | "none" | {"tex"}
interruptible: "off" | {"on"}
linestyle: {"-"} | "--" | "-." | ":" | "none"
See Section 15.4.2 [Line Styles], page 425.
linewidth: def. 0.50000
Width of the line object measured in points.
marker: "*" | "+" | "." | "<" | ">" | "^" | "d" | "diamond" | "h" | "hexagram" |
{"none"} | "o" | "p" | "pentagram" | "s" | "square" | "v" | "x"
Shape of the marker for each data point. See Section 15.4.3 [Marker Styles],
page 425.
markeredgecolor: {"auto"} | "none"
Color of the edge of the markers. When set to "auto", the marker edges have
the same color as the line. If set to "none", no marker edges are displayed. This
property can also be set to any color. See Section 15.4.1 [colorspec], page 425.

Chapter 15: Plotting

401

markerfacecolor: "auto" | {"none"}
Color of the face of the markers. When set to "auto", the marker faces have the
same color as the line. If set to "none", the marker faces are not displayed. This
property can also be set to any color. See Section 15.4.1 [colorspec], page 425.
markersize: scalar, def. 6
Size of the markers measured in points.
parent: graphics handle
Handle of the parent graphics object.
selected: {"off"} | "on"
selectionhighlight: "off" | {"on"}
tag: string, def. ""
A user-defined string to label the graphics object.
type (read-only): string
Class name of the graphics object. type is always "line"
uicontextmenu: graphics handle, def. [](0x0)
Graphics handle of the uicontextmenu object that is currently associated to
this line object.
userdata: Any Octave data, def. [](0x0)
User-defined data to associate with the graphics object.
visible: "off" | {"on"}
If visible is "off", the line is not rendered on screen.
xdata: vector, def. [0 1]
Vector of x data to be plotted.
xdatasource: string, def. ""
Name of a vector in the current base workspace to use as x data.
ydata: vector, def. [0 1]
Vector of y data to be plotted.
ydatasource: string, def. ""
Name of a vector in the current base workspace to use as y data.
zdata: vector, def. [](0x0)
Vector of z data to be plotted.
zdatasource: string, def. ""
Name of a vector in the current base workspace to use as z data.

15.3.3.5 Text Properties
The text properties are:
__modified__: "off" | {"on"}
backgroundcolor: colorspec, def. "none"
Background area is not yet implemented for text objects. backgroundcolor is
unused.

402

GNU Octave

beingdeleted: {"off"} | "on"
busyaction: "cancel" | {"queue"}
buttondownfcn: string | function handle, def. [](0x0)
For information on how to write graphics listener functions see Section 15.4.4
[Callbacks section], page 426.
children (read-only): vector of graphics handles, def. [](0x1)
children is unused.
clipping: "off" | {"on"}
If clipping is "on", the text is clipped in its parent axes limits.
color: colorspec, def. [0 0 0]
Color of the text. See Section 15.4.1 [colorspec], page 425.
createfcn: string | function handle, def. [](0x0)
Callback function executed immediately after text has been created.
Function is set by using default property on root object, e.g., set (0,
"defaulttextcreatefcn", ’disp ("text created!")’).
For information on how to write graphics listener functions see Section 15.4.4
[Callbacks section], page 426.
deletefcn: string | function handle, def. [](0x0)
Callback function executed immediately before text is deleted.
For information on how to write graphics listener functions see Section 15.4.4
[Callbacks section], page 426.
displayname: def. ""
edgecolor: colorspec, def. "none"
Background area is not yet implemented for text objects. edgecolor is unused.
editing: {"off"} | "on"
editing is unused.
extent (read-only): four-element vector, def. [0.000000 -0.005843 0.000000 0.032136]
Vector [x0 y0 width height] indicating the size and location of the text string.
fontangle: "italic" | {"normal"} | "oblique"
Control whether the font is italic or normal. fontangle is currently unused.
fontname: string, def. "*"
The font used for the text.
fontsize: scalar, def. 10
The font size of the text as measured in fontunits.
fontunits: "centimeters" | "inches" | "normalized" | "pixels" | {"points"}
The units used to interpret fontsize property.
fontweight: "bold" | "demi" | "light" | {"normal"}
Control variant of base font used: bold, light, normal, etc.
handlevisibility: "callback" | "off" | {"on"}
If handlevisibility is "off", the text’s handle is not visible in its parent’s
"children" property.

Chapter 15: Plotting

hittest: "off" | {"on"}
horizontalalignment: "center" | {"left"} | "right"
interpreter: "latex" | "none" | {"tex"}
Control the way the "string" property
[Use of the interpreter property], page 363.

403

is

interpreted.

See

interruptible: "off" | {"on"}
linestyle: {"-"} | "--" | "-." | ":" | "none"
Background area is not yet implemented for text objects. linestyle is unused.
linewidth: scalar, def. 0.50000
Background area is not yet implemented for text objects. linewidth is unused.
margin: scalar, def. 2
Background area is not yet implemented for text objects. margin is unused.
parent: graphics handle
Handle of the parent graphics object.
position: four-element vector, def. [0 0 0]
Vector [X0 Y0 Z0] where X0, Y0 and Z0 indicate the position of the text anchor
as defined by verticalalignment and horizontalalignment.
rotation: scalar, def. 0
The angle of rotation for the displayed text, measured in degrees.
selected: {"off"} | "on"
selectionhighlight: "off" | {"on"}
string: string, def. ""
The text object string content.
tag: string, def. ""
A user-defined string to label the graphics object.
type (read-only): string
Class name of the graphics object. type is always "text"
uicontextmenu: graphics handle, def. [](0x0)
Graphics handle of the uicontextmenu object that is currently associated to
this text object.
units: "centimeters" | {"data"} | "inches" | "normalized" | "pixels" | "points"
userdata: Any Octave data, def. [](0x0)
User-defined data to associate with the graphics object.
verticalalignment: "baseline" | "bottom" | "cap" | {"middle"} | "top"
visible: "off" | {"on"}
If visible is "off", the text is not rendered on screen.

15.3.3.6 Image Properties
The image properties are:
__modified__: "off" | {"on"}
alphadata: scalar | matrix, def. 1
Transparency is not yet implemented for image objects. alphadata is unused.

404

GNU Octave

alphadatamapping: "direct" | {"none"} | "scaled"
Transparency is not yet implemented for image objects. alphadatamapping is
unused.
beingdeleted: {"off"} | "on"
busyaction: "cancel" | {"queue"}
buttondownfcn: string | function handle, def. [](0x0)
For information on how to write graphics listener functions see Section 15.4.4
[Callbacks section], page 426.
cdata: matrix, def. 64-by-64 double
cdatamapping: {"direct"} | "scaled"
children (read-only): vector of graphics handles, def. [](0x1)
children is unused.
clipping: "off" | {"on"}
If clipping is "on", the image is clipped in its parent axes limits.
createfcn: string | function handle, def. [](0x0)
Callback function executed immediately after image has been created.
Function is set by using default property on root object, e.g., set (0,
"defaultimagecreatefcn", ’disp ("image created!")’).
For information on how to write graphics listener functions see Section 15.4.4
[Callbacks section], page 426.
deletefcn: string | function handle, def. [](0x0)
Callback function executed immediately before image is deleted.
For information on how to write graphics listener functions see Section 15.4.4
[Callbacks section], page 426.
displayname: string | cell array of strings, def. ""
Text for the legend entry corresponding to this image.
handlevisibility: "callback" | "off" | {"on"}
If handlevisibility is "off", the image’s handle is not visible in its parent’s
"children" property.
hittest: "off" | {"on"}
interruptible: "off" | {"on"}
parent: graphics handle
Handle of the parent graphics object.
selected: {"off"} | "on"
selectionhighlight: "off" | {"on"}
tag: string, def. ""
A user-defined string to label the graphics object.
type (read-only): string
Class name of the graphics object. type is always "image"
uicontextmenu: graphics handle, def. [](0x0)
Graphics handle of the uicontextmenu object that is currently associated to
this image object.

Chapter 15: Plotting

405

userdata: Any Octave data, def. [](0x0)
User-defined data to associate with the graphics object.
visible: "off" | {"on"}
If visible is "off", the image is not rendered on screen.
xdata: two-element vector, def. [1 64]
Two-element vector [xmin xmax] specifying the x coordinates of the first and
last columns of the image.
Setting xdata to the empty matrix ([]) will restore the default value of [1
columns(image)].
ydata: two-element vector, def. [1 64]
Two-element vector [ymin ymax] specifying the y coordinates of the first and
last rows of the image.
Setting ydata to the empty matrix ([]) will restore the default value of [1
rows(image)].

15.3.3.7 Patch Properties
The patch properties are:
__modified__: "off" | {"on"}
alphadatamapping: "direct" | "none" | {"scaled"}
Transparency is not yet implemented for patch objects. alphadatamapping is
unused.
ambientstrength: scalar, def. 0.30000
Strength of the ambient light. Value between 0.0 and 1.0
backfacelighting: "lit" | {"reverselit"} | "unlit"
"lit": The normals are used as is for lighting. "reverselit": The normals
are always oriented towards the point of view. "unlit": Faces with normals
pointing away from the point of view are unlit.
beingdeleted: {"off"} | "on"
busyaction: "cancel" | {"queue"}
buttondownfcn: string | function handle, def. [](0x0)
For information on how to write graphics listener functions see Section 15.4.4
[Callbacks section], page 426.
cdata: scalar | matrix, def. [](0x0)
Data defining the patch object color. Patch color can be defined for faces or
for vertices.
If cdata is a scalar index into the current colormap or a RGB triplet, it defines
the color of all faces.
If cdata is an N-by-1 vector of indices or an N-by-3 (RGB) matrix, it defines
the color of each one of the N faces.
If cdata is an N-by-M or an N-by-M-by-3 (RGB) matrix, it defines the color at
each vertex.

406

GNU Octave

cdatamapping: "direct" | {"scaled"}
children (read-only): vector of graphics handles, def. [](0x1)
children is unused.
clipping: "off" | {"on"}
If clipping is "on", the patch is clipped in its parent axes limits.
createfcn: string | function handle, def. [](0x0)
Callback function executed immediately after patch has been created.
Function is set by using default property on root object, e.g., set (0,
"defaultpatchcreatefcn", ’disp ("patch created!")’).
For information on how to write graphics listener functions see Section 15.4.4
[Callbacks section], page 426.
deletefcn: string | function handle, def. [](0x0)
Callback function executed immediately before patch is deleted.
For information on how to write graphics listener functions see Section 15.4.4
[Callbacks section], page 426.
diffusestrength: scalar, def. 0.60000
Strength of the diffuse reflex. Value between 0.0 (no diffuse reflex) and 1.0 (full
diffuse reflex).
displayname: def. ""
Text of the legend entry corresponding to this patch.
edgealpha: scalar | matrix, def. 1
Transparency is not yet implemented for patch objects. edgealpha is unused.
edgecolor: def. [0 0 0]
edgelighting: "flat" | "gouraud" | {"none"} | "phong"
When set to a value other than "none", the edges of the object are drawn with
light and shadow effects. Supported values are "none" (no lighting effects),
"flat" (facetted look) and "gouraud" (linear interpolation of the lighting effects between the vertices). "phong" is deprecated and has the same effect as
"gouraud".
facealpha: scalar | matrix, def. 1
Transparency is not yet implemented for patch objects. facealpha is unused.
facecolor: {colorspec} | "none" | "flat" | "interp", def. [0 0 0]
facelighting: {"flat"} | "gouraud" | "none" | "phong"
When set to a value other than "none", the faces of the object are drawn with
light and shadow effects. Supported values are "none" (no lighting effects),
"flat" (facetted look) and "gouraud" (linear interpolation of the lighting effects between the vertices). "phong" is deprecated and has the same effect as
"gouraud".

Chapter 15: Plotting

407

facenormals: def. [](0x0)
facenormalsmode: {"auto"} | "manual"
faces: def. [1 2 3]
facevertexalphadata: scalar | matrix, def. [](0x0)
Transparency is not yet implemented for patch objects. facevertexalphadata
is unused.
facevertexcdata: def. [](0x0)
handlevisibility: "callback" | "off" | {"on"}
If handlevisibility is "off", the patch’s handle is not visible in its parent’s
"children" property.
hittest: "off" | {"on"}
interpreter: "latex" | "none" | {"tex"}
interpreter is unused.
interruptible: "off" | {"on"}
linestyle: {"-"} | "--" | "-." | ":" | "none"
linewidth: def. 0.50000
marker: "*" | "+" | "." | "<" | ">" | "^" | "d" | "diamond" | "h" | "hexagram" |
{"none"} | "o" | "p" | "pentagram" | "s" | "square" | "v" | "x"
See [line marker property], page 400.
markeredgecolor: {"auto"} | "flat" | "none"
See [line markeredgecolor property], page 400.
markerfacecolor: "auto" | "flat" | {"none"}
See [line markerfacecolor property], page 400.
markersize: scalar, def. 6
See [line markersize property], page 401.
parent: graphics handle
Handle of the parent graphics object.
selected: {"off"} | "on"
selectionhighlight: "off" | {"on"}
specularcolorreflectance: scalar, def. 1
Reflectance for specular color. Value between 0.0 (color of underlying face) and
1.0 (color of light source).
specularexponent: scalar, def. 10
Exponent for the specular reflex. The lower the value, the more the reflex is
spread out.
specularstrength: scalar, def. 0.90000
Strength of the specular reflex. Value between 0.0 (no specular reflex) and 1.0
(full specular reflex).
tag: string, def. ""
A user-defined string to label the graphics object.
type (read-only): string
Class name of the graphics object. type is always "patch"

408

GNU Octave

uicontextmenu: graphics handle, def. [](0x0)
Graphics handle of the uicontextmenu object that is currently associated to
this patch object.
userdata: Any Octave data, def. [](0x0)
User-defined data to associate with the graphics object.
vertexnormals: def. [](0x0)
vertexnormalsmode: {"auto"} | "manual"
vertices: vector | matrix, def. 3-by-2 double
visible: "off" | {"on"}
If visible is "off", the patch is not rendered on screen.
xdata: vector | matrix, def. [0; 1; 0]
ydata: vector | matrix, def. [1; 1; 0]
zdata: vector | matrix, def. [](0x0)

15.3.3.8 Surface Properties
The surface properties are:
__modified__: "off" | {"on"}
alphadata: scalar | matrix, def. 1
Transparency is not yet implemented for surface objects. alphadata is unused.
alphadatamapping: "direct" | "none" | {"scaled"}
Transparency is not yet implemented for surface objects. alphadatamapping
is unused.
ambientstrength: scalar, def. 0.30000
Strength of the ambient light. Value between 0.0 and 1.0
backfacelighting: "lit" | {"reverselit"} | "unlit"
"lit": The normals are used as is for lighting. "reverselit": The normals
are always oriented towards the point of view. "unlit": Faces with normals
pointing away from the point of view are unlit.
beingdeleted: {"off"} | "on"
busyaction: "cancel" | {"queue"}
buttondownfcn: string | function handle, def. [](0x0)
For information on how to write graphics listener functions see Section 15.4.4
[Callbacks section], page 426.
cdata: matrix, def. 3-by-3 double
cdatamapping: "direct" | {"scaled"}
cdatasource: def. ""
children (read-only): vector of graphics handles, def. [](0x1)
children is unused.
clipping: "off" | {"on"}
If clipping is "on", the surface is clipped in its parent axes limits.

Chapter 15: Plotting

409

createfcn: string | function handle, def. [](0x0)
Callback function executed immediately after surface has been created.
Function is set by using default property on root object, e.g., set (0,
"defaultsurfacecreatefcn", ’disp ("surface created!")’).
For information on how to write graphics listener functions see Section 15.4.4
[Callbacks section], page 426.
deletefcn: string | function handle, def. [](0x0)
Callback function executed immediately before surface is deleted.
For information on how to write graphics listener functions see Section 15.4.4
[Callbacks section], page 426.
diffusestrength: scalar, def. 0.60000
Strength of the diffuse reflex. Value between 0.0 (no diffuse reflex) and 1.0 (full
diffuse reflex).
displayname: def. ""
Text for the legend entry corresponding to this surface.
edgealpha: scalar, def. 1
Transparency is not yet implemented for surface objects. edgealpha is unused.
edgecolor: def. [0 0 0]
edgelighting: "flat" | "gouraud" | {"none"} | "phong"
When set to a value other than "none", the edges of the object are drawn with
light and shadow effects. Supported values are "none" (no lighting effects),
"flat" (facetted look) and "gouraud" (linear interpolation of the lighting effects between the vertices). "phong" is deprecated and has the same effect as
"gouraud".
facealpha: scalar | matrix, def. 1
Transparency is not yet implemented for surface objects. facealpha is unused.
facecolor: {"flat"} | "interp" | "none" | "texturemap"
facelighting: {"flat"} | "gouraud" | "none" | "phong"
When set to a value other than "none", the faces of the object are drawn with
light and shadow effects. Supported values are "none" (no lighting effects),
"flat" (facetted look) and "gouraud" (linear interpolation of the lighting effects between the vertices). "phong" is deprecated and has the same effect as
"gouraud".
facenormals: def. [](0x0)
facenormalsmode: {"auto"} | "manual"
handlevisibility: "callback" | "off" | {"on"}
If handlevisibility is "off", the surface’s handle is not visible in its parent’s
"children" property.
hittest: "off" | {"on"}
interpreter: "latex" | "none" | {"tex"}
interruptible: "off" | {"on"}
linestyle: {"-"} | "--" | "-." | ":" | "none"
See Section 15.4.2 [Line Styles], page 425.

410

GNU Octave

linewidth: def. 0.50000
See [line linewidth property], page 400.
marker: "*" | "+" | "." | "<" | ">" | "^" | "d" | "diamond" | "h" | "hexagram" |
{"none"} | "o" | "p" | "pentagram" | "s" | "square" | "v" | "x"
See Section 15.4.3 [Marker Styles], page 425.
markeredgecolor: {"auto"} | "flat" | "none"
See [line markeredgecolor property], page 400.
markerfacecolor: "auto" | "flat" | {"none"}
See [line markerfacecolor property], page 400.
markersize: scalar, def. 6
See [line markersize property], page 401.
meshstyle: {"both"} | "column" | "row"
parent: graphics handle
Handle of the parent graphics object.
selected: {"off"} | "on"
selectionhighlight: "off" | {"on"}
specularcolorreflectance: scalar, def. 1
Reflectance for specular color. Value between 0.0 (color of underlying face) and
1.0 (color of light source).
specularexponent: scalar, def. 10
Exponent for the specular reflex. The lower the value, the more the reflex is
spread out.
specularstrength: scalar, def. 0.90000
Strength of the specular reflex. Value between 0.0 (no specular reflex) and 1.0
(full specular reflex).
tag: string, def. ""
A user-defined string to label the graphics object.
type (read-only): string
Class name of the graphics object. type is always "surface"
uicontextmenu: graphics handle, def. [](0x0)
Graphics handle of the uicontextmenu object that is currently associated to
this surface object.
userdata: Any Octave data, def. [](0x0)
User-defined data to associate with the graphics object.
vertexnormals: def. 3-by-3-by-3 double
vertexnormalsmode: {"auto"} | "manual"
visible: "off" | {"on"}
If visible is "off", the surface is not rendered on screen.

Chapter 15: Plotting

411

xdata: matrix, def. [1 2 3]
xdatasource: def. ""
ydata: matrix, def. [1; 2; 3]
ydatasource: def. ""
zdata: matrix, def. 3-by-3 double
zdatasource: def. ""

15.3.3.9 Light Properties
The light properties are:
__modified__: "off" | {"on"}
beingdeleted: {"off"} | "on"
busyaction: "cancel" | {"queue"}
buttondownfcn: string | function handle, def. [](0x0)
For information on how to write graphics listener functions see Section 15.4.4
[Callbacks section], page 426.
children (read-only): vector of graphics handles, def. [](0x1)
children is unused.
clipping: "off" | {"on"}
If clipping is "on", the light is clipped in its parent axes limits.
color: colorspec, def. [1 1 1]
Color of the light source. See Section 15.4.1 [colorspec], page 425.
createfcn: string | function handle, def. [](0x0)
Callback function executed immediately after light has been created.
Function is set by using default property on root object, e.g., set (0,
"defaultlightcreatefcn", ’disp ("light created!")’).
For information on how to write graphics listener functions see Section 15.4.4
[Callbacks section], page 426.
deletefcn: string | function handle, def. [](0x0)
Callback function executed immediately before light is deleted.
For information on how to write graphics listener functions see Section 15.4.4
[Callbacks section], page 426.
handlevisibility: "callback" | "off" | {"on"}
If handlevisibility is "off", the light’s handle is not visible in its parent’s
"children" property.
hittest: "off" | {"on"}
interruptible: "off" | {"on"}
parent: graphics handle
Handle of the parent graphics object.
position: def. [1 0 1]
Position of the light source.

412

GNU Octave

selected: {"off"} | "on"
selectionhighlight: "off" | {"on"}
style: {"infinite"} | "local"
This string defines whether the light emanates from a light source at infinite
distance ("infinite") or from a local point source ("local").
tag: string, def. ""
A user-defined string to label the graphics object.
type (read-only): string
Class name of the graphics object. type is always "light"
uicontextmenu: graphics handle, def. [](0x0)
Graphics handle of the uicontextmenu object that is currently associated to
this light object.
userdata: Any Octave data, def. [](0x0)
User-defined data to associate with the graphics object.
visible: "off" | {"on"}
If visible is "off", the light is not rendered on screen.

15.3.3.10 Uimenu Properties
The uimenu properties are:
__modified__: "off" | {"on"}
accelerator: def. ""
beingdeleted: {"off"} | "on"
busyaction: "cancel" | {"queue"}
buttondownfcn: string | function handle, def. [](0x0)
buttondownfcn is unused.
callback: def. [](0x0)
checked: {"off"} | "on"
children (read-only): vector of graphics handles, def. [](0x1)
Graphics handles of the uimenu’s children.
clipping: "off" | {"on"}
If clipping is "on", the uimenu is clipped in its parent axes limits.
createfcn: string | function handle, def. [](0x0)
Callback function executed immediately after uimenu has been created.
Function is set by using default property on root object, e.g., set (0,
"defaultuimenucreatefcn", ’disp ("uimenu created!")’).
For information on how to write graphics listener functions see Section 15.4.4
[Callbacks section], page 426.
deletefcn: string | function handle, def. [](0x0)
Callback function executed immediately before uimenu is deleted.
For information on how to write graphics listener functions see Section 15.4.4
[Callbacks section], page 426.

Chapter 15: Plotting

413

enable: "off" | {"on"}
foregroundcolor: def. [0 0 0]
handlevisibility: "callback" | "off" | {"on"}
If handlevisibility is "off", the uimenu’s handle is not visible in its parent’s
"children" property.
hittest: "off" | {"on"}
interruptible: "off" | {"on"}
label: def. ""
parent: graphics handle
Handle of the parent graphics object.
position: def. 1
selected: {"off"} | "on"
selectionhighlight: "off" | {"on"}
separator: {"off"} | "on"
tag: string, def. ""
A user-defined string to label the graphics object.
type (read-only): string
Class name of the graphics object. type is always "uimenu"
uicontextmenu: graphics handle, def. [](0x0)
Graphics handle of the uicontextmenu object that is currently associated to
this uimenu object.
userdata: Any Octave data, def. [](0x0)
User-defined data to associate with the graphics object.
visible: "off" | {"on"}
If visible is "off", the uimenu is not rendered on screen.

15.3.3.11 Uibuttongroup Properties
The uibuttongroup properties are:
__modified__: "off" | {"on"}
backgroundcolor: def. [0.93725 0.94118 0.94510]
beingdeleted: {"off"} | "on"
bordertype: "beveledin" | "beveledout" | {"etchedin"} | "etchedout" | "line" |
"none"
borderwidth: def. 1
busyaction: "cancel" | {"queue"}
buttondownfcn: string | function handle, def. [](0x0)
For information on how to write graphics listener functions see Section 15.4.4
[Callbacks section], page 426.
children (read-only): vector of graphics handles, def. [](0x1)
Graphics handles of the uibuttongroup’s children.
clipping: "off" | {"on"}
If clipping is "on", the uibuttongroup is clipped in its parent axes limits.

414

GNU Octave

createfcn: string | function handle, def. [](0x0)
Callback function executed immediately after uibuttongroup has been
created. Function is set by using default property on root object, e.g.,
set (0, "defaultuibuttongroupcreatefcn", ’disp ("uibuttongroup
created!")’).
For information on how to write graphics listener functions see Section 15.4.4
[Callbacks section], page 426.
deletefcn: string | function handle, def. [](0x0)
Callback function executed immediately before uibuttongroup is deleted.
For information on how to write graphics listener functions see Section 15.4.4
[Callbacks section], page 426.
fontangle: "italic" | {"normal"} | "oblique"
fontname: def. "*"
fontsize: def. 10
fontunits: "centimeters" | "inches" | "normalized" | "pixels" | {"points"}
fontweight: "bold" | "demi" | "light" | {"normal"}
foregroundcolor: def. [0.19216 0.21176 0.23137]
handlevisibility: "callback" | "off" | {"on"}
If handlevisibility is "off", the uibuttongroup’s handle is not visible in its
parent’s "children" property.
highlightcolor: def. [1 1 1]
hittest: "off" | {"on"}
interruptible: "off" | {"on"}
parent: graphics handle
Handle of the parent graphics object.
position: def. [0 0 1 1]
resizefcn: def. [](0x0)
selected: {"off"} | "on"
selectedobject: def. [](0x0)
selectionchangedfcn: def. [](0x0)
selectionhighlight: "off" | {"on"}
shadowcolor: def. [0.53371 0.55532 0.57606]
sizechangedfcn: def. [](0x0)
tag: string, def. ""
A user-defined string to label the graphics object.
title: def. ""
titleposition: "centerbottom" | "centertop" | "leftbottom" | {"lefttop"} |
"rightbottom" | "righttop"
type (read-only): string
Class name of the graphics object. type is always "uibuttongroup"
uicontextmenu: graphics handle, def. [](0x0)
Graphics handle of the uicontextmenu object that is currently associated to
this uibuttongroup object.

Chapter 15: Plotting

415

units: "centimeters" | "characters" | "inches" | {"normalized"} | "pixels" |
"points"
userdata: Any Octave data, def. [](0x0)
User-defined data to associate with the graphics object.
visible: "off" | {"on"}
If visible is "off", the uibuttongroup is not rendered on screen.

15.3.3.12 Uicontextmenu Properties
The uicontextmenu properties are:
__modified__: "off" | {"on"}
beingdeleted: {"off"} | "on"
busyaction: "cancel" | {"queue"}
buttondownfcn: string | function handle, def. [](0x0)
buttondownfcn is unused.
callback: def. [](0x0)
children (read-only): vector of graphics handles, def. [](0x1)
Graphics handles of the uicontextmenu’s children.
clipping: "off" | {"on"}
If clipping is "on", the uicontextmenu is clipped in its parent axes limits.
createfcn: string | function handle, def. [](0x0)
Callback function executed immediately after uicontextmenu has been
created. Function is set by using default property on root object, e.g.,
set (0, "defaultuicontextmenucreatefcn", ’disp ("uicontextmenu
created!")’).
For information on how to write graphics listener functions see Section 15.4.4
[Callbacks section], page 426.
deletefcn: string | function handle, def. [](0x0)
Callback function executed immediately before uicontextmenu is deleted.
For information on how to write graphics listener functions see Section 15.4.4
[Callbacks section], page 426.
handlevisibility: "callback" | "off" | {"on"}
If handlevisibility is "off", the uicontextmenu’s handle is not visible in its
parent’s "children" property.
hittest: "off" | {"on"}
interruptible: "off" | {"on"}
parent: graphics handle
Handle of the parent graphics object.
position: def. [0 0]
selected: {"off"} | "on"
selectionhighlight: "off" | {"on"}
tag: string, def. ""
A user-defined string to label the graphics object.

416

GNU Octave

type (read-only): string
Class name of the graphics object. type is always "uicontextmenu"
uicontextmenu: graphics handle, def. [](0x0)
Graphics handle of the uicontextmenu object that is currently associated to
this uicontextmenu object.
userdata: Any Octave data, def. [](0x0)
User-defined data to associate with the graphics object.
visible: "off" | {"on"}
If visible is "off", the uicontextmenu is not rendered on screen.

15.3.3.13 Uipanel Properties
The uipanel properties are:
__modified__: "off" | {"on"}
backgroundcolor: def. [0.93725 0.94118 0.94510]
beingdeleted: {"off"} | "on"
bordertype: "beveledin" | "beveledout" | {"etchedin"} | "etchedout" | "line" |
"none"
borderwidth: def. 1
busyaction: "cancel" | {"queue"}
buttondownfcn: string | function handle, def. [](0x0)
For information on how to write graphics listener functions see Section 15.4.4
[Callbacks section], page 426.
children (read-only): vector of graphics handles, def. [](0x1)
Graphics handles of the uipanel’s children.
clipping: "off" | {"on"}
If clipping is "on", the uipanel is clipped in its parent axes limits.
createfcn: string | function handle, def. [](0x0)
Callback function executed immediately after uipanel has been created.
Function is set by using default property on root object, e.g., set (0,
"defaultuipanelcreatefcn", ’disp ("uipanel created!")’).
For information on how to write graphics listener functions see Section 15.4.4
[Callbacks section], page 426.
deletefcn: string | function handle, def. [](0x0)
Callback function executed immediately before uipanel is deleted.
For information on how to write graphics listener functions see Section 15.4.4
[Callbacks section], page 426.

Chapter 15: Plotting

417

fontangle: "italic" | {"normal"} | "oblique"
fontname: def. "*"
fontsize: def. 10
fontunits: "centimeters" | "inches" | "normalized" | "pixels" | {"points"}
fontweight: "bold" | "demi" | "light" | {"normal"}
foregroundcolor: def. [0.19216 0.21176 0.23137]
handlevisibility: "callback" | "off" | {"on"}
If handlevisibility is "off", the uipanel’s handle is not visible in its parent’s
"children" property.
highlightcolor: def. [1 1 1]
hittest: "off" | {"on"}
interruptible: "off" | {"on"}
parent: graphics handle
Handle of the parent graphics object.
position: def. [0 0 1 1]
resizefcn: def. [](0x0)
selected: {"off"} | "on"
selectionhighlight: "off" | {"on"}
shadowcolor: def. [0.53371 0.55532 0.57606]
tag: string, def. ""
A user-defined string to label the graphics object.
title: def. ""
titleposition: "centerbottom" | "centertop" | "leftbottom" | {"lefttop"} |
"rightbottom" | "righttop"
type (read-only): string
Class name of the graphics object. type is always "uipanel"
uicontextmenu: graphics handle, def. [](0x0)
Graphics handle of the uicontextmenu object that is currently associated to
this uipanel object.
units: "centimeters" | "characters" | "inches" | {"normalized"} | "pixels" |
"points"
userdata: Any Octave data, def. [](0x0)
User-defined data to associate with the graphics object.
visible: "off" | {"on"}
If visible is "off", the uipanel is not rendered on screen.

15.3.3.14 Uicontrol Properties
The uicontrol properties are:

418

GNU Octave

__modified__: "off" | {"on"}
backgroundcolor: def. [0.93725 0.94118 0.94510]
beingdeleted: {"off"} | "on"
busyaction: "cancel" | {"queue"}
buttondownfcn: string | function handle, def. [](0x0)
For information on how to write graphics listener functions see Section 15.4.4
[Callbacks section], page 426.
callback: def. [](0x0)
cdata: def. [](0x0)
children (read-only): vector of graphics handles, def. [](0x1)
Graphics handles of the uicontrol’s children.
clipping: "off" | {"on"}
If clipping is "on", the uicontrol is clipped in its parent axes limits.
createfcn: string | function handle, def. [](0x0)
Callback function executed immediately after uicontrol has been created.
Function is set by using default property on root object, e.g., set (0,
"defaultuicontrolcreatefcn", ’disp ("uicontrol created!")’).
For information on how to write graphics listener functions see Section 15.4.4
[Callbacks section], page 426.
deletefcn: string | function handle, def. [](0x0)
Callback function executed immediately before uicontrol is deleted.
For information on how to write graphics listener functions see Section 15.4.4
[Callbacks section], page 426.
enable: "inactive" | "off" | {"on"}
extent (read-only): def. [0 0 0 0]
fontangle: "italic" | {"normal"} | "oblique"
fontname: def. "*"
fontsize: def. 10
fontunits: "centimeters" | "inches" | "normalized" | "pixels" | {"points"}
fontweight: "bold" | "demi" | "light" | {"normal"}
foregroundcolor: def. [0.19216 0.21176 0.23137]
handlevisibility: "callback" | "off" | {"on"}
If handlevisibility is "off", the uicontrol’s handle is not visible in its parent’s "children" property.
hittest: "off" | {"on"}
horizontalalignment: {"center"} | "left" | "right"
interruptible: "off" | {"on"}
keypressfcn: def. [](0x0)
listboxtop: def. 1
max: def. 1
min: def. 0
parent: graphics handle
Handle of the parent graphics object.

Chapter 15: Plotting

419

position: def. [0 0 80 30]
selected: {"off"} | "on"
selectionhighlight: "off" | {"on"}
sliderstep: def. [0.010000 0.100000]
string: def. ""
style: "checkbox" | "edit" | "frame" | "listbox" | "popupmenu" | {"pushbutton"}
| "radiobutton" | "slider" | "text" | "togglebutton"
tag: string, def. ""
A user-defined string to label the graphics object.
tooltipstring: def. ""
type (read-only): string
Class name of the graphics object. type is always "uicontrol"
uicontextmenu: graphics handle, def. [](0x0)
Graphics handle of the uicontextmenu object that is currently associated to
this uicontrol object.
units: "centimeters" | "characters" | "inches" | "normalized" | {"pixels"} |
"points"
userdata: Any Octave data, def. [](0x0)
User-defined data to associate with the graphics object.
value: def. 0
verticalalignment: "bottom" | {"middle"} | "top"
visible: "off" | {"on"}
If visible is "off", the uicontrol is not rendered on screen.

15.3.3.15 Uitoolbar Properties
The uitoolbar properties are:
__modified__: "off" | {"on"}
beingdeleted: {"off"} | "on"
busyaction: "cancel" | {"queue"}
buttondownfcn: string | function handle, def. [](0x0)
buttondownfcn is unused.
children (read-only): vector of graphics handles, def. [](0x1)
Graphics handles of the uitoolbar’s children.
clipping: "off" | {"on"}
If clipping is "on", the uitoolbar is clipped in its parent axes limits.
createfcn: string | function handle, def. [](0x0)
Callback function executed immediately after uitoolbar has been created.
Function is set by using default property on root object, e.g., set (0,
"defaultuitoolbarcreatefcn", ’disp ("uitoolbar created!")’).
For information on how to write graphics listener functions see Section 15.4.4
[Callbacks section], page 426.
deletefcn: string | function handle, def. [](0x0)
Callback function executed immediately before uitoolbar is deleted.

420

GNU Octave

For information on how to write graphics listener functions see Section 15.4.4
[Callbacks section], page 426.
handlevisibility: "callback" | "off" | {"on"}
If handlevisibility is "off", the uitoolbar’s handle is not visible in its parent’s "children" property.
hittest: "off" | {"on"}
interruptible: "off" | {"on"}
parent: graphics handle
Handle of the parent graphics object.
selected: {"off"} | "on"
selectionhighlight: "off" | {"on"}
tag: string, def. ""
A user-defined string to label the graphics object.
type (read-only): string
Class name of the graphics object. type is always "uitoolbar"
uicontextmenu: graphics handle, def. [](0x0)
Graphics handle of the uicontextmenu object that is currently associated to
this uitoolbar object.
userdata: Any Octave data, def. [](0x0)
User-defined data to associate with the graphics object.
visible: "off" | {"on"}
If visible is "off", the uitoolbar is not rendered on screen.

15.3.3.16 Uipushtool Properties
The uipushtool properties are:
__modified__: "off" | {"on"}
beingdeleted: {"off"} | "on"
busyaction: "cancel" | {"queue"}
buttondownfcn: string | function handle, def. [](0x0)
buttondownfcn is unused.
cdata: def. [](0x0)
children (read-only): vector of graphics handles, def. [](0x1)
Graphics handles of the uipushtool’s children.
clickedcallback: def. [](0x0)
clipping: "off" | {"on"}
If clipping is "on", the uipushtool is clipped in its parent axes limits.
createfcn: string | function handle, def. [](0x0)
Callback function executed immediately after uipushtool has been created.
Function is set by using default property on root object, e.g., set (0,
"defaultuipushtoolcreatefcn", ’disp ("uipushtool created!")’).
For information on how to write graphics listener functions see Section 15.4.4
[Callbacks section], page 426.

Chapter 15: Plotting

421

deletefcn: string | function handle, def. [](0x0)
Callback function executed immediately before uipushtool is deleted.
For information on how to write graphics listener functions see Section 15.4.4
[Callbacks section], page 426.
enable: "off" | {"on"}
handlevisibility: "callback" | "off" | {"on"}
If handlevisibility is "off", the uipushtool’s handle is not visible in its
parent’s "children" property.
hittest: "off" | {"on"}
interruptible: "off" | {"on"}
parent: graphics handle
Handle of the parent graphics object.
selected: {"off"} | "on"
selectionhighlight: "off" | {"on"}
separator: {"off"} | "on"
tag: string, def. ""
A user-defined string to label the graphics object.
tooltipstring: def. ""
type (read-only): string
Class name of the graphics object. type is always "uipushtool"
uicontextmenu: graphics handle, def. [](0x0)
Graphics handle of the uicontextmenu object that is currently associated to
this uipushtool object.
userdata: Any Octave data, def. [](0x0)
User-defined data to associate with the graphics object.
visible: "off" | {"on"}
If visible is "off", the uipushtool is not rendered on screen.

15.3.3.17 Uitoggletool Properties
The uitoggletool properties are:
__modified__: "off" | {"on"}
beingdeleted: {"off"} | "on"
busyaction: "cancel" | {"queue"}
buttondownfcn: string | function handle, def. [](0x0)
buttondownfcn is unused.
cdata: def. [](0x0)
children (read-only): vector of graphics handles, def. [](0x1)
Graphics handles of the uitoggletool’s children.
clickedcallback: def. [](0x0)
clipping: "off" | {"on"}
If clipping is "on", the uitoggletool is clipped in its parent axes limits.

422

GNU Octave

createfcn: string | function handle, def. [](0x0)
Callback function executed immediately after uitoggletool has been created.
Function is set by using default property on root object, e.g., set (0,
"defaultuitoggletoolcreatefcn", ’disp ("uitoggletool created!")’).
For information on how to write graphics listener functions see Section 15.4.4
[Callbacks section], page 426.
deletefcn: string | function handle, def. [](0x0)
Callback function executed immediately before uitoggletool is deleted.
For information on how to write graphics listener functions see Section 15.4.4
[Callbacks section], page 426.
enable: "off" | {"on"}
handlevisibility: "callback" | "off" | {"on"}
If handlevisibility is "off", the uitoggletool’s handle is not visible in its
parent’s "children" property.
hittest: "off" | {"on"}
interruptible: "off" | {"on"}
offcallback: def. [](0x0)
oncallback: def. [](0x0)
parent: graphics handle
Handle of the parent graphics object.
selected: {"off"} | "on"
selectionhighlight: "off" | {"on"}
separator: {"off"} | "on"
state: {"off"} | "on"
tag: string, def. ""
A user-defined string to label the graphics object.
tooltipstring: def. ""
type (read-only): string
Class name of the graphics object. type is always "uitoggletool"
uicontextmenu: graphics handle, def. [](0x0)
Graphics handle of the uicontextmenu object that is currently associated to
this uitoggletool object.
userdata: Any Octave data, def. [](0x0)
User-defined data to associate with the graphics object.
visible: "off" | {"on"}
If visible is "off", the uitoggletool is not rendered on screen.

15.3.4 Searching Properties
h = findobj ()
h = findobj (prop_name, prop_value, . . . )
h = findobj (prop_name, prop_value, "-logical_op", prop_name,
prop_value)

Chapter 15: Plotting

h
h
h
h
h

=
=
=
=
=

423

("-property", prop_name)
("-regexp", prop_name, pattern)
(hlist, . . . )
(hlist, "flat", . . . )
(hlist, "-depth", d, . . . )
Find graphics objects with specified properties.

findobj
findobj
findobj
findobj
findobj

When called without arguments, return all graphic objects beginning with the root
object (0) and including all of its descendants.
The simplest form for narrowing the results is
findobj (prop_name, prop_value)
which returns the handles of all objects which have a property named prop name
that has the value prop value. If multiple property/value pairs are specified then
only objects meeting all of the conditions (equivalent to -and) are returned.
The search can be limited to a particular set of objects and their descendants, by
passing a handle or set of handles hlist as the first argument.
The depth of the object hierarchy to search can be limited with the "-depth" argument. An example of searching through only three generations of children is:
findobj (hlist, "-depth", 3, prop_name, prop_value)
Specifying a depth d of 0 limits the search to the set of objects passed in hlist. A
depth of 0 is also equivalent to the "flat" argument. The default depth value is Inf
which includes all descendants.
A specified logical operator may be used between prop name, prop value pairs. The
supported logical operators are: "-and", "-or", "-xor", "-not". Example code to
locate all figure and axes objects is
findobj ("type", "figure", "-or", "type", "axes")
Objects may also be matched by comparing a regular expression to the property values, where property values that match regexp (prop_value, pattern) are returned.
Finally, objects which have a property name can be found with the "-property"
option. For example, code to locate objects with a "meshstyle" property is
findobj ("-property", "meshstyle")
Implementation Note: The search only includes objects with visible handles
(HandleVisibility = "on"). See [findall], page 423, to search for all objects including
hidden ones.
See also: [findall], page 423, [allchild], page 385, [get], page 383, [set], page 383.

h = findall ()
h = findall (prop_name, prop_value, . . . )
h = findall (prop_name, prop_value, "-logical_op", prop_name,
prop_value)
h = findall ("-property", prop_name)
h = findall ("-regexp", prop_name, pattern)
h = findall (hlist, . . . )
h = findall (hlist, "flat", . . . )

424

GNU Octave

h = findall (hlist, "-depth", d, . . . )
Find graphics object, including hidden ones, with specified properties.
The return value h is a list of handles to the found graphic objects.
findall performs the same search as findobj, but it includes hidden objects (HandleVisibility = "off"). For full documentation, see [findobj], page 422.
See also: [findobj], page 422, [allchild], page 385, [get], page 383, [set], page 383.

15.3.5 Managing Default Properties
Object properties have two classes of default values, factory defaults (the initial values) and
user-defined defaults, which may override the factory defaults.
Although default values may be set for any object, they are set in parent objects and
apply to child objects, of the specified object type. For example, setting the default color
property of line objects to "green", for the root object, will result in all line objects
inheriting the color "green" as the default value.
set (0, "defaultlinecolor", "green");
sets the default line color for all objects. The rule for constructing the property name to
set a default value is
default + object-type + property-name
This rule can lead to some strange looking names, for example defaultlinelinewidth"
specifies the default linewidth property for line objects.
The example above used the root figure object, 0, so the default property value will
apply to all line objects. However, default values are hierarchical, so defaults set in a figure
objects override those set in the root figure object. Likewise, defaults set in axes objects
override those set in figure or root figure objects. For example,
subplot (2, 1, 1);
set (0, "defaultlinecolor", "red");
set (1, "defaultlinecolor", "green");
set (gca (), "defaultlinecolor", "blue");
line (1:10, rand (1, 10));
subplot (2, 1, 2);
line (1:10, rand (1, 10));
figure (2)
line (1:10, rand (1, 10));
produces two figures. The line in first subplot window of the first figure is blue because it
inherits its color from its parent axes object. The line in the second subplot window of the
first figure is green because it inherits its color from its parent figure object. The line in the
second figure window is red because it inherits its color from the global root figure parent
object.
To remove a user-defined default setting, set the default property to the value "remove".
For example,
set (gca (), "defaultlinecolor", "remove");
removes the user-defined default line color setting from the current axes object. To quickly
remove all user-defined defaults use the reset function.

Chapter 15: Plotting

425

reset (h)
Reset the properties of the graphic object h to their default values.
For figures, the properties "position", "units", "windowstyle", and "paperunits"
are not affected. For axes, the properties "position" and "units" are not affected.
The input h may also be a vector of graphic handles in which case each individual
object will be reset.
See also: [cla], page 362, [clf], page 362, [newplot], page 360.
Getting the "default" property of an object returns a list of user-defined defaults set
for the object. For example,
get (gca (), "default");
returns a list of user-defined default values for the current axes object.
Factory default values are stored in the root figure object. The command
get (0, "factory");
returns a list of factory defaults.

15.4 Advanced Plotting
15.4.1 Colors
Colors may be specified as RGB triplets with values ranging from zero to one, or by name.
Recognized color names include "blue", "black", "cyan", "green", "magenta", "red",
"white", and "yellow".

15.4.2 Line Styles
Line styles are specified by the following properties:
linestyle
May be one of
"-"

Solid line. [default]

"--"

Dashed line.

":"

Dotted line.

"-."

A dash-dot line.

"none"

No line. Points will still be marked using the current Marker Style.

linewidth
A number specifying the width of the line. The default is 1. A value of 2 is
twice as wide as the default, etc.

15.4.3 Marker Styles
Marker styles are specified by the following properties:
marker

A character indicating a plot marker to be place at each data point, or "none",
meaning no markers should be displayed.

426

GNU Octave

markeredgecolor
The color of the edge around the marker, or "auto", meaning that the edge
color is the same as the face color. See Section 15.4.1 [Colors], page 425.
markerfacecolor
The color of the marker, or "none" to indicate that the marker should not be
filled. See Section 15.4.1 [Colors], page 425.
markersize
A number specifying the size of the marker. The default is 1. A value of 2 is
twice as large as the default, etc.
The colstyle function will parse a plot-style specification and will return the color,
line, and marker values that would result.

[style, color, marker, msg] = colstyle (linespec)
Parse linespec and return the line style, color, and markers given.
In the case of an error, the string msg will return the text of the error.

15.4.4 Callbacks
Callback functions can be associated with graphics objects and triggered after certain events
occur. The basic structure of all callback function is
function mycallback (hsrc, evt)
...
endfunction
where hsrc is a handle to the source of the callback, and evt gives some event specific data.
The function can be provided as a function handle to a plain Octave function, as an
anonymous function or as a string representing an Octvae command. The latter syntax
is not recommended since syntax errors will only occur when the string is evaluated. See
Section 11.11 [Function Handles section], page 211.
This can then be associated with an object either at the objects creation or later with
the set function. For example,
plot (x, "DeleteFcn", @(h, e) disp ("Window Deleted"))
where at the moment that the plot is deleted, the message "Window Deleted" will be
displayed.
Additional user arguments can be passed to callback functions, and will be passed after
the 2 default arguments. For example:
plot (x, "DeleteFcn", {@mycallback, "1"})
...
function mycallback (h, e, a1)
fprintf ("Closing plot %d\n", a1);
endfunction
The basic callback functions that are available for all graphics objects are
• CreateFcn: called at the moment of the objects creation. It is not called if the object
is altered in any way, and so it only makes sense to define this callback in the function
call that defines the object. Callbacks that are added to CreateFcn later with the set
function will never be executed.

Chapter 15: Plotting

427

• DeleteFcn: called at the moment an object is deleted.

• ButtonDownFcn: called if a mouse button is pressed while the pointer is over this
object. Note, that the gnuplot interface does not respect this callback.
Caution: the second evt argument in callback functions is only loosely implemented in
the Qt graphics toolkit:
• Mouse click events: evt is a class double value, 1 for left, 2 for middle and 3 for right
click.
• Key press events: evt is a structure with fields Key (string), Character (string) and
Modifier (cell array of strings).
• Other events: evt is a class double empty matrix.
The object and figure that the event occurred in that resulted in the callback being
called can be found with the gcbo and gcbf functions.

h = gcbo ()
[h, fig] = gcbo ()
Return a handle to the object whose callback is currently executing.
If no callback is executing, this function returns the empty matrix. This handle is
obtained from the root object property "CallbackObject".
When called with a second output argument, return the handle of the figure containing
the object whose callback is currently executing. If no callback is executing the second
output is also set to the empty matrix.
See also: [gcbf], page 427, [gco], page 382, [gca], page 382, [gcf], page 381, [get],
page 383, [set], page 383.

fig = gcbf ()
Return a handle to the figure containing the object whose callback is currently executing.
If no callback is executing, this function returns the empty matrix. The handle
returned by this function is the same as the second output argument of gcbo.
See also: [gcbo], page 427, [gcf], page 381, [gco], page 382, [gca], page 382, [get],
page 383, [set], page 383.
Callbacks can equally be added to properties with the addlistener function described
below.

15.4.5 Application-defined Data
Octave has a provision for attaching application-defined data to a graphics handle. The
data can be anything which is meaningful to the application, and will be completely ignored
by Octave.

setappdata (h, name, value)
setappdata (h, name1, value1, name2, value3, . . . )
Set the application data name to value for the graphics object with handle h.

428

GNU Octave

h may also be a vector of graphics handles. If the application data with the specified
name does not exist, it is created. Multiple name/value pairs can be specified at a
time.
See also: [getappdata], page 428, [isappdata], page 428, [rmappdata], page 428,
[guidata], page 814, [get], page 383, [set], page 383, [getpref], page 816, [setpref],
page 816.

value = getappdata (h, name)
appdata = getappdata (h)
Return the value of the application data name for the graphics object with handle h.
h may also be a vector of graphics handles. If no second argument name is given then
getappdata returns a structure, appdata, whose fields correspond to the appdata
properties.
See also: [setappdata], page 427, [isappdata], page 428, [rmappdata], page 428,
[guidata], page 814, [get], page 383, [set], page 383, [getpref], page 816, [setpref],
page 816.

rmappdata (h, name)
rmappdata (h, name1, name2, . . . )
Delete the application data name from the graphics object with handle h.
h may also be a vector of graphics handles. Multiple application data names may be
supplied to delete several properties at once.
See also: [setappdata], page 427, [getappdata], page 428, [isappdata], page 428.

valid = isappdata (h, name)
Return true if the named application data, name, exists for the graphics object with
handle h.
h may also be a vector of graphics handles.
See also: [getappdata], page 428, [setappdata], page 427, [rmappdata], page 428,
[guidata], page 814, [get], page 383, [set], page 383, [getpref], page 816, [setpref],
page 816.

15.4.6 Object Groups
A number of Octave high level plot functions return groups of other graphics objects or
they return graphics objects that have their properties linked in such a way that changes to
one of the properties results in changes in the others. A graphic object that groups other
objects is an hggroup

hggroup ()
hggroup (hax)
hggroup ( . . . , property, value, . . . )
h = hggroup ( . . . )
Create handle graphics group object with axes parent hax.
If no parent is specified, the group is created in the current axes.
Multiple property/value pairs may be specified for the hggroup, but they must appear
in pairs.

Chapter 15: Plotting

429

The optional return value h is a graphics handle to the created hggroup object.
Programming Note: An hggroup is a way to group base graphics objects such as line
objects or patch objects into a single unit which can react appropriately. For example,
the individual lines of a contour plot are collected into a single hggroup so that they
can be made visible/invisible with a single command, set (hg_handle, "visible",
"off").
See also: [addproperty], page 429, [addlistener], page 430.
For example a simple use of a hggroup might be
x = 0:0.1:10;
hg = hggroup ();
plot (x, sin (x), "color", [1, 0, 0], "parent", hg);
hold on
plot (x, cos (x), "color", [0, 1, 0], "parent", hg);
set (hg, "visible", "off");
which groups the two plots into a single object and controls their visibility directly. The
default properties of an hggroup are the same as the set of common properties for the other
graphics objects. Additional properties can be added with the addproperty function.

addproperty (name, h, type)
addproperty (name, h, type, arg, . . . )
Create a new property named name in graphics object h.
type determines the type of the property to create. args usually contains the default
value of the property, but additional arguments might be given, depending on the
type of the property.
The supported property types are:
string

A string property. arg contains the default string value.

any

An un-typed property. This kind of property can hold any octave value.
args contains the default value.

radio

A string property with a limited set of accepted values. The first argument must be a string with all accepted values separated by a vertical
bar (’|’). The default value can be marked by enclosing it with a ’{’ ’}’
pair. The default value may also be given as an optional second string
argument.

boolean

A boolean property. This property type is equivalent to a radio property
with "on|off" as accepted values. arg contains the default property value.

double

A scalar double property. arg contains the default value.

handle

A handle property. This kind of property holds the handle of a graphics
object. arg contains the default handle value. When no default value is
given, the property is initialized to the empty matrix.

data

A data (matrix) property. arg contains the default data value. When no
default value is given, the data is initialized to the empty matrix.

430

GNU Octave

A color property. arg contains the default color value. When no default
color is given, the property is set to black. An optional second string
argument may be given to specify an additional set of accepted string
values (like a radio property).

color

type may also be the concatenation of a core object type and a valid property name
for that object type. The property created then has the same characteristics as the
referenced property (type, possible values, hidden state. . . ). This allows one to clone
an existing property into the graphics object h.
Examples:
addproperty ("my_property", gcf, "string", "a string value");
addproperty ("my_radio", gcf, "radio", "val_1|val_2|{val_3}");
addproperty ("my_style", gcf, "linelinestyle", "--");
See also: [addlistener], page 430, [hggroup], page 428.
Once a property in added to an hggroup, it is not linked to any other property of either
the children of the group, or any other graphics object. Add so to control the way in which
this newly added property is used, the addlistener function is used to define a callback
function that is executed when the property is altered.

addlistener (h, prop, fcn)
Register fcn as listener for the property prop of the graphics object h.
Property listeners are executed (in order of registration) when the property is set.
The new value is already available when the listeners are executed.
prop must be a string naming a valid property in h.
fcn can be a function handle, a string or a cell array whose first element is a function
handle. If fcn is a function handle, the corresponding function should accept at least
2 arguments, that will be set to the object handle and the empty matrix respectively.
If fcn is a string, it must be any valid octave expression. If fcn is a cell array, the first
element must be a function handle with the same signature as described above. The
next elements of the cell array are passed as additional arguments to the function.
Example:
function my_listener (h, dummy, p1)
fprintf ("my_listener called with p1=%s\n", p1);
endfunction
addlistener (gcf, "position", {@my_listener, "my string"})
See also: [dellistener], page 430, [addproperty], page 429, [hggroup], page 428.

dellistener (h, prop, fcn)
Remove the registration of fcn as a listener for the property prop of the graphics
object h.
The function fcn must be the same variable (not just the same value), as was passed
to the original call to addlistener.
If fcn is not defined then all listener functions of prop are removed.

Chapter 15: Plotting

431

Example:
function my_listener (h, dummy, p1)
fprintf ("my_listener called with p1=%s\n", p1);
endfunction
c = {@my_listener, "my string"};
addlistener (gcf, "position", c);
dellistener (gcf, "position", c);
See also: [addlistener], page 430.
An example of the use of these two functions might be
x = 0:0.1:10;
hg = hggroup ();
h = plot (x, sin (x), "color", [1, 0, 0], "parent", hg);
addproperty ("linestyle", hg, "linelinestyle", get (h, "linestyle"));
addlistener (hg, "linestyle", @update_props);
hold on
plot (x, cos (x), "color", [0, 1, 0], "parent", hg);
function update_props (h, d)
set (get (h, "children"), "linestyle", get (h, "linestyle"));
endfunction
that adds a linestyle property to the hggroup and propagating any changes its value to
the children of the group. The linkprop function can be used to simplify the above to be
x = 0:0.1:10;
hg = hggroup ();
h1 = plot (x, sin (x), "color", [1, 0, 0], "parent", hg);
addproperty ("linestyle", hg, "linelinestyle", get (h, "linestyle"));
hold on
h2 = plot (x, cos (x), "color", [0, 1, 0], "parent", hg);
hlink = linkprop ([hg, h1, h2], "color");

hlink = linkprop (h, "prop")
hlink = linkprop (h, {"prop1", "prop2", . . . })

Link graphic object properties, such that a change in one is propagated to the others.
The input h is a vector of graphic handles to link.
prop may be a string when linking a single property, or a cell array of strings for
multiple properties. During the linking process all properties in prop will initially be
set to the values that exist on the first object in the list h.
The function returns hlink which is a special object describing the link. As long as
the reference hlink exists the link between graphic objects will be active. This means
that hlink must be preserved in a workspace variable, a global variable, or otherwise
stored using a function such as setappdata, guidata. To unlink properties, execute
clear hlink.

432

GNU Octave

An example of the use of linkprop is
x = 0:0.1:10;
subplot (1,2,1);
h1 = plot (x, sin (x));
subplot (1,2,2);
h2 = plot (x, cos (x));
hlink = linkprop ([h1, h2], {"color","linestyle"});
set (h1, "color", "green");
set (h2, "linestyle", "--");
See also: [linkaxes], page 432.

linkaxes (hax)
linkaxes (hax, optstr)
Link the axis limits of 2-D plots such that a change in one is propagated to the others.
The axes handles to be linked are passed as the first argument hax.
The optional second argument is a string which defines which axis limits will be linked.
The possible values for optstr are:
"x"

Link x-axes

"y"

Link y-axes

"xy" (default)
Link both axes
"off"

Turn off linking

If unspecified the default is to link both X and Y axes.
When linking, the limits from the first axes in hax are applied to the other axes in
the list. Subsequent changes to any one of the axes will be propagated to the others.
See also: [linkprop], page 431, [addproperty], page 429.
These capabilities are used in a number of basic graphics objects. The hggroup objects
created by the functions of Octave contain one or more graphics object and are used to:
• group together multiple graphics objects,
• create linked properties between different graphics objects, and
• to hide the nominal user data, from the actual data of the objects.
For example the stem function creates a stem series where each hggroup of the stem series
contains two line objects representing the body and head of the stem. The ydata property
of the hggroup of the stem series represents the head of the stem, whereas the body of the
stem is between the baseline and this value. For example
h = stem (1:4)
get (h, "xdata")
⇒ [ 1
2
3
4]’
get (get (h, "children")(1), "xdata")
⇒ [ 1
1 NaN
2
2 NaN
3
3 NaN
4
4 NaN]’
shows the difference between the xdata of the hggroup of a stem series object and the
underlying line.

Chapter 15: Plotting

433

The basic properties of such group objects is that they consist of one or more linked
hggroup, and that changes in certain properties of these groups are propagated to other
members of the group. Whereas, certain properties of the members of the group only apply
to the current member.
In addition the members of the group can also be linked to other graphics objects through
callback functions. For example the baseline of the bar or stem functions is a line object,
whose length and position are automatically adjusted, based on changes to the corresponding hggroup elements.

15.4.6.1 Data Sources in Object Groups
All of the group objects contain data source parameters. There are string parameters that
contain an expression that is evaluated to update the relevant data property of the group
when the refreshdata function is called.

refreshdata ()
refreshdata (h)
refreshdata (h, workspace)
Evaluate any ‘datasource’ properties of the current figure and update the plot if the
corresponding data has changed.
If the first argument h is a list of graphic handles, then operate on these objects rather
than the current figure returned by gcf.
The optional second argument workspace can take the following values:
"base"

Evaluate the datasource properties in the base workspace. (default).

"caller"

Evaluate the datasource properties in the workspace of the function that
called refreshdata.

An example of the use of refreshdata is:
x = 0:0.1:10;
y = sin (x);
plot (x, y, "ydatasource", "y");
for i = 1 : 100
pause (0.1);
y = sin (x + 0.1*i);
refreshdata ();
endfor

15.4.6.2 Area Series
Area series objects are created by the area function. Each of the hggroup elements contains
a single patch object. The properties of the area series are
basevalue
The value where the base of the area plot is drawn.
linewidth
linestyle
The line width and style of the edge of the patch objects making up the areas.
See Section 15.4.2 [Line Styles], page 425.

434

GNU Octave

edgecolor
facecolor
The line and fill color of the patch objects making up the areas.
Section 15.4.1 [Colors], page 425.
xdata
ydata

See

The x and y coordinates of the original columns of the data passed to area
prior to the cumulative summation used in the area function.

xdatasource
ydatasource
Data source variables.

15.4.6.3 Bar Series
Bar series objects are created by the bar or barh functions. Each hggroup element contains
a single patch object. The properties of the bar series are
showbaseline
baseline
basevalue
The property showbaseline flags whether the baseline of the bar series is displayed (default is "on"). The handle of the graphics object representing the
baseline is given by the baseline property and the y-value of the baseline by
the basevalue property.
Changes to any of these properties are propagated to the other members of the
bar series and to the baseline itself. Equally, changes in the properties of the
base line itself are propagated to the members of the corresponding bar series.
barwidth
barlayout
horizontal
The property barwidth is the width of the bar corresponding to the width variable passed to bar or barh. Whether the bar series is "grouped" or "stacked"
is determined by the barlayout property and whether the bars are horizontal
or vertical by the horizontal property.
Changes to any of these property are propagated to the other members of the
bar series.
linewidth
linestyle
The line width and style of the edge of the patch objects making up the bars.
See Section 15.4.2 [Line Styles], page 425.
edgecolor
facecolor
The line and fill color of the patch objects making up the bars. See Section 15.4.1
[Colors], page 425.
xdata

The nominal x positions of the bars. Changes in this property and propagated
to the other members of the bar series.

Chapter 15: Plotting

ydata

435

The y value of the bars in the hggroup.

xdatasource
ydatasource
Data source variables.

15.4.6.4 Contour Groups
Contour group objects are created by the contour, contourf, and contour3 functions.
The are also one of the handles returned by the surfc and meshc functions. The properties
of the contour group are
contourmatrix
A read only property that contains the data return by contourc used to create
the contours of the plot.
fill

A radio property that can have the values "on" or "off" that flags whether the
contours to plot are to be filled.

zlevelmode
zlevel
The radio property zlevelmode can have the values "none", "auto", or
"manual". When its value is "none" there is no z component to the plotted
contours. When its value is "auto" the z value of the plotted contours is at
the same value as the contour itself. If the value is "manual", then the z value
at which to plot the contour is determined by the zlevel property.
levellistmode
levellist
levelstepmode
levelstep
If levellistmode is "manual", then the levels at which to plot the contours
is determined by levellist. If levellistmode is set to "auto", then the
distance between contours is determined by levelstep. If both levellistmode
and levelstepmode are set to "auto", then there are assumed to be 10 equal
spaced contours.
textlistmode
textlist
textstepmode
textstep If textlistmode is "manual", then the labeled contours is determined by
textlist. If textlistmode is set to "auto", then the distance between labeled
contours is determined by textstep. If both textlistmode and textstepmode
are set to "auto", then there are assumed to be 10 equal spaced labeled contours.
showtext

Flag whether the contour labels are shown or not.

labelspacing
The distance between labels on a single contour in points.

436

GNU Octave

linewidth
linestyle
linecolor
The properties of the contour lines. The properties linewidth and linestyle
are similar to the corresponding properties for lines. The property linecolor
is a color property (see Section 15.4.1 [Colors], page 425), that can also have
the values of "none" or "auto". If linecolor is "none", then no contour line
is drawn. If linecolor is "auto" then the line color is determined by the
colormap.
xdata
ydata
zdata

The original x, y, and z data of the contour lines.

xdatasource
ydatasource
zdatasource
Data source variables.

15.4.6.5 Error Bar Series
Error bar series are created by the errorbar function. Each hggroup element contains two
line objects representing the data and the errorbars separately. The properties of the error
bar series are
color

The RGB color or color name of the line objects of the error bars.
Section 15.4.1 [Colors], page 425.

See

linewidth
linestyle
The line width and style of the line objects of the error bars. See Section 15.4.2
[Line Styles], page 425.
marker
markeredgecolor
markerfacecolor
markersize
The line and fill color of the markers on the error bars. See Section 15.4.1
[Colors], page 425.
xdata
ydata
ldata
udata
xldata
xudata

The original x, y, l, u, xl, xu data of the error bars.

Chapter 15: Plotting

437

xdatasource
ydatasource
ldatasource
udatasource
xldatasource
xudatasource
Data source variables.

15.4.6.6 Line Series
Line series objects are created by the plot and plot3 functions and are of the type line.
The properties of the line series with the ability to add data sources.
color

The RGB color or color name of the line objects. See Section 15.4.1 [Colors],
page 425.

linewidth
linestyle
The line width and style of the line objects. See Section 15.4.2 [Line Styles],
page 425.
marker
markeredgecolor
markerfacecolor
markersize
The line and fill color of the markers. See Section 15.4.1 [Colors], page 425.
xdata
ydata
zdata

The original x, y and z data.

xdatasource
ydatasource
zdatasource
Data source variables.

15.4.6.7 Quiver Group
Quiver series objects are created by the quiver or quiver3 functions. Each hggroup element
of the series contains three line objects as children representing the body and head of the
arrow, together with a marker as the point of origin of the arrows. The properties of the
quiver series are
autoscale
autoscalefactor
Flag whether the length of the arrows is scaled or defined directly from
the u, v and w data. If the arrow length is flagged as being scaled by the
autoscale property, then the length of the autoscaled arrow is controlled by
the autoscalefactor.
maxheadsize
This property controls the size of the head of the arrows in the quiver series.
The default value is 0.2.

438

GNU Octave

showarrowhead
Flag whether the arrow heads are displayed in the quiver plot.
color

The RGB color or color name of the line objects of the quiver. See Section 15.4.1
[Colors], page 425.

linewidth
linestyle
The line width and style of the line objects of the quiver. See Section 15.4.2
[Line Styles], page 425.
marker
markerfacecolor
markersize
The line and fill color of the marker objects at the original of the arrows. See
Section 15.4.1 [Colors], page 425.
xdata
ydata
zdata

The origins of the values of the vector field.

udata
vdata
wdata

The values of the vector field to plot.

xdatasource
ydatasource
zdatasource
udatasource
vdatasource
wdatasource
Data source variables.

15.4.6.8 Scatter Group
Scatter series objects are created by the scatter or scatter3 functions. A single hggroup
element contains as many children as there are points in the scatter plot, with each child
representing one of the points. The properties of the stem series are
linewidth
The line width of the line objects of the points. See Section 15.4.2 [Line Styles],
page 425.
marker
markeredgecolor
markerfacecolor
The line and fill color of the markers of the points. See Section 15.4.1 [Colors],
page 425.
xdata
ydata
zdata

The original x, y and z data of the stems.

Chapter 15: Plotting

439

cdata

The color data for the points of the plot. Each point can have a separate color,
or a unique color can be specified.

sizedata

The size data for the points of the plot. Each point can its own size or a unique
size can be specified.

xdatasource
ydatasource
zdatasource
cdatasource
sizedatasource
Data source variables.

15.4.6.9 Stair Group
Stair series objects are created by the stair function. Each hggroup element of the series
contains a single line object as a child representing the stair. The properties of the stair
series are
color

The RGB color or color name of the line objects of the stairs. See Section 15.4.1
[Colors], page 425.

linewidth
linestyle
The line width and style of the line objects of the stairs. See Section 15.4.2
[Line Styles], page 425.
marker
markeredgecolor
markerfacecolor
markersize
The line and fill color of the markers on the stairs. See Section 15.4.1 [Colors],
page 425.
xdata
ydata

The original x and y data of the stairs.

xdatasource
ydatasource
Data source variables.

15.4.6.10 Stem Series
Stem series objects are created by the stem or stem3 functions. Each hggroup element
contains a single line object as a child representing the stems. The properties of the stem
series are
showbaseline
baseline
basevalue
The property showbaseline flags whether the baseline of the stem series is
displayed (default is "on"). The handle of the graphics object representing
the baseline is given by the baseline property and the y-value (or z-value for
stem3) of the baseline by the basevalue property.

440

GNU Octave

Changes to any of these property are propagated to the other members of the
stem series and to the baseline itself. Equally changes in the properties of the
base line itself are propagated to the members of the corresponding stem series.
color

The RGB color or color name of the line objects of the stems. See Section 15.4.1
[Colors], page 425.

linewidth
linestyle
The line width and style of the line objects of the stems. See Section 15.4.2
[Line Styles], page 425.
marker
markeredgecolor
markerfacecolor
markersize
The line and fill color of the markers on the stems. See Section 15.4.1 [Colors],
page 425.
xdata
ydata
zdata

The original x, y and z data of the stems.

xdatasource
ydatasource
zdatasource
Data source variables.

15.4.6.11 Surface Group
Surface group objects are created by the surf or mesh functions, but are equally one of
the handles returned by the surfc or meshc functions. The surface group is of the type
surface.
The properties of the surface group are
edgecolor
facecolor
The RGB color or color name of the edges or faces of the surface.
Section 15.4.1 [Colors], page 425.

See

linewidth
linestyle
The line width and style of the lines on the surface. See Section 15.4.2 [Line
Styles], page 425.
marker
markeredgecolor
markerfacecolor
markersize
The line and fill color of the markers on the surface. See Section 15.4.1 [Colors],
page 425.

Chapter 15: Plotting

xdata
ydata
zdata
cdata

441

The original x, y, z and c data.

xdatasource
ydatasource
zdatasource
cdatasource
Data source variables.

15.4.7 Graphics Toolkits
name = graphics_toolkit ()
name = graphics_toolkit (hlist)
graphics_toolkit (name)
graphics_toolkit (hlist, name)
Query or set the default graphics toolkit which is assigned to new figures.
With no inputs, return the current default graphics toolkit. If the input is a list of
figure graphic handles, hlist, then return the name of the graphics toolkit in use for
each figure.
When called with a single input name set the default graphics toolkit to name. If the
toolkit is not already loaded, it is initialized by calling the function __init_name__.
If the first input is a list of figure handles, hlist, then the graphics toolkit is set to
name for these figures only.
See also: [available graphics toolkits], page 441.

available_graphics_toolkits ()
Return a cell array of registered graphics toolkits.
See also: [graphics toolkit], page 441, [register graphics toolkit], page 441.

loaded_graphics_toolkits ()
Return a cell array of the currently loaded graphics toolkits.
See also: [available graphics toolkits], page 441.

register_graphics_toolkit (toolkit)
List toolkit as an available graphics toolkit.
See also: [available graphics toolkits], page 441.

15.4.7.1 Customizing Toolkit Behavior
The specific behavior of the backend toolkit may be modified using the following utility
functions. Note: Not all functions apply to every graphics toolkit.

[prog, args] = gnuplot_binary ()
[old_prog, old_args] = gnuplot_binary (new_prog, arg1, . . . )
Query or set the name of the program invoked by the plot command when the graphics
toolkit is set to "gnuplot".

442

GNU Octave

Additional arguments to pass to the external plotting program may also be given.
The default value is "gnuplot" with no additional arguments. See Appendix E [Installation], page 947.
See also: [graphics toolkit], page 441.

443

16 Matrix Manipulation
There are a number of functions available for checking to see if the elements of a matrix
meet some condition, and for rearranging the elements of a matrix. For example, Octave
can easily tell you if all the elements of a matrix are finite, or are less than some specified
value. Octave can also rotate the elements, extract the upper- or lower-triangular parts, or
sort the columns of a matrix.

16.1 Finding Elements and Checking Conditions
The functions any and all are useful for determining whether any or all of the elements
of a matrix satisfy some condition. The find function is also useful in determining which
elements of a matrix meet a specified condition.

any (x)
any (x, dim)
For a vector argument, return true (logical 1) if any element of the vector is nonzero.
For a matrix argument, return a row vector of logical ones and zeros with each element
indicating whether any of the elements of the corresponding column of the matrix are
nonzero. For example:
any (eye (2, 4))
⇒ [ 1, 1, 0, 0 ]
If the optional argument dim is supplied, work along dimension dim. For example:
any (eye (2, 4), 2)
⇒ [ 1; 1 ]
See also: [all], page 443.

all (x)
all (x, dim)
For a vector argument, return true (logical 1) if all elements of the vector are nonzero.
For a matrix argument, return a row vector of logical ones and zeros with each element
indicating whether all of the elements of the corresponding column of the matrix are
nonzero. For example:
all ([2, 3; 1, 0])
⇒ [ 1, 0 ]
If the optional argument dim is supplied, work along dimension dim.
See also: [any], page 443.
Since the comparison operators (see Section 8.4 [Comparison Ops], page 148) return
matrices of ones and zeros, it is easy to test a matrix for many things, not just whether the
elements are nonzero. For example,
all (all (rand (5) < 0.9))
⇒ 0
tests a random 5 by 5 matrix to see if all of its elements are less than 0.9.
Note that in conditional contexts (like the test clause of if and while statements) Octave
treats the test as if you had typed all (all (condition)).

444

GNU Octave

z = xor (x, y)
z = xor (x1, x2, . . . )
Return the exclusive or of x and y.
For boolean expressions x and y, xor (x, y) is true if and only if one of x or y is
true. Otherwise, if x and y are both true or both false, xor returns false.
The truth table for the xor operation is
x y z
- 0 0 0
1 0 1
0 1 1
1 1 0
If more than two arguments are given the xor operation is applied cumulatively from
left to right:
(...((x1 XOR x2) XOR x3) XOR ...)
See also: [and], page 150, [or], page 150, [not], page 150.

diff (x)
diff (x, k)
diff (x, k, dim)
If x is a vector of length n, diff (x) is the vector of first differences x2 − x1 , . . . , xn −
xn−1 .
If x is a matrix, diff (x) is the matrix of column differences along the first nonsingleton dimension.
The second argument is optional. If supplied, diff (x, k), where k is a non-negative
integer, returns the k-th differences. It is possible that k is larger than the first nonsingleton dimension of the matrix. In this case, diff continues to take the differences
along the next non-singleton dimension.
The dimension along which to take the difference can be explicitly stated with the
optional variable dim. In this case the k-th order differences are calculated along this
dimension. In the case where k exceeds size (x, dim) an empty matrix is returned.
See also: [sort], page 452, [merge], page 152.

isinf (x)
Return a logical array which is true where the elements of x are infinite and false
where they are not.
For example:
isinf ([13, Inf, NA, NaN])
⇒ [ 0, 1, 0, 0 ]

See also: [isfinite], page 445, [isnan], page 444, [isna], page 43.

isnan (x)
Return a logical array which is true where the elements of x are NaN values and false
where they are not.

Chapter 16: Matrix Manipulation

445

NA values are also considered NaN values. For example:
isnan ([13, Inf, NA, NaN])
⇒ [ 0, 0, 1, 1 ]
See also: [isna], page 43, [isinf], page 444, [isfinite], page 445.

isfinite (x)
Return a logical array which is true where the elements of x are finite values and false
where they are not.
For example:
isfinite ([13, Inf, NA, NaN])
⇒ [ 1, 0, 0, 0 ]
See also: [isinf], page 444, [isnan], page 444, [isna], page 43.

[err, yi, ...] = common_size (xi, . . . )
Determine if all input arguments are either scalar or of common size.
If true, err is zero, and yi is a matrix of the common size with all entries equal to xi
if this is a scalar or xi otherwise. If the inputs cannot be brought to a common size,
err is 1, and yi is xi. For example:
[err, a, b]
⇒ err
⇒ a =
⇒ b =

=
=
[
[

common_size ([1 2; 3 4], 5)
0
1, 2; 3, 4 ]
5, 5; 5, 5 ]

This is useful for implementing functions where arguments can either be scalars or of
common size.
See also: [size], page 45, [size equal], page 46, [numel], page 44, [ndims], page 44.

idx
idx
idx
[i,
[i,

= find (x)
= find (x, n)
= find (x, n, direction)
j] = find ( . . . )
j, v] = find ( . . . )
Return a vector of indices of nonzero elements of a matrix, as a row if x is a row
vector or as a column otherwise.
To obtain a single index for each matrix element, Octave pretends that the columns
of a matrix form one long vector (like Fortran arrays are stored). For example:
find (eye (2))
⇒ [ 1; 4 ]

If two inputs are given, n indicates the maximum number of elements to find from
the beginning of the matrix or vector.
If three inputs are given, direction should be one of "first" or "last", requesting
only the first or last n indices, respectively. However, the indices are always returned
in ascending order.

446

GNU Octave

If two outputs are requested, find returns the row and column indices of nonzero
elements of a matrix. For example:
[i, j] = find (2 * eye (2))
⇒ i = [ 1; 2 ]
⇒ j = [ 1; 2 ]
If three outputs are requested, find also returns a vector containing the nonzero
values. For example:
[i, j, v] = find (3 * eye (2))
⇒ i = [ 1; 2 ]
⇒ j = [ 1; 2 ]
⇒ v = [ 3; 3 ]
Note that this function is particularly useful for sparse matrices, as it extracts the
nonzero elements as vectors, which can then be used to create the original matrix.
For example:
sz = size (a);
[i, j, v] = find (a);
b = sparse (i, j, v, sz(1), sz(2));
See also: [nonzeros], page 570.

idx = lookup (table, y)
idx = lookup (table, y, opt)
Lookup values in a sorted table.
This function is usually used as a prelude to interpolation.
If table is increasing and idx = lookup (table, y), then table(idx(i)) <= y(i) <
table(idx(i+1)) for all y(i) within the table. If y(i) < table(1) then idx(i) is
0. If y(i) >= table(end) or isnan (y(i)) then idx(i) is n.
If the table is decreasing, then the tests are reversed. For non-strictly monotonic
tables, empty intervals are always skipped. The result is undefined if table is not
monotonic, or if table contains a NaN.
The complexity of the lookup is O(M*log(N)) where N is the size of table and
M is the size of y. In the special case when y is also sorted, the complexity is
O(min(M*log(N),M+N)).
table and y can also be cell arrays of strings (or y can be a single string). In this
case, string lookup is performed using lexicographical comparison.
If opts is specified, it must be a string with letters indicating additional options.
m

table(idx(i)) == y(i) if y(i) occurs in table; otherwise, idx(i) is
zero.

b

idx(i) is a logical 1 or 0, indicating whether val(i) is contained in table
or not.

l

For numeric lookups the leftmost subinterval shall be extended to infinity
(i.e., all indices at least 1)

r

For numeric lookups the rightmost subinterval shall be extended to infinity (i.e., all indices at most n-1).

Chapter 16: Matrix Manipulation

447

If you wish to check if a variable exists at all, instead of properties its elements may
have, consult Section 7.3 [Status of Variables], page 129.

16.2 Rearranging Matrices
fliplr (x)
Flip array left to right.
Return a copy of x with the order of the columns reversed. In other words, x is
flipped left-to-right about a vertical axis. For example:
fliplr ([1, 2; 3, 4])
⇒ 2 1
4 3
See also: [flipud], page 447, [flip], page 447, [rot90], page 448, [rotdim], page 448.

flipud (x)
Flip array upside down.
Return a copy of x with the order of the rows reversed. In other words, x is flipped
upside-down about a horizontal axis. For example:
flipud ([1, 2; 3, 4])
⇒ 3 4
1 2
See also: [fliplr], page 447, [flip], page 447, [rot90], page 448, [rotdim], page 448.

flip (x)
flip (x, dim)
Flip array across dimension dim.
Return a copy of x flipped about the dimension dim. dim defaults to the first nonsingleton dimension. For example:
flip ([1 2 3 4])
⇒ 4 3 2 1
flip ([1; 2; 3; 4])
⇒ 4
3
2
1
flip ([1 2; 3 4])
⇒ 3 4
1 2
flip ([1 2; 3 4], 2)
⇒ 2 1
4 3
See also: [fliplr], page 447, [flipud], page 447, [rot90], page 448, [rotdim], page 448,
[permute], page 449, [transpose], page 147.

448

GNU Octave

rot90 (A)
rot90 (A, k)
Rotate array by 90 degree increments.
Return a copy of A with the elements rotated counterclockwise in 90-degree increments.
The second argument is optional, and specifies how many 90-degree rotations are
to be applied (the default value is 1). Negative values of k rotate the matrix in a
clockwise direction. For example,
rot90 ([1, 2; 3, 4], -1)
⇒ 3 1
4 2
rotates the given matrix clockwise by 90 degrees. The following are all equivalent
statements:
rot90 ([1, 2; 3, 4], -1)
rot90 ([1, 2; 3, 4], 3)
rot90 ([1, 2; 3, 4], 7)
The rotation is always performed on the plane of the first two dimensions, i.e., rows
and columns. To perform a rotation on any other plane, use rotdim.
See also: [rotdim], page 448, [fliplr], page 447, [flipud], page 447, [flip], page 447.

rotdim (x)
rotdim (x, n)
rotdim (x, n, plane)
Return a copy of x with the elements rotated counterclockwise in 90-degree increments.
The second argument n is optional, and specifies how many 90-degree rotations are
to be applied (the default value is 1). Negative values of n rotate the matrix in a
clockwise direction.
The third argument is also optional and defines the plane of the rotation. If present,
plane is a two element vector containing two different valid dimensions of the matrix.
When plane is not given the first two non-singleton dimensions are used.
For example,
rotdim ([1, 2; 3, 4], -1, [1, 2])
⇒ 3 1
4 2
rotates the given matrix clockwise by 90 degrees. The following are all equivalent
statements:
rotdim ([1, 2; 3, 4], -1, [1, 2])
rotdim ([1, 2; 3, 4], 3, [1, 2])
rotdim ([1, 2; 3, 4], 7, [1, 2])
See also: [rot90], page 448, [fliplr], page 447, [flipud], page 447, [flip], page 447.

Chapter 16: Matrix Manipulation

449

cat (dim, array1, array2, . . . , arrayN)
Return the concatenation of N-D array objects, array1, array2, . . . , arrayN along
dimension dim.
A = ones (2, 2);
B = zeros (2, 2);
cat (2, A, B)
⇒ 1 1 0 0
1 1 0 0
Alternatively, we can concatenate A and B along the second dimension in the following
way:
[A, B]
dim can be larger than the dimensions of the N-D array objects and the result will
thus have dim dimensions as the following example shows:
cat (4, ones (2, 2), zeros (2, 2))
⇒ ans(:,:,1,1) =
1 1
1 1
ans(:,:,1,2) =
0 0
0 0
See also: [horzcat], page 449, [vertcat], page 449.

horzcat (array1, array2, . . . , arrayN)
Return the horizontal concatenation of N-D array objects, array1, array2, . . . , arrayN
along dimension 2.
Arrays may also be concatenated horizontally using the syntax for creating new matrices. For example:
hcat = [ array1, array2, ... ]
See also: [cat], page 449, [vertcat], page 449.

vertcat (array1, array2, . . . , arrayN)
Return the vertical concatenation of N-D array objects, array1, array2, . . . , arrayN
along dimension 1.
Arrays may also be concatenated vertically using the syntax for creating new matrices.
For example:
vcat = [ array1; array2; ... ]
See also: [cat], page 449, [horzcat], page 449.

permute (A, perm)
Return the generalized transpose for an N-D array object A.

450

GNU Octave

The permutation vector perm must contain the elements 1:ndims (A) (in any order,
but each element must appear only once). The N th dimension of A gets remapped
to dimension PERM(N). For example:
x = zeros ([2, 3, 5, 7]);
size (x)
⇒ 2
3
5
7
size (permute (x, [2, 1, 3, 4]))
⇒ 3
2
5
7
size (permute (x, [1, 3, 4, 2]))
⇒ 2
5
7
3
## The identity permutation
size (permute (x, [1, 2, 3, 4]))
⇒ 2
3
5
7
See also: [ipermute], page 450.

ipermute (A, iperm)
The inverse of the permute function.
The expression
ipermute (permute (A, perm), perm)
returns the original array A.
See also: [permute], page 449.
(A, m, n, . . . )
(A, [m n . . . ])
(A, . . . , [], . . . )
(A, size)
Return a matrix with the specified dimensions (m, n, . . . ) whose elements are taken
from the matrix A.

reshape
reshape
reshape
reshape

The elements of the matrix are accessed in column-major order (like Fortran arrays
are stored).
The following code demonstrates reshaping a 1x4 row vector into a 2x2 square matrix.
reshape ([1, 2, 3, 4], 2, 2)
⇒ 1 3
2 4
Note that the total number of elements in the original matrix (prod (size (A)))
must match the total number of elements in the new matrix (prod ([m n ...])).
A single dimension of the return matrix may be left unspecified and Octave will
determine its size automatically. An empty matrix ([]) is used to flag the unspecified
dimension.
See also: [resize], page 451, [vec], page 455, [postpad], page 456, [cat], page 449,
[squeeze], page 46.

Chapter 16: Matrix Manipulation

451

resize (x, m)
resize (x, m, n, . . . )
resize (x, [m n . . . ])
Resize x cutting off elements as necessary.
In the result, element with certain indices is equal to the corresponding element of x
if the indices are within the bounds of x; otherwise, the element is set to zero.
In other words, the statement
y = resize (x, dv)
is equivalent to the following code:
y = zeros (dv, class (x));
sz = min (dv, size (x));
for i = 1:length (sz)
idx{i} = 1:sz(i);
endfor
y(idx{:}) = x(idx{:});
but is performed more efficiently.
If only m is supplied, and it is a scalar, the dimension of the result is m-by-m. If m,
n, . . . are all scalars, then the dimensions of the result are m-by-n-by-. . . . If given a
vector as input, then the dimensions of the result are given by the elements of that
vector.
An object can be resized to more dimensions than it has; in such case the missing
dimensions are assumed to be 1. Resizing an object to fewer dimensions is not possible.
See also: [reshape], page 450, [postpad], page 456, [prepad], page 455, [cat], page 449.

y = circshift (x, n)
y = circshift (x, n, dim)
Circularly shift the values of the array x.
n must be a vector of integers no longer than the number of dimensions in x. The
values of n can be either positive or negative, which determines the direction in which
the values of x are shifted. If an element of n is zero, then the corresponding dimension
of x will not be shifted.
If a scalar dim is given then operate along the specified dimension. In this case n
must be a scalar as well.
Examples:

452

GNU Octave

x = [1, 2, 3;
circshift (x,
⇒ 7, 8, 9
1, 2, 3
4, 5, 6
circshift (x,
⇒ 7, 8, 9
1, 2, 3
4, 5, 6
circshift (x,
⇒ 3, 1, 2
6, 4, 5
9, 7, 8

4, 5, 6; 7, 8, 9];
1)

-2)

[0,1])

See also: [permute], page 449, [ipermute], page 450, [shiftdim], page 452.

shift (x, b)
shift (x, b, dim)
If x is a vector, perform a circular shift of length b of the elements of x.
If x is a matrix, do the same for each column of x.
If the optional dim argument is given, operate along this dimension.

y = shiftdim (x, n)
[y, ns] = shiftdim (x)
Shift the dimensions of x by n, where n must be an integer scalar.
When n is positive, the dimensions of x are shifted to the left, with the leading
dimensions circulated to the end. If n is negative, then the dimensions of x are
shifted to the right, with n leading singleton dimensions added.
Called with a single argument, shiftdim, removes the leading singleton dimensions,
returning the number of dimensions removed in the second output argument ns.
For example:
x = ones (1, 2, 3);
size (shiftdim (x, -1))
⇒ [1, 1, 2, 3]
size (shiftdim (x, 1))
⇒ [2, 3]
[b, ns] = shiftdim (x)
⇒ b = [1, 1, 1; 1, 1, 1]
⇒ ns = 1
See also: [reshape], page 450, [permute], page 449, [ipermute], page 450, [circshift],
page 451, [squeeze], page 46.

[s,
[s,
[s,
[s,

(x)
(x, dim)
(x, mode)
(x, dim, mode)
Return a copy of x with the elements arranged in increasing order.

i]
i]
i]
i]

=
=
=
=

sort
sort
sort
sort

Chapter 16: Matrix Manipulation

453

For matrices, sort orders the elements within columns
For example:
sort ([1,
⇒ 1
2
3

2; 2, 3; 3, 1])
1
2
3

If the optional argument dim is given, then the matrix is sorted along the dimension
defined by dim. The optional argument mode defines the order in which the values
will be sorted. Valid values of mode are "ascend" or "descend".
The sort function may also be used to produce a matrix containing the original row
indices of the elements in the sorted matrix. For example:
[s, i] = sort ([1, 2; 2, 3; 3, 1])
⇒ s = 1 1
2 2
3 3
⇒ i = 1 3
2 1
3 2
For equal elements, the indices are such that equal elements are listed in the order in
which they appeared in the original list.
Sorting of complex entries is done first by magnitude (abs (z)) and for any ties by
phase angle (angle (z)). For example:
sort ([1+i; 1; 1-i])
⇒ 1 + 0i
1 - 1i
1 + 1i
NaN values are treated as being greater than any other value and are sorted to the
end of the list.
The sort function may also be used to sort strings and cell arrays of strings, in which
case ASCII dictionary order (uppercase ’A’ precedes lowercase ’a’) of the strings is
used.
The algorithm used in sort is optimized for the sorting of partially ordered lists.
See also: [sortrows], page 453, [issorted], page 454.

[s, i] = sortrows (A)
[s, i] = sortrows (A, c)
Sort the rows of the matrix A according to the order of the columns specified in c.
By default (c omitted, or a particular column unspecified in c) an ascending sort
order is used. However, if elements of c are negative then the corresponding column
is sorted in descending order. If the elements of A are strings then a lexicographical
sort is used.
Example: sort by column 2 in descending order, then 3 in ascending order

454

GNU Octave

x = [ 7,
8,
9,
sortrows
⇒ 8
9
7

1, 4;
3, 5;
3, 6 ];
(x, [-2, 3])
3 5
3 6
1 4

See also: [sort], page 452.

issorted (a)
issorted (a, mode)
issorted (a, "rows", mode)
Return true if the array is sorted according to mode, which may be either
"ascending", "descending", or "either".
By default, mode is "ascending". NaNs are treated in the same manner as sort.
If the optional argument "rows" is supplied, check whether the array is sorted by
rows as output by the function sortrows (with no options).
This function does not support sparse matrices.
See also: [sort], page 452, [sortrows], page 453.

nth_element (x, n)
nth_element (x, n, dim)
Select the n-th smallest element of a vector, using the ordering defined by sort.
The result is equivalent to sort(x)(n).
n can also be a contiguous range, either ascending l:u or descending u:-1:l, in which
case a range of elements is returned.
If x is an array, nth_element operates along the dimension defined by dim, or the
first non-singleton dimension if dim is not given.
Programming Note: nth element encapsulates the C++ standard library algorithms
nth element and partial sort. On average, the complexity of the operation is
O(M*log(K)), where M = size (x, dim) and K = length (n). This function is
intended for cases where the ratio K/M is small; otherwise, it may be better to use
sort.
See also: [sort], page 452, [min], page 485, [max], page 485.

tril
tril
tril
triu
triu
triu

(A)
(A, k)
(A, k, pack)
(A)
(A, k)
(A, k, pack)
Return a new matrix formed by extracting the lower (tril) or upper (triu) triangular
part of the matrix A, and setting all other elements to zero.
The second argument is optional, and specifies how many diagonals above or below
the main diagonal should also be set to zero.

Chapter 16: Matrix Manipulation

455

The default value of k is zero, so that triu and tril normally include the main
diagonal as part of the result.
If the value of k is nonzero integer, the selection of elements starts at an offset of
k diagonals above or below the main diagonal; above for positive k and below for
negative k.
The absolute value of k must not be greater than the number of subdiagonals or
superdiagonals.
For example:
tril (ones (3), -1)
⇒ 0 0 0
1 0 0
1 1 0
and
tril (ones (3), 1)
⇒ 1 1 0
1 1 1
1 1 1
If the option "pack" is given as third argument, the extracted elements are not
inserted into a matrix, but rather stacked column-wise one above other.
See also: [diag], page 456.

v = vec (x)
v = vec (x, dim)
Return the vector obtained by stacking the columns of the matrix x one above the
other.
Without dim this is equivalent to x(:).
If dim is supplied, the dimensions of v are set to dim with all elements along the last
dimension. This is equivalent to shiftdim (x(:), 1-dim).
See also: [vech], page 455, [resize], page 451, [cat], page 449.

vech (x)
Return the vector obtained by eliminating all superdiagonal elements of the square
matrix x and stacking the result one column above the other.
This has uses in matrix calculus where the underlying matrix is symmetric and it
would be pointless to keep values above the main diagonal.
See also: [vec], page 455.

prepad (x, l)
prepad (x, l, c)
prepad (x, l, c, dim)
Prepend the scalar value c to the vector x until it is of length l. If c is not given, a
value of 0 is used.
If length (x) > l, elements from the beginning of x are removed until a vector of
length l is obtained.

456

GNU Octave

If x is a matrix, elements are prepended or removed from each row.
If the optional argument dim is given, operate along this dimension.
If dim is larger than the dimensions of x, the result will have dim dimensions.
See also: [postpad], page 456, [cat], page 449, [resize], page 451.

postpad (x, l)
postpad (x, l, c)
postpad (x, l, c, dim)
Append the scalar value c to the vector x until it is of length l. If c is not given, a
value of 0 is used.
If length (x) > l, elements from the end of x are removed until a vector of length l
is obtained.
If x is a matrix, elements are appended or removed from each row.
If the optional argument dim is given, operate along this dimension.
If dim is larger than the dimensions of x, the result will have dim dimensions.
See also: [prepad], page 455, [cat], page 449, [resize], page 451.

M
M
M
v
v

=
=
=
=
=

(v)
(v, k)
(v, m, n)
(M)
(M, k)
Return a diagonal matrix with vector v on diagonal k.

diag
diag
diag
diag
diag

The second argument is optional. If it is positive, the vector is placed on the k-th
superdiagonal. If it is negative, it is placed on the -k-th subdiagonal. The default
value of k is 0, and the vector is placed on the main diagonal. For example:
diag ([1,
⇒ 0
0
0
0

2,
1
0
0
0

3], 1)
0 0
2 0
0 3
0 0

The 3-input form returns a diagonal matrix with vector v on the main diagonal and
the resulting matrix being of size m rows x n columns.
Given a matrix argument, instead of a vector, diag extracts the k-th diagonal of the
matrix.

blkdiag (A, B, C, . . . )
Build a block diagonal matrix from A, B, C, . . .
All arguments must be numeric and either two-dimensional matrices or scalars. If
any argument is of type sparse, the output will also be sparse.
See also: [diag], page 456, [horzcat], page 449, [vertcat], page 449, [sparse], page 568.

Chapter 16: Matrix Manipulation

457

16.3 Special Utility Matrices
eye
eye
eye
eye

(n)
(m, n)
([m n])
( . . . , class)
Return an identity matrix.
If invoked with a single scalar argument n, return a square NxN identity matrix.
If supplied two scalar arguments (m, n), eye takes them to be the number of rows
and columns. If given a vector with two elements, eye uses the values of the elements
as the number of rows and columns, respectively. For example:
eye (3)
⇒ 1 0
0 1
0 0

0
0
1

The following expressions all produce the same result:
eye (2)
≡
eye (2, 2)
≡
eye (size ([1, 2; 3, 4]))
The optional argument class, allows eye to return an array of the specified type, like
val = zeros (n,m, "uint8")
Calling eye with no arguments is equivalent to calling it with an argument of 1. Any
negative dimensions are treated as zero. These odd definitions are for compatibility
with matlab.
See also: [speye], page 566, [ones], page 457, [zeros], page 458.

ones
ones
ones
ones
ones

(n)
(m, n)
(m, n, k, . . . )
([m n . . . ])
( . . . , class)
Return a matrix or N-dimensional array whose elements are all 1.
If invoked with a single scalar integer argument n, return a square NxN matrix.
If invoked with two or more scalar integer arguments, or a vector of integer values,
return an array with the given dimensions.
To create a constant matrix whose values are all the same use an expression such as
val_matrix = val * ones (m, n)
The optional argument class specifies the class of the return array and defaults to
double. For example:
val = ones (m,n, "uint8")
See also: [zeros], page 458.

458

GNU Octave

(n)
(m, n)
(m, n, k, . . . )
([m n . . . ])
( . . . , class)
Return a matrix or N-dimensional array whose elements are all 0.

zeros
zeros
zeros
zeros
zeros

If invoked with a single scalar integer argument, return a square NxN matrix.
If invoked with two or more scalar integer arguments, or a vector of integer values,
return an array with the given dimensions.
The optional argument class specifies the class of the return array and defaults to
double. For example:
val = zeros (m,n, "uint8")
See also: [ones], page 457.
(A, m)
(A, m, n)
(A, m, n, p . . . )
(A, [m n])
(A, [m n p . . . ])
Repeat matrix or N-D array.

repmat
repmat
repmat
repmat
repmat

Form a block matrix of size m by n, with a copy of matrix A as each element.
If n is not specified, form an m by m block matrix. For copying along more than two
dimensions, specify the number of times to copy across each dimension m, n, p, . . . ,
in a vector in the second argument.
See also: [bsxfun], page 536, [kron], page 528, [repelems], page 458.

repelems (x, r)
Construct a vector of repeated elements from x.
r is a 2xN integer matrix specifying which elements to repeat and how often to
repeat each element. Entries in the first row, r(1,j), select an element to repeat. The
corresponding entry in the second row, r(2,j), specifies the repeat count. If x is a
matrix then the columns of x are imagined to be stacked on top of each other for
purposes of the selection index. A row vector is always returned.
Conceptually the result is calculated as follows:
y = [];
for i = 1:columns (r)
y = [y, x(r(1,i)*ones(1, r(2,i)))];
endfor
See also: [repmat], page 458, [cat], page 449.
The functions linspace and logspace make it very easy to create vectors with evenly
or logarithmically spaced elements. See Section 4.2 [Ranges], page 52.

Chapter 16: Matrix Manipulation

459

linspace (start, end)
linspace (start, end, n)
Return a row vector with n linearly spaced elements between start and end.
If the number of elements is greater than one, then the endpoints start and end are
always included in the range. If start is greater than end, the elements are stored in
decreasing order. If the number of points is not specified, a value of 100 is used.
The linspace function returns a row vector when both start and end are scalars.
If one, or both, inputs are vectors, then linspace transforms them to column
vectors and returns a matrix where each row is an independent sequence between
start(row_n), end(row_n).
For compatibility with matlab, return the second argument (end) when only a single
value (n = 1) is requested.
See also: [colon], page 784, [logspace], page 459.

logspace (a, b)
logspace (a, b, n)
logspace (a, pi, n)
Return a row vector with n elements logarithmically spaced from 10a to 10b .
If n is unspecified it defaults to 50.
If b is equal to π, the points are between 10a and π, not 10a and 10π , in order to be
compatible with the corresponding matlab function.
Also for compatibility with matlab, return the right-hand side of the range (10b )
when just a single value is requested.
See also: [linspace], page 458.

rand (n)
rand (m, n, . . . )
rand ([m n . . . ])
v = rand ("state")
rand ("state", v)
rand ("state", "reset")
v = rand ("seed")
rand ("seed", v)
rand ("seed", "reset")
rand ( . . . , "single")
rand ( . . . , "double")
Return a matrix with random elements uniformly distributed on the interval (0, 1).
The arguments are handled the same as the arguments for eye.
You can query the state of the random number generator using the form
v = rand ("state")
This returns a column vector v of length 625. Later, you can restore the random
number generator to the state v using the form
rand ("state", v)

460

GNU Octave

You may also initialize the state vector from an arbitrary vector of length ≤ 625 for
v. This new state will be a hash based on the value of v, not v itself.
By default, the generator is initialized from /dev/urandom if it is available, otherwise
from CPU time, wall clock time, and the current fraction of a second. Note that this
differs from matlab, which always initializes the state to the same state at startup.
To obtain behavior comparable to matlab, initialize with a deterministic state vector
in Octave’s startup files (see Section 2.1.2 [Startup Files], page 19).
To compute the pseudo-random sequence, rand uses the Mersenne Twister with a
period of 219937 − 1 (See M. Matsumoto and T. Nishimura, Mersenne Twister: A
623-dimensionally equidistributed uniform pseudorandom number generator, ACM
Trans. on Modeling and Computer Simulation Vol. 8, No. 1, pp. 3–30, January
1998, http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/emt.html). Do
not use for cryptography without securely hashing several returned values together,
otherwise the generator state can be learned after reading 624 consecutive values.
Older versions of Octave used a different random number generator. The new generator is used by default as it is significantly faster than the old generator, and produces
random numbers with a significantly longer cycle time. However, in some circumstances it might be desirable to obtain the same random sequences as produced by
the old generators. To do this the keyword "seed" is used to specify that the old
generators should be used, as in
rand ("seed", val)
which sets the seed of the generator to val. The seed of the generator can be queried
with
s = rand ("seed")
However, it should be noted that querying the seed will not cause rand to use the
old generators, only setting the seed will. To cause rand to once again use the new
generators, the keyword "state" should be used to reset the state of the rand.
The state or seed of the generator can be reset to a new random value using the
"reset" keyword.
The class of the value returned can be controlled by a trailing "double" or "single"
argument. These are the only valid classes.
See also: [randn], page 461, [rande], page 461, [randg], page 463, [randp], page 462.
(imax)
(imax, n)
(imax, m, n, . . . )
([imin imax], . . . )
( . . . , "class")
Return random integers in the range 1:imax.

randi
randi
randi
randi
randi

Additional arguments determine the shape of the return matrix. When no arguments
are specified a single random integer is returned. If one argument n is specified then
a square matrix (n x n) is returned. Two or more arguments will return a multidimensional matrix (m x n x . . . ).

Chapter 16: Matrix Manipulation

461

The integer range may optionally be described by a two element matrix with a
lower and upper bound in which case the returned integers will be on the interval
[imin, imax].
The optional argument class will return a matrix of the requested type. The default
is "double".
The following example returns 150 integers in the range 1–10.
ri = randi (10, 150, 1)
Implementation Note: randi relies internally on rand which uses class "double" to
represent numbers. This limits the maximum integer (imax) and range (imax - imin)
to the value returned by the flintmax function. For IEEE floating point numbers
this value is 253 − 1.
See also: [rand], page 459.

randn (n)
randn (m, n, . . . )
randn ([m n . . . ])
v = randn ("state")
randn ("state", v)
randn ("state", "reset")
v = randn ("seed")
randn ("seed", v)
randn ("seed", "reset")
randn ( . . . , "single")
randn ( . . . , "double")
Return a matrix with normally distributed random elements having zero mean and
variance one.
The arguments are handled the same as the arguments for rand.
By default, randn uses the Marsaglia and Tsang “Ziggurat technique” to transform
from a uniform to a normal distribution.
The class of the value returned can be controlled by a trailing "double" or "single"
argument. These are the only valid classes.
Reference: G. Marsaglia and W.W. Tsang, Ziggurat Method for Generating Random
Variables, J. Statistical Software, vol 5, 2000, http://www.jstatsoft.org/v05/
i08/
See also: [rand], page 459, [rande], page 461, [randg], page 463, [randp], page 462.

rande (n)
rande (m, n, . . . )
rande ([m n . . . ])
v = rande ("state")
rande ("state", v)
rande ("state", "reset")
v = rande ("seed")
rande ("seed", v)
rande ("seed", "reset")

462

GNU Octave

rande ( . . . , "single")
rande ( . . . , "double")
Return a matrix with exponentially distributed random elements.
The arguments are handled the same as the arguments for rand.
By default, rande uses the Marsaglia and Tsang “Ziggurat technique” to transform
from a uniform to an exponential distribution.
The class of the value returned can be controlled by a trailing "double" or "single"
argument. These are the only valid classes.
Reference: G. Marsaglia and W.W. Tsang, Ziggurat Method for Generating Random
Variables, J. Statistical Software, vol 5, 2000, http://www.jstatsoft.org/v05/
i08/
See also: [rand], page 459, [randn], page 461, [randg], page 463, [randp], page 462.

randp (l, n)
randp (l, m, n, . . . )
randp (l, [m n . . . ])
v = randp ("state")
randp ("state", v)
randp ("state", "reset")
v = randp ("seed")
randp ("seed", v)
randp ("seed", "reset")
randp ( . . . , "single")
randp ( . . . , "double")
Return a matrix with Poisson distributed random elements with mean value parameter given by the first argument, l.
The arguments are handled the same as the arguments for rand, except for the argument l.
Five different algorithms are used depending on the range of l and whether or not l
is a scalar or a matrix.
For scalar l ≤ 12, use direct method.
W.H. Press, et al., Numerical Recipes in C, Cambridge University Press,
1992.
For scalar l > 12, use rejection method.[1]
W.H. Press, et al., Numerical Recipes in C, Cambridge University Press,
1992.
For matrix l ≤ 10, use inversion method.[2]
E. Stadlober, et al., WinRand source code, available via FTP.
For matrix l > 10, use patchwork rejection method.
E. Stadlober, et al., WinRand source code, available via FTP, or H. Zechner, Efficient sampling from continuous and discrete unimodal distributions, Doctoral Dissertation, 156pp., Technical University Graz, Austria,
1994.

Chapter 16: Matrix Manipulation

463

For l > 1e8, use normal approximation.
L. Montanet, et al., Review of Particle Properties, Physical Review D 50
p1284, 1994.
The class of the value returned can be controlled by a trailing "double" or "single"
argument. These are the only valid classes.
See also: [rand], page 459, [randn], page 461, [rande], page 461, [randg], page 463.

randg (a, n)
randg (a, m, n, . . . )
randg (a, [m n . . . ])
v = randg ("state")
randg ("state", v)
randg ("state", "reset")
v = randg ("seed")
randg ("seed", v)
randg ("seed", "reset")
randg ( . . . , "single")
randg ( . . . , "double")
Return a matrix with gamma (a,1) distributed random elements.
The arguments are handled the same as the arguments for rand, except for the argument a.
This can be used to generate many distributions:
gamma (a, b) for a > -1, b > 0
r = b * randg (a)
beta (a, b) for a > -1, b > -1
r1 = randg (a, 1)
r = r1 / (r1 + randg (b, 1))
Erlang (a, n)
r = a * randg (n)
chisq (df) for df > 0
r = 2 * randg (df / 2)
t (df) for 0 < df < inf (use randn if df is infinite)
r = randn () / sqrt (2 * randg (df / 2) / df)
F (n1, n2) for 0 < n1, 0 < n2
## r1 equals 1
r1 = 2 * randg
## r2 equals 1
r2 = 2 * randg
r = r1 / r2

if n1
(n1 /
if n2
(n2 /

is
2)
is
2)

infinite
/ n1
infinite
/ n2

negative binomial (n, p) for n > 0, 0 < p <= 1
r = randp ((1 - p) / p * randg (n))

464

GNU Octave

non-central chisq (df, L), for df >= 0 and L > 0
(use chisq if L = 0)
r = randp (L / 2)
r(r > 0) = 2 * randg (r(r > 0))
r(df > 0) += 2 * randg (df(df > 0)/2)
Dirichlet (a1, ... ak)
r = (randg (a1), ..., randg (ak))
r = r / sum (r)
The class of the value returned can be controlled by a trailing "double" or "single"
argument. These are the only valid classes.
See also: [rand], page 459, [randn], page 461, [rande], page 461, [randp], page 462.
The generators operate in the new or old style together, it is not possible to mix the
two. Initializing any generator with "state" or "seed" causes the others to switch to the
same style for future calls.
The state of each generator is independent and calls to different generators can be
interleaved without affecting the final result. For example,
rand ("state", [11, 22, 33]);
randn ("state", [44, 55, 66]);
u = rand (100, 1);
n = randn (100, 1);
and
rand ("state", [11, 22, 33]);
randn ("state", [44, 55, 66]);
u = zeros (100, 1);
n = zeros (100, 1);
for i = 1:100
u(i) = rand ();
n(i) = randn ();
end
produce equivalent results. When the generators are initialized in the old style with "seed"
only rand and randn are independent, because the old rande, randg and randp generators
make calls to rand and randn.
The generators are initialized with random states at start-up, so that the sequences of
random numbers are not the same each time you run Octave.1 If you really do need to
reproduce a sequence of numbers exactly, you can set the state or seed to a specific value.
If invoked without arguments, rand and randn return a single element of a random
sequence.
The original rand and randn functions use Fortran code from ranlib, a library of Fortran
routines for random number generation, compiled by Barry W. Brown and James Lovato
of the Department of Biomathematics at The University of Texas, M.D. Anderson Cancer
Center, Houston, TX 77030.
1

The old versions of rand and randn obtain their initial seeds from the system clock.

Chapter 16: Matrix Manipulation

465

randperm (n)
randperm (n, m)
Return a row vector containing a random permutation of 1:n.
If m is supplied, return m unique entries, sampled without replacement from 1:n.
The complexity is O(n) in memory and O(m) in time, unless m < n/5, in which case
O(m) memory is used as well. The randomization is performed using rand(). All
permutations are equally likely.
See also: [perms], page 657.

16.4 Famous Matrices
The following functions return famous matrix forms.

gallery (name)
gallery (name, args)
Create interesting matrices for testing.

c = gallery ("cauchy", x)
c = gallery ("cauchy", x, y)
Create a Cauchy matrix.

c = gallery ("chebspec", n)
c = gallery ("chebspec", n, k)
Create a Chebyshev spectral differentiation matrix.

c = gallery ("chebvand", p)
c = gallery ("chebvand", m, p)
Create a Vandermonde-like matrix for the Chebyshev polynomials.

a = gallery ("chow", n)
a = gallery ("chow", n, alpha)
a = gallery ("chow", n, alpha, delta)
Create a Chow matrix – a singular Toeplitz lower Hessenberg matrix.

c = gallery ("circul", v)
Create a circulant matrix.

a = gallery ("clement", n)
a = gallery ("clement", n, k)
Create a tridiagonal matrix with zero diagonal entries.

c = gallery ("compar", a)
c = gallery ("compar", a, k)
Create a comparison matrix.

a = gallery ("condex", n)
a = gallery ("condex", n, k)
a = gallery ("condex", n, k, theta)
Create a ‘counterexample’ matrix to a condition estimator.

466

GNU Octave

a = gallery ("cycol", [m n])
a = gallery ("cycol", n)
a = gallery ( . . . , k)
Create a matrix whose columns repeat cyclically.

[c, d, e] = gallery ("dorr", n)
[c, d, e] = gallery ("dorr", n, theta)
a = gallery ("dorr", . . . )
Create a diagonally dominant, ill-conditioned, tridiagonal matrix.

a = gallery ("dramadah", n)
a = gallery ("dramadah", n, k)
Create a (0, 1) matrix whose inverse has large integer entries.

a = gallery ("fiedler", c)
Create a symmetric Fiedler matrix.

a = gallery ("forsythe", n)
a = gallery ("forsythe", n, alpha)
a = gallery ("forsythe", n, alpha, lambda)
Create a Forsythe matrix (a perturbed Jordan block).

f = gallery ("frank", n)
f = gallery ("frank", n, k)
Create a Frank matrix (ill-conditioned eigenvalues).

c = gallery ("gcdmat", n)
Create a greatest common divisor matrix.
c is an n-by-n matrix whose values correspond to the greatest common divisor of its
coordinate values, i.e., c(i,j) correspond gcd (i, j).

a = gallery ("gearmat", n)
a = gallery ("gearmat", n, i)
a = gallery ("gearmat", n, i, j)
Create a Gear matrix.

g = gallery ("grcar", n)
g = gallery ("grcar", n, k)
Create a Toeplitz matrix with sensitive eigenvalues.

a = gallery ("hanowa", n)
a = gallery ("hanowa", n, d)
Create a matrix whose eigenvalues lie on a vertical line in the complex plane.

v = gallery ("house", x)
[v, beta] = gallery ("house", x)
Create a householder matrix.

Chapter 16: Matrix Manipulation

a
a
a
a
a

=
=
=
=
=

467

("integerdata", imax, [M N . . . ], j)
("integerdata", imax, M, N, . . . , j)
("integerdata", [imin, imax], [M N . . . ], j)
("integerdata", [imin, imax], M, N, . . . , j)
("integerdata", . . . , "class")
Create a matrix with random integers in the range [1, imax]. If imin is given then
the integers are in the range [imin, imax].
The second input is a matrix of dimensions describing the size of the output. The
dimensions can also be input as comma-separated arguments.
The input j is an integer index in the range [0, 2^32-1]. The values of the output
matrix are always exactly the same (reproducibility) for a given size input and j index.
The final optional argument determines the class of the resulting matrix. Possible
values for class: "uint8", "uint16", "uint32", "int8", "int16", int32", "single",
"double". The default is "double".

gallery
gallery
gallery
gallery
gallery

a = gallery ("invhess", x)
a = gallery ("invhess", x, y)
Create the inverse of an upper Hessenberg matrix.

a = gallery ("invol", n)
Create an involutory matrix.

a = gallery ("ipjfact", n)
a = gallery ("ipjfact", n, k)
Create a Hankel matrix with factorial elements.

a = gallery ("jordbloc", n)
a = gallery ("jordbloc", n, lambda)
Create a Jordan block.

u = gallery ("kahan", n)
u = gallery ("kahan", n, theta)
u = gallery ("kahan", n, theta, pert)
Create a Kahan matrix (upper trapezoidal).

a = gallery ("kms", n)
a = gallery ("kms", n, rho)
Create a Kac-Murdock-Szego Toeplitz matrix.

b = gallery ("krylov", a)
b = gallery ("krylov", a, x)
b = gallery ("krylov", a, x, j)
Create a Krylov matrix.

a = gallery ("lauchli", n)
a = gallery ("lauchli", n, mu)
Create a Lauchli matrix (rectangular).

a = gallery ("lehmer", n)
Create a Lehmer matrix (symmetric positive definite).

468

GNU Octave

t = gallery ("lesp", n)
Create a tridiagonal matrix with real, sensitive eigenvalues.

a = gallery ("lotkin", n)
Create a Lotkin matrix.

a = gallery ("minij", n)
Create a symmetric positive definite matrix MIN(i,j).

a = gallery ("moler", n)
a = gallery ("moler", n, alpha)
Create a Moler matrix (symmetric positive definite).

[a, t] = gallery ("neumann", n)
Create a singular matrix from the discrete Neumann problem (sparse).

a = gallery ("normaldata", [M N . . . ], j)
a = gallery ("normaldata", M, N, . . . , j)
a = gallery ("normaldata", . . . , "class")
Create a matrix with random samples from the standard normal distribution (mean
= 0, std = 1).
The first input is a matrix of dimensions describing the size of the output. The
dimensions can also be input as comma-separated arguments.
The input j is an integer index in the range [0, 2^32-1]. The values of the output
matrix are always exactly the same (reproducibility) for a given size input and j index.
The final optional argument determines the class of the resulting matrix. Possible
values for class: "single", "double". The default is "double".

q = gallery ("orthog", n)
q = gallery ("orthog", n, k)
Create orthogonal and nearly orthogonal matrices.

a = gallery ("parter", n)
Create a Parter matrix (a Toeplitz matrix with singular values near pi).

p = gallery ("pei", n)
p = gallery ("pei", n, alpha)
Create a Pei matrix.

a = gallery ("Poisson", n)
Create a block tridiagonal matrix from Poisson’s equation (sparse).

a = gallery ("prolate", n)
a = gallery ("prolate", n, w)
Create a prolate matrix (symmetric, ill-conditioned Toeplitz matrix).

h = gallery ("randhess", x)
Create a random, orthogonal upper Hessenberg matrix.

Chapter 16: Matrix Manipulation

469

a = gallery ("rando", n)
a = gallery ("rando", n, k)
Create a random matrix with elements -1, 0 or 1.

a
a
a
a
a

=
=
=
=
=

("randsvd", n)
("randsvd", n, kappa)
("randsvd", n, kappa, mode)
("randsvd", n, kappa, mode, kl)
("randsvd", n, kappa, mode, kl, ku)
Create a random matrix with pre-assigned singular values.

gallery
gallery
gallery
gallery
gallery

a = gallery ("redheff", n)
Create a zero and ones matrix of Redheffer associated with the Riemann hypothesis.

a = gallery ("riemann", n)
Create a matrix associated with the Riemann hypothesis.

a = gallery ("ris", n)
Create a symmetric Hankel matrix.

a = gallery ("smoke", n)
a = gallery ("smoke", n, k)
Create a complex matrix, with a ‘smoke ring’ pseudospectrum.
("toeppd", n)
("toeppd", n, m)
("toeppd", n, m, w)
("toeppd", n, m, w, theta)
Create a symmetric positive definite Toeplitz matrix.

t
t
t
t

=
=
=
=

gallery
gallery
gallery
gallery

p
p
p
p
p
p

=
=
=
=
=
=

gallery
gallery
gallery
gallery
gallery
gallery

("toeppen", n)
("toeppen", n, a)
("toeppen", n, a, b)
("toeppen", n, a, b, c)
("toeppen", n, a, b, c, d)
("toeppen", n, a, b, c, d, e)
Create a pentadiagonal Toeplitz matrix (sparse).

a = gallery ("tridiag", x, y, z)
a = gallery ("tridiag", n)
a = gallery ("tridiag", n, c, d, e)
Create a tridiagonal matrix (sparse).

t = gallery ("triw", n)
t = gallery ("triw", n, alpha)
t = gallery ("triw", n, alpha, k)
Create an upper triangular matrix discussed by Kahan, Golub, and Wilkinson.

a = gallery ("uniformdata", [M N . . . ], j)
a = gallery ("uniformdata", M, N, . . . , j)

470

GNU Octave

a = gallery ("uniformdata", . . . , "class")
Create a matrix with random samples from the standard uniform distribution (range
[0,1]).
The first input is a matrix of dimensions describing the size of the output. The
dimensions can also be input as comma-separated arguments.
The input j is an integer index in the range [0, 2^32-1]. The values of the output
matrix are always exactly the same (reproducibility) for a given size input and j index.
The final optional argument determines the class of the resulting matrix. Possible
values for class: "single", "double". The default is "double".

a = gallery ("wathen", nx, ny)
a = gallery ("wathen", nx, ny, k)
Create the Wathen matrix.

[a, b] = gallery ("wilk", n)
Create various specific matrices devised/discussed by Wilkinson.

hadamard (n)
Construct a Hadamard matrix (Hn) of size n-by-n.
The size n must be of the form 2k ∗ p in which p is one of 1, 12, 20 or 28. The returned
matrix is normalized, meaning Hn(:,1) == 1 and Hn(1,:) == 1.
Some of the properties of Hadamard matrices are:
• kron (Hm, Hn) is a Hadamard matrix of size m-by-n.
• Hn * Hn’ = n * eye (n).

• The rows of Hn are orthogonal.

• det (A) <= abs (det (Hn)) for all A with abs (A(i, j)) <= 1.

• Multiplying any row or column by -1 and the matrix will remain a Hadamard
matrix.
See also: [compan], page 690, [hankel], page 470, [toeplitz], page 472.

hankel (c)
hankel (c, r)
Return the Hankel matrix constructed from the first column c, and (optionally) the
last row r.
If the last element of c is not the same as the first element of r, the last element of c
is used. If the second argument is omitted, it is assumed to be a vector of zeros with
the same size as c.
A Hankel matrix formed from an m-vector c, and an n-vector r, has the elements
H(i, j) =



ci+j−1 , i + j − 1 ≤ m;
ri+j−m , otherwise.

See also: [hadamard], page 470, [toeplitz], page 472.

Chapter 16: Matrix Manipulation

471

hilb (n)
Return the Hilbert matrix of order n.
The i, j element of a Hilbert matrix is defined as
H(i, j) =

1
(i + j − 1)

Hilbert matrices are close to being singular which make them difficult to invert with
numerical routines. Comparing the condition number of a random matrix 5x5 matrix
with that of a Hilbert matrix of order 5 reveals just how difficult the problem is.
cond (rand (5))
⇒ 14.392
cond (hilb (5))
⇒ 4.7661e+05
See also: [invhilb], page 471.

invhilb (n)
Return the inverse of the Hilbert matrix of order n.
This can be computed exactly using
Aij = −1
=

i+j

(i + j − 1)

p(i)p(j)
(i + j − 1)



n+i−1
n−j

where
p(k) = −1

k





n+j−1
n−i

k+n−1
k−1



n
k



i+j−2
i−2

2



The validity of this formula can easily be checked by expanding the binomial coefficients in both formulas as factorials. It can be derived more directly via the theory
of Cauchy matrices. See J. W. Demmel, Applied Numerical Linear Algebra, p. 92.
Compare this with the numerical calculation of inverse (hilb (n)), which suffers
from the ill-conditioning of the Hilbert matrix, and the finite precision of your computer’s floating point arithmetic.
See also: [hilb], page 471.

magic (n)
Create an n-by-n magic square.
A magic square is an arrangement of the integers 1:n^2 such that the row sums,
column sums, and diagonal sums are all equal to the same value.
Note: n must be a scalar greater than or equal to 3. If you supply n less than 3,
magic returns either a nonmagic square, or else the degenerate magic squares 1 and
[].

472

GNU Octave

pascal (n)
pascal (n, t)
Return the Pascal matrix of order n if t = 0.
The default value of t is 0.
When t = 1, return the pseudo-lower triangular Cholesky factor of the Pascal matrix
(The sign of some columns may be negative). This matrix is its own inverse, that is
pascal (n, 1) ^ 2 == eye (n).
If t = -1, return the true Cholesky factor with strictly positive values on the diagonal.
If t = 2, return a transposed and permuted version of pascal (n, 1), which is the
cube root of the identity matrix. That is, pascal (n, 2) ^ 3 == eye (n).
See also: [chol], page 515.

rosser ()
Return the Rosser matrix.
This is a difficult test case used to evaluate eigenvalue algorithms.
See also: [wilkinson], page 473, [eig], page 509.

toeplitz (c)
toeplitz (c, r)
Return the Toeplitz matrix constructed from the first column c, and (optionally) the
first row r.
If the first element of r is not the same as the first element of c, the first element of
c is used. If the second argument is omitted, the first row is taken to be the same as
the first column.
A square Toeplitz matrix has the form:


c0
 c1

 c2

 .
 ..

cn

r1
c0
c1
..
.

r2
r1
c0
..
.

···
···
···
..
.

cn−1

cn−2

...

rn



rn−1 

rn−2 

.. 
. 
c0

See also: [hankel], page 470.

vander (c)
vander (c, n)
Return the Vandermonde matrix whose next to last column is c.
If n is specified, it determines the number of columns; otherwise, n is taken to be
equal to the length of c.
A Vandermonde matrix has the form:
 n−1

c1
· · · c21 c1 1
 cn−1
· · · c22 c2 1 
 2

 .
.
.. .. 
..
 ..
. ..
. .
cn−1
n

See also: [polyfit], page 695.

···

c2n

cn

1

Chapter 16: Matrix Manipulation

473

wilkinson (n)
Return the Wilkinson matrix of order n.
Wilkinson matrices are symmetric and tridiagonal with pairs of nearly, but not exactly, equal eigenvalues. They are useful in testing the behavior and performance of
eigenvalue solvers.
See also: [rosser], page 472, [eig], page 509.

475

17 Arithmetic
Unless otherwise noted, all of the functions described in this chapter will work for real and
complex scalar, vector, or matrix arguments. Functions described as mapping functions
apply the given operation individually to each element when given a matrix argument. For
example:
sin ([1, 2; 3, 4])
⇒ 0.84147
0.90930
0.14112 -0.75680

17.1 Exponents and Logarithms
exp (x)
Compute ex for each element of x.
To compute the matrix exponential, see Chapter 18 [Linear Algebra], page 507.
See also: [log], page 475.

expm1 (x)
Compute ex − 1 accurately in the neighborhood of zero.
See also: [exp], page 475.

log (x)
Compute the natural logarithm, ln (x), for each element of x.
To compute the matrix logarithm, see Chapter 18 [Linear Algebra], page 507.
See also: [exp], page 475, [log1p], page 475, [log2], page 475, [log10], page 475,
[logspace], page 459.

reallog (x)
Return the real-valued natural logarithm of each element of x.
If any element results in a complex return value reallog aborts and issues an error.
See also: [log], page 475, [realpow], page 476, [realsqrt], page 476.

log1p (x)
Compute ln (1 + x) accurately in the neighborhood of zero.
See also: [log], page 475, [exp], page 475, [expm1], page 475.

log10 (x)
Compute the base-10 logarithm of each element of x.
See also: [log], page 475, [log2], page 475, [logspace], page 459, [exp], page 475.

log2 (x)
[f, e] = log2 (x)
Compute the base-2 logarithm of each element of x.
If called with two output arguments, split x into binary mantissa and exponent so
that 21 ≤ |f | < 1 and e is an integer. If x = 0, f = e = 0.
See also: [pow2], page 476, [log], page 475, [log10], page 475, [exp], page 475.

476

GNU Octave

pow2 (x)
pow2 (f, e)
With one input argument, compute 2x for each element of x.
With two input arguments, return f · 2e .

See also: [log2], page 475, [nextpow2], page 476, [power], page 147.

nextpow2 (x)
Compute the exponent for the smallest power of two larger than the input.
For each element in the input array x, return the first integer n such that 2n ≥ |x|.
See also: [pow2], page 476, [log2], page 475.

realpow (x, y)
Compute the real-valued, element-by-element power operator.
This is equivalent to x .^ y, except that realpow reports an error if any return value
is complex.
See also: [power], page 147, [reallog], page 475, [realsqrt], page 476.

sqrt (x)
Compute the square root of each element of x.
If x is negative, a complex result is returned.
To compute the matrix square root, see Chapter 18 [Linear Algebra], page 507.
See also: [realsqrt], page 476, [nthroot], page 476.

realsqrt (x)
Return the real-valued square root of each element of x.
If any element results in a complex return value realsqrt aborts and issues an error.
See also: [sqrt], page 476, [realpow], page 476, [reallog], page 475.

cbrt (x)
Compute the real cube root of each element of x.
Unlike x^(1/3), the result will be negative if x is negative.
See also: [nthroot], page 476.

nthroot (x, n)
Compute the real (non-complex) n-th root of x.
x must have all real entries and n must be a scalar. If n is an even integer and x has
negative entries then nthroot aborts and issues an error.
Example:
nthroot (-1, 3)
⇒ -1
(-1) ^ (1 / 3)
⇒ 0.50000 - 0.86603i
See also: [realsqrt], page 476, [sqrt], page 476, [cbrt], page 476.

Chapter 17: Arithmetic

477

17.2 Complex Arithmetic
In the descriptions
of the following functions, z is the complex number x + iy, where i is
√
defined as −1.

abs (z)
Compute the magnitude of z.
√
The magnitude is defined as |z| = x2 + y 2 .
For example:
abs (3 + 4i)
⇒ 5
See also: [arg], page 477.

arg (z)
angle (z)
Compute the argument, i.e., angle of z.
This is defined as, θ = atan2(y, x), in radians.
For example:
arg (3 + 4i)
⇒ 0.92730

See also: [abs], page 477.

conj (z)
Return the complex conjugate of z.
The complex conjugate is defined as z̄ = x − iy.
See also: [real], page 478, [imag], page 477.

cplxpair (z)
cplxpair (z, tol)
cplxpair (z, tol, dim)
Sort the numbers z into complex conjugate pairs ordered by increasing real part.
The negative imaginary complex numbers are placed first within each pair. All real
numbers (those with abs (imag (z) / z) < tol) are placed after the complex pairs.
tol is a weighting factor which determines the tolerance of matching. The default
value is 100 and the resulting tolerance for a given complex pair is 100 * eps (abs
(z(i))).
By default the complex pairs are sorted along the first non-singleton dimension of z.
If dim is specified, then the complex pairs are sorted along this dimension.
Signal an error if some complex numbers could not be paired. Signal an error if all
complex numbers are not exact conjugates (to within tol). Note that there is no
defined order for pairs with identical real parts but differing imaginary parts.
cplxpair (exp (2i*pi*[0:4]’/5)) == exp (2i*pi*[3; 2; 4; 1; 0]/5)

imag (z)
Return the imaginary part of z as a real number.
See also: [real], page 478, [conj], page 477.

478

GNU Octave

real (z)
Return the real part of z.
See also: [imag], page 477, [conj], page 477.

17.3 Trigonometry
Octave provides the following trigonometric functions where angles are specified in radians.
To convert from degrees to radians multiply by π/180 or use the deg2rad function. For
example, sin (30 * pi/180) returns the sine of 30 degrees. As an alternative, Octave
provides a number of trigonometric functions which work directly on an argument specified
in degrees. These functions are named after the base trigonometric function with a ‘d’ suffix.
As an example, sin expects an angle in radians while sind expects an angle in degrees.
Octave uses the C library trigonometric functions. It is expected that these functions
are defined by the ISO/IEC 9899 Standard. This Standard is available at: http://www.
open-std . org / jtc1 / sc22 / wg14 / www / docs / n1124 . pdf. Section F.9.1 deals with the
trigonometric functions. The behavior of most of the functions is relatively straightforward.
However, there are some exceptions to the standard behavior. Many of the exceptions
involve the behavior for -0. The most complex case is atan2. Octave exactly implements
the behavior given in the Standard. Including atan2(±0, −0) returns ±π.
It should be noted that matlab uses different definitions which apparently do not distinguish -0.

rad = deg2rad (deg)
Convert degrees to radians.
The input deg must be a scalar, vector, or N-dimensional array of double or single
floating point values. deg may be complex in which case the real and imaginary
components are converted separately.
The output rad is the same size and shape as deg with degrees converted to radians
using the conversion constant pi/180.
Example:
deg2rad ([0, 90, 180, 270, 360])
⇒ 0.00000
1.57080
3.14159
4.71239
6.28319
See also: [rad2deg], page 478.

deg = rad2deg (rad)
Convert radians to degrees.
The input rad must be a scalar, vector, or N-dimensional array of double or single
floating point values. rad may be complex in which case the real and imaginary
components are converted separately.
The output deg is the same size and shape as rad with radians converted to degrees
using the conversion constant 180/pi.
Example:
rad2deg ([0, pi/2, pi, 3/2*pi, 2*pi])
⇒ 0
90
180
270
360
See also: [deg2rad], page 478.

Chapter 17: Arithmetic

sin (x)
Compute the sine for each element of x in radians.
See also: [asin], page 479, [sind], page 481, [sinh], page 480.

cos (x)
Compute the cosine for each element of x in radians.
See also: [acos], page 479, [cosd], page 481, [cosh], page 480.

tan (z)
Compute the tangent for each element of x in radians.
See also: [atan], page 479, [tand], page 481, [tanh], page 480.

sec (x)
Compute the secant for each element of x in radians.
See also: [asec], page 479, [secd], page 481, [sech], page 480.

csc (x)
Compute the cosecant for each element of x in radians.
See also: [acsc], page 479, [cscd], page 481, [csch], page 480.

cot (x)
Compute the cotangent for each element of x in radians.
See also: [acot], page 479, [cotd], page 481, [coth], page 480.

asin (x)
Compute the inverse sine in radians for each element of x.
See also: [sin], page 479, [asind], page 481.

acos (x)
Compute the inverse cosine in radians for each element of x.
See also: [cos], page 479, [acosd], page 482.

atan (x)
Compute the inverse tangent in radians for each element of x.
See also: [tan], page 479, [atand], page 482.

asec (x)
Compute the inverse secant in radians for each element of x.
See also: [sec], page 479, [asecd], page 482.

acsc (x)
Compute the inverse cosecant in radians for each element of x.
See also: [csc], page 479, [acscd], page 482.

acot (x)
Compute the inverse cotangent in radians for each element of x.
See also: [cot], page 479, [acotd], page 482.

479

480

GNU Octave

sinh (x)
Compute the hyperbolic sine for each element of x.
See also: [asinh], page 480, [cosh], page 480, [tanh], page 480.

cosh (x)
Compute the hyperbolic cosine for each element of x.
See also: [acosh], page 480, [sinh], page 480, [tanh], page 480.

tanh (x)
Compute hyperbolic tangent for each element of x.
See also: [atanh], page 480, [sinh], page 480, [cosh], page 480.

sech (x)
Compute the hyperbolic secant of each element of x.
See also: [asech], page 480.

csch (x)
Compute the hyperbolic cosecant of each element of x.
See also: [acsch], page 480.

coth (x)
Compute the hyperbolic cotangent of each element of x.
See also: [acoth], page 480.

asinh (x)
Compute the inverse hyperbolic sine for each element of x.
See also: [sinh], page 480.

acosh (x)
Compute the inverse hyperbolic cosine for each element of x.
See also: [cosh], page 480.

atanh (x)
Compute the inverse hyperbolic tangent for each element of x.
See also: [tanh], page 480.

asech (x)
Compute the inverse hyperbolic secant of each element of x.
See also: [sech], page 480.

acsch (x)
Compute the inverse hyperbolic cosecant of each element of x.
See also: [csch], page 480.

acoth (x)
Compute the inverse hyperbolic cotangent of each element of x.
See also: [coth], page 480.

Chapter 17: Arithmetic

481

atan2 (y, x)
Compute atan (y / x) for corresponding elements of y and x.
y and x must match in size and orientation. The signs of elements of y and x are
used to determine the quadrants of each resulting value.
This function is equivalent to arg (complex (x, y)).
See also: [tan], page 479, [tand], page 481, [tanh], page 480, [atanh], page 480.
Octave provides the following trigonometric functions where angles are specified in degrees. These functions produce true zeros at the appropriate intervals rather than the small
round-off error that occurs when using radians. For example:
cosd (90)
⇒ 0
cos (pi/2)
⇒ 6.1230e-17

sind (x)
Compute the sine for each element of x in degrees.
Returns zero for elements where x/180 is an integer.
See also: [asind], page 481, [sin], page 479.

cosd (x)
Compute the cosine for each element of x in degrees.
Returns zero for elements where (x-90)/180 is an integer.
See also: [acosd], page 482, [cos], page 479.

tand (x)
Compute the tangent for each element of x in degrees.
Returns zero for elements where x/180 is an integer and Inf for elements where
(x-90)/180 is an integer.
See also: [atand], page 482, [tan], page 479.

secd (x)
Compute the secant for each element of x in degrees.
See also: [asecd], page 482, [sec], page 479.

cscd (x)
Compute the cosecant for each element of x in degrees.
See also: [acscd], page 482, [csc], page 479.

cotd (x)
Compute the cotangent for each element of x in degrees.
See also: [acotd], page 482, [cot], page 479.

asind (x)
Compute the inverse sine in degrees for each element of x.
See also: [sind], page 481, [asin], page 479.

482

GNU Octave

acosd (x)
Compute the inverse cosine in degrees for each element of x.
See also: [cosd], page 481, [acos], page 479.

atand (x)
Compute the inverse tangent in degrees for each element of x.
See also: [tand], page 481, [atan], page 479.

atan2d (y, x)
Compute atan (y / x) in degrees for corresponding elements from y and x.
See also: [tand], page 481, [atan2], page 481.

asecd (x)
Compute the inverse secant in degrees for each element of x.
See also: [secd], page 481, [asec], page 479.

acscd (x)
Compute the inverse cosecant in degrees for each element of x.
See also: [cscd], page 481, [acsc], page 479.

acotd (x)
Compute the inverse cotangent in degrees for each element of x.
See also: [cotd], page 481, [acot], page 479.

17.4 Sums and Products
sum
sum
sum
sum
sum

(x)
(x, dim)
( . . . , "native")
( . . . , "double")
( . . . , "extra")
Sum of elements along dimension dim.
If dim is omitted, it defaults to the first non-singleton dimension.
The optional "type" input determines the class of the variable used for calculations.
If the argument "native" is given, then the operation is performed in the same type
as the original argument, rather than the default double type.
For example:
sum ([true, true])
⇒ 2
sum ([true, true], "native")
⇒ true
On the contrary, if "double" is given, the sum is performed in double precision even
for single precision inputs.
For double precision inputs, the "extra" option will use a more accurate algorithm
than straightforward summation. For single precision inputs, "extra" is the same as
"double". Otherwise, "extra" has no effect.

Chapter 17: Arithmetic

483

See also: [cumsum], page 483, [sumsq], page 484, [prod], page 483.

prod
prod
prod
prod

(x)
(x, dim)
( . . . , "native")
( . . . , "double")
Product of elements along dimension dim.
If dim is omitted, it defaults to the first non-singleton dimension.
The optional "type" input determines the class of the variable used for calculations.
If the argument "native" is given, then the operation is performed in the same type
as the original argument, rather than the default double type.
For example:
prod ([true, true])
⇒ 1
prod ([true, true], "native")
⇒ true

On the contrary, if "double" is given, the operation is performed in double precision
even for single precision inputs.
See also: [cumprod], page 483, [sum], page 482.
(x)
(x, dim)
( . . . , "native")
( . . . , "double")
Cumulative sum of elements along dimension dim.

cumsum
cumsum
cumsum
cumsum

If dim is omitted, it defaults to the first non-singleton dimension. For example:
cumsum ([1, 2; 3, 4; 5, 6])
⇒ 1
2
4
6
9 12
See sum for an explanation of the optional parameters "native" and "double".
See also: [sum], page 482, [cumprod], page 483.

cumprod (x)
cumprod (x, dim)
Cumulative product of elements along dimension dim.
If dim is omitted, it defaults to the first non-singleton dimension. For example:
cumprod ([1, 2; 3, 4; 5, 6])
⇒ 1
2
3
8
15 48
See also: [prod], page 483, [cumsum], page 483.

484

GNU Octave

sumsq (x)
sumsq (x, dim)
Sum of squares of elements along dimension dim.
If dim is omitted, it defaults to the first non-singleton dimension.
This function is conceptually equivalent to computing
sum (x .* conj (x), dim)
but it uses less memory and avoids calling conj if x is real.
See also: [sum], page 482, [prod], page 483.

17.5 Utility Functions
ceil (x)
Return the smallest integer not less than x.
This is equivalent to rounding towards positive infinity.
If x is complex, return ceil (real (x)) + ceil (imag (x)) * I.
ceil ([-2.7, 2.7])
⇒ -2
3
See also: [floor], page 484, [round], page 484, [fix], page 484.

fix (x)
Truncate fractional portion of x and return the integer portion.
This is equivalent to rounding towards zero. If x is complex, return fix (real (x))
+ fix (imag (x)) * I.
fix ([-2.7, 2.7])
⇒ -2
2
See also: [ceil], page 484, [floor], page 484, [round], page 484.

floor (x)
Return the largest integer not greater than x.
This is equivalent to rounding towards negative infinity. If x is complex, return floor
(real (x)) + floor (imag (x)) * I.
floor ([-2.7, 2.7])
⇒ -3
2
See also: [ceil], page 484, [round], page 484, [fix], page 484.

round (x)
Return the integer nearest to x.
If x is complex, return round (real (x)) + round (imag (x)) * I. If there are two
nearest integers, return the one further away from zero.
round ([-2.7, 2.7])
⇒ -3
3
See also: [ceil], page 484, [floor], page 484, [fix], page 484, [roundb], page 485.

Chapter 17: Arithmetic

485

roundb (x)
Return the integer nearest to x. If there are two nearest integers, return the even one
(banker’s rounding).
If x is complex, return roundb (real (x)) + roundb (imag (x)) * I.
See also: [round], page 484.

max
max
[w,
max

(x)
(x, [], dim)

iw] = max (x)
(x, y)
Find maximum values in the array x.
For a vector argument, return the maximum value. For a matrix argument, return a
row vector with the maximum value of each column. For a multi-dimensional array,
max operates along the first non-singleton dimension.
If the optional third argument dim is present then operate along this dimension. In
this case the second argument is ignored and should be set to the empty matrix.
For two matrices (or a matrix and a scalar), return the pairwise maximum.
Thus,
max (max (x))
returns the largest element of the 2-D matrix x, and
max (2:5, pi)
⇒ 3.1416

3.1416

4.0000

5.0000

compares each element of the range 2:5 with pi, and returns a row vector of the
maximum values.
For complex arguments, the magnitude of the elements are used for comparison. If
the magnitudes are identical, then the results are ordered by phase angle in the range
(-pi, pi]. Hence,
max ([-1 i 1 -i])
⇒ -1

because all entries have magnitude 1, but -1 has the largest phase angle with value
pi.
If called with one input and two output arguments, max also returns the first index
of the maximum value(s). Thus,
[x, ix] = max ([1, 3, 5, 2, 5])
⇒ x = 5
ix = 3
See also: [min], page 485, [cummax], page 486, [cummin], page 487.

min
min
[w,
min

(x)
(x, [], dim)

iw] = min (x)
(x, y)
Find minimum values in the array x.

486

GNU Octave

For a vector argument, return the minimum value. For a matrix argument, return a
row vector with the minimum value of each column. For a multi-dimensional array,
min operates along the first non-singleton dimension.
If the optional third argument dim is present then operate along this dimension. In
this case the second argument is ignored and should be set to the empty matrix.
For two matrices (or a matrix and a scalar), return the pairwise minimum.
Thus,
min (min (x))
returns the smallest element of the 2-D matrix x, and
min (2:5, pi)
⇒ 2.0000

3.0000

3.1416

3.1416

compares each element of the range 2:5 with pi, and returns a row vector of the
minimum values.
For complex arguments, the magnitude of the elements are used for comparison. If
the magnitudes are identical, then the results are ordered by phase angle in the range
(-pi, pi]. Hence,
min ([-1 i 1 -i])
⇒ -i

because all entries have magnitude 1, but -i has the smallest phase angle with value
-pi/2.
If called with one input and two output arguments, min also returns the first index
of the minimum value(s). Thus,
[x, ix] = min ([1, 3, 0, 2, 0])
⇒ x = 0
ix = 3
See also: [max], page 485, [cummin], page 487, [cummax], page 486.

cummax (x)
cummax (x, dim)
[w, iw] = cummax ( . . . )
Return the cumulative maximum values along dimension dim.
If dim is unspecified it defaults to column-wise operation. For example:
cummax ([1 3 2 6 4 5])
⇒ 1 3 3 6 6 6

If called with two output arguments the index of the maximum value is also returned.
[w, iw] = cummax ([1 3 2 6 4 5])
⇒
w = 1 3 3 6 6 6
iw = 1 2 2 4 4 4
See also: [cummin], page 487, [max], page 485, [min], page 485.

Chapter 17: Arithmetic

487

cummin (x)
cummin (x, dim)
[w, iw] = cummin (x)
Return the cumulative minimum values along dimension dim.
If dim is unspecified it defaults to column-wise operation. For example:
cummin ([5 4 6 2 3 1])
⇒ 5 4 4 2 2 1
If called with two output arguments the index of the minimum value is also returned.
[w, iw] = cummin ([5 4 6 2 3 1])
⇒
w = 5 4 4 2 2 1
iw = 1 2 2 4 4 6
See also: [cummax], page 486, [min], page 485, [max], page 485.

hypot (x, y)
hypot (x, y, z, . . . )
Compute the element-by-element square root of the sum of the squares of x and y.
This is equivalent to sqrt (x.^2 + y.^2), but is calculated in a manner that avoids
overflows for large values of x or y.
hypot can also be called with more than 2 arguments; in this case, the arguments are
accumulated from left to right:
hypot (hypot (x, y), z)
hypot (hypot (hypot (x, y), z), w), etc.

dx = gradient (m)
[dx, dy, dz, ...] =
[...] = gradient (m,
[...] = gradient (m,
[...] = gradient (f,
[...] = gradient (f,
[...] = gradient (f,

gradient (m)
s)
x, y, z, . . . )
x0)
x0, s)
x0, x, y, . . . )

Calculate the gradient of sampled data or a function.
If m is a vector, calculate the one-dimensional gradient of m. If m is a matrix the
gradient is calculated for each dimension.
[dx, dy] = gradient (m) calculates the one-dimensional gradient for x and y direction if m is a matrix. Additional return arguments can be use for multi-dimensional
matrices.
A constant spacing between two points can be provided by the s parameter. If s is a
scalar, it is assumed to be the spacing for all dimensions. Otherwise, separate values
of the spacing can be supplied by the x, . . . arguments. Scalar values specify an
equidistant spacing. Vector values for the x, . . . arguments specify the coordinate for
that dimension. The length must match their respective dimension of m.
At boundary points a linear extrapolation is applied. Interior points are calculated
with the first approximation of the numerical gradient
y’(i) = 1/(x(i+1)-x(i-1)) * (y(i-1)-y(i+1)).

488

GNU Octave

If the first argument f is a function handle, the gradient of the function at the points
in x0 is approximated using central difference. For example, gradient (@cos, 0)
approximates the gradient of the cosine function in the point x0 = 0. As with sampled
data, the spacing values between the points from which the gradient is estimated can
be set via the s or dx, dy, . . . arguments. By default a spacing of 1 is used.
See also: [diff], page 444, [del2], page 489.

dot (x, y, dim)
Compute the dot product of two vectors.
If x and y are matrices, calculate the dot products along the first non-singleton
dimension.
If the optional argument dim is given, calculate the dot products along this dimension.
This is equivalent to sum (conj (X) .* Y, dim), but avoids forming a temporary
array and is faster. When X and Y are column vectors, the result is equivalent to X’
* Y.
See also: [cross], page 488, [divergence], page 488.

cross (x, y)
cross (x, y, dim)
Compute the vector cross product of two 3-dimensional vectors x and y.
If x and y are matrices, the cross product is applied along the first dimension with
three elements.
The optional argument dim forces the cross product to be calculated along the specified dimension.
Example Code:
cross ([1,1,0], [0,1,1])
⇒ [ 1; -1; 1 ]

See also: [dot], page 488, [curl], page 488, [divergence], page 488.

div
div
div
div

(x, y, z, fx, fy, fz)
(fx, fy, fz)
(x, y, fx, fy)
(fx, fy)
Calculate divergence of a vector field given by the arrays fx, fy, and fz or fx, fy
respectively.

=
=
=
=

divergence
divergence
divergence
divergence

divF (x, y, z) = ∂x F + ∂y F + ∂z F
The coordinates of the vector field can be given by the arguments x, y, z or x, y
respectively.
See also: [curl], page 488, [gradient], page 487, [del2], page 489, [dot], page 488.

[cx, cy, cz, v] = curl (x, y, z, fx, fy, fz)
[cz, v] = curl (x, y, fx, fy)
[...] = curl (fx, fy, fz)

Chapter 17: Arithmetic

489

[...] = curl (fx, fy)
v = curl ( . . . )
Calculate curl of vector field given by the arrays fx, fy, and fz or fx, fy respectively.
curlF (x, y, z) =



∂d
∂d
∂d
∂d
∂d
∂d
Fz −
Fy , Fx −
Fz ,
Fy −
Fx
∂y
∂z
∂z
∂x
∂x
∂y



The coordinates of the vector field can be given by the arguments x, y, z or x, y
respectively. v calculates the scalar component of the angular velocity vector in
direction of the z-axis for two-dimensional input. For three-dimensional input the
scalar rotation is calculated at each grid point in direction of the vector field at that
point.
See also: [divergence], page 488, [gradient], page 487, [del2], page 489, [cross],
page 488.

d = del2 (M)
d = del2 (M, h)
d = del2 (M, dx, dy, . . . )
Calculate the discrete Laplace operator (∇2 ).
For a 2-dimensional matrix M this is defined as
d=

1
4



d2
d2
M
(x,
y)
+
M (x, y)
dx2
dy 2



For N-dimensional arrays the sum in parentheses is expanded to include second derivatives over the additional higher dimensions.
The spacing between evaluation points may be defined by h, which is a scalar defining
the equidistant spacing in all dimensions. Alternatively, the spacing in each dimension may be defined separately by dx, dy, etc. A scalar spacing argument defines
equidistant spacing, whereas a vector argument can be used to specify variable spacing. The length of the spacing vectors must match the respective dimension of M.
The default spacing value is 1.
At least 3 data points are needed for each dimension. Boundary points are calculated
from the linear extrapolation of interior points.
See also: [gradient], page 487, [diff], page 444.

factorial (n)
Return the factorial of n where n is a real non-negative integer.
If n is a scalar, this is equivalent to prod (1:n). For vector or matrix arguments,
return the factorial of each element in the array.
For non-integers see the generalized factorial function gamma. Note that the factorial
function grows large quite quickly, and even with double precision values overflow will
occur if n > 171. For such cases consider gammaln.
See also: [prod], page 483, [gamma], page 498, [gammaln], page 499.

490

GNU Octave

pf = factor (q)
[pf, n] = factor (q)
Return the prime factorization of q.
The prime factorization is defined as prod (pf) == q where every element of pf is a
prime number. If q == 1, return 1.
With two output arguments, return the unique prime factors pf and their multiplicities. That is, prod (pf .^ n) == q.
Implementation Note: The input q must be less than flintmax (9.0072e+15) in order
to factor correctly.
See also: [gcd], page 490, [lcm], page 490, [isprime], page 64, [primes], page 491.

g = gcd (a1, a2, . . . )
[g, v1, ...] = gcd (a1, a2, . . . )
Compute the greatest common divisor of a1, a2, . . . .
If more than one argument is given then all arguments must be the same size or scalar.
In this case the greatest common divisor is calculated for each element individually.
All elements must be ordinary or Gaussian (complex) integers. Note that for Gaussian
integers, the gcd is only unique up to a phase factor (multiplication by 1, -1, i, or -i),
so an arbitrary greatest common divisor among the four possible is returned.
Optional return arguments v1, . . . , contain integer vectors such that,
g = v1 a1 + v2 a2 + · · ·

Example code:

gcd ([15, 9], [20, 18])
⇒ 5 9
See also: [lcm], page 490, [factor], page 490, [isprime], page 64.

lcm (x, y)
lcm (x, y, . . . )
Compute the least common multiple of x and y, or of the list of all arguments.
All elements must be numeric and of the same size or scalar.
See also: [factor], page 490, [gcd], page 490, [isprime], page 64.

chop (x, ndigits, base)
Truncate elements of x to a length of ndigits such that the resulting numbers are
exactly divisible by base.
If base is not specified it defaults to 10.
format long
chop (-pi, 5, 10)
⇒ -3.14200000000000
chop (-pi, 5, 5)
⇒ -3.14150000000000

rem (x, y)
Return the remainder of the division x / y.

Chapter 17: Arithmetic

491

The remainder is computed using the expression
x - y .* fix (x ./ y)
An error message is printed if the dimensions of the arguments do not agree, or if
either argument is complex.
Programming Notes: Floating point numbers within a few eps of an integer will be
rounded to an integer before computation for compatibility with matlab.
By convention,
rem (x, 0) = NaN
rem (x, 0) = 0
rem (x, y)

if x is a floating point variable
if x is an integer variable
returns a value with the signbit from x

For the opposite conventions see the mod function. In general, rem is best when computing the remainder after division of two positive numbers. For negative numbers,
or when the values are periodic, mod is a better choice.
See also: [mod], page 491.

mod (x, y)
Compute the modulo of x and y.
Conceptually this is given by
x - y .* floor (x ./ y)
and is written such that the correct modulus is returned for integer types. This
function handles negative values correctly. That is, mod (-1, 3) is 2, not -1, as
rem (-1, 3) returns.
An error results if the dimensions of the arguments do not agree, or if either of the
arguments is complex.
Programming Notes: Floating point numbers within a few eps of an integer will be
rounded to an integer before computation for compatibility with matlab.
By convention,
mod (x, 0) = x
mod (x, y)

returns a value with the signbit from y

For the opposite conventions see the rem function. In general, mod is a better choice
than rem when any of the inputs are negative numbers or when the values are periodic.
See also: [rem], page 490.

primes (n)
Return all primes up to n.
The output data class (double, single, uint32, etc.) is the same as the input class of
n. The algorithm used is the Sieve of Eratosthenes.
Notes: If you need a specific number of primes you can use the fact that the distance
from one prime to the next is, on average, proportional to the logarithm of the prime.
Integrating, one finds that there are about k primes less than k log(5k).
See also list_primes if you need a specific number n of primes.
See also: [list primes], page 492, [isprime], page 64.

492

GNU Octave

list_primes ()
list_primes (n)
List the first n primes.
If n is unspecified, the first 25 primes are listed.
See also: [primes], page 491, [isprime], page 64.

sign (x)
Compute the signum function.
This is defined as
sign(x) =


 1,


0,
−1,

x > 0;
x = 0;
x < 0.

For complex arguments, sign returns x ./ abs (x).
Note that sign (-0.0) is 0. Although IEEE 754 floating point allows zero to be
signed, 0.0 and -0.0 compare equal. If you must test whether zero is signed, use the
signbit function.
See also: [signbit], page 492.

signbit (x)
Return logical true if the value of x has its sign bit set and false otherwise.
This behavior is consistent with the other logical functions. See Section 4.6 [Logical
Values], page 60. The behavior differs from the C language function which returns
nonzero if the sign bit is set.
This is not the same as x < 0.0, because IEEE 754 floating point allows zero to
be signed. The comparison -0.0 < 0.0 is false, but signbit (-0.0) will return a
nonzero value.
See also: [sign], page 492.

17.6 Special Functions
[a, ierr] = airy (k, z, opt)
Compute Airy functions of the first and second kind, and their derivatives.
K
Function
Scale factor (if "opt" is supplied)
--- ---------------------------------------------0
Ai (Z)
exp ((2/3) * Z * sqrt (Z))
1
dAi(Z)/dZ exp ((2/3) * Z * sqrt (Z))
2
Bi (Z)
exp (-abs (real ((2/3) * Z * sqrt (Z))))
3
dBi(Z)/dZ exp (-abs (real ((2/3) * Z * sqrt (Z))))
The function call airy (z) is equivalent to airy (0, z).
The result is the same size as z.
If requested, ierr contains the following status information and is the same size as the
result.
0. Normal return.
1. Input error, return NaN.

Chapter 17: Arithmetic

493

2. Overflow, return Inf.
3. Loss of significance by argument reduction results in less than half of machine
accuracy.
4. Complete loss of significance by argument reduction, return NaN.
5. Error—no computation, algorithm termination condition not met, return NaN.

[j,
[y,
[i,
[k,
[h,

(alpha, x, opt)
(alpha, x, opt)
(alpha, x, opt)
(alpha, x, opt)
(alpha, k, x, opt)
Compute Bessel or Hankel functions of various kinds:

ierr]
ierr]
ierr]
ierr]
ierr]

=
=
=
=
=

besselj
bessely
besseli
besselk
besselh

besselj

Bessel functions of the first kind. If the argument opt is 1 or true, the
result is multiplied by exp (-abs (imag (x))).

bessely

Bessel functions of the second kind. If the argument opt is 1 or true, the
result is multiplied by exp (-abs (imag (x))).

besseli
Modified Bessel functions of the first kind. If the argument opt is 1 or
true, the result is multiplied by exp (-abs (real (x))).
besselk
Modified Bessel functions of the second kind. If the argument opt is 1 or
true, the result is multiplied by exp (x).
besselh

Compute Hankel functions of the first (k = 1) or second (k = 2) kind. If
the argument opt is 1 or true, the result is multiplied by exp (-I*x) for
k = 1 or exp (I*x) for k = 2.

If alpha is a scalar, the result is the same size as x. If x is a scalar, the result is the
same size as alpha. If alpha is a row vector and x is a column vector, the result is a
matrix with length (x) rows and length (alpha) columns. Otherwise, alpha and x
must conform and the result will be the same size.
The value of alpha must be real. The value of x may be complex.
If requested, ierr contains the following status information and is the same size as the
result.
0. Normal return.
1. Input error, return NaN.
2. Overflow, return Inf.
3. Loss of significance by argument reduction results in less than half of machine
accuracy.
4. Complete loss of significance by argument reduction, return NaN.
5. Error—no computation, algorithm termination condition not met, return NaN.

494

GNU Octave

beta (a, b)
Compute the Beta function for real inputs a and b.
The Beta function definition is
Γ(a)Γ(b)
.
Γ(a + b)

B(a, b) =

The Beta function can grow quite large and it is often more useful to work with the
logarithm of the output rather than the function directly. See [betaln], page 494, for
computing the logarithm of the Beta function in an efficient manner.
See also: [betaln], page 494, [betainc], page 494, [betaincinv], page 494.

betainc (x, a, b)
Compute the regularized incomplete Beta function.
The regularized incomplete Beta function is defined by
1
I(x, a, b) =
B(a, b)

Z

0

x

t(a−z) (1 − t)(b−1) dt.

If x has more than one component, both a and b must be scalars. If x is a scalar, a
and b must be of compatible dimensions.
See also: [betaincinv], page 494, [beta], page 494, [betaln], page 494.

betaincinv (y, a, b)
Compute the inverse of the incomplete Beta function.
The inverse is the value x such that
y == betainc (x, a, b)
See also: [betainc], page 494, [beta], page 494, [betaln], page 494.

betaln (a, b)
Compute the natural logarithm of the Beta function for real inputs a and b.
betaln is defined as
betaln(a, b) = ln(B(a, b)) ≡ ln(

Γ(a)Γ(b)
).
Γ(a + b)

and is calculated in a way to reduce the occurrence of underflow.
The Beta function can grow quite large and it is often more useful to work with the
logarithm of the output rather than the function directly.
See also: [beta], page 494, [betainc], page 494, [betaincinv], page 494, [gammaln],
page 499.

bincoeff (n, k)
Return the binomial coefficient of n and k, defined as
n
k

!

=

n(n − 1)(n − 2) · · · (n − k + 1)
k!

Chapter 17: Arithmetic

495

For example:
bincoeff (5, 2)
⇒ 10
In most cases, the nchoosek function is faster for small scalar integer arguments. It
also warns about loss of precision for big arguments.
See also: [nchoosek], page 657.

commutation_matrix (m, n)
Return the commutation matrix Km,n which is the unique mn × mn matrix such that
Km,n · vec(A) = vec(AT ) for all m × n matrices A.
If only one argument m is given, Km,m is returned.
See Magnus and Neudecker (1988), Matrix Differential Calculus with Applications in
Statistics and Econometrics.

duplication_matrix (n)
Return the duplication matrix Dn which is the unique n2 × n(n + 1)/2 matrix such
that Dn ∗ vech(A) = vec(A) for all symmetric n × n matrices A.
See Magnus and Neudecker (1988), Matrix Differential Calculus with Applications in
Statistics and Econometrics.

dawson (z)
Compute the Dawson (scaled imaginary error) function.
The Dawson function is defined as
√
√
π −z2
π −z2
e erfi(z) ≡ −i
e erf(iz)
2
2
See also: [erfc], page 496, [erf], page 496, [erfcx], page 497, [erfi], page 497, [erfinv],
page 497, [erfcinv], page 497.

[sn, cn, dn, err] = ellipj (u, m)
[sn, cn, dn, err] = ellipj (u, m, tol)
Compute the Jacobi elliptic functions sn, cn, and dn of complex argument u and real
parameter m.
If m is a scalar, the results are the same size as u. If u is a scalar, the results are the
same size as m. If u is a column vector and m is a row vector, the results are matrices
with length (u) rows and length (m) columns. Otherwise, u and m must conform
in size and the results will be the same size as the inputs.
The value of u may be complex. The value of m must be 0 ≤ m ≤ 1.
The optional input tol is currently ignored (matlab uses this to allow faster, less
accurate approximation).
If requested, err contains the following status information and is the same size as the
result.
0. Normal return.
1. Error—no computation, algorithm termination condition not met, return NaN.

496

GNU Octave

Reference: Milton Abramowitz and Irene A Stegun, Handbook of Mathematical Functions, Chapter 16 (Sections 16.4, 16.13, and 16.15), Dover, 1965.
See also: [ellipke], page 496.

k = ellipke (m)
k = ellipke (m, tol)
[k, e] = ellipke ( . . . )
Compute complete elliptic integrals of the first K(m) and second E(m) kind.
m must be a scalar or real array with -Inf ≤ m ≤ 1.

The optional input tol controls the stopping tolerance of the algorithm and defaults to
eps (class (m)). The tolerance can be increased to compute a faster, less accurate
approximation.
When called with one output only elliptic integrals of the first kind are returned.
Mathematical Note:
Elliptic integrals of the first kind are defined as
K(m) =

Z

1

0

dt

p

t2 )(1

Z

√
1 − mt2
√
dt
1 − t2

(1 −

− mt2 )

Elliptic integrals of the second kind are defined as
E(m) =

0

1

Reference: Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical
Functions, Chapter 17, Dover, 1965.
See also: [ellipj], page 495.

erf (z)
Compute the error function.
The error function is defined as
2
erf(z) = √
π

Z

z

2

e−t dt

0

See also: [erfc], page 496, [erfcx], page 497, [erfi], page 497, [dawson], page 495,
[erfinv], page 497, [erfcinv], page 497.

erfc (z)
Compute the complementary error function.
The complementary error function is defined as 1 − erf(z).
See also: [erfcinv], page 497, [erfcx], page 497, [erfi], page 497, [dawson], page 495,
[erf], page 496, [erfinv], page 497.

Chapter 17: Arithmetic

497

erfcx (z)
Compute the scaled complementary error function.
The scaled complementary error function is defined as
2

2

ez erfc(z) ≡ ez (1 − erf(z))
See also: [erfc], page 496, [erf], page 496, [erfi], page 497, [dawson], page 495, [erfinv],
page 497, [erfcinv], page 497.

erfi (z)
Compute the imaginary error function.
The imaginary error function is defined as
−ierf(iz)
See also: [erfc], page 496, [erf], page 496, [erfcx], page 497, [dawson], page 495, [erfinv],
page 497, [erfcinv], page 497.

erfinv (x)
Compute the inverse error function.
The inverse error function is defined such that
erf (y) == x
See also: [erf], page 496, [erfc], page 496, [erfcx], page 497, [erfi], page 497, [dawson],
page 495, [erfcinv], page 497.

erfcinv (x)
Compute the inverse complementary error function.
The inverse complementary error function is defined such that
erfc (y) == x
See also: [erfc], page 496, [erf], page 496, [erfcx], page 497, [erfi], page 497, [dawson],
page 495, [erfinv], page 497.

expint (x)
Compute the exponential integral:
E1 (x) =

Z

∞

x

e−t
dt
t

Note: For compatibility, this functions uses the matlab definition of the exponential
integral. Most other sources refer to this particular value as E1 (x), and the exponential integral as
Z ∞ −t
e
Ei(x) = −
dt.
t
−x
The two definitions are related, for positive real values of x, by E1 (−x) = −Ei(x)−iπ.

498

GNU Octave

gamma (z)
Compute the Gamma function.
The Gamma function is defined as
Γ(z) =

Z

∞

tz−1 e−t dt.

0

Programming Note: The gamma function can grow quite large even for small input
values. In many cases it may be preferable to use the natural logarithm of the gamma
function (gammaln) in calculations to minimize loss of precision. The final result is
then exp (result_using_gammaln).
See also: [gammainc], page 498, [gammaln], page 499, [factorial], page 489.

gammainc (x, a)
gammainc (x, a, "lower")
gammainc (x, a, "upper")
Compute the normalized incomplete gamma function.
This is defined as
Z x
1
ta−1 e−t dt
γ(x, a) =
Γ(a) 0
with the limiting value of 1 as x approaches infinity. The standard notation is P (a, x),
e.g., Abramowitz and Stegun (6.5.1).
If a is scalar, then gammainc (x, a) is returned for each element of x and vice versa.
If neither x nor a is scalar, the sizes of x and a must agree, and gammainc is applied
element-by-element.
By default the incomplete gamma function integrated from 0 to x is computed. If
"upper" is given then the complementary function integrated from x to infinity is
calculated. It should be noted that
gammainc (x, a) ≡ 1 - gammainc (x, a, "upper")
See also: [gamma], page 498, [gammaln], page 499.

l = legendre (n, x)
l = legendre (n, x, normalization)
Compute the associated Legendre function of degree n and order m = 0 . . . n.
The value n must be a real non-negative integer.
x is a vector with real-valued elements in the range [-1, 1].
The optional argument normalization may be one of "unnorm", "sch", or "norm".
The default if no normalization is given is "unnorm".
When the optional argument normalization is "unnorm", compute the associated Legendre function of degree n and order m and return all values for m = 0 . . . n. The
return value has one dimension more than x.
The associated Legendre function of degree n and order m:
Pnm (x) = (−1)m (1 − x2 )m/2

dm
Pn (x)
dxm

Chapter 17: Arithmetic

499

with Legendre polynomial of degree n:
P (x) =



dn 2
1
(x − 1)n
2n n! dxn



legendre (3, [-1.0, -0.9, -0.8]) returns the matrix:
x |
-1.0
|
-0.9
|
-0.8
-----------------------------------m=0 | -1.00000 | -0.47250 | -0.08000
m=1 | 0.00000 | -1.99420 | -1.98000
m=2 | 0.00000 | -2.56500 | -4.32000
m=3 | 0.00000 | -1.24229 | -3.24000
When the optional argument normalization is "sch", compute the Schmidt seminormalized associated Legendre function. The Schmidt semi-normalized associated
Legendre function is related to the unnormalized Legendre functions by the following:
For Legendre functions of degree n and order 0:
SPn0 (x) = Pn0 (x)
For Legendre functions of degree n and order m:
SPnm (x)

=

Pnm (x)(−1)m



2(n − m)!
(n + m)!

0.5

When the optional argument normalization is "norm", compute the fully normalized
associated Legendre function. The fully normalized associated Legendre function is
related to the unnormalized associated Legendre functions by the following:
For Legendre functions of degree n and order m
N Pnm (x)

=

Pnm (x)(−1)m



(n + 0.5)(n − m)!
(n + m)!

0.5

gammaln (x)
lgamma (x)
Return the natural logarithm of the gamma function of x.
See also: [gamma], page 498, [gammainc], page 498.

psi (z)
psi (k, z)
Compute the psi (polygamma) function.
The polygamma functions are the kth derivative of the logarithm of the gamma
function. If unspecified, k defaults to zero. A value of zero computes the digamma
function, a value of 1, the trigamma function, and so on.
The digamma function is defined:
Ψ(z) =

d(log(Γ(z)))
dx

500

GNU Octave

When computing the digamma function (when k equals zero), z can have any value
real or complex value. However, for polygamma functions (k higher than 0), z must
be real and non-negative.
See also: [gamma], page 498, [gammainc], page 498, [gammaln], page 499.

17.7 Rational Approximations
s = rat (x, tol)
[n, d] = rat (x, tol)
Find a rational approximation to x within the tolerance defined by tol using a continued fraction expansion.
For example:
rat (pi) = 3 + 1/(7 + 1/16) = 355/113
rat (e) = 3 + 1/(-4 + 1/(2 + 1/(5 + 1/(-2 + 1/(-7)))))
= 1457/536
When called with two output arguments return the numerator and denominator separately as two matrices.
See also: [rats], page 500.

rats (x, len)
Convert x into a rational approximation represented as a string.
The string can be converted back into a matrix as follows:
r = rats (hilb (4));
x = str2num (r)
The optional second argument defines the maximum length of the string representing
the elements of x. By default len is 9.
If the length of the smallest possible rational approximation exceeds len, an asterisk
(*) padded with spaces will be returned instead.
See also: [format], page 246, [rat], page 500.

17.8 Coordinate Transformations
[theta, r] = cart2pol (x, y)
[theta, r, z] = cart2pol (x, y, z)
[theta, r] = cart2pol (C)
[theta, r, z] = cart2pol (C)
P = cart2pol ( . . . )
Transform Cartesian coordinates to polar or cylindrical coordinates.
The inputs x, y (, and z) must be the same shape, or scalar. If called with a single
matrix argument then each row of C represents the Cartesian coordinate (x, y (, z)).
theta describes the angle relative to the positive x-axis.
r is the distance to the z-axis (0, 0, z).
If only a single return argument is requested then return a matrix P where each row
represents one polar/(cylindrical) coordinate (theta, phi (, z)).
See also: [pol2cart], page 501, [cart2sph], page 501, [sph2cart], page 501.

Chapter 17: Arithmetic

[x,
[x,
[x,
[x,
C =

501

y] = pol2cart (theta, r)
y, z] = pol2cart (theta, r, z)
y] = pol2cart (P)
y, z] = pol2cart (P)
pol2cart ( . . . )
Transform polar or cylindrical coordinates to Cartesian coordinates.
The inputs theta, r, (and z) must be the same shape, or scalar. If called with a single
matrix argument then each row of P represents the polar/(cylindrical) coordinate
(theta, r (, z)).
theta describes the angle relative to the positive x-axis.
r is the distance to the z-axis (0, 0, z).
If only a single return argument is requested then return a matrix C where each row
represents one Cartesian coordinate (x, y (, z)).
See also: [cart2pol], page 500, [sph2cart], page 501, [cart2sph], page 501.

[theta, phi, r] = cart2sph (x, y, z)
[theta, phi, r] = cart2sph (C)
S = cart2sph ( . . . )
Transform Cartesian coordinates to spherical coordinates.
The inputs x, y, and z must be the same shape, or scalar. If called with a single
matrix argument then each row of C represents the Cartesian coordinate (x, y, z).
theta describes the angle relative to the positive x-axis.
phi is the angle relative to the xy-plane.
r is the distance to the origin (0, 0, 0).
If only a single return argument is requested then return a matrix S where each row
represents one spherical coordinate (theta, phi, r).
See also: [sph2cart], page 501, [cart2pol], page 500, [pol2cart], page 501.

[x, y, z] = sph2cart (theta, phi, r)
[x, y, z] = sph2cart (S)
C = sph2cart ( . . . )
Transform spherical coordinates to Cartesian coordinates.
The inputs theta, phi, and r must be the same shape, or scalar. If called with a single
matrix argument then each row of S represents the spherical coordinate (theta, phi,
r).
theta describes the angle relative to the positive x-axis.
phi is the angle relative to the xy-plane.
r is the distance to the origin (0, 0, 0).
If only a single return argument is requested then return a matrix C where each row
represents one Cartesian coordinate (x, y, z).
See also: [cart2sph], page 501, [pol2cart], page 501, [cart2pol], page 500.

502

GNU Octave

17.9 Mathematical Constants
e
e
e
e
e

(n)
(n, m)
(n, m, k, . . . )
( . . . , class)
Return a scalar, matrix, or N-dimensional array whose elements are all equal to the
base of natural logarithms.
The constant e satisfies the equation log(e) = 1.
When called with no arguments, return a scalar with the value e.
When called with a single argument, return a square matrix with the dimension
specified.
When called with more than one scalar argument the first two arguments are taken as
the number of rows and columns and any further arguments specify additional matrix
dimensions.
The optional argument class specifies the return type and may be either "double" or
"single".
See also: [log], page 475, [exp], page 475, [pi], page 502, [I], page 502.

pi
pi
pi
pi
pi

(n)
(n, m)
(n, m, k, . . . )
( . . . , class)
Return a scalar, matrix, or N-dimensional array whose elements are all equal to the
ratio of the circumference of a circle to its diameter(π).
Internally, pi is computed as ‘4.0 * atan (1.0)’.
When called with no arguments, return a scalar with the value of π.
When called with a single argument, return a square matrix with the dimension
specified.
When called with more than one scalar argument the first two arguments are taken as
the number of rows and columns and any further arguments specify additional matrix
dimensions.
The optional argument class specifies the return type and may be either "double" or
"single".
See also: [e], page 502, [I], page 502.

I
I
I
I
I

(n)
(n, m)
(n, m, k, . . . )
( . . . , class)
Return a scalar, matrix, or N-dimensional
array whose elements are all equal to the
√
pure imaginary unit, defined as −1.

Chapter 17: Arithmetic

503

I, and its equivalents i, j, and J, are functions so any of the names may be reused for
other purposes (such as i for a counter variable).
When called with no arguments, return a scalar with the value i.
When called with a single argument, return a square matrix with the dimension
specified.
When called with more than one scalar argument the first two arguments are taken as
the number of rows and columns and any further arguments specify additional matrix
dimensions.
The optional argument class specifies the return type and may be either "double" or
"single".
See also: [e], page 502, [pi], page 502, [log], page 475, [exp], page 475.

Inf
Inf
Inf
Inf
Inf

(n)
(n, m)
(n, m, k, . . . )
( . . . , class)
Return a scalar, matrix or N-dimensional array whose elements are all equal to the
IEEE representation for positive infinity.
Infinity is produced when results are too large to be represented using the IEEE
floating point format for numbers. Two common examples which produce infinity are
division by zero and overflow.
[ 1/0 e^800 ]
⇒ Inf
Inf

When called with no arguments, return a scalar with the value ‘Inf’.
When called with a single argument, return a square matrix with the dimension
specified.
When called with more than one scalar argument the first two arguments are taken as
the number of rows and columns and any further arguments specify additional matrix
dimensions.
The optional argument class specifies the return type and may be either "double" or
"single".
See also: [isinf], page 444, [NaN], page 503.

NaN
NaN
NaN
NaN
NaN

(n)
(n, m)
(n, m, k, . . . )
( . . . , class)
Return a scalar, matrix, or N-dimensional array whose elements are all equal to the
IEEE symbol NaN (Not a Number).
NaN is the result of operations which do not produce a well defined numerical result.
Common operations which produce a NaN are arithmetic with infinity (∞ − ∞), zero
divided by zero (0/0), and any operation involving another NaN value (5 + NaN).

504

GNU Octave

Note that NaN always compares not equal to NaN (NaN != NaN). This behavior is
specified by the IEEE standard for floating point arithmetic. To find NaN values, use
the isnan function.
When called with no arguments, return a scalar with the value ‘NaN’.
When called with a single argument, return a square matrix with the dimension
specified.
When called with more than one scalar argument the first two arguments are taken as
the number of rows and columns and any further arguments specify additional matrix
dimensions.
The optional argument class specifies the return type and may be either "double" or
"single".
See also: [isnan], page 444, [Inf], page 503.

eps
eps
eps
eps
eps

(x)
(n, m)
(n, m, k, . . . )
( . . . , class)
Return a scalar, matrix or N-dimensional array whose elements are all eps, the machine precision.
More precisely, eps is the relative spacing between any two adjacent numbers in the
machine’s floating point system. This number is obviously system dependent. On
machines that support IEEE floating point arithmetic, eps is approximately 2.2204 ×
10−16 for double precision and 1.1921 × 10−7 for single precision.

When called with no arguments, return a scalar with the value eps (1.0).

Given a single argument x, return the distance between x and the next largest value.
When called with more than one argument the first two arguments are taken as the
number of rows and columns and any further arguments specify additional matrix
dimensions. The optional argument class specifies the return type and may be either
"double" or "single".
See also: [realmax], page 504, [realmin], page 505, [intmax], page 56, [flintmax],
page 56.

realmax
realmax
realmax
realmax
realmax

(n)
(n, m)
(n, m, k, . . . )
( . . . , class)
Return a scalar, matrix, or N-dimensional array whose elements are all equal to the
largest floating point number that is representable.
The actual value is system dependent. On machines that support IEEE floating
point arithmetic, realmax is approximately 1.7977 × 10308 for double precision and
3.4028 × 1038 for single precision.
When called with no arguments, return a scalar with the value realmax ("double").

Chapter 17: Arithmetic

505

When called with a single argument, return a square matrix with the dimension
specified.
When called with more than one scalar argument the first two arguments are taken as
the number of rows and columns and any further arguments specify additional matrix
dimensions.
The optional argument class specifies the return type and may be either "double" or
"single".
See also: [realmin], page 505, [intmax], page 56, [flintmax], page 56, [eps], page 504.

realmin
realmin
realmin
realmin
realmin

(n)
(n, m)
(n, m, k, . . . )
( . . . , class)
Return a scalar, matrix, or N-dimensional array whose elements are all equal to the
smallest normalized floating point number that is representable.
The actual value is system dependent. On machines that support IEEE floating
point arithmetic, realmin is approximately 2.2251 × 10−308 for double precision and
1.1755 × 10−38 for single precision.
When called with no arguments, return a scalar with the value realmin ("double").
When called with a single argument, return a square matrix with the dimension
specified.
When called with more than one scalar argument the first two arguments are taken as
the number of rows and columns and any further arguments specify additional matrix
dimensions.
The optional argument class specifies the return type and may be either "double" or
"single".
See also: [realmax], page 504, [intmin], page 56, [eps], page 504.

507

18 Linear Algebra
This chapter documents the linear algebra functions provided in Octave. Reference material
for many of these functions may be found in Golub and Van Loan, Matrix Computations,
2nd Ed., Johns Hopkins, 1989, and in the lapack Users’ Guide, SIAM, 1992. The lapack
Users’ Guide is available at: http://www.netlib.org/lapack/lug/
A common text for engineering courses is G. Strang, Linear Algebra and Its Applications,
4th Edition. It has become a widespread reference for linear algebra. An alternative is
P. Lax Linear Algebra and Its Applications, and also is a good choice. It claims to be
suitable for high school students with substantial mathematical interests as well as firstyear undergraduates.

18.1 Techniques Used for Linear Algebra
Octave includes a polymorphic solver that selects an appropriate matrix factorization depending on the properties of the matrix itself. Generally, the cost of determining the matrix
type is small relative to the cost of factorizing the matrix itself. In any case the matrix type
is cached once it is calculated so that it is not re-determined each time it is used in a linear
equation.
The selection tree for how the linear equation is solved or a matrix inverse is formed is
given by:
1. If the matrix is upper or lower triangular sparse use a forward or backward substitution
using the lapack xTRTRS function, and goto 4.
2. If the matrix is square, Hermitian with a real positive diagonal, attempt Cholesky factorization using the lapack xPOTRF function.
3. If the Cholesky factorization failed or the matrix is not Hermitian with a real positive
diagonal, and the matrix is square, factorize using the lapack xGETRF function.
4. If the matrix is not square, or any of the previous solvers flags a singular or near
singular matrix, find a least squares solution using the lapack xGELSD function.
The user can force the type of the matrix with the matrix_type function. This overcomes
the cost of discovering the type of the matrix. However, it should be noted that identifying
the type of the matrix incorrectly will lead to unpredictable results, and so matrix_type
should be used with care.
It should be noted that the test for whether a matrix is a candidate for Cholesky factorization, performed above, and by the matrix_type function, does not make certain that
the matrix is Hermitian. However, the attempt to factorize the matrix will quickly detect
a non-Hermitian matrix.

18.2 Basic Matrix Functions
AA = balance (A)
AA = balance (A, opt)
[DD, AA] = balance (A, opt)
[D, P, AA] = balance (A, opt)

508

GNU Octave

[CC, DD, AA, BB] = balance (A, B, opt)
Balance the matrix A to reduce numerical errors in future calculations.
Compute AA = DD \ A * DD in which AA is a matrix whose row and column norms
are roughly equal in magnitude, and DD = P * D, in which P is a permutation matrix
and D is a diagonal matrix of powers of two. This allows the equilibration to be
computed without round-off. Results of eigenvalue calculation are typically improved
by balancing first.
If two output values are requested, balance returns the diagonal D and the permutation P separately as vectors. In this case, DD = eye(n)(:,P) * diag (D), where n
is the matrix size.
If four output values are requested, compute AA = CC*A*DD and BB = CC*B*DD, in
which AA and BB have nonzero elements of approximately the same magnitude and
CC and DD are permuted diagonal matrices as in DD for the algebraic eigenvalue
problem.
The eigenvalue balancing option opt may be one of:
"noperm", "S"
Scale only; do not permute.
"noscal", "P"
Permute only; do not scale.
Algebraic eigenvalue balancing uses standard lapack routines.
Generalized eigenvalue problem balancing uses Ward’s algorithm (SIAM Journal on
Scientific and Statistical Computing, 1981).

bw = bandwidth (A, type)
[lower, upper] = bandwidth (A)
Compute the bandwidth of A.
The type argument is the string "lower" for the lower bandwidth and "upper" for the
upper bandwidth. If no type is specified return both the lower and upper bandwidth
of A.
The lower/upper bandwidth of a matrix is the number of subdiagonals/superdiagonals
with nonzero entries.
See also: [isbanded], page 64, [isdiag], page 64, [istril], page 64, [istriu], page 64.

cond (A)
cond (A, p)
Compute the p-norm condition number of a matrix with respect to inversion.
cond (A) is defined as k A kp ∗ k A−1 kp .
By default, p = 2 is used which implies a (relatively slow) singular value decomposition. Other possible selections are p = 1, Inf, "fro" which are generally faster. See
norm for a full discussion of possible p values.
The condition number of a matrix quantifies the sensitivity of the matrix inversion
operation when small changes are made to matrix elements. Ideally the condition
number will be close to 1. When the number is large this indicates small changes
(such as underflow or round-off error) will produce large changes in the resulting

Chapter 18: Linear Algebra

509

output. In such cases the solution results from numerical computing are not likely to
be accurate.
See also: [condest], page 586, [rcond], page 515, [condeig], page 509, [norm], page 513,
[svd], page 525.

c = condeig (a)
[v, lambda, c] = condeig (a)
Compute condition numbers of a matrix with respect to eigenvalues.
The condition numbers are the reciprocals of the cosines of the angles between the
left and right eigenvectors; Large values indicate that the matrix has multiple distinct
eigenvalues.
The input a must be a square numeric matrix.
The outputs are:
• c is a vector of condition numbers for the eigenvalues of a.
• v is the matrix of right eigenvectors of a. The result is equivalent to calling [v,
lambda] = eig (a).
• lambda is the diagonal matrix of eigenvalues of a. The result is equivalent to
calling [v, lambda] = eig (a).
Example
a = [1, 2; 3, 4];
c = condeig (a)
⇒ [1.0150; 1.0150]

See also: [eig], page 509, [cond], page 508, [balance], page 507.

det (A)
[d, rcond] = det (A)
Compute the determinant of A.
Return an estimate of the reciprocal condition number if requested.
Programming Notes: Routines from lapack are used for full matrices and code from
umfpack is used for sparse matrices.
The determinant should not be used to check a matrix for singularity. For that, use
any of the condition number functions: cond, condest, rcond.
See also: [cond], page 508, [condest], page 586, [rcond], page 515.

lambda = eig (A)
lambda = eig (A, B)
[V, lambda] = eig (A)
[V, lambda] = eig (A, B)
[V, lambda, W] = eig (A)
[V, lambda, W] = eig (A, B)
[...] = eig (A, balanceOption)
[...] = eig (A, B, algorithm)
[...] = eig ( . . . , eigvalOption)
Compute the eigenvalues (lambda) and optionally the right eigenvectors (V ) and the
left eigenvectors (W ) of a matrix or a pair of matrices.

510

GNU Octave

The flag balanceOption can be one of:
"balance"
Preliminary balancing is on. (default)
"nobalance"
Disables preliminary balancing.
The flag eigvalOption can be one of:
"matrix"

Return the eigenvalues in a diagonal matrix. (default if 2 or 3 outputs
are specified)

"vector"

Return the eigenvalues in a column vector. (default if 1 output is specified, e.g. lambda = eig (A))

The flag algorithm can be one of:
"chol"

Uses the Cholesky factorization of B. (default if A is symmetric (Hermitian) and B is symmetric (Hermitian) positive definite)

"qz"

Uses the QZ algorithm. (When A or B are not symmetric always the QZ
algorithm will be used)

both are symmetric
at least one is not symmetric

no flag
"chol"
"qz"

chol
"chol"
"qz"

qz
"qz"
"qz"

The eigenvalues returned by eig are not ordered.
See also: [eigs], page 590, [svd], page 525.

G = givens (x, y)
[c, s] = givens (x, y)
Compute the Givens rotation matrix G.
The Givens matrix is a 2 × 2 orthogonal matrix

such that



c
G=
−s0

s
c



 

 

x
∗
G
=
y
0

with x and y scalars.
If two output arguments are requested, return the factors c and s rather than the
Givens rotation matrix.
For example:
givens (1, 1)
⇒
0.70711
-0.70711

0.70711
0.70711

See also: [planerot], page 511.

Chapter 18: Linear Algebra

511

[G, y] = planerot (x)
Given a two-element column vector, return the 2 × 2 orthogonal matrix G such that
y = g * x and y(2) = 0.
See also: [givens], page 510.

x = inv (A)
[x, rcond] = inv (A)
Compute the inverse of the square matrix A.
Return an estimate of the reciprocal condition number if requested, otherwise warn
of an ill-conditioned matrix if the reciprocal condition number is small.
In general it is best to avoid calculating the inverse of a matrix directly. For example,
it is both faster and more accurate to solve systems of equations (A*x = b) with y =
A \ b, rather than y = inv (A) * b.
If called with a sparse matrix, then in general x will be a full matrix requiring significantly more storage. Avoid forming the inverse of a sparse matrix if possible.
See also: [ldivide], page 146, [rdivide], page 147, [pinv], page 514.

x = linsolve (A, b)
x = linsolve (A, b, opts)
[x, R] = linsolve ( . . . )
Solve the linear system A*x = b.
With no options, this function is equivalent to the left division operator (x = A \ b)
or the matrix-left-divide function (x = mldivide (A, b)).
Octave ordinarily examines the properties of the matrix A and chooses a solver that
best matches the matrix. By passing a structure opts to linsolve you can inform Octave directly about the matrix A. In this case Octave will skip the matrix examination
and proceed directly to solving the linear system.
Warning: If the matrix A does not have the properties listed in the opts structure
then the result will not be accurate AND no warning will be given. When in doubt,
let Octave examine the matrix and choose the appropriate solver as this step takes
little time and the result is cached so that it is only done once per linear system.
Possible opts fields (set value to true/false):
LT

A is lower triangular

UT

A is upper triangular

UHESS

A is upper Hessenberg (currently makes no difference)

SYM

A is symmetric or complex Hermitian (currently makes no difference)

POSDEF

A is positive definite

RECT

A is general rectangular (currently makes no difference)

TRANSA

Solve A’*x = b by transpose (A) \ b

The optional second output R is the inverse condition number of A (zero if matrix is
singular).
See also: [mldivide], page 146, [matrix type], page 512, [rcond], page 515.

512

GNU Octave

(A)
(A, "nocompute")
type)
"upper", perm)
"lower", perm)
"banded", nl, nu)
Identify the matrix type or mark a matrix as a particular type.
This allows more rapid solutions of linear equations involving A to be performed.
Called with a single argument, matrix_type returns the type of the matrix and caches
it for future use.
Called with more than one argument, matrix_type allows the type of the matrix to
be defined.
If the option "nocompute" is given, the function will not attempt to guess the type
if it is still unknown. This is useful for debugging purposes.
The possible matrix types depend on whether the matrix is full or sparse, and can be
one of the following

type = matrix_type
type = matrix_type
A = matrix_type (A,
A = matrix_type (A,
A = matrix_type (A,
A = matrix_type (A,

"unknown"
Remove any previously cached matrix type, and mark type as unknown.
"full"

Mark the matrix as full.

"positive definite"
Probable full positive definite matrix.
"diagonal"
Diagonal matrix. (Sparse matrices only)
"permuted diagonal"
Permuted Diagonal matrix. The permutation does not need to be specifically indicated, as the structure of the matrix explicitly gives this. (Sparse
matrices only)
"upper"

Upper triangular. If the optional third argument perm is given, the matrix is assumed to be a permuted upper triangular with the permutations
defined by the vector perm.

"lower"

Lower triangular. If the optional third argument perm is given, the matrix
is assumed to be a permuted lower triangular with the permutations
defined by the vector perm.

"banded"
"banded positive definite"
Banded matrix with the band size of nl below the diagonal and nu above
it. If nl and nu are 1, then the matrix is tridiagonal and treated with
specialized code. In addition the matrix can be marked as probably a
positive definite. (Sparse matrices only)
"singular"
The matrix is assumed to be singular and will be treated with a minimum
norm solution.

Chapter 18: Linear Algebra

513

Note that the matrix type will be discovered automatically on the first attempt to
solve a linear equation involving A. Therefore matrix_type is only useful to give
Octave hints of the matrix type. Incorrectly defining the matrix type will result in
incorrect results from solutions of linear equations; it is entirely the responsibility of
the user to correctly identify the matrix type.
Also, the test for positive definiteness is a low-cost test for a Hermitian matrix with
a real positive diagonal. This does not guarantee that the matrix is positive definite, but only that it is a probable candidate. When such a matrix is factorized, a
Cholesky factorization is first attempted, and if that fails the matrix is then treated
with an LU factorization. Once the matrix has been factorized, matrix_type will
return the correct classification of the matrix.

norm (A)
norm (A, p)
norm (A, p, opt)
Compute the p-norm of the matrix A.
If the second argument is not given, p = 2 is used.
If A is a matrix (or sparse matrix):
p=1

1-norm, the largest column sum of the absolute values of A.

p=2

Largest singular value of A.

p = Inf or "inf"
Infinity norm, the largest row sum of the absolute values of A.
p = "fro"
Frobenius norm of A, sqrt (sum (diag (A’ * A))).
other p, p > 1
maximum norm (A*x, p) such that norm (x, p) == 1
If A is a vector or a scalar:
p = Inf or "inf"
max (abs (A)).
p = -Inf

min (abs (A)).

p = "fro"
Frobenius norm of A, sqrt (sumsq (abs (A))).
p=0

Hamming norm—the number of nonzero elements.

other p, p > 1
p-norm of A, (sum (abs (A) .^ p)) ^ (1/p).
other p p < 1
the p-pseudonorm defined as above.
If opt is the value "rows", treat each row as a vector and compute its norm. The
result is returned as a column vector. Similarly, if opt is "columns" or "cols" then
compute the norms of each column and return a row vector.
See also: [normest], page 585, [normest1], page 585, [cond], page 508, [svd], page 525.

514

GNU Octave

null (A)
null (A, tol)
Return an orthonormal basis of the null space of A.
The dimension of the null space is taken as the number of singular values of A not
greater than tol. If the argument tol is missing, it is computed as
max (size (A)) * max (svd (A)) * eps
See also: [orth], page 514.

orth (A)
orth (A, tol)
Return an orthonormal basis of the range space of A.
The dimension of the range space is taken as the number of singular values of A
greater than tol. If the argument tol is missing, it is computed as
max (size (A)) * max (svd (A)) * eps
See also: [null], page 514.

[y, h] = mgorth (x, v)
Orthogonalize a given column vector x with respect to a set of orthonormal vectors
comprising the columns of v using the modified Gram-Schmidt method.
On exit, y is a unit vector such that:
norm (y) = 1
v’ * y = 0
x = [v, y]*h’

pinv (x)
pinv (x, tol)
Return the Moore-Penrose pseudoinverse of x.
Singular values less than tol are ignored.
If the second argument is omitted, it is taken to be
tol = max ([rows(x), columns(x)]) * norm (x) * eps
See also: [inv], page 511, [ldivide], page 146.

rank (A)
rank (A, tol)
Compute the rank of matrix A, using the singular value decomposition.
The rank is taken to be the number of singular values of A that are greater than the
specified tolerance tol. If the second argument is omitted, it is taken to be
tol = max (size (A)) * sigma(1) * eps;
where eps is machine precision and sigma(1) is the largest singular value of A.
The rank of a matrix is the number of linearly independent rows or columns and
determines how many particular solutions exist to a system of equations. Use null
for finding the remaining homogenous solutions.

Chapter 18: Linear Algebra

515

Example:
x = [1 2 3
4 5 6
7 8 9];
rank (x)
⇒ 2

The number of linearly independent rows is only 2 because the final row is a linear
combination of -1*row1 + 2*row2.
See also: [null], page 514, [sprank], page 588, [svd], page 525.

c = rcond (A)
Compute the 1-norm estimate of the reciprocal condition number as returned by
lapack.
If the matrix is well-conditioned then c will be near 1 and if the matrix is poorly
conditioned it will be close to 0.
The matrix A must not be sparse. If the matrix is sparse then condest (A) or rcond
(full (A)) should be used instead.
See also: [cond], page 508, [condest], page 586.

trace (A)
Compute the trace of A, the sum of the elements along the main diagonal.
The implementation is straightforward: sum (diag (A)).
See also: [eig], page 509.

rref (A)
rref (A, tol)
[r, k] = rref ( . . . )
Return the reduced row echelon form of A.
tol defaults to eps * max (size (A)) * norm (A, inf).
The optional return argument k contains the vector of "bound variables", which are
those columns on which elimination has been performed.

18.3 Matrix Factorizations
R =
[R,
[R,
[R,
[L,
[R,

chol (A)
p] = chol (A)
p, Q] = chol (A)
p, Q] = chol (A, "vector")
...] = chol ( . . . , "lower")
...] = chol ( . . . , "upper")
Compute the upper Cholesky factor, R, of the real symmetric or complex Hermitian
positive definite matrix A.
The upper Cholesky factor R is computed by using the upper triangular part of matrix
A and is defined by RT R = A.

516

GNU Octave

Calling chol using the optional "upper" flag has the same behavior. In contrast, using
the optional "lower" flag, chol returns the lower triangular factorization, computed
by using the lower triangular part of matrix A, such that LLT = A.
Called with one output argument chol fails if matrix A is not positive definite. Note
that if matrix A is not real symmetric or complex Hermitian then the lower triangular
part is considered to be the (complex conjugate) transpose of the upper triangular
part, or vice versa, given the "lower" flag.
Called with two or more output arguments p flags whether the matrix A was positive
definite and chol does not fail. A zero value of p indicates that matrix A is positive
definite and R gives the factorization. Otherwise, p will have a positive value.
If called with three output arguments matrix A must be sparse and a sparsity preserving row/column permutation is applied to matrix A prior to the factorization.
That is R is the factorization of A(Q,Q) such that RT R = QT AQ.
The sparsity preserving permutation is generally returned as a matrix. However, given
the optional flag "vector", Q will be returned as a vector such that RT R = A(Q, Q).
In general the lower triangular factorization is significantly faster for sparse matrices.
See also: [hess], page 517, [lu], page 518, [qr], page 519, [qz], page 522, [schur],
page 523, [svd], page 525, [ichol], page 597, [cholinv], page 516, [chol2inv], page 516,
[cholupdate], page 516, [cholinsert], page 517, [choldelete], page 517, [cholshift],
page 517.

cholinv (A)
Compute the inverse of the symmetric positive definite matrix A using the
Cholesky factorization.
See also: [chol], page 515, [chol2inv], page 516, [inv], page 511.

chol2inv (U)
Invert a symmetric, positive definite square matrix from its Cholesky decomposition,
U.
Note that U should be an upper-triangular matrix with positive diagonal elements.
chol2inv (U) provides inv (U’*U) but it is much faster than using inv.
See also: [chol], page 515, [cholinv], page 516, [inv], page 511.

[R1, info] = cholupdate (R, u, op)
Update or downdate a Cholesky factorization.
Given an upper triangular matrix R and a column vector u, attempt to determine
another upper triangular matrix R1 such that
• R1’*R1 = R’*R + u*u’ if op is "+"
• R1’*R1 = R’*R - u*u’ if op is "-"
If op is "-", info is set to
• 0 if the downdate was successful,
• 1 if R’*R - u*u’ is not positive definite,
• 2 if R is singular.

Chapter 18: Linear Algebra

517

If info is not present, an error message is printed in cases 1 and 2.
See also: [chol], page 515, [cholinsert], page 517, [choldelete], page 517, [cholshift],
page 517.

R1 = cholinsert (R, j, u)
[R1, info] = cholinsert (R, j, u)
Given a Cholesky factorization of a real symmetric or complex Hermitian positive
definite matrix A = R’*R, R upper triangular, return the Cholesky factorization of
A1, where A1(p,p) = A, A1(:,j) = A1(j,:)’ = u and p = [1:j-1,j+1:n+1]. u(j) should
be positive.
On return, info is set to
• 0 if the insertion was successful,
• 1 if A1 is not positive definite,
• 2 if R is singular.
If info is not present, an error message is printed in cases 1 and 2.
See also: [chol], page 515, [cholupdate], page 516, [choldelete], page 517, [cholshift],
page 517.

R1 = choldelete (R, j)
Given a Cholesky factorization of a real symmetric or complex Hermitian positive
definite matrix A = R’*R, R upper triangular, return the Cholesky factorization of
A(p,p), where p = [1:j-1,j+1:n+1].
See also: [chol], page 515, [cholupdate], page 516, [cholinsert], page 517, [cholshift],
page 517.

R1 = cholshift (R, i, j)
Given a Cholesky factorization of a real symmetric or complex Hermitian positive
definite matrix A = R’*R, R upper triangular, return the Cholesky factorization of
A(p,p), where p is the permutation
p = [1:i-1, shift(i:j, 1), j+1:n] if i < j
or
p = [1:j-1, shift(j:i,-1), i+1:n] if j < i.
See also: [chol], page 515, [cholupdate], page 516, [cholinsert], page 517, [choldelete],
page 517.

H = hess (A)
[P, H] = hess (A)
Compute the Hessenberg decomposition of the matrix A.
The Hessenberg decomposition is
A = P HP T
where P is a square unitary matrix (P T P = I), and H is upper Hessenberg (Hi,j =
0, ∀i > j + 1).

518

GNU Octave

The Hessenberg decomposition is usually used as the first step in an eigenvalue computation, but has other applications as well (see Golub, Nash, and Van Loan, IEEE
Transactions on Automatic Control, 1979).
See also: [eig], page 509, [chol], page 515, [lu], page 518, [qr], page 519, [qz], page 522,
[schur], page 523, [svd], page 525.

[L, U] = lu (A)
[L, U, P] = lu (A)
[L, U, P, Q] = lu (S)
[L, U, P, Q, R] = lu (S)
[...] = lu (S, thres)
y = lu ( . . . )
[...] = lu ( . . . , "vector")
Compute the LU decomposition of A.
If A is full then subroutines from lapack are used, and if A is sparse then umfpack
is used.
The result is returned in a permuted form, according to the optional return value P.
For example, given the matrix a = [1, 2; 3, 4],
[l, u, p] = lu (a)
returns
l =
1.00000
0.33333

0.00000
1.00000

u =
3.00000
0.00000

4.00000
0.66667

p =
0 1
1 0
The matrix is not required to be square.
When called with two or three output arguments and a sparse input matrix, lu does
not attempt to perform sparsity preserving column permutations. Called with a fourth
output argument, the sparsity preserving column transformation Q is returned, such
that P * A * Q = L * U.
Called with a fifth output argument and a sparse input matrix, lu attempts to use a
scaling factor R on the input matrix such that P * (R \ A) * Q = L * U. This typically
leads to a sparser and more stable factorization.
An additional input argument thres, that defines the pivoting threshold can be given.
thres can be a scalar, in which case it defines the umfpack pivoting tolerance for
both symmetric and unsymmetric cases. If thres is a 2-element vector, then the

Chapter 18: Linear Algebra

519

first element defines the pivoting tolerance for the unsymmetric umfpack pivoting
strategy and the second for the symmetric strategy. By default, the values defined by
spparms are used ([0.1, 0.001]).
Given the string argument "vector", lu returns the values of P and Q as vector
values, such that for full matrix, A(P,:) = L * U, and R(P,:) * A(:,Q) = L * U.
With two output arguments, returns the permuted forms of the upper and lower
triangular matrices, such that A = L * U. With one output argument y, then the
matrix returned by the lapack routines is returned. If the input matrix is sparse
then the matrix L is embedded into U to give a return value similar to the full case.
For both full and sparse matrices, lu loses the permutation information.
See also: [luupdate], page 519, [ilu], page 599, [chol], page 515, [hess], page 517, [qr],
page 519, [qz], page 522, [schur], page 523, [svd], page 525.

[L, U] = luupdate (L, U, x, y)
[L, U, P] = luupdate (L, U, P, x, y)
Given an LU factorization of a real or complex matrix A = L*U , L lower unit trapezoidal and U upper trapezoidal, return the LU factorization of A + x*y.’, where x
and y are column vectors (rank-1 update) or matrices with equal number of columns
(rank-k update).
Optionally, row-pivoted updating can be used by supplying a row permutation (pivoting) matrix P; in that case, an updated permutation matrix is returned. Note that
if L, U, P is a pivoted LU factorization as obtained by lu:
[L, U, P] = lu (A);
then a factorization of A+x*y.’ can be obtained either as
[L1, U1] = lu (L, U, P*x, y)
or
[L1, U1, P1] = lu (L, U, P, x, y)
The first form uses the unpivoted algorithm, which is faster, but less stable. The
second form uses a slower pivoted algorithm, which is more stable.
The matrix case is done as a sequence of rank-1 updates; thus, for large enough k, it
will be both faster and more accurate to recompute the factorization from scratch.
See also: [lu], page 518, [cholupdate], page 516, [qrupdate], page 521.

[Q, R] = qr (A)
[Q, R, P] = qr (A) # non-sparse A
X = qr (A)
R = qr (A) # sparse A
[C, R] = qr (A, B)
[...] = qr ( . . . , 0)
[...] = qr ( . . . , ’vector’)
[...] = qr ( . . . , ’matrix’)
Compute the QR factorization of A, using standard lapack subroutines. The QR factorization is QR = A where Q is an orthogonal matrix and R is upper triangular.
For example, given the matrix A = [1, 2; 3, 4],
[Q, R] = qr (A)

520

GNU Octave

returns
Q =
-0.31623
-0.94868

-0.94868
0.31623

R =
-3.16228
0.00000

-4.42719
-0.63246

The qr factorization has applications in the solution of least squares problems
min kAx − bk2
x

for overdetermined systems of equations (i.e., A is a tall, thin matrix).
If only a single return value is requested, then it is either R if A is sparse, or X such
that R = triu (X) if A is full. (Note: Unlike most commands, the single return value
is not the first return value when multiple are requested.)
If the matrix A is full, the permuted QR factorization [Q, R, P] = qr (A) forms the
QR factorization such that the diagonal entries of R are decreasing in magnitude
order. For example, given the matrix a = [1, 2; 3, 4],
[Q, R, P] = qr (A)
returns
Q =
-0.44721
-0.89443

-0.89443
0.44721

R =
-4.47214
0.00000

-3.13050
0.44721

P =
0
1

1
0

The permuted qr factorization [Q, R, P] = qr (A) factorization allows the construction of an orthogonal basis of span (A).
If the matrix A is sparse, then the sparse QR factorization of A is computed using
CSparse. As the matrix Q is in general a full matrix, it is recommended to request
only one return value, which is the Q-less factorization R of A, such that R = chol
(A’ * A).

Chapter 18: Linear Algebra

521

If an additional matrix B is supplied and two return values are requested, then qr
returns C, where C = Q’ * B. This allows the least squares approximation of A \ B to
be calculated as
[C, R] = qr (A, B)
x = R \ C
If the final argument is the scalar 0 and the number of rows is larger than the number
of columns, then an "economy" factorization is returned, omitting zeroes of R and
the corresponding columns of Q. That is, R will have only size (A,1) rows. In this
case, P is a vector rather than a matrix.
If the final argument is the string "vector" then P is a permutation vector instead
of a permutation matrix.
See also: [chol], page 515, [hess], page 517, [lu], page 518, [qz], page 522, [schur],
page 523, [svd], page 525, [qrupdate], page 521, [qrinsert], page 521, [qrdelete],
page 521, [qrshift], page 522.

[Q1, R1] = qrupdate (Q, R, u, v)
Given a QR factorization of a real or complex matrix A = Q*R, Q unitary and
R upper trapezoidal, return the QR factorization of A + u*v’, where u and v are
column vectors (rank-1 update) or matrices with equal number of columns (rank-k
update). Notice that the latter case is done as a sequence of rank-1 updates; thus, for
k large enough, it will be both faster and more accurate to recompute the factorization
from scratch.
The QR factorization supplied may be either full (Q is square) or economized (R is
square).
See also: [qr], page 519, [qrinsert], page 521, [qrdelete], page 521, [qrshift], page 522.

[Q1, R1] = qrinsert (Q, R, j, x, orient)
Given a QR factorization of a real or complex matrix A = Q*R, Q unitary and
R upper trapezoidal, return the QR factorization of [A(:,1:j-1) x A(:,j:n)], where u is
a column vector to be inserted into A (if orient is "col"), or the QR factorization
of [A(1:j-1,:);x;A(:,j:n)], where x is a row vector to be inserted into A (if orient is
"row").
The default value of orient is "col". If orient is "col", u may be a matrix and
j an index vector resulting in the QR factorization of a matrix B such that B(:,j)
gives u and B(:,j) = [] gives A. Notice that the latter case is done as a sequence of
k insertions; thus, for k large enough, it will be both faster and more accurate to
recompute the factorization from scratch.
If orient is "col", the QR factorization supplied may be either full (Q is square) or
economized (R is square).
If orient is "row", full factorization is needed.
See also: [qr], page 519, [qrupdate], page 521, [qrdelete], page 521, [qrshift], page 522.

[Q1, R1] = qrdelete (Q, R, j, orient)
Given a QR factorization of a real or complex matrix A = Q*R, Q unitary and
R upper trapezoidal, return the QR factorization of [A(:,1:j-1), U, A(:,j:n)], where u

522

GNU Octave

is a column vector to be inserted into A (if orient is "col"), or the QR factorization
of [A(1:j-1,:);X;A(:,j:n)], where x is a row orient is "row"). The default value of orient
is "col".
If orient is "col", j may be an index vector resulting in the QR factorization of
a matrix B such that A(:,j) = [] gives B. Notice that the latter case is done as a
sequence of k deletions; thus, for k large enough, it will be both faster and more
accurate to recompute the factorization from scratch.
If orient is "col", the QR factorization supplied may be either full (Q is square) or
economized (R is square).
If orient is "row", full factorization is needed.
See also: [qr], page 519, [qrupdate], page 521, [qrinsert], page 521, [qrshift], page 522.

[Q1, R1] = qrshift (Q, R, i, j)
Given a QR factorization of a real or complex matrix A = Q*R, Q unitary and
R upper trapezoidal, return the QR factorization of A(:,p), where p is the permutation
p = [1:i-1, shift(i:j, 1), j+1:n] if i < j
or
p = [1:j-1, shift(j:i,-1), i+1:n] if j < i.
See also: [qr], page 519, [qrupdate], page 521, [qrinsert], page 521, [qrdelete],
page 521.

lambda =
[AA, BB,
[AA, BB,
[AA, BB,

qz
Q,
Z]
Z,

(A, B)

Z, V, W, lambda] = qz (A, B)
= qz (A, B, opt)
lambda] = qz (A, B, opt)

QZ decomposition of the generalized eigenvalue problem
Ax = λBx
There are three calling forms of the function:
1. lambda = qz (A, B)
Compute the generalized eigenvalues λ.
2. [AA, BB, Q, Z, V, W, lambda] = qz (A, B)
Compute QZ decomposition, generalized eigenvectors, and generalized eigenvalues.
AV = BV diag(λ)
W T A = diag(λ)W T B
AA = QT AZ, BB = QT BZ
with Q and Z orthogonal (unitary for complex case).
3. [AA, BB, Z {, lambda}] = qz (A, B, opt)
As in form 2 above, but allows ordering of generalized eigenpairs for, e.g., solution of discrete time algebraic Riccati equations. Form 3 is not available for

Chapter 18: Linear Algebra

523

complex matrices, and does not compute the generalized eigenvectors V, W, nor
the orthogonal matrix Q.
opt

for ordering eigenvalues of the GEP pencil. The leading block of the
revised pencil contains all eigenvalues that satisfy:
"N"

unordered (default)

"S"

small: leading block has all |λ| < 1

"B"

big: leading block has all |λ| ≥ 1

"-"

negative real part: leading block has all eigenvalues in
the open left half-plane

"+"

non-negative real part: leading block has all eigenvalues
in the closed right half-plane

Note: qz performs permutation balancing, but not scaling (see [balance], page 507),
which may be lead to less accurate results than eig. The order of output arguments
was selected for compatibility with matlab.
See also: [eig], page 509, [balance], page 507, [lu], page 518, [chol], page 515, [hess],
page 517, [qr], page 519, [qzhess], page 523, [schur], page 523, [svd], page 525.

[aa, bb, q, z] = qzhess (A, B)
Compute the Hessenberg-triangular decomposition of the matrix pencil (A, B), returning aa = q * A * z, bb = q * B * z, with q and z orthogonal.
For example:
[aa, bb, q, z] = qzhess ([1, 2; 3, 4], [5, 6; 7, 8])
⇒ aa = [ -3.02244, -4.41741; 0.92998, 0.69749 ]
⇒ bb = [ -8.60233, -9.99730; 0.00000, -0.23250 ]
⇒ q = [ -0.58124, -0.81373; -0.81373, 0.58124 ]
⇒ z = [ 1, 0; 0, 1 ]
The Hessenberg-triangular decomposition is the first step in Moler and Stewart’s
QZ decomposition algorithm.
Algorithm taken from Golub and Van Loan, Matrix Computations, 2nd edition.
See also: [lu], page 518, [chol], page 515, [hess], page 517, [qr], page 519, [qz], page 522,
[schur], page 523, [svd], page 525.

S =
S =
S =
S =
[U,

schur (A)
schur (A, "real")
schur (A, "complex")
schur (A, opt)
S] = schur ( . . . )
Compute the Schur decomposition of A.
The Schur decomposition is defined as
S = U T AU
where U is a unitary matrix (U T U is identity) and S is upper triangular. The eigenvalues of A (and S) are the diagonal elements of S. If the matrix A is real, then the

524

GNU Octave

real Schur decomposition is computed, in which the matrix U is orthogonal and S
is block upper triangular with blocks of size at most 2 × 2 along the diagonal. The
diagonal elements of S (or the eigenvalues of the 2 × 2 blocks, when appropriate) are
the eigenvalues of A and S.
The default for real matrices is a real Schur decomposition. A complex decomposition
may be forced by passing the flag "complex".
The eigenvalues are optionally ordered along the diagonal according to the value of
opt. opt = "a" indicates that all eigenvalues with negative real parts should be moved
to the leading block of S (used in are), opt = "d" indicates that all eigenvalues with
magnitude less than one should be moved to the leading block of S (used in dare),
and opt = "u", the default, indicates that no ordering of eigenvalues should occur.
The leading k columns of U always span the A-invariant subspace corresponding to
the k leading eigenvalues of S.
The Schur decomposition is used to compute eigenvalues of a square matrix, and has
applications in the solution of algebraic Riccati equations in control (see are and
dare).
See also: [rsf2csf], page 524, [ordschur], page 524, [lu], page 518, [chol], page 515,
[hess], page 517, [qr], page 519, [qz], page 522, [svd], page 525.

[U, T] = rsf2csf (UR, TR)
Convert a real, upper quasi-triangular Schur form TR to a complex, upper triangular
Schur form T.
Note that the following relations hold:
U R · T R · U RT = U T U † and U † U is the identity matrix I.
Note also that U and T are not unique.
See also: [schur], page 523.

[UR, SR] = ordschur (U, S, select)
Reorders the real Schur factorization (U,S) obtained with the schur function, so that
selected eigenvalues appear in the upper left diagonal blocks of the quasi triangular
Schur matrix.
The logical vector select specifies the selected eigenvalues as they appear along S’s
diagonal.
For example, given the matrix A = [1, 2; 3, 4], and its Schur decomposition
[U, S] = schur (A)
which returns
U =
-0.82456
0.56577

-0.56577
-0.82456

S =
-0.37228
0.00000

-1.00000
5.37228

Chapter 18: Linear Algebra

525

It is possible to reorder the decomposition so that the positive eigenvalue is in the
upper left corner, by doing:
[U, S] = ordschur (U, S, [0,1])
See also: [schur], page 523.

angle = subspace (A, B)
Determine the largest principal angle between two subspaces spanned by the columns
of matrices A and B.

s = svd (A)
[U, S, V] = svd (A)
[U, S, V] = svd (A, econ)
Compute the singular value decomposition of A
A = U SV †
The function svd normally returns only the vector of singular values. When called
with three return values, it computes U , S, and V . For example,
svd (hilb (3))
returns
ans =
1.4083189
0.1223271
0.0026873
and
[u, s, v] = svd (hilb (3))
returns
u =
-0.82704
-0.45986
-0.32330

0.54745
-0.52829
-0.64901

0.12766
-0.71375
0.68867

s =
1.40832
0.00000
0.00000

0.00000
0.12233
0.00000

0.00000
0.00000
0.00269

v =
-0.82704
-0.45986
-0.32330

0.54745
-0.52829
-0.64901

0.12766
-0.71375
0.68867

526

GNU Octave

If given a second argument, svd returns an economy-sized decomposition, eliminating
the unnecessary rows or columns of U or V.
See also: [svd driver], page 526, [svds], page 592, [eig], page 509, [lu], page 518, [chol],
page 515, [hess], page 517, [qr], page 519, [qz], page 522.

val = svd_driver ()
old_val = svd_driver (new_val)
svd_driver (new_val, "local")
Query or set the underlying lapack driver used by svd.
Currently recognized values are "gesvd" and "gesdd". The default is "gesvd".
When called from inside a function with the "local" option, the variable is changed
locally for the function and any subroutines it calls. The original variable value is
restored when exiting the function.
See also: [svd], page 525.

[housv, beta, zer] = housh (x, j, z)
Compute Householder reflection vector housv to reflect x to be the j-th column of
identity, i.e.,
(I - beta*housv*housv’)x = norm (x)*e(j) if x(j) < 0,
(I - beta*housv*housv’)x = -norm (x)*e(j) if x(j) >= 0
Inputs
x

vector

j

index into vector

z

threshold for zero (usually should be the number 0)

Outputs (see Golub and Van Loan):
beta

If beta = 0, then no reflection need be applied (zer set to 0)

housv

householder vector

[u, h, nu] = krylov (A, V, k, eps1, pflg)
Construct an orthogonal basis u of block Krylov subspace
[v a*v a^2*v ... a^(k+1)*v]
using Householder reflections to guard against loss of orthogonality.
If V is a vector, then h contains the Hessenberg matrix such that a*u == u*h+rk*ek’,
in which rk = a*u(:,k)-u*h(:,k), and ek’ is the vector [0, 0, ..., 1] of length
k. Otherwise, h is meaningless.
If V is a vector and k is greater than length (A) - 1, then h contains the Hessenberg
matrix such that a*u == u*h.
The value of nu is the dimension of the span of the Krylov subspace (based on eps1).
If b is a vector and k is greater than m-1, then h contains the Hessenberg decomposition of A.
The optional parameter eps1 is the threshold for zero. The default value is 1e-12.

Chapter 18: Linear Algebra

527

If the optional parameter pflg is nonzero, row pivoting is used to improve numerical
behavior. The default value is 0.
Reference: A. Hodel, P. Misra, Partial Pivoting in the Computation of Krylov Subspaces of Large Sparse Systems, Proceedings of the 42nd IEEE Conference on Decision
and Control, December 2003.

18.4 Functions of a Matrix
expm (A)
Return the exponential of a matrix.
The matrix exponential is defined as the infinite Taylor series
exp(A) = I + A +

A2 A3
+
+ ···
2!
3!

However, the Taylor series is not the way to compute the matrix exponential; see
Moler and Van Loan, Nineteen Dubious Ways to Compute the Exponential of a
Matrix, SIAM Review, 1978. This routine uses Ward’s diagonal Padé approximation
method with three step preconditioning (SIAM Journal on Numerical Analysis, 1977).
Diagonal Padé approximations are rational polynomials of matrices Dq (A)−1 Nq (A)
whose Taylor series matches the first 2q + 1 terms of the Taylor series above; direct
evaluation of the Taylor series (with the same preconditioning steps) may be desirable
in lieu of the Padé approximation when Dq (A) is ill-conditioned.
See also: [logm], page 527, [sqrtm], page 527.

s = logm (A)
s = logm (A, opt_iters)
[s, iters] = logm ( . . . )
Compute the matrix logarithm of the square matrix A.
The implementation utilizes a Padé approximant and the identity
logm (A) = 2^k * logm (A^(1 / 2^k))
The optional input opt iters is the maximum number of square roots to compute and
defaults to 100.
The optional output iters is the number of square roots actually computed.
See also: [expm], page 527, [sqrtm], page 527.

s = sqrtm (A)
[s, error_estimate] = sqrtm (A)
Compute the matrix square root of the square matrix A.
Ref: N.J. Higham. A New sqrtm for matlab. Numerical Analysis Report No. 336,
Manchester Centre for Computational Mathematics, Manchester, England, January
1999.
See also: [expm], page 527, [logm], page 527.

528

GNU Octave

kron (A, B)
kron (A1, A2, . . . )
Form the Kronecker product of two or more matrices.
This is defined block by block as
x = [ a(i,j)*b ]
For example:
kron (1:4, ones (3, 1))
⇒ 1 2 3 4
1 2 3 4
1 2 3 4
If there are more than two input arguments A1, A2, . . . , An the Kronecker product
is computed as
kron (kron (A1, A2), ..., An)
Since the Kronecker product is associative, this is well-defined.

blkmm (A, B)
Compute products of matrix blocks.
The blocks are given as 2-dimensional subarrays of the arrays A, B. The size of A
must have the form [m,k,...] and size of B must be [k,n,...]. The result is then
of size [m,n,...] and is computed as follows:
for i = 1:prod (size (A)(3:end))
C(:,:,i) = A(:,:,i) * B(:,:,i)
endfor

X = sylvester (A, B, C)
Solve the Sylvester equation
AX + XB = C
using standard lapack subroutines.
For example:
sylvester ([1, 2; 3, 4], [5, 6; 7, 8], [9, 10; 11, 12])
⇒ [ 0.50000, 0.66667; 0.66667, 0.50000 ]

18.5 Specialized Solvers
x = bicg (A, b, rtol, maxit, M1, M2, x0)
x = bicg (A, b, rtol, maxit, P)
[x, flag, relres, iter, resvec] = bicg (A, b, . . . )
Solve A x = b using the Bi-conjugate gradient iterative method.
− rtol is the relative tolerance, if not given or set to [] the default value 1e-6 is used.
− maxit the maximum number of outer iterations, if not given or set to [] the default
value min (20, numel (b)) is used.
− x0 the initial guess, if not given or set to [] the default value zeros (size (b))
is used.

Chapter 18: Linear Algebra

529

A can be passed as a matrix or as a function handle or inline function f such that
f(x, "notransp") = A*x and f(x, "transp") = A’*x.
The preconditioner P is given as P = M1 * M2. Both M1 and M2 can be passed as a
matrix or as a function handle or inline function g such that g(x, "notransp") = M1 \
x or g(x, "notransp") = M2 \ x and g(x, "transp") = M1’ \ x or g(x, "transp")
= M2’ \ x.
If called with more than one output parameter
− flag indicates the exit status:
− 0: iteration converged to the within the chosen tolerance
− 1: the maximum number of iterations was reached before convergence
− 3: the algorithm reached stagnation
(the value 2 is unused but skipped for compatibility).
− relres is the final value of the relative residual.
− iter is the number of iterations performed.
− resvec is a vector containing the relative residual at each iteration.
See also: [bicgstab], page 529, [cgs], page 530, [gmres], page 530, [pcg], page 593,
[qmr], page 531.

x = bicgstab (A, b, rtol, maxit, M1, M2, x0)
x = bicgstab (A, b, rtol, maxit, P)
[x, flag, relres, iter, resvec] = bicgstab (A, b, . . . )
Solve A x = b using the stabilizied Bi-conjugate gradient iterative method.
− rtol is the relative tolerance, if not given or set to [] the default value 1e-6 is used.
− maxit the maximum number of outer iterations, if not given or set to [] the default
value min (20, numel (b)) is used.
− x0 the initial guess, if not given or set to [] the default value zeros (size (b))
is used.
A can be passed as a matrix or as a function handle or inline function f such that
f(x) = A*x.
The preconditioner P is given as P = M1 * M2. Both M1 and M2 can be passed as a
matrix or as a function handle or inline function g such that g(x) = M1 \ x or g(x)
= M2 \ x.
If called with more than one output parameter
− flag indicates the exit status:
− 0: iteration converged to the within the chosen tolerance
− 1: the maximum number of iterations was reached before convergence
− 3: the algorithm reached stagnation
(the value 2 is unused but skipped for compatibility).
− relres is the final value of the relative residual.
− iter is the number of iterations performed.
− resvec is a vector containing the relative residual at each iteration.

530

GNU Octave

See also: [bicg], page 528, [cgs], page 530, [gmres], page 530, [pcg], page 593, [qmr],
page 531.

x = cgs (A, b, rtol, maxit, M1, M2, x0)
x = cgs (A, b, rtol, maxit, P)
[x, flag, relres, iter, resvec] = cgs (A, b, . . . )
Solve A x = b, where A is a square matrix, using the Conjugate Gradients Squared
method.
− rtol is the relative tolerance, if not given or set to [] the default value 1e-6 is used.
− maxit the maximum number of outer iterations, if not given or set to [] the default
value min (20, numel (b)) is used.
− x0 the initial guess, if not given or set to [] the default value zeros (size (b))
is used.
A can be passed as a matrix or as a function handle or inline function f such that
f(x) = A*x.
The preconditioner P is given as P = M1 * M2. Both M1 and M2 can be passed as a
matrix or as a function handle or inline function g such that g(x) = M1 \ x or g(x)
= M2 \ x.
If called with more than one output parameter
− flag
−
−
−

indicates the exit status:
0: iteration converged to the within the chosen tolerance
1: the maximum number of iterations was reached before convergence
3: the algorithm reached stagnation

(the value 2 is unused but skipped for compatibility).
− relres is the final value of the relative residual.
− iter is the number of iterations performed.
− resvec is a vector containing the relative residual at each iteration.
See also: [pcg], page 593, [bicgstab], page 529, [bicg], page 528, [gmres], page 530,
[qmr], page 531.

x = gmres (A, b, m, rtol, maxit, M1, M2, x0)
x = gmres (A, b, m, rtol, maxit, P)
[x, flag, relres, iter, resvec] = gmres ( . . . )
Solve A x = b using the Preconditioned GMRES iterative method with restart, a.k.a.
PGMRES(m).
− rtol is the relative tolerance, if not given or set to [] the default value 1e-6 is used.
− maxit is the maximum number of outer iterations, if not given or set to [] the
default value min (10, numel (b) / restart) is used.
− x0 is the initial guess, if not given or set to [] the default value zeros (size (b))
is used.
− m is the restart parameter, if not given or set to [] the default value numel (b)
is used.

Chapter 18: Linear Algebra

531

Argument A can be passed as a matrix, function handle, or inline function f such
that f(x) = A*x.
The preconditioner P is given as P = M1 * M2. Both M1 and M2 can be passed as a
matrix, function handle, or inline function g such that g(x) = M1\x or g(x) = M2\x.
Besides the vector x, additional outputs are:
− flag indicates the exit status:

0 : iteration converged to within the specified tolerance
1 : maximum number of iterations exceeded
2 : unused, but skipped for compatibility
3 : algorithm reached stagnation (no change between iterations)
− relres is the final value of the relative residual.
− iter is a vector containing the number of outer iterations and total iterations
performed.
− resvec is a vector containing the relative residual at each iteration.

See also: [bicg], page 528, [bicgstab], page 529, [cgs], page 530, [pcg], page 593, [pcr],
page 595, [qmr], page 531.

x = qmr (A, b, rtol, maxit, M1, M2, x0)
x = qmr (A, b, rtol, maxit, P)
[x, flag, relres, iter, resvec] = qmr (A, b, . . . )
Solve A x = b using the Quasi-Minimal Residual iterative method (without lookahead).
− rtol is the relative tolerance, if not given or set to [] the default value 1e-6 is used.
− maxit the maximum number of outer iterations, if not given or set to [] the default
value min (20, numel (b)) is used.
− x0 the initial guess, if not given or set to [] the default value zeros (size (b))
is used.
A can be passed as a matrix or as a function handle or inline function f such that
f(x, "notransp") = A*x and f(x, "transp") = A’*x.
The preconditioner P is given as P = M1 * M2. Both M1 and M2 can be passed as a
matrix or as a function handle or inline function g such that g(x, "notransp") = M1 \
x or g(x, "notransp") = M2 \ x and g(x, "transp") = M1’ \ x or g(x, "transp")
= M2’ \ x.
If called with more than one output parameter
− flag indicates the exit status:
− 0: iteration converged to the within the chosen tolerance
− 1: the maximum number of iterations was reached before convergence
− 3: the algorithm reached stagnation
(the value 2 is unused but skipped for compatibility).
− relres is the final value of the relative residual.
− iter is the number of iterations performed.
− resvec is a vector containing the residual norms at each iteration.

532

GNU Octave

References:
1. R. Freund and N. Nachtigal, QMR: a quasi-minimal residual method for nonHermitian linear systems, Numerische Mathematik, 1991, 60, pp. 315-339.
2. R. Barrett, M. Berry, T. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhour,
R. Pozo, C. Romine, and H. van der Vorst, Templates for the solution of linear
systems: Building blocks for iterative methods, SIAM, 2nd ed., 1994.
See also: [bicg], page 528, [bicgstab], page 529, [cgs], page 530, [gmres], page 530,
[pcg], page 593.

533

19 Vectorization and Faster Code Execution
Vectorization is a programming technique that uses vector operations instead of element-byelement loop-based operations. Besides frequently producing more succinct Octave code,
vectorization also allows for better optimization in the subsequent implementation. The
optimizations may occur either in Octave’s own Fortran, C, or C++ internal implementation,
or even at a lower level depending on the compiler and external numerical libraries used to
build Octave. The ultimate goal is to make use of your hardware’s vector instructions if
possible or to perform other optimizations in software.
Vectorization is not a concept unique to Octave, but it is particularly important because
Octave is a matrix-oriented language. Vectorized Octave code will see a dramatic speed up
(10X–100X) in most cases.
This chapter discusses vectorization and other techniques for writing faster code.

19.1 Basic Vectorization
To a very good first approximation, the goal in vectorization is to write code that avoids
loops and uses whole-array operations. As a trivial example, consider
for i = 1:n
for j = 1:m
c(i,j) = a(i,j) + b(i,j);
endfor
endfor
compared to the much simpler
c = a + b;
This isn’t merely easier to write; it is also internally much easier to optimize. Octave delegates this operation to an underlying implementation which, among other optimizations,
may use special vector hardware instructions or could conceivably even perform the additions in parallel. In general, if the code is vectorized, the underlying implementation has
more freedom about the assumptions it can make in order to achieve faster execution.
This is especially important for loops with "cheap" bodies. Often it suffices to vectorize
just the innermost loop to get acceptable performance. A general rule of thumb is that the
"order" of the vectorized body should be greater or equal to the "order" of the enclosing
loop.
As a less trivial example, instead of
for i = 1:n-1
a(i) = b(i+1) - b(i);
endfor
write
a = b(2:n) - b(1:n-1);
This shows an important general concept about using arrays for indexing instead of
looping over an index variable. See Section 8.1 [Index Expressions], page 137. Also use

534

GNU Octave

boolean indexing generously. If a condition needs to be tested, this condition can also be
written as a boolean index. For instance, instead of
for i = 1:n
if (a(i) > 5)
a(i) -= 20
endif
endfor
write
a(a>5) -= 20;
which exploits the fact that a > 5 produces a boolean index.
Use elementwise vector operators whenever possible to avoid looping (operators like .*
and .^). See Section 8.3 [Arithmetic Ops], page 144. For simple inline functions, the
vectorize function can do this automatically.

vectorize (fun)
Create a vectorized version of the inline function fun by replacing all occurrences of
*, /, etc., with .*, ./, etc.
This may be useful, for example, when using inline functions with numerical integration or optimization where a vector-valued function is expected.
fcn = vectorize (inline ("x^2 - 1"))
⇒ fcn = f(x) = x.^2 - 1
quadv (fcn, 0, 3)
⇒ 6
See also: [inline], page 214, [formula], page 214, [argnames], page 214.

Also exploit broadcasting in these elementwise operators both to avoid looping and
unnecessary intermediate memory allocations. See Section 19.2 [Broadcasting], page 535.
Use built-in and library functions if possible. Built-in and compiled functions are very
fast. Even with an m-file library function, chances are good that it is already optimized, or
will be optimized more in a future release.
For instance, even better than
a = b(2:n) - b(1:n-1);
is
a = diff (b);
Most Octave functions are written with vector and array arguments in mind. If you
find yourself writing a loop with a very simple operation, chances are that such a function
already exists. The following functions occur frequently in vectorized code:
• Index manipulation
• find
• sub2ind
• ind2sub
• sort
• unique

Chapter 19: Vectorization and Faster Code Execution

535

• lookup
• ifelse / merge
• Repetition
• repmat
• repelems
• Vectorized arithmetic
• sum
• prod
• cumsum
• cumprod
• sumsq
• diff
• dot
• cummax
• cummin
• Shape of higher dimensional arrays
• reshape
• resize
• permute
• squeeze
• deal

19.2 Broadcasting
Broadcasting refers to how Octave binary operators and functions behave when their matrix
or array operands or arguments differ in size. Since version 3.6.0, Octave now automatically broadcasts vectors, matrices, and arrays when using elementwise binary operators and
functions. Broadly speaking, smaller arrays are “broadcast” across the larger one, until
they have a compatible shape. The rule is that corresponding array dimensions must either
1. be equal, or
2. one of them must be 1.
In case all dimensions are equal, no broadcasting occurs and ordinary element-by-element
arithmetic takes place. For arrays of higher dimensions, if the number of dimensions isn’t
the same, then missing trailing dimensions are treated as 1. When one of the dimensions is
1, the array with that singleton dimension gets copied along that dimension until it matches
the dimension of the other array. For example, consider
x = [1 2 3;
4 5 6;
7 8 9];
y = [10 20 30];
x + y

536

GNU Octave

Without broadcasting, x + y would be an error because the dimensions do not agree. However, with broadcasting it is as if the following operation were performed:
x = [1 2 3
4 5 6
7 8 9];
y = [10 20 30
10 20 30
10 20 30];
x + y
⇒
11
22
33
14
25
36
17
28
39
That is, the smaller array of size [1 3] gets copied along the singleton dimension (the
number of rows) until it is [3 3]. No actual copying takes place, however. The internal implementation reuses elements along the necessary dimension in order to achieve the desired
effect without copying in memory.
Both arrays can be broadcast across each other, for example, all pairwise differences of
the elements of a vector with itself:
y - y’
⇒
0
10
20
-10
0
10
-20 -10
0
Here the vectors of size [1 3] and [3 1] both get broadcast into matrices of size [3 3]
before ordinary matrix subtraction takes place.
A special case of broadcasting that may be familiar is when all dimensions of the array
being broadcast are 1, i.e., the array is a scalar. Thus for example, operations like x - 42
and max (x, 2) are basic examples of broadcasting.
For a higher-dimensional example, suppose img is an RGB image of size [m n 3] and we
wish to multiply each color by a different scalar. The following code accomplishes this with
broadcasting,
img .*= permute ([0.8, 0.9, 1.2], [1, 3, 2]);
Note the usage of permute to match the dimensions of the [0.8, 0.9, 1.2] vector with
img.
For functions that are not written with broadcasting semantics, bsxfun can be useful
for coercing them to broadcast.

bsxfun (f, A, B)
The binary singleton expansion function performs broadcasting, that is, it applies a
binary function f element-by-element to two array arguments A and B, and expands
as necessary singleton dimensions in either input argument.
f is a function handle, inline function, or string containing the name of the function to
evaluate. The function f must be capable of accepting two column-vector arguments
of equal length, or one column vector argument and a scalar.

Chapter 19: Vectorization and Faster Code Execution

537

The dimensions of A and B must be equal or singleton. The singleton dimensions of
the arrays will be expanded to the same dimensionality as the other array.
See also: [arrayfun], page 538, [cellfun], page 540.
Broadcasting is only applied if either of the two broadcasting conditions hold. As usual,
however, broadcasting does not apply when two dimensions differ and neither is 1:
x = [1 2 3
4 5 6];
y = [10 20
30 40];
x + y
This will produce an error about nonconformant arguments.
Besides common arithmetic operations, several functions of two arguments also broadcast. The full list of functions and operators that broadcast is
plus
minus
times
rdivide
ldivide
power
lt
le
eq
gt
ge
ne
and
or
atan2
hypot
max
min
mod
rem
xor
+=

-=

+ .+
- ..*
./
.\
.^ .**
<
<=
==
>
>=
!= ~=
&
|

.+=

.-=

.*=

./=

.\=

.^=

.**=

&=

|=

Beware of resorting to broadcasting if a simpler operation will suffice. For matrices a
and b, consider the following:
c = sum (permute (a, [1, 3, 2]) .* permute (b, [3, 2, 1]), 3);
This operation broadcasts the two matrices with permuted dimensions across each other
during elementwise multiplication in order to obtain a larger 3-D array, and this array is
then summed along the third dimension. A moment of thought will prove that this operation
is simply the much faster ordinary matrix multiplication, c = a*b;.

538

GNU Octave

A note on terminology: “broadcasting” is the term popularized by the Numpy numerical
environment in the Python programming language. In other programming languages and
environments, broadcasting may also be known as binary singleton expansion (BSX, in
matlab, and the origin of the name of the bsxfun function), recycling (R programming
language), single-instruction multiple data (SIMD), or replication.

19.2.1 Broadcasting and Legacy Code
The new broadcasting semantics almost never affect code that worked in previous versions
of Octave. Consequently, all code inherited from matlab that worked in previous versions
of Octave should still work without change in Octave. The only exception is code such as
try
c = a.*b;
catch
c = a.*a;
end_try_catch
that may have relied on matrices of different size producing an error. Because such operation
is now valid Octave syntax, this will no longer produce an error. Instead, the following code
should be used:
if (isequal (size (a), size (b)))
c = a .* b;
else
c = a .* a;
endif

19.3 Function Application
As a general rule, functions should already be written with matrix arguments in mind
and should consider whole matrix operations in a vectorized manner. Sometimes, writing
functions in this way appears difficult or impossible for various reasons. For those situations,
Octave provides facilities for applying a function to each element of an array, cell, or struct.

arrayfun (func, A)
x = arrayfun (func, A)
x = arrayfun (func, A, b, . . . )
[x, y, ...] = arrayfun (func, A, . . . )
arrayfun ( . . . , "UniformOutput", val)
arrayfun ( . . . , "ErrorHandler", errfunc)
Execute a function on each element of an array.
This is useful for functions that do not accept array arguments. If the function does
accept array arguments it is better to call the function directly.
The first input argument func can be a string, a function handle, an inline function, or
an anonymous function. The input argument A can be a logic array, a numeric array,
a string array, a structure array, or a cell array. By a call of the function arrayfun
all elements of A are passed on to the named function func individually.
The named function can also take more than two input arguments, with the input
arguments given as third input argument b, fourth input argument c, . . . If given

Chapter 19: Vectorization and Faster Code Execution

539

more than one array input argument then all input arguments must have the same
sizes, for example:
arrayfun (@atan2, [1, 0], [0, 1])
⇒ [ 1.57080
0.00000 ]

If the parameter val after a further string input argument "UniformOutput" is set
true (the default), then the named function func must return a single element which
then will be concatenated into the return value and is of type matrix. Otherwise, if
that parameter is set to false, then the outputs are concatenated in a cell array. For
example:
arrayfun (@(x,y) x:y, "abc", "def", "UniformOutput", false)
⇒
{
[1,1] = abcd
[1,2] = bcde
[1,3] = cdef
}
If more than one output arguments are given then the named function must return
the number of return values that also are expected, for example:
[A, B, C] = arrayfun (@find, [10; 0], "UniformOutput", false)
⇒
A =
{
[1,1] = 1
[2,1] = [](0x0)
}
B =
{
[1,1] = 1
[2,1] = [](0x0)
}
C =
{
[1,1] = 10
[2,1] = [](0x0)
}
If the parameter errfunc after a further string input argument "ErrorHandler" is
another string, a function handle, an inline function, or an anonymous function,
then errfunc defines a function to call in the case that func generates an error. The
definition of the function must be of the form
function [...] = errfunc (s, ...)
where there is an additional input argument to errfunc relative to func, given by
s. This is a structure with the elements "identifier", "message", and "index"
giving, respectively, the error identifier, the error message, and the index of the array
elements that caused the error. The size of the output argument of errfunc must have

540

GNU Octave

the same size as the output argument of func, otherwise a real error is thrown. For
example:
function y = ferr (s, x), y = "MyString"; endfunction
arrayfun (@str2num, [1234],
"UniformOutput", false, "ErrorHandler", @ferr)
⇒
{
[1,1] = MyString
}
See also: [spfun], page 540, [cellfun], page 540, [structfun], page 542.

y = spfun (f, S)
Compute f(S) for the nonzero values of S.
This results in a sparse matrix with the same structure as S. The function f can be
passed as a string, a function handle, or an inline function.
See also: [arrayfun], page 538, [cellfun], page 540, [structfun], page 542.

cellfun (name, C)
cellfun ("size", C, k)
cellfun ("isclass", C, class)
cellfun (func, C)
cellfun (func, C, D)
[a, ...] = cellfun ( . . . )
cellfun ( . . . , "ErrorHandler", errfunc)
cellfun ( . . . , "UniformOutput", val)
Evaluate the function named name on the elements of the cell array C.
Elements in C are passed on to the named function individually. The function name
can be one of the functions
isempty

Return 1 for empty elements.

islogical
Return 1 for logical elements.
isnumeric
Return 1 for numeric elements.
isreal

Return 1 for real elements.

length

Return a vector of the lengths of cell elements.

ndims

Return the number of dimensions of each element.

numel
prodofsize
Return the number of elements contained within each cell element. The
number is the product of the dimensions of the object at each cell element.
size

Return the size along the k-th dimension.

isclass

Return 1 for elements of class.

Chapter 19: Vectorization and Faster Code Execution

541

Additionally, cellfun accepts an arbitrary function func in the form of an inline
function, function handle, or the name of a function (in a character string). The
function can take one or more arguments, with the inputs arguments given by C, D,
etc. Equally the function can return one or more output arguments. For example:
cellfun ("atan2", {1, 0}, {0, 1})
⇒ [ 1.57080
0.00000 ]

The number of output arguments of cellfun matches the number of output arguments of the function. The outputs of the function will be collected into the output
arguments of cellfun like this:
function [a, b] = twoouts (x)
a = x;
b = x*x;
endfunction
[aa, bb] = cellfun (@twoouts, {1, 2, 3})
⇒
aa =
1 2 3
bb =
1 4 9
Note that per default the output argument(s) are arrays of the same size as the input
arguments. Input arguments that are singleton (1x1) cells will be automatically
expanded to the size of the other arguments.
If the parameter "UniformOutput" is set to true (the default), then the function
must return scalars which will be concatenated into the return array(s). If
"UniformOutput" is false, the outputs are concatenated into a cell array (or cell
arrays). For example:
cellfun ("tolower", {"Foo", "Bar", "FooBar"},
"UniformOutput", false)
⇒ {"foo", "bar", "foobar"}

Given the parameter "ErrorHandler", then errfunc defines a function to call in case
func generates an error. The form of the function is
function [...] = errfunc (s, ...)
where there is an additional input argument to errfunc relative to func, given by
s. This is a structure with the elements "identifier", "message", and "index"
giving respectively the error identifier, the error message, and the index into the
input arguments of the element that caused the error. For example:
function y = foo (s, x), y = NaN; endfunction
cellfun ("factorial", {-1,2}, "ErrorHandler", @foo)
⇒ [NaN 2]

Use cellfun intelligently. The cellfun function is a useful tool for avoiding loops.
It is often used with anonymous function handles; however, calling an anonymous
function involves an overhead quite comparable to the overhead of an m-file function.
Passing a handle to a built-in function is faster, because the interpreter is not involved
in the internal loop. For example:

542

GNU Octave

a = {...}
v = cellfun (@(x) det (x), a); # compute determinants
v = cellfun (@det, a); # faster
See also: [arrayfun], page 538, [structfun], page 542, [spfun], page 540.

structfun (func, S)
[A, ...] = structfun ( . . . )
structfun ( . . . , "ErrorHandler", errfunc)
structfun ( . . . , "UniformOutput", val)
Evaluate the function named name on the fields of the structure S. The fields of S
are passed to the function func individually.
structfun accepts an arbitrary function func in the form of an inline function, function handle, or the name of a function (in a character string). In the case of a
character string argument, the function must accept a single argument named x, and
it must return a string value. If the function returns more than one argument, they
are returned as separate output variables.
If the parameter "UniformOutput" is set to true (the default), then the function
must return a single element which will be concatenated into the return value. If
"UniformOutput" is false, the outputs are placed into a structure with the same
fieldnames as the input structure.
s.name1 = "John Smith";
s.name2 = "Jill Jones";
structfun (@(x) regexp (x, ’(\w+)$’, "matches"){1}, s,
"UniformOutput", false)
⇒
{
name1 = Smith
name2 = Jones
}
Given the parameter "ErrorHandler", errfunc defines a function to call in case func
generates an error. The form of the function is
function [...] = errfunc (se, ...)
where there is an additional input argument to errfunc relative to func, given by
se. This is a structure with the elements "identifier", "message" and "index",
giving respectively the error identifier, the error message, and the index into the input
arguments of the element that caused the error. For an example on how to use an
error handler, see [cellfun], page 540.
See also: [cellfun], page 540, [arrayfun], page 538, [spfun], page 540.
Consistent with earlier advice, seek to use Octave built-in functions whenever possible
for the best performance. This advice applies especially to the four functions above. For
example, when adding two arrays together element-by-element one could use a handle to
the built-in addition function @plus or define an anonymous function @(x,y) x + y. But,
the anonymous function is 60% slower than the first method. See Section 34.4.2 [Operator Overloading], page 786, for a list of basic functions which might be used in place of
anonymous ones.

Chapter 19: Vectorization and Faster Code Execution

543

19.4 Accumulation
Whenever it’s possible to categorize according to indices the elements of an array when
performing a computation, accumulation functions can be useful.

accumarray (subs, vals, sz, func, fillval, issparse)
accumarray (subs, vals, . . . )
Create an array by accumulating the elements of a vector into the positions defined
by their subscripts.
The subscripts are defined by the rows of the matrix subs and the values by vals.
Each row of subs corresponds to one of the values in vals. If vals is a scalar, it will
be used for each of the row of subs. If subs is a cell array of vectors, all vectors must
be of the same length, and the subscripts in the kth vector must correspond to the
kth dimension of the result.
The size of the matrix will be determined by the subscripts themselves. However, if
sz is defined it determines the matrix size. The length of sz must correspond to the
number of columns in subs. An exception is if subs has only one column, in which
case sz may be the dimensions of a vector and the subscripts of subs are taken as the
indices into it.
The default action of accumarray is to sum the elements with the same subscripts.
This behavior can be modified by defining the func function. This should be a function
or function handle that accepts a column vector and returns a scalar. The result of
the function should not depend on the order of the subscripts.
The elements of the returned array that have no subscripts associated with them are
set to zero. Defining fillval to some other value allows these values to be defined. This
behavior changes, however, for certain values of func. If func is @min (respectively,
@max) then the result will be filled with the minimum (respectively, maximum) integer
if vals is of integral type, logical false (respectively, logical true) if vals is of logical
type, zero if fillval is zero and all values are non-positive (respectively, non-negative),
and NaN otherwise.
By default accumarray returns a full matrix. If issparse is logically true, then a sparse
matrix is returned instead.
The following accumarray example constructs a frequency table that in the first
column counts how many occurrences each number in the second column has, taken
from the vector x. Note the usage of unique for assigning to all repeated elements of
x the same index (see [unique], page 685).
x = [91, 92, 90, 92, 90, 89, 91, 89, 90, 100, 100, 100];
[u, ~, j] = unique (x);
[accumarray(j’, 1), u’]
⇒ 2
89
3
90
2
91
2
92
3
100
Another example, where the result is a multi-dimensional 3-D array and the default
value (zero) appears in the output:

544

GNU Octave

accumarray ([1,
2,
2,
2,
2,
⇒ ans(:,:,1) =
⇒ ans(:,:,2) =

1, 1;
1, 2;
3, 2;
1, 2;
3, 2], 101:105)
[101, 0, 0; 0, 0, 0]
[0, 0, 0; 206, 0, 208]

The sparse option can be used as an alternative to the sparse constructor (see [sparse],
page 568). Thus
sparse (i, j, sv)
can be written with accumarray as
accumarray ([i, j], sv’, [], [], 0, true)
For repeated indices, sparse adds the corresponding value. To take the minimum
instead, use min as an accumulator function:
accumarray ([i, j], sv’, [], @min, 0, true)
The complexity of accumarray in general for the non-sparse case is generally O(M+N),
where N is the number of subscripts and M is the maximum subscript (linearized in
multi-dimensional case). If func is one of @sum (default), @max, @min or @(x) {x}, an
optimized code path is used. Note that for general reduction function the interpreter
overhead can play a major part and it may be more efficient to do multiple accumarray
calls and compute the results in a vectorized manner.
See also: [accumdim], page 544, [unique], page 685, [sparse], page 568.

accumdim (subs, vals, dim, n, func, fillval)
Create an array by accumulating the slices of an array into the positions defined by
their subscripts along a specified dimension.
The subscripts are defined by the index vector subs. The dimension is specified by
dim. If not given, it defaults to the first non-singleton dimension. The length of subs
must be equal to size (vals, dim).
The extent of the result matrix in the working dimension will be determined by the
subscripts themselves. However, if n is defined it determines this extent.
The default action of accumdim is to sum the subarrays with the same subscripts. This
behavior can be modified by defining the func function. This should be a function or
function handle that accepts an array and a dimension, and reduces the array along
this dimension. As a special exception, the built-in min and max functions can be
used directly, and accumdim accounts for the middle empty argument that is used in
their calling.
The slices of the returned array that have no subscripts associated with them are set
to zero. Defining fillval to some other value allows these values to be defined.
An example of the use of accumdim is:

Chapter 19: Vectorization and Faster Code Execution

545

accumdim ([1, 2, 1, 2, 1], [ 7, -10,
4;
-5, -12,
8;
-12,
2,
8;
-10,
9, -3;
-5, -3, -13])
⇒ [-10,-11,-1;-15,-3,5]
See also: [accumarray], page 543.

19.5 JIT Compiler
Vectorization is the preferred technique for eliminating loops and speeding up code. Nevertheless, it is not always possible to replace every loop. In such situations it may be worth
trying Octave’s experimental Just-In-Time (JIT) compiler.
A JIT compiler works by analyzing the body of a loop, translating the Octave statements
into another language, compiling the new code segment into an executable, and then running
the executable and collecting any results. The process is not simple and there is a significant
amount of work to perform for each step. It can still make sense, however, if the number
of loop iterations is large. Because Octave is an interpreted language every time through
a loop Octave must parse the statements in the loop body before executing them. With a
JIT compiler this is done just once when the body is translated to another language.
The JIT compiler is a very new feature in Octave and not all valid Octave statements
can currently be accelerated. However, if no other technique is available it may be worth
benchmarking the code with JIT enabled. The function jit_enable is used to turn compilation on or off. The function jit_startcnt sets the threshold for acceleration. Loops with
iteration counts above jit_startcnt will be accelerated. The functions jit_failcnt and
debug_jit are not likely to be of use to anyone not working directly on the implementation
of the JIT compiler.

val = jit_enable ()
old_val = jit_enable (new_val)
jit_enable (new_val, "local")
Query or set the internal variable that enables Octave’s JIT compiler.
When called from inside a function with the "local" option, the variable is changed
locally for the function and any subroutines it calls. The original variable value is
restored when exiting the function.
See also: [jit startcnt], page 545, [debug jit], page 546.

val = jit_startcnt ()
old_val = jit_startcnt (new_val)
jit_startcnt (new_val, "local")
Query or set the internal variable that determines whether JIT compilation will take
place for a specific loop.
Because compilation is a costly operation it does not make sense to employ JIT when
the loop count is low. By default only loops with greater than 1000 iterations will be
accelerated.

546

GNU Octave

When called from inside a function with the "local" option, the variable is changed
locally for the function and any subroutines it calls. The original variable value is
restored when exiting the function.
See also: [jit enable], page 545, [jit failcnt], page 546, [debug jit], page 546.

val = jit_failcnt ()
old_val = jit_failcnt (new_val)
jit_failcnt (new_val, "local")
Query or set the internal variable that counts the number of JIT fail exceptions for
Octave’s JIT compiler.
When called from inside a function with the "local" option, the variable is changed
locally for the function and any subroutines it calls. The original variable value is
restored when exiting the function.
See also: [jit enable], page 545, [jit startcnt], page 545, [debug jit], page 546.

val = debug_jit ()
old_val = debug_jit (new_val)
debug_jit (new_val, "local")
Query or set the internal variable that determines whether debugging/tracing is enabled for Octave’s JIT compiler.
When called from inside a function with the "local" option, the variable is changed
locally for the function and any subroutines it calls. The original variable value is
restored when exiting the function.
See also: [jit enable], page 545, [jit startcnt], page 545.

19.6 Miscellaneous Techniques
Here are some other ways of improving the execution speed of Octave programs.
• Avoid computing costly intermediate results multiple times. Octave currently does
not eliminate common subexpressions. Also, certain internal computation results are
cached for variables. For instance, if a matrix variable is used multiple times as an
index, checking the indices (and internal conversion to integers) is only done once.
• Be aware of lazy copies (copy-on-write). When a copy of an object is created, the data
is not immediately copied, but rather shared. The actual copying is postponed until
the copied data needs to be modified. For example:
a = zeros (1000); # create a 1000x1000 matrix
b = a; # no copying done here
b(1) = 1; # copying done here
Lazy copying applies to whole Octave objects such as matrices, cells, struct, and also
individual cell or struct elements (not array elements).
Additionally, index expressions also use lazy copying when Octave can determine that
the indexed portion is contiguous in memory. For example:
a = zeros (1000); # create a 1000x1000 matrix
b = a(:,10:100); # no copying done here
b = a(10:100,:); # copying done here

Chapter 19: Vectorization and Faster Code Execution

547

This applies to arrays (matrices), cell arrays, and structs indexed using ‘()’. Index
expressions generating comma-separated lists can also benefit from shallow copying in
some cases. In particular, when a is a struct array, expressions like {a.x}, {a(:,2).x}
will use lazy copying, so that data can be shared between a struct array and a cell array.
Most indexing expressions do not live longer than their parent objects. In rare cases,
however, a lazily copied slice outlasts its parent, in which case it becomes orphaned,
still occupying unnecessarily more memory than needed. To provide a remedy working
in most real cases, Octave checks for orphaned lazy slices at certain situations, when a
value is stored into a "permanent" location, such as a named variable or cell or struct
element, and possibly economizes them. For example:
a = zeros (1000); # create a 1000x1000 matrix
b = a(:,10:100); # lazy slice
a = []; # the original "a" array is still allocated
c{1} = b; # b is reallocated at this point
• Avoid deep recursion. Function calls to m-file functions carry a relatively significant
overhead, so rewriting a recursion as a loop often helps. Also, note that the maximum
level of recursion is limited.
• Avoid resizing matrices unnecessarily. When building a single result matrix from a
series of calculations, set the size of the result matrix first, then insert values into it.
Write
result = zeros (big_n, big_m)
for i = over:and_over
ridx = ...
cidx = ...
result(ridx, cidx) = new_value ();
endfor
instead of
result = [];
for i = ever:and_ever
result = [ result, new_value() ];
endfor
Sometimes the number of items can not be computed in advance, and stack-like operations are needed. When elements are being repeatedly inserted or removed from the
end of an array, Octave detects it as stack usage and attempts to use a smarter memory
management strategy by pre-allocating the array in bigger chunks. This strategy is also
applied to cell and struct arrays.
a = [];
while (condition)
...
a(end+1) = value; # "push" operation
...
a(end) = []; # "pop" operation
...
endwhile

548

GNU Octave

• Avoid calling eval or feval excessively. Parsing input or looking up the name of a
function in the symbol table are relatively expensive operations.
If you are using eval merely as an exception handling mechanism, and not because you
need to execute some arbitrary text, use the try statement instead. See Section 10.9
[The try Statement], page 172.
• Use ignore_function_time_stamp when appropriate. If you are calling lots of functions, and none of them will need to change during your run, set the variable ignore_
function_time_stamp to "all". This will stop Octave from checking the time stamp
of a function file to see if it has been updated while the program is being run.

19.7 Examples
The following are examples of vectorization questions asked by actual users of Octave and
their solutions.
• For a vector A, the following loop
n = length (A);
B = zeros (n, 2);
for i = 1:length (A)
## this will be two columns, the first is the difference and
## the second the mean of the two elements used for the diff.
B(i,:) = [A(i+1)-A(i), (A(i+1) + A(i))/2];
endfor
can be turned into the following one-liner:
B = [diff(A)(:), 0.5*(A(1:end-1)+A(2:end))(:)]
Note the usage of colon indexing to flatten an intermediate result into a column vector.
This is a common vectorization trick.

549

20 Nonlinear Equations
20.1 Solvers
Octave can solve sets of nonlinear equations of the form
f (x) = 0
using the function fsolve, which is based on the minpack subroutine hybrd. This is an
iterative technique so a starting point must be provided. This also has the consequence
that convergence is not guaranteed even if a solution exists.

fsolve (fcn, x0, options)
[x, fvec, info, output, fjac] = fsolve (fcn, . . . )
Solve a system of nonlinear equations defined by the function fcn.
fcn should accept a vector (array) defining the unknown variables, and return a vector
of left-hand sides of the equations. Right-hand sides are defined to be zeros. In
other words, this function attempts to determine a vector x such that fcn (x) gives
(approximately) all zeros.
x0 determines a starting guess. The shape of x0 is preserved in all calls to fcn, but
otherwise it is treated as a column vector.
options is a structure specifying additional options.
Currently, fsolve
recognizes these options: "FunValCheck", "OutputFcn", "TolX", "TolFun",
"MaxIter", "MaxFunEvals", "Jacobian", "Updating", "ComplexEqn" "TypicalX",
"AutoScaling" and "FinDiffType".
If "Jacobian" is "on", it specifies that fcn, called with 2 output arguments also returns the Jacobian matrix of right-hand sides at the requested point. "TolX" specifies
the termination tolerance in the unknown variables, while "TolFun" is a tolerance for
equations. Default is 1e-7 for both "TolX" and "TolFun".
If "AutoScaling" is on, the variables will be automatically scaled according to the
column norms of the (estimated) Jacobian. As a result, TolF becomes scalingindependent. By default, this option is off because it may sometimes deliver unexpected (though mathematically correct) results.
If "Updating" is "on", the function will attempt to use Broyden updates to update
the Jacobian, in order to reduce the amount of Jacobian calculations. If your user
function always calculates the Jacobian (regardless of number of output arguments)
then this option provides no advantage and should be set to false.
"ComplexEqn" is "on", fsolve will attempt to solve complex equations in complex
variables, assuming that the equations possess a complex derivative (i.e., are holomorphic). If this is not what you want, you should unpack the real and imaginary
parts of the system to get a real system.
For description of the other options, see optimset.
On return, fval contains the value of the function fcn evaluated at x.
info may be one of the following values:
1

Converged to a solution point. Relative residual error is less than specified
by TolFun.

550

GNU Octave

2

Last relative step size was less that TolX.

3

Last relative decrease in residual was less than TolF.

0

Iteration limit exceeded.

-3

The trust region radius became excessively small.

Note: If you only have a single nonlinear equation of one variable, using fzero is
usually a much better idea.
Note about user-supplied Jacobians: As an inherent property of the algorithm, a
Jacobian is always requested for a solution vector whose residual vector is already
known, and it is the last accepted successful step. Often this will be one of the last
two calls, but not always. If the savings by reusing intermediate results from residual
calculation in Jacobian calculation are significant, the best strategy is to employ
OutputFcn: After a vector is evaluated for residuals, if OutputFcn is called with that
vector, then the intermediate results should be saved for future Jacobian evaluation,
and should be kept until a Jacobian evaluation is requested or until OutputFcn is
called with a different vector, in which case they should be dropped in favor of this
most recent vector. A short example how this can be achieved follows:
function [fvec, fjac] = user_func (x, optimvalues, state)
persistent sav = [], sav0 = [];
if (nargin == 1)
## evaluation call
if (nargout == 1)
sav0.x = x; # mark saved vector
## calculate fvec, save results to sav0.
elseif (nargout == 2)
## calculate fjac using sav.
endif
else
## outputfcn call.
if (all (x == sav0.x))
sav = sav0;
endif
## maybe output iteration status, etc.
endif
endfunction
## ...
fsolve (@user_func, x0, optimset ("OutputFcn", @user_func, ...))
See also: [fzero], page 551, [optimset], page 645.
The following is a complete example. To solve the set of equations
−2x2 + 3xy + 4 sin(y) − 6 = 0
3x2 − 2xy 2 + 3 cos(x) + 4 = 0

Chapter 20: Nonlinear Equations

551

you first need to write a function to compute the value of the given function. For example:
function y = f (x)
y = zeros (2, 1);
y(1) = -2*x(1)^2 + 3*x(1)*x(2)
+ 4*sin(x(2)) - 6;
y(2) = 3*x(1)^2 - 2*x(1)*x(2)^2 + 3*cos(x(1)) + 4;
endfunction
Then, call fsolve with a specified initial condition to find the roots of the system of
equations. For example, given the function f defined above,
[x, fval, info] = fsolve (@f, [1; 2])
results in the solution
x =
0.57983
2.54621
fval =
-5.7184e-10
5.5460e-10
info = 1
A value of info = 1 indicates that the solution has converged.
When no Jacobian is supplied (as in the example above) it is approximated numerically.
This requires more function evaluations, and hence is less efficient. In the example above
we could compute the Jacobian analytically as


∂f1
∂x1
∂f2
∂x1

∂f1
∂x2
∂f2
∂x2





3x2 − 4x1
=
2
−2x2 − 3 sin(x1 ) + 6x1

4 cos(x2 ) + 3x1
−4x1 x2



and compute it with the following Octave function
function [y, jac] = f (x)
y = zeros (2, 1);
y(1) = -2*x(1)^2 + 3*x(1)*x(2)
+ 4*sin(x(2)) - 6;
y(2) = 3*x(1)^2 - 2*x(1)*x(2)^2 + 3*cos(x(1)) + 4;
if (nargout == 2)
jac = zeros (2, 2);
jac(1,1) = 3*x(2) - 4*x(1);
jac(1,2) = 4*cos(x(2)) + 3*x(1);
jac(2,1) = -2*x(2)^2 - 3*sin(x(1)) + 6*x(1);
jac(2,2) = -4*x(1)*x(2);
endif
endfunction
The Jacobian can then be used with the following call to fsolve:
[x, fval, info] = fsolve (@f, [1; 2], optimset ("jacobian", "on"));
which gives the same solution as before.

552

GNU Octave

fzero (fun, x0)
fzero (fun, x0, options)
[x, fval, info, output] = fzero ( . . . )
Find a zero of a univariate function.
fun is a function handle, inline function, or string containing the name of the function
to evaluate.
x0 should be a two-element vector specifying two points which bracket a zero. In
other words, there must be a change in sign of the function between x0(1) and x0(2).
More mathematically, the following must hold
sign (fun(x0(1))) * sign (fun(x0(2))) <= 0
If x0 is a single scalar then several nearby and distant values are probed in an attempt
to obtain a valid bracketing. If this is not successful, the function fails.
options is a structure specifying additional options. Currently, fzero recognizes these
options: "FunValCheck", "OutputFcn", "TolX", "MaxIter", "MaxFunEvals". For a
description of these options, see [optimset], page 645.
On exit, the function returns x, the approximate zero point and fval, the function
value thereof.
info is an exit flag that can have these values:
• 1 The algorithm converged to a solution.
• 0 Maximum number of iterations or function evaluations has been reached.
• -1 The algorithm has been terminated from user output function.
• -5 The algorithm may have converged to a singular point.
output is a structure containing runtime information about the fzero algorithm.
Fields in the structure are:
• iterations Number of iterations through loop.
• nfev Number of function evaluations.
• bracketx A two-element vector with the final bracketing of the zero along the
x-axis.
• brackety A two-element vector with the final bracketing of the zero along the
y-axis.
See also: [optimset], page 645, [fsolve], page 549.

20.2 Minimizers
Often it is useful to find the minimum value of a function rather than just the zeroes
where it crosses the x-axis. fminbnd is designed for the simpler, but very common, case of a
univariate function where the interval to search is bounded. For unbounded minimization of
a function with potentially many variables use fminunc or fminsearch. The two functions
use different internal algorithms and some knowledge of the objective function is required.
For functions which can be differentiated, fminunc is appropriate. For functions with
discontinuities, or for which a gradient search would fail, use fminsearch. See Chapter 25
[Optimization], page 633, for minimization with the presence of constraint functions. Note
that searches can be made for maxima by simply inverting the objective function (Fmax =
−Fmin ).

Chapter 20: Nonlinear Equations

553

[x, fval, info, output] = fminbnd (fun, a, b, options)
Find a minimum point of a univariate function.
fun should be a function handle or name. a, b specify a starting interval. options
is a structure specifying additional options. Currently, fminbnd recognizes these
options: "FunValCheck", "OutputFcn", "TolX", "MaxIter", "MaxFunEvals". For a
description of these options, see [optimset], page 645.
On exit, the function returns x, the approximate minimum point and fval, the function
value thereof.
info is an exit flag that can have these values:
• 1 The algorithm converged to a solution.
• 0 Maximum number of iterations or function evaluations has been exhausted.
• -1 The algorithm has been terminated from user output function.
Notes: The search for a minimum is restricted to be in the interval bound by a and
b. If you only have an initial point to begin searching from you will need to use
an unconstrained minimization algorithm such as fminunc or fminsearch. fminbnd
internally uses a Golden Section search strategy.
See also: [fzero], page 551, [fminunc], page 553, [fminsearch], page 554, [optimset],
page 645.

fminunc (fcn, x0)
fminunc (fcn, x0, options)
[x, fval, info, output, grad, hess] = fminunc (fcn, . . . )
Solve an unconstrained optimization problem defined by the function fcn.
fcn should accept a vector (array) defining the unknown variables, and return the
objective function value, optionally with gradient. fminunc attempts to determine a
vector x such that fcn (x) is a local minimum.
x0 determines a starting guess. The shape of x0 is preserved in all calls to fcn, but
otherwise is treated as a column vector.
options is a structure specifying additional options. Currently, fminunc recognizes
these options: "FunValCheck", "OutputFcn", "TolX", "TolFun", "MaxIter",
"MaxFunEvals", "GradObj", "FinDiffType", "TypicalX", "AutoScaling".
If "GradObj" is "on", it specifies that fcn, when called with two output arguments,
also returns the Jacobian matrix of partial first derivatives at the requested point.
TolX specifies the termination tolerance for the unknown variables x, while TolFun is
a tolerance for the objective function value fval. The default is 1e-7 for both options.
For a description of the other options, see optimset.
On return, x is the location of the minimum and fval contains the value of the objective
function at x.
info may be one of the following values:
1

Converged to a solution point. Relative gradient error is less than specified by TolFun.

2

Last relative step size was less than TolX.

554

GNU Octave

3

Last relative change in function value was less than TolFun.

0

Iteration limit exceeded—either maximum number of algorithm iterations
MaxIter or maximum number of function evaluations MaxFunEvals.

-1

Algorithm terminated by OutputFcn.

-3

The trust region radius became excessively small.

Optionally, fminunc can return a structure with convergence statistics (output), the
output gradient (grad) at the solution x, and approximate Hessian (hess) at the
solution x.
Application Notes: If the objective function is a single nonlinear equation of one
variable then using fminbnd is usually a better choice.
The algorithm used by fminunc is a gradient search which depends on the objective
function being differentiable. If the function has discontinuities it may be better to
use a derivative-free algorithm such as fminsearch.
See also: [fminbnd], page 552, [fminsearch], page 554, [optimset], page 645.

x = fminsearch (fun, x0)
x = fminsearch (fun, x0, options)
[x, fval] = fminsearch ( . . . )
Find a value of x which minimizes the function fun.
The search begins at the point x0 and iterates using the Nelder & Mead Simplex
algorithm (a derivative-free method). This algorithm is better-suited to functions
which have discontinuities or for which a gradient-based search such as fminunc fails.
Options for the search are provided in the parameter options using the function
optimset. Currently, fminsearch accepts the options: "TolX", "MaxFunEvals",
"MaxIter", "Display". For a description of these options, see optimset.
On exit, the function returns x, the minimum point, and fval, the function value
thereof.
Example usages:
fminsearch (@(x) (x(1)-5).^2+(x(2)-8).^4, [0;0])
fminsearch (inline ("(x(1)-5).^2+(x(2)-8).^4", "x"), [0;0])
See also: [fminbnd], page 552, [fminunc], page 553, [optimset], page 645.

555

21 Diagonal and Permutation Matrices
21.1 Creating and Manipulating Diagonal/Permutation
Matrices
A diagonal matrix is defined as a matrix that has zero entries outside the main diagonal;
that is, Dij = 0 if i 6= j Most often, square diagonal matrices are considered; however, the
definition can equally be applied to non-square matrices, in which case we usually speak of
a rectangular diagonal matrix.
A permutation matrix is defined as a square matrix that has a single element equal to
unity in each row and each column; all other elements are zero. That is, there exists a
permutation (vector) p such that Pij = 1 if j = pi and Pij = 0 otherwise.
Octave provides special treatment of real and complex rectangular diagonal matrices,
as well as permutation matrices. They are stored as special objects, using efficient storage
and algorithms, facilitating writing both readable and efficient matrix algebra expressions
in the Octave language. The special treatment may be disabled by using the functions
disable diagonal matrix and disable permutation matrix.

val = disable_diagonal_matrix ()
old_val = disable_diagonal_matrix (new_val)
disable_diagonal_matrix (new_val, "local")
Query or set the internal variable that controls whether diagonal matrices are stored
in a special space-efficient format.
The default value is true. If this option is disabled Octave will store diagonal matrices
as full matrices.
When called from inside a function with the "local" option, the variable is changed
locally for the function and any subroutines it calls. The original variable value is
restored when exiting the function.
See also: [disable range], page 53, [disable permutation matrix], page 555.

val = disable_permutation_matrix ()
old_val = disable_permutation_matrix (new_val)
disable_permutation_matrix (new_val, "local")
Query or set the internal variable that controls whether permutation matrices are
stored in a special space-efficient format.
The default value is true. If this option is disabled Octave will store permutation
matrices as full matrices.
When called from inside a function with the "local" option, the variable is changed
locally for the function and any subroutines it calls. The original variable value is
restored when exiting the function.
See also: [disable range], page 53, [disable diagonal matrix], page 555.
The space savings are significant as demonstrated by the following code.

556

GNU Octave

x = diag (rand (10, 1));
xf = full (x);
sizeof (x)
⇒ 80
sizeof (xf)
⇒ 800

21.1.1 Creating Diagonal Matrices
The most common and easiest way to create a diagonal matrix is using the built-in function
diag. The expression diag (v), with v a vector, will create a square diagonal matrix with
elements on the main diagonal given by the elements of v, and size equal to the length of v.
diag (v, m, n) can be used to construct a rectangular diagonal matrix. The result of these
expressions will be a special diagonal matrix object, rather than a general matrix object.
Diagonal matrix with unit elements can be created using eye. Some other built-in functions can also return diagonal matrices. Examples include balance or inv.
Example:
diag (1:4)
⇒
Diagonal Matrix
1
0
0
0

0
2
0
0

0
0
3
0

0
0
0
4

diag (1:3,5,3)
⇒
Diagonal Matrix
1
0
0
0
0

0
2
0
0
0

0
0
3
0
0

21.1.2 Creating Permutation Matrices
For creating permutation matrices, Octave does not introduce a new function, but rather
overrides an existing syntax: permutation matrices can be conveniently created by indexing
an identity matrix by permutation vectors. That is, if q is a permutation vector of length
n, the expression
P = eye (n) (:, q);
will create a permutation matrix - a special matrix object.
eye (n) (q, :)
will also work (and create a row permutation matrix), as well as

Chapter 21: Diagonal and Permutation Matrices

557

eye (n) (q1, q2).
For example:
eye (4) ([1,3,2,4],:)
⇒
Permutation Matrix
1
0
0
0

0
0
1
0

0
1
0
0

0
0
0
1

eye (4) (:,[1,3,2,4])
⇒
Permutation Matrix
1
0
0
0

0
0
1
0

0
1
0
0

0
0
0
1

Mathematically, an identity matrix is both diagonal and permutation matrix. In Octave, eye (n) returns a diagonal matrix, because a matrix can only have one class. You
can convert this diagonal matrix to a permutation matrix by indexing it by an identity
permutation, as shown below. This is a special property of the identity matrix; indexing
other diagonal matrices generally produces a full matrix.
eye (3)
⇒
Diagonal Matrix
1
0
0

0
1
0

0
0
1

eye(3)(1:3,:)
⇒
Permutation Matrix
1
0
0

0
1
0

0
0
1

Some other built-in functions can also return permutation matrices. Examples include
inv or lu.

21.1.3 Explicit and Implicit Conversions
The diagonal and permutation matrices are special objects in their own right. A number of
operations and built-in functions are defined for these matrices to use special, more efficient

558

GNU Octave

code than would be used for a full matrix in the same place. Examples are given in further
sections.
To facilitate smooth mixing with full matrices, backward compatibility, and compatibility
with matlab, the diagonal and permutation matrices should allow any operation that works
on full matrices, and will either treat it specially, or implicitly convert themselves to full
matrices.
Instances include matrix indexing, except for extracting a single element or a leading
submatrix, indexed assignment, or applying most mapper functions, such as exp.
An explicit conversion to a full matrix can be requested using the built-in function full.
It should also be noted that the diagonal and permutation matrix objects will cache the
result of the conversion after it is first requested (explicitly or implicitly), so that subsequent
conversions will be very cheap.

21.2 Linear Algebra with Diagonal/Permutation Matrices
As has been already said, diagonal and permutation matrices make it possible to use efficient
algorithms while preserving natural linear algebra syntax. This section describes in detail
the operations that are treated specially when performed on these special matrix objects.

21.2.1 Expressions Involving Diagonal Matrices
Assume D is a diagonal matrix. If M is a full matrix, then D*M will scale the rows of M.
That means, if S = D*M, then for each pair of indices i,j it holds
Sij = Dii Mij
Similarly, M*D will do a column scaling.
The matrix D may also be rectangular, m-by-n where m != n. If m < n, then the expression D*M is equivalent to
D(:,1:m) * M(1:m,:),
i.e., trailing n-m rows of M are ignored. If m > n, then D*M is equivalent to
[D(1:n,:) * M; zeros(m-n, columns (M))],
i.e., null rows are appended to the result. The situation for right-multiplication M*D is
analogous.
The expressions D \ M and M / D perform inverse scaling. They are equivalent to solving a diagonal (or rectangular diagonal) in a least-squares minimum-norm sense. In exact
arithmetic, this is equivalent to multiplying by a pseudoinverse. The pseudoinverse of a rectangular diagonal matrix is again a rectangular diagonal matrix with swapped dimensions,
where each nonzero diagonal element is replaced by its reciprocal. The matrix division
algorithms do, in fact, use division rather than multiplication by reciprocals for better numerical accuracy; otherwise, they honor the above definition. Note that a diagonal matrix
is never truncated due to ill-conditioning; otherwise, it would not be of much use for scaling.
This is typically consistent with linear algebra needs. A full matrix that only happens to
be diagonal (and is thus not a special object) is of course treated normally.
Multiplication and division by diagonal matrices work efficiently also when combined
with sparse matrices, i.e., D*S, where D is a diagonal matrix and S is a sparse matrix scales

Chapter 21: Diagonal and Permutation Matrices

559

the rows of the sparse matrix and returns a sparse matrix. The expressions S*D, D\S, S/D
work analogically.
If D1 and D2 are both diagonal matrices, then the expressions
D1 + D2
D1 - D2
D1 * D2
D1 / D2
D1 \ D2
again produce diagonal matrices, provided that normal dimension matching rules are
obeyed. The relations used are same as described above.
Also, a diagonal matrix D can be multiplied or divided by a scalar, or raised to a scalar
power if it is square, producing diagonal matrix result in all cases.
A diagonal matrix can also be transposed or conjugate-transposed, giving the expected
result. Extracting a leading submatrix of a diagonal matrix, i.e., D(1:m,1:n), will produce
a diagonal matrix, other indexing expressions will implicitly convert to full matrix.
Adding a diagonal matrix to a full matrix only operates on the diagonal elements. Thus,
A = A + eps * eye (n)
is an efficient method of augmenting the diagonal of a matrix. Subtraction works analogically.
When involved in expressions with other element-by-element operators, .*, ./, .\ or .^,
an implicit conversion to full matrix will take place. This is not always strictly necessary
but chosen to facilitate better consistency with matlab.

21.2.2 Expressions Involving Permutation Matrices
If P is a permutation matrix and M a matrix, the expression P*M will permute the rows of
M. Similarly, M*P will yield a column permutation. Matrix division P\M and M/P can be
used to do inverse permutation.
The previously described syntax for creating permutation matrices can actually help an
user to understand the connection between a permutation matrix and a permuting vector.
Namely, the following holds, where I = eye (n) is an identity matrix:
I(p,:) * M = (I*M) (p,:) = M(p,:)
Similarly,
M * I(:,p) = (M*I) (:,p) = M(:,p)
The expressions I(p,:) and I(:,p) are permutation matrices.
A permutation matrix can be transposed (or conjugate-transposed, which is the same,
because a permutation matrix is never complex), inverting the permutation, or equivalently,
turning a row-permutation matrix into a column-permutation one. For permutation matrices, transpose is equivalent to inversion, thus P\M is equivalent to P’*M. Transpose of a
permutation matrix (or inverse) is a constant-time operation, flipping only a flag internally,
and thus the choice between the two above equivalent expressions for inverse permuting is
completely up to the user’s taste.
Multiplication and division by permutation matrices works efficiently also when combined with sparse matrices, i.e., P*S, where P is a permutation matrix and S is a sparse

560

GNU Octave

matrix permutes the rows of the sparse matrix and returns a sparse matrix. The expressions
S*P, P\S, S/P work analogically.
Two permutation matrices can be multiplied or divided (if their sizes match), performing
a composition of permutations. Also a permutation matrix can be indexed by a permutation
vector (or two vectors), giving again a permutation matrix. Any other operations do not
generally yield a permutation matrix and will thus trigger the implicit conversion.

21.3 Functions That Are Aware of These Matrices
This section lists the built-in functions that are aware of diagonal and permutation matrices
on input, or can return them as output. Passed to other functions, these matrices will in
general trigger an implicit conversion. (Of course, user-defined dynamically linked functions
may also work with diagonal or permutation matrices).

21.3.1 Diagonal Matrix Functions
inv and pinv can be applied to a diagonal matrix, yielding again a diagonal matrix. det will
use an efficient straightforward calculation when given a diagonal matrix, as well as cond.
The following mapper functions can be applied to a diagonal matrix without converting it
to a full one: abs, real, imag, conj, sqrt. A diagonal matrix can also be returned from the
balance and svd functions. The sparse function will convert a diagonal matrix efficiently
to a sparse matrix.

21.3.2 Permutation Matrix Functions
inv and pinv will invert a permutation matrix, preserving its specialness. det can be applied
to a permutation matrix, efficiently calculating the sign of the permutation (which is equal
to the determinant).
A permutation matrix can also be returned from the built-in functions lu and qr, if a
pivoted factorization is requested.
The sparse function will convert a permutation matrix efficiently to a sparse matrix.
The find function will also work efficiently with a permutation matrix, making it possible
to conveniently obtain the permutation indices.

21.4 Examples of Usage
The following can be used to solve a linear system A*x = b using the pivoted LU factorization:
[L, U, P] = lu (A); ## now L*U = P*A
x = U \ (L \ P) * b;
This is one way to normalize columns of a matrix X to unit norm:
s = norm (X, "columns");
X /= diag (s);
The same can also be accomplished with broadcasting (see Section 19.2 [Broadcasting],
page 535):
s = norm (X, "columns");
X ./= s;

Chapter 21: Diagonal and Permutation Matrices

561

The following expression is a way to efficiently calculate the sign of a permutation, given
by a permutation vector p. It will also work in earlier versions of Octave, but slowly.
det (eye (length (p))(p, :))
Finally, here’s how to solve a linear system A*x = b with Tikhonov regularization (ridge
regression) using SVD (a skeleton only):
m = rows (A); n = columns (A);
[U, S, V] = svd (A);
## determine the regularization factor alpha
## alpha = ...
## transform to orthogonal basis
b = U’*b;
## Use the standard formula, replacing A with S.
## S is diagonal, so the following will be very fast and accurate.
x = (S’*S + alpha^2 * eye (n)) \ (S’ * b);
## transform to solution basis
x = V*x;

21.5 Differences in Treatment of Zero Elements
Making diagonal and permutation matrices special matrix objects in their own right and
the consequent usage of smarter algorithms for certain operations implies, as a side effect,
small differences in treating zeros. The contents of this section apply also to sparse matrices,
discussed in the following chapter. (see Chapter 22 [Sparse Matrices], page 563)
The IEEE floating point standard defines the result of the expressions 0*Inf and 0*NaN
as NaN. This is widely agreed to be a good compromise. Numerical software dealing with
structured and sparse matrices (including Octave) however, almost always makes a distinction between a "numerical zero" and an "assumed zero". A "numerical zero" is a zero
value occurring in a place where any floating-point value could occur. It is normally stored
somewhere in memory as an explicit value. An "assumed zero", on the contrary, is a zero
matrix element implied by the matrix structure (diagonal, triangular) or a sparsity pattern;
its value is usually not stored explicitly anywhere, but is implied by the underlying data
structure.
The primary distinction is that an assumed zero, when multiplied by any number, or
divided by any nonzero number, yields always a zero, even when, e.g., multiplied by Inf
or divided by NaN. The reason for this behavior is that the numerical multiplication is not
actually performed anywhere by the underlying algorithm; the result is just assumed to be
zero. Equivalently, one can say that the part of the computation involving assumed zeros
is performed symbolically, not numerically.
This behavior not only facilitates the most straightforward and efficient implementation
of algorithms, but also preserves certain useful invariants, like:
• scalar * diagonal matrix is a diagonal matrix
• sparse matrix / scalar preserves the sparsity pattern
• permutation matrix * matrix is equivalent to permuting rows
all of these natural mathematical truths would be invalidated by treating assumed zeros
as numerical ones.

562

GNU Octave

Note that matlab does not strictly follow this principle and converts assumed zeros to
numerical zeros in certain cases, while not doing so in other cases. As of today, there are
no intentions to mimic such behavior in Octave.
Examples of effects of assumed zeros vs. numerical zeros:
Inf * eye (3)
⇒
Inf
0
0
0
Inf
0
0
0
Inf
Inf * speye (3)
⇒
Compressed Column Sparse (rows = 3, cols = 3, nnz = 3 [33%])
(1, 1) -> Inf
(2, 2) -> Inf
(3, 3) -> Inf
Inf * full (eye (3))
⇒
Inf
NaN
NaN
NaN
Inf
NaN
NaN
NaN
Inf
diag (1:3) * [NaN; 1; 1]
⇒
NaN
2
3
sparse (1:3,1:3,1:3) * [NaN; 1; 1]
⇒
NaN
2
3
[1,0,0;0,2,0;0,0,3] * [NaN; 1; 1]
⇒
NaN
NaN
NaN

563

22 Sparse Matrices
22.1 Creation and Manipulation of Sparse Matrices
The size of mathematical problems that can be treated at any particular time is generally
limited by the available computing resources. Both, the speed of the computer and its
available memory place limitation on the problem size.
There are many classes of mathematical problems which give rise to matrices, where a
large number of the elements are zero. In this case it makes sense to have a special matrix
type to handle this class of problems where only the nonzero elements of the matrix are
stored. Not only does this reduce the amount of memory to store the matrix, but it also
means that operations on this type of matrix can take advantage of the a priori knowledge
of the positions of the nonzero elements to accelerate their calculations.
A matrix type that stores only the nonzero elements is generally called sparse. It is the
purpose of this document to discuss the basics of the storage and creation of sparse matrices
and the fundamental operations on them.

22.1.1 Storage of Sparse Matrices
It is not strictly speaking necessary for the user to understand how sparse matrices are
stored. However, such an understanding will help to get an understanding of the size of
sparse matrices. Understanding the storage technique is also necessary for those users
wishing to create their own oct-files.
There are many different means of storing sparse matrix data. What all of the methods
have in common is that they attempt to reduce the complexity and storage given a priori
knowledge of the particular class of problems that will be solved. A good summary of
the available techniques for storing sparse matrix is given by Saad1 . With full matrices,
knowledge of the point of an element of the matrix within the matrix is implied by its
position in the computers memory. However, this is not the case for sparse matrices, and
so the positions of the nonzero elements of the matrix must equally be stored.
An obvious way to do this is by storing the elements of the matrix as triplets, with two
elements being their position in the array (rows and column) and the third being the data
itself. This is conceptually easy to grasp, but requires more storage than is strictly needed.
The storage technique used within Octave is the compressed column format. It is similar
to the Yale format.2 In this format the position of each element in a row and the data are
stored as previously. However, if we assume that all elements in the same column are stored
adjacent in the computers memory, then we only need to store information on the number
of nonzero elements in each column, rather than their positions. Thus assuming that the
matrix has more nonzero elements than there are columns in the matrix, we win in terms
of the amount of memory used.
In fact, the column index contains one more element than the number of columns, with
the first element always being zero. The advantage of this is a simplification in the code, in
1
2

Y. Saad "SPARSKIT: A basic toolkit for sparse matrix computation", 1994, http://www-users.cs.
umn.edu/~saad/software/SPARSKIT/paper.ps
http://en.wikipedia.org/wiki/Sparse_matrix#Yale_format

564

GNU Octave

that there is no special case for the first or last columns. A short example, demonstrating
this in C is.
for (j = 0; j < nc; j++)
for (i = cidx(j); i < cidx(j+1); i++)
printf ("nonzero element (%i,%i) is %d\n",
ridx(i), j, data(i));
A clear understanding might be had by considering an example of how the above applies
to an example matrix. Consider the matrix
1
0
0

2
0
0

0
0
0

0
3
4

The nonzero elements of this matrix are
(1,
(1,
(2,
(3,

1)
2)
4)
4)

⇒
⇒
⇒
⇒

1
2
3
4

This will be stored as three vectors cidx, ridx and data, representing the column indexing,
row indexing and data respectively. The contents of these three vectors for the above matrix
will be
cidx = [0, 1, 2, 2, 4]
ridx = [0, 0, 1, 2]
data = [1, 2, 3, 4]
Note that this is the representation of these elements with the first row and column
assumed to start at zero, while in Octave itself the row and column indexing starts at one.
Thus the number of elements in the i-th column is given by cidx (i + 1) - cidx (i).
Although Octave uses a compressed column format, it should be noted that compressed
row formats are equally possible. However, in the context of mixed operations between
mixed sparse and dense matrices, it makes sense that the elements of the sparse matrices
are in the same order as the dense matrices. Octave stores dense matrices in column major
ordering, and so sparse matrices are equally stored in this manner.
A further constraint on the sparse matrix storage used by Octave is that all elements in
the rows are stored in increasing order of their row index, which makes certain operations
faster. However, it imposes the need to sort the elements on the creation of sparse matrices.
Having disordered elements is potentially an advantage in that it makes operations such as
concatenating two sparse matrices together easier and faster, however it adds complexity
and speed problems elsewhere.

22.1.2 Creating Sparse Matrices
There are several means to create sparse matrix.
Returned from a function
There are many functions that directly return sparse matrices. These include
speye, sprand, diag, etc.

Chapter 22: Sparse Matrices

565

Constructed from matrices or vectors
The function sparse allows a sparse matrix to be constructed from three vectors
representing the row, column and data. Alternatively, the function spconvert
uses a three column matrix format to allow easy importation of data from
elsewhere.
Created and then filled
The function sparse or spalloc can be used to create an empty matrix that is
then filled by the user
From a user binary program
The user can directly create the sparse matrix within an oct-file.
There are several basic functions to return specific sparse matrices. For example the
sparse identity matrix, is a matrix that is often needed. It therefore has its own function to
create it as speye (n) or speye (r, c), which creates an n-by-n or r-by-c sparse identity
matrix.
Another typical sparse matrix that is often needed is a random distribution of random
elements. The functions sprand and sprandn perform this for uniform and normal random
distributions of elements. They have exactly the same calling convention, where sprand
(r, c, d), creates an r-by-c sparse matrix with a density of filled elements of d.
Other functions of interest that directly create sparse matrices, are diag or its generalization spdiags, that can take the definition of the diagonals of the matrix and create the
sparse matrix that corresponds to this. For example,
s = diag (sparse (randn (1,n)), -1);
creates a sparse (n+1)-by-(n+1) sparse matrix with a single diagonal defined.

B =
[B,
B =
A =
A =

spdiags (A)
d] = spdiags (A)
spdiags (A, d)
spdiags (v, d, A)
spdiags (v, d, m, n)
A generalization of the function diag.
Called with a single input argument, the nonzero diagonals d of A are extracted.
With two arguments the diagonals to extract are given by the vector d.
The other two forms of spdiags modify the input matrix by replacing the diagonals.
They use the columns of v to replace the diagonals represented by the vector d. If the
sparse matrix A is defined then the diagonals of this matrix are replaced. Otherwise
a matrix of m by n is created with the diagonals given by the columns of v.
Negative values of d represent diagonals below the main diagonal, and positive values
of d diagonals above the main diagonal.
For example:
spdiags (reshape (1:12, 4, 3), [-1 0 1], 5, 4)
⇒ 5 10 0 0
1 6 11 0
0 2 7 12
0 0 3 8
0 0 0 4

566

GNU Octave

See also: [diag], page 456.

s = speye (m, n)
s = speye (m)
s = speye (sz)
Return a sparse identity matrix of size mxn.
The implementation is significantly more efficient than sparse (eye (m)) as the full
matrix is not constructed.
Called with a single argument a square matrix of size m-by-m is created. If called
with a single vector argument sz, this argument is taken to be the size of the matrix
to create.
See also: [sparse], page 568, [spdiags], page 565, [eye], page 457.

r = spones (S)
Replace the nonzero entries of S with ones.
This creates a sparse matrix with the same structure as S.
See also: [sparse], page 568, [sprand], page 566, [sprandn], page 566, [sprandsym],
page 567, [spfun], page 540, [spy], page 572.

sprand (m, n, d)
sprand (m, n, d, rc)
sprand (s)
Generate a sparse matrix with uniformly distributed random values.
The size of the matrix is mxn with a density of values d. d must be between 0 and 1.
Values will be uniformly distributed on the interval (0, 1).
If called with a single matrix argument, a sparse matrix is generated with random
values wherever the matrix s is nonzero.
If called with a scalar fourth argument rc, a random sparse matrix with reciprocal
condition number rc is generated. If rc is a vector, then it specifies the first singular
values of the generated matrix (length (rc) <= min (m, n)).
See also: [sprandn], page 566, [sprandsym], page 567, [rand], page 459.

sprandn (m, n, d)
sprandn (m, n, d, rc)
sprandn (s)
Generate a sparse matrix with normally distributed random values.
The size of the matrix is mxn with a density of values d. d must be between 0 and 1.
Values will be normally distributed with a mean of 0 and a variance of 1.
If called with a single matrix argument, a sparse matrix is generated with random
values wherever the matrix s is nonzero.
If called with a scalar fourth argument rc, a random sparse matrix with reciprocal
condition number rc is generated. If rc is a vector, then it specifies the first singular
values of the generated matrix (length (rc) <= min (m, n)).
See also: [sprand], page 566, [sprandsym], page 567, [randn], page 461.

Chapter 22: Sparse Matrices

567

sprandsym (n, d)
sprandsym (s)
Generate a symmetric random sparse matrix.
The size of the matrix will be nxn, with a density of values given by d. d must be
between 0 and 1 inclusive. Values will be normally distributed with a mean of zero
and a variance of 1.
If called with a single matrix argument, a random sparse matrix is generated wherever
the matrix s is nonzero in its lower triangular part.
See also: [sprand], page 566, [sprandn], page 566, [spones], page 566, [sparse],
page 568.
The recommended way for the user to create a sparse matrix, is to create two vectors
containing the row and column index of the data and a third vector of the same size
containing the data to be stored. For example,
ri = ci = d = [];
for j = 1:c
ri = [ri; randperm(r,n)’];
ci = [ci; j*ones(n,1)];
d = [d; rand(n,1)];
endfor
s = sparse (ri, ci, d, r, c);
creates an r-by-c sparse matrix with a random distribution of n ( 1
-> 2
-> 3

An example of creating and filling a matrix might be
k = 5;
nz = r * k;
s = spalloc (r, c, nz)
for j = 1:c
idx = randperm (r);
s (:, j) = [zeros(r - k, 1); ...
rand(k, 1)] (idx);
endfor

568

GNU Octave

It should be noted, that due to the way that the Octave assignment functions are written
that the assignment will reallocate the memory used by the sparse matrix at each iteration
of the above loop. Therefore the spalloc function ignores the nz argument and does not
pre-assign the memory for the matrix. Therefore, it is vitally important that code using
to above structure should be vectorized as much as possible to minimize the number of
assignments and reduce the number of memory allocations.

FM = full (SM)
Return a full storage matrix from a sparse, diagonal, or permutation matrix, or a
range.
See also: [sparse], page 568, [issparse], page 570.

s = spalloc (m, n, nz)
Create an m-by-n sparse matrix with pre-allocated space for at most nz nonzero
elements.
This is useful for building a matrix incrementally by a sequence of indexed assignments. Subsequent indexed assignments after spalloc will reuse the pre-allocated
memory, provided they are of one of the simple forms
• s(I:J) = x
• s(:,I:J) = x
• s(K:L,I:J) = x
and
•
•
•

that the following conditions are met:
the assignment does not decrease nnz (S).
after the assignment, nnz (S) does not exceed nz.
no index is out of bounds.

Partial movement of data may still occur, but in general the assignment will be more
memory and time efficient under these circumstances. In particular, it is possible to
efficiently build a pre-allocated sparse matrix from a contiguous block of columns.
The amount of pre-allocated memory for a given matrix may be queried using the
function nzmax.
See also: [nzmax], page 570, [sparse], page 568.

s
s
s
s
s
s

=
=
=
=
=
=

(a)
(i, j, sv, m, n)
(i, j, sv)
(m, n)
(i, j, s, m, n, "unique")
(i, j, sv, m, n, nzmax)
Create a sparse matrix from a full matrix, or row, column, value triplets.
If a is a full matrix, convert it to a sparse matrix representation, removing all zero
values in the process.
Given the integer index vectors i and j, and a 1-by-nnz vector of real or complex values
sv, construct the sparse matrix S(i(k),j(k)) = sv(k) with overall dimensions m and
n. If any of sv, i or j are scalars, they are expanded to have a common size.

sparse
sparse
sparse
sparse
sparse
sparse

Chapter 22: Sparse Matrices

569

If m or n are not specified their values are derived from the maximum index in the
vectors i and j as given by m = max (i), n = max (j).
Note: if multiple values are specified with the same i, j indices, the corresponding
value in s will be the sum of the values at the repeated location. See accumarray
for an example of how to produce different behavior, such as taking the minimum
instead.
If the option "unique" is given, and more than one value is specified at the same i, j
indices, then the last specified value will be used.
sparse (m, n) will create an empty mxn sparse matrix and is equivalent to sparse
([], [], [], m, n)
The argument nzmax is ignored but accepted for compatibility with matlab.
Example 1 (sum at repeated indices):
i = [1 1 2]; j = [1 1 2]; sv = [3 4 5];
sparse (i, j, sv, 3, 4)
⇒
Compressed Column Sparse (rows = 3, cols = 4, nnz = 2 [17%])
(1, 1) ->
(2, 2) ->

7
5

Example 2 ("unique" option):
i = [1 1 2]; j = [1 1 2]; sv = [3 4 5];
sparse (i, j, sv, 3, 4, "unique")
⇒
Compressed Column Sparse (rows = 3, cols = 4, nnz = 2 [17%])
(1, 1) ->
(2, 2) ->

4
5

See also: [full], page 568, [accumarray], page 543, [spalloc], page 568, [spdiags],
page 565, [speye], page 566, [spones], page 566, [sprand], page 566, [sprandn],
page 566, [sprandsym], page 567, [spconvert], page 569, [spfun], page 540.

x = spconvert (m)
Convert a simple sparse matrix format easily generated by other programs into Octave’s internal sparse format.
The input m is either a 3 or 4 column real matrix, containing the row, column, real,
and imaginary parts of the elements of the sparse matrix. An element with a zero
real and imaginary part can be used to force a particular matrix size.
See also: [sparse], page 568.
The above problem of memory reallocation can be avoided in oct-files. However, the
construction of a sparse matrix from an oct-file is more complex than can be discussed
here. See Appendix A [External Code Interface], page 875, for a full description of the
techniques involved.

570

GNU Octave

22.1.3 Finding Information about Sparse Matrices
There are a number of functions that allow information concerning sparse matrices to be
obtained. The most basic of these is issparse that identifies whether a particular Octave
object is in fact a sparse matrix.
Another very basic function is nnz that returns the number of nonzero entries there are
in a sparse matrix, while the function nzmax returns the amount of storage allocated to the
sparse matrix. Note that Octave tends to crop unused memory at the first opportunity for
sparse objects. There are some cases of user created sparse objects where the value returned
by nzmax will not be the same as nnz, but in general they will give the same result. The
function spstats returns some basic statistics on the columns of a sparse matrix including
the number of elements, the mean and the variance of each column.

issparse (x)
Return true if x is a sparse matrix.
See also: [ismatrix], page 63.

n = nnz (a)
Return the number of nonzero elements in a.
See also: [nzmax], page 570, [nonzeros], page 570, [find], page 445.

nonzeros (s)
Return a vector of the nonzero values of the sparse matrix s.
See also: [find], page 445, [nnz], page 570.

n = nzmax (SM)
Return the amount of storage allocated to the sparse matrix SM.
Note that Octave tends to crop unused memory at the first opportunity for sparse
objects. Thus, in general the value of nzmax will be the same as nnz except for some
cases of user-created sparse objects.
See also: [nnz], page 570, [spalloc], page 568, [sparse], page 568.

[count, mean, var] = spstats (S)
[count, mean, var] = spstats (S, j)
Return the stats for the nonzero elements of the sparse matrix S.
count is the number of nonzeros in each column, mean is the mean of the nonzeros in
each column, and var is the variance of the nonzeros in each column.
Called with two input arguments, if S is the data and j is the bin number for the
data, compute the stats for each bin. In this case, bins can contain data values of
zero, whereas with spstats (S) the zeros may disappear.
When solving linear equations involving sparse matrices Octave determines the means to
solve the equation based on the type of the matrix (see Section 22.2 [Sparse Linear Algebra],
page 584). Octave probes the matrix type when the div (/) or ldiv (\) operator is first used
with the matrix and then caches the type. However the matrix type function can be used

Chapter 22: Sparse Matrices

571

to determine the type of the sparse matrix prior to use of the div or ldiv operators. For
example,
a = tril (sprandn (1024, 1024, 0.02), -1) ...
+ speye (1024);
matrix_type (a);
ans = Lower
shows that Octave correctly determines the matrix type for lower triangular matrices. matrix type can also be used to force the type of a matrix to be a particular type. For
example:
a = matrix_type (tril (sprandn (1024, ...
1024, 0.02), -1) + speye (1024), "Lower");
This allows the cost of determining the matrix type to be avoided. However, incorrectly
defining the matrix type will result in incorrect results from solutions of linear equations,
and so it is entirely the responsibility of the user to correctly identify the matrix type
There are several graphical means of finding out information about sparse matrices. The
first is the spy command, which displays the structure of the nonzero elements of the matrix.
See Figure 22.1, for an example of the use of spy. More advanced graphical information can
be obtained with the treeplot, etreeplot and gplot commands.

0

50

100

150

200
0

50

100

150

200

Figure 22.1: Structure of simple sparse matrix.
One use of sparse matrices is in graph theory, where the interconnections between nodes
are represented as an adjacency matrix. That is, if the i-th node in a graph is connected to
the j-th node. Then the ij-th node (and in the case of undirected graphs the ji-th node) of the
sparse adjacency matrix is nonzero. If each node is then associated with a set of coordinates,
then the gplot command can be used to graphically display the interconnections between
nodes.
As a trivial example of the use of gplot consider the example,

572

GNU Octave

A = sparse ([2,6,1,3,2,4,3,5,4,6,1,5],
[1,1,2,2,3,3,4,4,5,5,6,6],1,6,6);
xy = [0,4,8,6,4,2;5,0,5,7,5,7]’;
gplot (A,xy)
which creates an adjacency matrix A where node 1 is connected to nodes 2 and 6, node 2
with nodes 1 and 3, etc. The coordinates of the nodes are given in the n-by-2 matrix xy.
See Figure 22.2.

7
6
5
4
3
2
1
0
0

2

4

6

8

Figure 22.2: Simple use of the gplot command.
The dependencies between the nodes of a Cholesky factorization can be calculated in
linear time without explicitly needing to calculate the Cholesky factorization by the etree
command. This command returns the elimination tree of the matrix and can be displayed
graphically by the command treeplot (etree (A)) if A is symmetric or treeplot (etree
(A+A’)) otherwise.

spy (x)
spy ( . . . , markersize)
spy ( . . . , line_spec)
Plot the sparsity pattern of the sparse matrix x.
If the argument markersize is given as a scalar value, it is used to determine the point
size in the plot.
If the string line spec is given it is passed to plot and determines the appearance of
the plot.
See also: [plot], page 288, [gplot], page 573.

p = etree (S)
p = etree (S, typ)
[p, q] = etree (S, typ)
Return the elimination tree for the matrix S.

Chapter 22: Sparse Matrices

573

By default S is assumed to be symmetric and the symmetric elimination tree is returned. The argument typ controls whether a symmetric or column elimination tree is
returned. Valid values of typ are "sym" or "col", for symmetric or column elimination
tree respectively.
Called with a second argument, etree also returns the postorder permutations on
the tree.

etreeplot (A)
etreeplot (A, node_style, edge_style)
Plot the elimination tree of the matrix A or A+A’ if A in not symmetric.
The optional parameters node style and edge style define the output style.
See also: [treeplot], page 573, [gplot], page 573.

gplot (A, xy)
gplot (A, xy, line_style)
[x, y] = gplot (A, xy)
Plot a graph defined by A and xy in the graph theory sense.
A is the adjacency matrix of the array to be plotted and xy is an n-by-2 matrix
containing the coordinates of the nodes of the graph.
The optional parameter line style defines the output style for the plot. Called with
no output arguments the graph is plotted directly. Otherwise, return the coordinates
of the plot in x and y.
See also: [treeplot], page 573, [etreeplot], page 573, [spy], page 572.

treeplot (tree)
treeplot (tree, node_style, edge_style)
Produce a graph of tree or forest.
The first argument is vector of predecessors.
The optional parameters node style and edge style define the output plot style.
The complexity of the algorithm is O(n) in terms of is time and memory requirements.
See also: [etreeplot], page 573, [gplot], page 573.

treelayout (tree)
treelayout (tree, permutation)
treelayout lays out a tree or a forest.
The first argument tree is a vector of predecessors.
The parameter permutation is an optional postorder permutation.
The complexity of the algorithm is O(n) in terms of time and memory requirements.
See also: [etreeplot], page 573, [gplot], page 573, [treeplot], page 573.

22.1.4 Basic Operators and Functions on Sparse Matrices

574

GNU Octave

22.1.4.1 Sparse Functions
Many Octave functions have been overloaded to work with either sparse or full matrices.
There is no difference in calling convention when using an overloaded function with a sparse
matrix, however, there is also no access to potentially sparse-specific features. At any time
the sparse matrix specific version of a function can be used by explicitly calling its function
name.
The table below lists all of the sparse functions of Octave. Note that the names of the
specific sparse forms of the functions are typically the same as the general versions with a
sp prefix. In the table below, and in the rest of this article, the specific sparse versions of
functions are used.
Generate sparse matrices:
spalloc, spdiags, speye, sprand, sprandn, sprandsym
Sparse matrix conversion:
full, sparse, spconvert
Manipulate sparse matrices
issparse, nnz, nonzeros, nzmax, spfun, spones, spy
Graph Theory:
etree, etreeplot, gplot, treeplot
Sparse matrix reordering:
amd, ccolamd, colamd, colperm, csymamd, dmperm, symamd, randperm, symrcm
Linear algebra:
condest, eigs, matrix type, normest, normest1, sprank, spaugment, svds
Iterative techniques:
ichol, ilu, pcg, pcr
Miscellaneous:
spparms, symbfact, spstats
In addition all of the standard Octave mapper functions (i.e., basic math functions that
take a single argument) such as abs, etc. can accept sparse matrices. The reader is referred
to the documentation supplied with these functions within Octave itself for further details.

22.1.4.2 Return Types of Operators and Functions
The two basic reasons to use sparse matrices are to reduce the memory usage and to not have
to do calculations on zero elements. The two are closely related in that the computation
time on a sparse matrix operator or function is roughly linear with the number of nonzero
elements.
Therefore, there is a certain density of nonzero elements of a matrix where it no longer
makes sense to store it as a sparse matrix, but rather as a full matrix. For this reason
operators and functions that have a high probability of returning a full matrix will always
return one. For example adding a scalar constant to a sparse matrix will almost always
make it a full matrix, and so the example,

Chapter 22: Sparse Matrices

575

speye (3) + 0
⇒
1 0 0
0 1 0
0 0 1
returns a full matrix as can be seen.
Additionally, if sparse_auto_mutate is true, all sparse functions test the amount of
memory occupied by the sparse matrix to see if the amount of storage used is larger than
the amount used by the full equivalent. Therefore speye (2) * 1 will return a full matrix
as the memory used is smaller for the full version than the sparse version.
As all of the mixed operators and functions between full and sparse matrices exist, in
general this does not cause any problems. However, one area where it does cause a problem
is where a sparse matrix is promoted to a full matrix, where subsequent operations would
resparsify the matrix. Such cases are rare, but can be artificially created, for example
(fliplr (speye (3)) + speye (3)) - speye (3) gives a full matrix when it should give a
sparse one. In general, where such cases occur, they impose only a small memory penalty.
There is however one known case where this behavior of Octave’s sparse matrices will
cause a problem. That is in the handling of the diag function. Whether diag returns a
sparse or full matrix depending on the type of its input arguments. So
a = diag (sparse ([1,2,3]), -1);
should return a sparse matrix. To ensure this actually happens, the sparse function, and
other functions based on it like speye, always returns a sparse matrix, even if the memory
used will be larger than its full representation.

val = sparse_auto_mutate ()
old_val = sparse_auto_mutate (new_val)
sparse_auto_mutate (new_val, "local")
Query or set the internal variable that controls whether Octave will automatically
mutate sparse matrices to full matrices to save memory.
For example:
s = speye (3);
sparse_auto_mutate (false);
s(:, 1) = 1;
typeinfo (s)
⇒ sparse matrix
sparse_auto_mutate (true);
s(1, :) = 1;
typeinfo (s)
⇒ matrix

When called from inside a function with the "local" option, the variable is changed
locally for the function and any subroutines it calls. The original variable value is
restored when exiting the function.
Note that the sparse_auto_mutate option is incompatible with matlab, and so it is
off by default.

576

GNU Octave

22.1.4.3 Mathematical Considerations
The attempt has been made to make sparse matrices behave in exactly the same manner
as there full counterparts. However, there are certain differences and especially differences
with other products sparse implementations.
First, the "./" and ".^" operators must be used with care. Consider what the examples
s = speye
a1 = s .^
a2 = s .^
a3 = s .^
a4 = s ./
a5 = 2 ./
a6 = s ./

(4);
2;
s;
-2;
2;
s;
s;

will give. The first example of s raised to the power of 2 causes no problems. However s
raised element-wise to itself involves a large number of terms 0 .^ 0 which is 1. There s .^
s is a full matrix.
Likewise s .^ -2 involves terms like 0 .^ -2 which is infinity, and so s .^ -2 is equally
a full matrix.
For the "./" operator s ./ 2 has no problems, but 2 ./ s involves a large number of
infinity terms as well and is equally a full matrix. The case of s ./ s involves terms like 0
./ 0 which is a NaN and so this is equally a full matrix with the zero elements of s filled
with NaN values.
The above behavior is consistent with full matrices, but is not consistent with sparse
implementations in other products.
A particular problem of sparse matrices comes about due to the fact that as the zeros
are not stored, the sign-bit of these zeros is equally not stored. In certain cases the sign-bit
of zero is important. For example:
a = 0 ./ [-1, 1; 1, -1];
b = 1 ./ a
⇒ -Inf
Inf
Inf
-Inf
c = 1 ./ sparse (a)
⇒ Inf
Inf
Inf
Inf
To correct this behavior would mean that zero elements with a negative sign-bit would
need to be stored in the matrix to ensure that their sign-bit was respected. This is not done
at this time, for reasons of efficiency, and so the user is warned that calculations where the
sign-bit of zero is important must not be done using sparse matrices.
In general any function or operator used on a sparse matrix will result in a sparse
matrix with the same or a larger number of nonzero elements than the original matrix.
This is particularly true for the important case of sparse matrix factorizations. The usual
way to address this is to reorder the matrix, such that its factorization is sparser than
the factorization of the original matrix. That is the factorization of L * U = P * S * Q has
sparser terms L and U than the equivalent factorization L * U = S.

Chapter 22: Sparse Matrices

577

Several functions are available to reorder depending on the type of the matrix to be
factorized. If the matrix is symmetric positive-definite, then symamd or csymamd should be
used. Otherwise amd, colamd or ccolamd should be used. For completeness the reordering
functions colperm and randperm are also available.

See Figure 22.3, for an example of the structure of a simple positive definite matrix.

0

50

100

150

200
0

50

100

150

200

Figure 22.3: Structure of simple sparse matrix.

The standard Cholesky factorization of this matrix can be obtained by the same command that would be used for a full matrix. This can be visualized with the command r =
chol (A); spy (r);. See Figure 22.4. The original matrix had 598 nonzero terms, while
this Cholesky factorization has 10200, with only half of the symmetric matrix being stored.
This is a significant level of fill in, and although not an issue for such a small test case, can
represents a large overhead in working with other sparse matrices.

The appropriate sparsity preserving permutation of the original matrix is given by
symamd and the factorization using this reordering can be visualized using the command
q = symamd (A); r = chol (A(q,q)); spy (r). This gives 399 nonzero terms which is a
significant improvement.

The Cholesky factorization itself can be used to determine the appropriate sparsity
preserving reordering of the matrix during the factorization, In that case this might be
obtained with three return arguments as [r, p, q] = chol (A); spy (r).

578

GNU Octave

0

50

100

150

200
0

50

100

150

200

Figure 22.4: Structure of the unpermuted Cholesky factorization of the above matrix.

0

50

100

150

200
0

50

100

150

200

Figure 22.5: Structure of the permuted Cholesky factorization of the above matrix.
In the case of an asymmetric matrix, the appropriate sparsity preserving permutation is
colamd and the factorization using this reordering can be visualized using the command q
= colamd (A); [l, u, p] = lu (A(:,q)); spy (l+u).
Finally, Octave implicitly reorders the matrix when using the div (/) and ldiv (\) operators, and so no the user does not need to explicitly reorder the matrix to maximize
performance.

Chapter 22: Sparse Matrices

579

p = amd (S)
p = amd (S, opts)
Return the approximate minimum degree permutation of a matrix.
This is a permutation such that the Cholesky factorization of S (p, p) tends to be
sparser than the Cholesky factorization of S itself. amd is typically faster than symamd
but serves a similar purpose.
The optional parameter opts is a structure that controls the behavior of amd. The
fields of the structure are
opts.dense Determines what amd considers to be a dense row or column of the input
matrix. Rows or columns with more than max (16, (dense * sqrt (n)))
entries, where n is the order of the matrix S, are ignored by amd during
the calculation of the permutation. The value of dense must be a positive
scalar and the default value is 10.0
opts.aggressive
If this value is a nonzero scalar, then amd performs aggressive absorption.
The default is not to perform aggressive absorption.
The author of the code itself is Timothy A. Davis davis@cise.ufl.edu, University
of Florida (see http://www.cise.ufl.edu/research/sparse/amd).
See also: [symamd], page 582, [colamd], page 580.

p =
p =
p =
[p,

ccolamd (S)
ccolamd (S, knobs)
ccolamd (S, knobs, cmember)
stats] = ccolamd ( . . . )
Constrained column approximate minimum degree permutation.
p = ccolamd (S) returns the column approximate minimum degree permutation vector for the sparse matrix S. For a non-symmetric matrix S, S(:, p) tends to have
sparser LU factors than S. chol (S(:, p)’ * S(:, p)) also tends to be sparser than
chol (S’ * S). p = ccolamd (S, 1) optimizes the ordering for lu (S(:, p)). The
ordering is followed by a column elimination tree post-ordering.
knobs is an optional 1-element to 5-element input vector, with a default value of [0
10 10 1 0] if not present or empty. Entries not present are set to their defaults.
knobs(1)

if nonzero, the ordering is optimized for lu (S(:, p)). It will be a poor
ordering for chol (S(:, p)’ * S(:, p)). This is the most important
knob for ccolamd.

knobs(2)

if S is m-by-n, rows with more than max (16, knobs(2) * sqrt (n)) entries are ignored.

knobs(3)

columns with more than max (16, knobs(3) * sqrt (min (m, n))) entries are ignored and ordered last in the output permutation (subject to
the cmember constraints).

knobs(4)

if nonzero, aggressive absorption is performed.

knobs(5)

if nonzero, statistics and knobs are printed.

580

GNU Octave

cmember is an optional vector of length n. It defines the constraints on the column
ordering. If cmember(j) = c, then column j is in constraint set c (c must be in the
range 1 to n). In the output permutation p, all columns in set 1 appear first, followed
by all columns in set 2, and so on. cmember = ones (1,n) if not present or empty.
ccolamd (S, [], 1 : n) returns 1 : n
p = ccolamd (S) is about the same as p = colamd (S). knobs and its default values
differ. colamd always does aggressive absorption, and it finds an ordering suitable for
both lu (S(:, p)) and chol (S(:, p)’ * S(:, p)); it cannot optimize its ordering
for lu (S(:, p)) to the extent that ccolamd (S, 1) can.
stats is an optional 20-element output vector that provides data about the ordering
and the validity of the input matrix S. Ordering statistics are in stats(1 : 3).
stats(1) and stats(2) are the number of dense or empty rows and columns ignored
by ccolamd and stats(3) is the number of garbage collections performed on the
internal data structure used by ccolamd (roughly of size 2.2 * nnz (S) + 4 * m + 7
* n integers).
stats(4 : 7) provide information if CCOLAMD was able to continue. The matrix is
OK if stats(4) is zero, or 1 if invalid. stats(5) is the rightmost column index that
is unsorted or contains duplicate entries, or zero if no such column exists. stats(6)
is the last seen duplicate or out-of-order row index in the column index given by
stats(5), or zero if no such row index exists. stats(7) is the number of duplicate
or out-of-order row indices. stats(8 : 20) is always zero in the current version of
ccolamd (reserved for future use).
The authors of the code itself are S. Larimore, T. Davis (Univ. of Florida) and
S. Rajamanickam in collaboration with J. Bilbert and E. Ng. Supported by the
National Science Foundation (DMS-9504974, DMS-9803599, CCR-0203270), and a
grant from Sandia National Lab. See http://www.cise.ufl.edu/research/sparse
for ccolamd, csymamd, amd, colamd, symamd, and other related orderings.
See also: [colamd], page 580, [csymamd], page 581.

p =
p =
[p,
[p,

colamd
colamd
stats]
stats]

(S)
(S, knobs)

= colamd (S)
= colamd (S, knobs)

Compute the column approximate minimum degree permutation.
p = colamd (S) returns the column approximate minimum degree permutation vector
for the sparse matrix S. For a non-symmetric matrix S, S(:,p) tends to have sparser
LU factors than S. The Cholesky factorization of S(:,p)’ * S(:,p) also tends to be
sparser than that of S’ * S.
knobs is an optional one- to three-element input vector. If S is m-by-n, then rows
with more than max(16,knobs(1)*sqrt(n)) entries are ignored. Columns with more
than max (16,knobs(2)*sqrt(min(m,n))) entries are removed prior to ordering, and
ordered last in the output permutation p. Only completely dense rows or columns
are removed if knobs(1) and knobs(2) are < 0, respectively. If knobs(3) is nonzero,
stats and knobs are printed. The default is knobs = [10 10 0]. Note that knobs
differs from earlier versions of colamd.

Chapter 22: Sparse Matrices

581

stats is an optional 20-element output vector that provides data about the ordering and the validity of the input matrix S. Ordering statistics are in stats(1:3).
stats(1) and stats(2) are the number of dense or empty rows and columns ignored
by colamd and stats(3) is the number of garbage collections performed on the
internal data structure used by colamd (roughly of size 2.2 * nnz(S) + 4 * m + 7 *
n integers).
Octave built-in functions are intended to generate valid sparse matrices, with no
duplicate entries, with ascending row indices of the nonzeros in each column, with a
non-negative number of entries in each column (!) and so on. If a matrix is invalid,
then colamd may or may not be able to continue. If there are duplicate entries (a
row index appears two or more times in the same column) or if the row indices in
a column are out of order, then colamd can correct these errors by ignoring the
duplicate entries and sorting each column of its internal copy of the matrix S (the
input matrix S is not repaired, however). If a matrix is invalid in other ways then
colamd cannot continue, an error message is printed, and no output arguments (p
or stats) are returned. colamd is thus a simple way to check a sparse matrix to see
if it’s valid.
stats(4:7) provide information if colamd was able to continue. The matrix is OK
if stats(4) is zero, or 1 if invalid. stats(5) is the rightmost column index that is
unsorted or contains duplicate entries, or zero if no such column exists. stats(6)
is the last seen duplicate or out-of-order row index in the column index given by
stats(5), or zero if no such row index exists. stats(7) is the number of duplicate
or out-of-order row indices. stats(8:20) is always zero in the current version of
colamd (reserved for future use).
The ordering is followed by a column elimination tree post-ordering.
The authors of the code itself are Stefan I. Larimore and Timothy A. Davis
davis@cise.ufl.edu, University of Florida. The algorithm was developed in
collaboration with John Gilbert, Xerox PARC, and Esmond Ng, Oak Ridge National
Laboratory. (see http://www.cise.ufl.edu/research/sparse/colamd)
See also: [colperm], page 581, [symamd], page 582, [ccolamd], page 579.

p = colperm (s)
Return the column permutations such that the columns of s (:, p) are ordered in
terms of increasing number of nonzero elements.
If s is symmetric, then p is chosen such that s (p, p) orders the rows and columns
with increasing number of nonzeros elements.

p =
p =
p =
[p,

csymamd (S)
csymamd (S, knobs)
csymamd (S, knobs, cmember)
stats] = csymamd ( . . . )
For a symmetric positive definite matrix S, return the permutation vector p such that
S(p,p) tends to have a sparser Cholesky factor than S.
Sometimes csymamd works well for symmetric indefinite matrices too. The matrix S
is assumed to be symmetric; only the strictly lower triangular part is referenced. S
must be square. The ordering is followed by an elimination tree post-ordering.

582

GNU Octave

knobs is an optional 1-element to 3-element input vector, with a default value of [10
1 0]. Entries not present are set to their defaults.
knobs(1)

If S is n-by-n, then rows and columns with more than
max(16,knobs(1)*sqrt(n)) entries are ignored, and ordered
last in the output permutation (subject to the cmember constraints).

knobs(2)

If nonzero, aggressive absorption is performed.

knobs(3)

If nonzero, statistics and knobs are printed.

cmember is an optional vector of length n. It defines the constraints on the ordering.
If cmember(j) = S, then row/column j is in constraint set c (c must be in the range
1 to n). In the output permutation p, rows/columns in set 1 appear first, followed by
all rows/columns in set 2, and so on. cmember = ones (1,n) if not present or empty.
csymamd (S,[],1:n) returns 1:n.
p = csymamd (S) is about the same as p = symamd (S). knobs and its default values
differ.
stats(4:7) provide information if CCOLAMD was able to continue. The matrix is
OK if stats(4) is zero, or 1 if invalid. stats(5) is the rightmost column index that
is unsorted or contains duplicate entries, or zero if no such column exists. stats(6)
is the last seen duplicate or out-of-order row index in the column index given by
stats(5), or zero if no such row index exists. stats(7) is the number of duplicate
or out-of-order row indices. stats(8:20) is always zero in the current version of
ccolamd (reserved for future use).
The authors of the code itself are S. Larimore, T. Davis (Univ. of Florida) and
S. Rajamanickam in collaboration with J. Bilbert and E. Ng. Supported by the
National Science Foundation (DMS-9504974, DMS-9803599, CCR-0203270), and a
grant from Sandia National Lab. See http://www.cise.ufl.edu/research/sparse
for ccolamd, csymamd, amd, colamd, symamd, and other related orderings.
See also: [symamd], page 582, [ccolamd], page 579.

p = dmperm (S)
[p, q, r, S] = dmperm (S)
Perform a Dulmage-Mendelsohn permutation of the sparse matrix S.
With a single output argument dmperm performs the row permutations p such that
S(p,:) has no zero elements on the diagonal.
Called with two or more output arguments, returns the row and column permutations,
such that S(p, q) is in block triangular form. The values of r and S define the
boundaries of the blocks. If S is square then r == S.
The method used is described in: A. Pothen & C.-J. Fan. Computing the Block
Triangular Form of a Sparse Matrix. ACM Trans. Math. Software, 16(4):303-324,
1990.
See also: [colamd], page 580, [ccolamd], page 579.

p = symamd (S)
p = symamd (S, knobs)
[p, stats] = symamd (S)

Chapter 22: Sparse Matrices

583

[p, stats] = symamd (S, knobs)
For a symmetric positive definite matrix S, returns the permutation vector p such
that S(p, p) tends to have a sparser Cholesky factor than S.
Sometimes symamd works well for symmetric indefinite matrices too. The matrix S
is assumed to be symmetric; only the strictly lower triangular part is referenced. S
must be square.
knobs is an optional one- to two-element input vector. If S is n-by-n, then rows
and columns with more than max (16,knobs(1)*sqrt(n)) entries are removed prior
to ordering, and ordered last in the output permutation p. No rows/columns are
removed if knobs(1) < 0. If knobs (2) is nonzero, stats and knobs are printed. The
default is knobs = [10 0]. Note that knobs differs from earlier versions of symamd.
stats is an optional 20-element output vector that provides data about the ordering and the validity of the input matrix S. Ordering statistics are in stats(1:3).
stats(1) = stats(2) is the number of dense or empty rows and columns ignored by
SYMAMD and stats(3) is the number of garbage collections performed on the internal data structure used by SYMAMD (roughly of size 8.4 * nnz (tril (S, -1))
+ 9 * n integers).
Octave built-in functions are intended to generate valid sparse matrices, with no
duplicate entries, with ascending row indices of the nonzeros in each column, with a
non-negative number of entries in each column (!) and so on. If a matrix is invalid,
then SYMAMD may or may not be able to continue. If there are duplicate entries
(a row index appears two or more times in the same column) or if the row indices in
a column are out of order, then SYMAMD can correct these errors by ignoring the
duplicate entries and sorting each column of its internal copy of the matrix S (the
input matrix S is not repaired, however). If a matrix is invalid in other ways then
SYMAMD cannot continue, an error message is printed, and no output arguments (p
or stats) are returned. SYMAMD is thus a simple way to check a sparse matrix to
see if it’s valid.
stats(4:7) provide information if SYMAMD was able to continue. The matrix is
OK if stats (4) is zero, or 1 if invalid. stats(5) is the rightmost column index that
is unsorted or contains duplicate entries, or zero if no such column exists. stats(6)
is the last seen duplicate or out-of-order row index in the column index given by
stats(5), or zero if no such row index exists. stats(7) is the number of duplicate
or out-of-order row indices. stats(8:20) is always zero in the current version of
SYMAMD (reserved for future use).
The ordering is followed by a column elimination tree post-ordering.
The authors of the code itself are Stefan I. Larimore and Timothy A. Davis
davis@cise.ufl.edu, University of Florida. The algorithm was developed in
collaboration with John Gilbert, Xerox PARC, and Esmond Ng, Oak Ridge National
Laboratory. (see http://www.cise.ufl.edu/research/sparse/colamd)
See also: [colperm], page 581, [colamd], page 580.

p = symrcm (S)
Return the symmetric reverse Cuthill-McKee permutation of S.

584

GNU Octave

p is a permutation vector such that S(p, p) tends to have its diagonal elements closer
to the diagonal than S. This is a good preordering for LU or Cholesky factorization
of matrices that come from “long, skinny” problems. It works for both symmetric
and asymmetric S.
The algorithm represents a heuristic approach to the NP-complete bandwidth minimization problem. The implementation is based in the descriptions found in
E. Cuthill, J. McKee. Reducing the Bandwidth of Sparse Symmetric Matrices. Proceedings of the 24th ACM National Conference, 157–172 1969, Brandon Press, New
Jersey.
A. George, J.W.H. Liu. Computer Solution of Large Sparse Positive Definite Systems,
Prentice Hall Series in Computational Mathematics, ISBN 0-13-165274-5, 1981.
See also: [colperm], page 581, [colamd], page 580, [symamd], page 582.

22.2 Linear Algebra on Sparse Matrices
Octave includes a polymorphic solver for sparse matrices, where the exact solver used to
factorize the matrix, depends on the properties of the sparse matrix itself. Generally, the
cost of determining the matrix type is small relative to the cost of factorizing the matrix
itself, but in any case the matrix type is cached once it is calculated, so that it is not
re-determined each time it is used in a linear equation.
The selection tree for how the linear equation is solve is
1. If the matrix is diagonal, solve directly and goto 8
2. If the matrix is a permuted diagonal, solve directly taking into account the permutations. Goto 8
3. If the matrix is square, banded and if the band density is less than that given by
spparms ("bandden") continue, else goto 4.
a. If the matrix is tridiagonal and the right-hand side is not sparse continue, else goto
3b.
1. If the matrix is Hermitian, with a positive real diagonal, attempt Cholesky factorization using lapack xPTSV.
2. If the above failed or the matrix is not Hermitian with a positive real diagonal
use Gaussian elimination with pivoting using lapack xGTSV, and goto 8.
b. If the matrix is Hermitian with a positive real diagonal, attempt Cholesky factorization using lapack xPBTRF.
c. if the above failed or the matrix is not Hermitian with a positive real diagonal use
Gaussian elimination with pivoting using lapack xGBTRF, and goto 8.
4. If the matrix is upper or lower triangular perform a sparse forward or backward substitution, and goto 8
5. If the matrix is an upper triangular matrix with column permutations or lower triangular matrix with row permutations, perform a sparse forward or backward substitution,
and goto 8
6. If the matrix is square, Hermitian with a real positive diagonal, attempt sparse
Cholesky factorization using cholmod.

Chapter 22: Sparse Matrices

585

7. If the sparse Cholesky factorization failed or the matrix is not Hermitian with a real
positive diagonal, and the matrix is square, factorize using umfpack.
8. If the matrix is not square, or any of the previous solvers flags a singular or near
singular matrix, find a minimum norm solution using cxsparse3 .
The band density is defined as the number of nonzero values in the band divided by the
total number of values in the full band. The banded matrix solvers can be entirely disabled
by using spparms to set bandden to 1 (i.e., spparms ("bandden", 1)).
The QR solver factorizes the problem with a Dulmage-Mendelsohn decomposition, to
separate the problem into blocks that can be treated as over-determined, multiple well
determined blocks, and a final over-determined block. For matrices with blocks of strongly
connected nodes this is a big win as LU decomposition can be used for many blocks. It also
significantly improves the chance of finding a solution to over-determined problems rather
than just returning a vector of NaN ’s.
All of the solvers above, can calculate an estimate of the condition number. This can
be used to detect numerical stability problems in the solution and force a minimum norm
solution to be used. However, for narrow banded, triangular or diagonal matrices, the cost
of calculating the condition number is significant, and can in fact exceed the cost of factoring
the matrix. Therefore the condition number is not calculated in these cases, and Octave
relies on simpler techniques to detect singular matrices or the underlying lapack code in
the case of banded matrices.
The user can force the type of the matrix with the matrix_type function. This overcomes
the cost of discovering the type of the matrix. However, it should be noted that identifying
the type of the matrix incorrectly will lead to unpredictable results, and so matrix_type
should be used with care.

nest = normest (A)
nest = normest (A, tol)
[nest, iter] = normest ( . . . )
Estimate the 2-norm of the matrix A using a power series analysis.
This is typically used for large matrices, where the cost of calculating norm (A) is
prohibitive and an approximation to the 2-norm is acceptable.
tol is the tolerance to which the 2-norm is calculated. By default tol is 1e-6.
The optional output iter returns the number of iterations that were required for
normest to converge.
See also: [normest1], page 585, [norm], page 513, [cond], page 508, [condest],
page 586.

nest =
nest =
nest =
nest =
[nest,
[nest,
3

normest1 (A)
normest1 (A, t)
normest1 (A, t, x0)
normest1 (Afun, t, x0, p1, p2, . . . )
v] = normest1 (A, . . . )
v, w] = normest1 (A, . . . )

The cholmod, umfpack and cxsparse packages were written by Tim Davis and are available at http://
www.cise.ufl.edu/research/sparse/

586

GNU Octave

[nest, v, w, iter] = normest1 (A, . . . )
Estimate the 1-norm of the matrix A using a block algorithm.
normest1 is best for large sparse matrices where only an estimate of the norm is
required. For small to medium sized matrices, consider using norm (A, 1). In addition, normest1 can be used for the estimate of the 1-norm of a linear operator A
when matrix-vector products A * x and A’ * x can be cheaply computed. In this case,
instead of the matrix A, a function Afun (flag, x) is used; it must return:
•
•
•
•

the dimension n of A, if flag is "dim"
true if A is a real operator, if flag is "real"
the result A * x, if flag is "notransp"
the result A’ * x, if flag is "transp"

A typical case is A defined by b ^ m, in which the result A * x can be computed
without even forming explicitly b ^ m by:
y = x;
for i = 1:m
y = b * y;
endfor
The parameters p1, p2, . . . are arguments of Afun (flag, x, p1, p2, ...).
The default value for t is 2. The algorithm requires matrix-matrix products with sizes
n x n and n x t.
The initial matrix x0 should have columns of unit 1-norm. The default initial matrix
x0 has the first column ones (n, 1) / n and, if t > 1, the remaining columns with
random elements -1 / n, 1 / n, divided by n.
On output, nest is the desired estimate, v and w are vectors such that w = A * v, with
norm (w, 1) = c * norm (v, 1). iter contains in iter(1) the number of iterations
(the maximum is hardcoded to 5) and in iter(2) the total number of products A *
x or A’ * x performed by the algorithm.
Algorithm Note: normest1 uses random numbers during evaluation. Therefore, if
consistent results are required, the "state" of the random generator should be fixed
before invoking normest1.
Reference: N. J. Higham and F. Tisseur, A block algorithm for matrix 1-norm estimation, with and application to 1-norm pseudospectra, SIAM J. Matrix Anal. Appl.,
pp. 1185–1201, Vol 21, No. 4, 2000.
See also: [normest], page 585, [norm], page 513, [cond], page 508, [condest], page 586.

cest =
cest =
cest =
cest =
[cest,

condest (A)
condest (A, t)
condest (A, solvefun, t, p1, p2, . . . )
condest (Afcn, solvefun, t, p1, p2, . . . )
v] = condest ( . . . )

Estimate the 1-norm condition number of a square matrix A using t test vectors and
a randomized 1-norm estimator.
The optional input t specifies the number of test vectors (default 5).

Chapter 22: Sparse Matrices

587

If the matrix is not explicit, e.g., when estimating the condition number of A given
an LU factorization, condest uses the following functions:
− Afcn which must return
• the dimension n of a, if flag is "dim"
• true if a is a real operator, if flag is "real"
• the result a * x, if flag is "notransp"
• the result a’ * x, if flag is "transp"
− solvefun which must return
• the dimension n of a, if flag is "dim"
• true if a is a real operator, if flag is "real"
• the result a \ x, if flag is "notransp"
• the result a’ \ x, if flag is "transp"

The parameters p1, p2, . . . are arguments of Afcn (flag, x, p1, p2, ...) and
solvefcn (flag, x, p1, p2, ...).
The principal output is the 1-norm condition number estimate cest.
The optional second output is an approximate null vector when cest is large; it satisfies
the equation norm (A*v, 1) == norm (A, 1) * norm (v, 1) / est.
Algorithm Note: condest uses a randomized algorithm to approximate the 1-norms.
Therefore, if consistent results are required, the "state" of the random generator
should be fixed before invoking condest.
References:
• N.J. Higham and F. Tisseur, A Block Algorithm for Matrix 1-Norm Estimation,
with an Application to 1-Norm Pseudospectra. SIMAX vol 21, no 4, pp 11851201. http://dx.doi.org/10.1137/S0895479899356080
• N.J. Higham and F. Tisseur, A Block Algorithm for Matrix 1-Norm Estimation,
with an Application to 1-Norm Pseudospectra. http://citeseer.ist.psu.
edu/223007.html
See also: [cond], page 508, [norm], page 513, [normest1], page 585, [normest],
page 585.

spparms ()
vals = spparms ()
[keys, vals] = spparms ()
val = spparms (key)
spparms (vals)
spparms ("default")
spparms ("tight")
spparms (key, val)
Query or set the parameters used by the sparse solvers and factorization functions.
The first four calls above get information about the current settings, while the others
change the current settings. The parameters are stored as pairs of keys and values,
where the values are all floats and the keys are one of the following strings:
‘spumoni’

Printing level of debugging information of the solvers (default 0)

588

GNU Octave

‘ths_rel’

Included for compatibility. Not used. (default 1)

‘ths_abs’

Included for compatibility. Not used. (default 1)

‘exact_d’

Included for compatibility. Not used. (default 0)

‘supernd’

Included for compatibility. Not used. (default 3)

‘rreduce’

Included for compatibility. Not used. (default 3)

‘wh_frac’

Included for compatibility. Not used. (default 0.5)

‘autommd’

Flag whether the LU/QR and the ’\’ and ’/’ operators will automatically
use the sparsity preserving mmd functions (default 1)

‘autoamd’

Flag whether the LU and the ’\’ and ’/’ operators will automatically use
the sparsity preserving amd functions (default 1)

‘piv_tol’

The pivot tolerance of the umfpack solvers (default 0.1)

‘sym_tol’

The pivot tolerance of the umfpack symmetric solvers (default 0.001)

‘bandden’

The density of nonzero elements in a banded matrix before it is treated
by the lapack banded solvers (default 0.5)

‘umfpack’

Flag whether the umfpack or mmd solvers are used for the LU, ’\’ and
’/’ operations (default 1)

The value of individual keys can be set with spparms (key, val). The default values
can be restored with the special keyword "default". The special keyword "tight"
can be used to set the mmd solvers to attempt a sparser solution at the potential cost
of longer running time.
See also: [chol], page 515, [colamd], page 580, [lu], page 518, [qr], page 519, [symamd],
page 582.

p = sprank (S)
Calculate the structural rank of the sparse matrix S.
Note that only the structure of the matrix is used in this calculation based on a
Dulmage-Mendelsohn permutation to block triangular form. As such the numerical
rank of the matrix S is bounded by sprank (S) >= rank (S). Ignoring floating point
errors sprank (S) == rank (S).
See also: [dmperm], page 582.

[count, h, parent, post, R] = symbfact (S)
[...] = symbfact (S, typ)
[...] = symbfact (S, typ, mode)
Perform a symbolic factorization analysis of the sparse matrix S.
The input variables are
S

S is a real or complex sparse matrix.

typ

Is the type of the factorization and can be one of
"sym" (default)
Factorize S. Assumes S is symmetric and uses the upper
triangular portion of the matrix.

Chapter 22: Sparse Matrices

mode

589

"col"

Factorize S’ * S.

"row"

Factorize S * S’.

"lo"

Factorize S’. Assumes S is symmetric and uses the lower
triangular portion of the matrix.

When mode is unspecified return the Cholesky factorization for R. If
mode is "lower" or "L" then return the conjugate transpose R’ which is
a lower triangular factor. The conjugate transpose version is faster and
uses less memory, but still returns the same values for all other outputs:
count, h, parent, and post.

The output variables are:
count

The row counts of the Cholesky factorization as determined by typ. The
computational difficulty of performing the true factorization using chol
is sum (count .^ 2).

h

The height of the elimination tree.

parent

The elimination tree itself.

post

A sparse boolean matrix whose structure is that of the Cholesky factorization as determined by typ.

See also: [chol], page 515, [etree], page 572, [treelayout], page 573.
For non square matrices, the user can also utilize the spaugment function to find a least
squares solution to a linear equation.

s = spaugment (A, c)
Create the augmented matrix of A.
This is given by
[c * eye(m, m), A;
A’, zeros(n, n)]
This is related to the least squares solution of A \ b, by
s * [ r / c; x] = [ b, zeros(n, columns(b)) ]
where r is the residual error
r = b - A * x
As the matrix s is symmetric indefinite it can be factorized with lu, and the minimum
norm solution can therefore be found without the need for a qr factorization. As the
residual error will be zeros (m, m) for underdetermined problems, and example can
be
m = 11; n = 10; mn = max (m, n);
A = spdiags ([ones(mn,1), 10*ones(mn,1), -ones(mn,1)],
[-1, 0, 1], m, n);
x0 = A \ ones (m,1);
s = spaugment (A);
[L, U, P, Q] = lu (s);
x1 = Q * (U \ (L \ (P * [ones(m,1); zeros(n,1)])));
x1 = x1(end - n + 1 : end);

590

GNU Octave

To find the solution of an overdetermined problem needs an estimate of the residual
error r and so it is more complex to formulate a minimum norm solution using the
spaugment function.
In general the left division operator is more stable and faster than using the spaugment
function.
See also: [mldivide], page 146.
Finally, the function eigs can be used to calculate a limited number of eigenvalues and
eigenvectors based on a selection criteria and likewise for svds which calculates a limited
number of singular values and vectors.

d =
d =
d =
d =
d =
d =
d =
d =
d =
d =
d =
d =
d =
d =
d =
d =
[V,
[V,
[V,
[V,

eigs (A)
eigs (A, k)
eigs (A, k, sigma)
eigs (A, k, sigma, opts)
eigs (A, B)
eigs (A, B, k)
eigs (A, B, k, sigma)
eigs (A, B, k, sigma, opts)
eigs (af, n)
eigs (af, n, B)
eigs (af, n, k)
eigs (af, n, B, k)
eigs (af, n, k, sigma)
eigs (af, n, B, k, sigma)
eigs (af, n, k, sigma, opts)
eigs (af, n, B, k, sigma, opts)
d] = eigs (A, . . . )
d] = eigs (af, n, . . . )
d, flag] = eigs (A, . . . )
d, flag] = eigs (af, n, . . . )
Calculate a limited number of eigenvalues and eigenvectors of A, based on a selection
criteria.
The number of eigenvalues and eigenvectors to calculate is given by k and defaults to
6.
By default, eigs solve the equation Aν = λν, where λ is a scalar representing one of
the eigenvalues, and ν is the corresponding eigenvector. If given the positive definite
matrix B then eigs solves the general eigenvalue equation Aν = λBν.
The argument sigma determines which eigenvalues are returned. sigma can be either
a scalar or a string. When sigma is a scalar, the k eigenvalues closest to sigma are
returned. If sigma is a string, it must have one of the following values.
"lm"

Largest Magnitude (default).

"sm"

Smallest Magnitude.

"la"

Largest Algebraic (valid only for real symmetric problems).

"sa"

Smallest Algebraic (valid only for real symmetric problems).

Chapter 22: Sparse Matrices

591

"be"

Both Ends, with one more from the high-end if k is odd (valid only for
real symmetric problems).

"lr"

Largest Real part (valid only for complex or unsymmetric problems).

"sr"

Smallest Real part (valid only for complex or unsymmetric problems).

"li"

Largest Imaginary part (valid only for complex or unsymmetric problems).

"si"

Smallest Imaginary part (valid only for complex or unsymmetric problems).

If opts is given, it is a structure defining possible options that eigs should use. The
fields of the opts structure are:
issym

If af is given, then flags whether the function af defines a symmetric
problem. It is ignored if A is given. The default is false.

isreal

If af is given, then flags whether the function af defines a real problem.
It is ignored if A is given. The default is true.

tol

Defines the required convergence tolerance, calculated as tol * norm (A).
The default is eps.

maxit

The maximum number of iterations. The default is 300.

p

The number of Lanzcos basis vectors to use. More vectors will result in
faster convergence, but a greater use of memory. The optimal value of
p is problem dependent and should be in the range k to n. The default
value is 2 * k.

v0

The starting vector for the algorithm. An initial vector close to the final
vector will speed up convergence. The default is for arpack to randomly
generate a starting vector. If specified, v0 must be an n-by-1 vector where
n = rows (A)

disp

The level of diagnostic printout (0|1|2). If disp is 0 then diagnostics are
disabled. The default value is 0.

cholB

Flag if chol (B) is passed rather than B. The default is false.

permB

The permutation vector of the Cholesky factorization of B if cholB is true.
It is obtained by [R, ~, permB] = chol (B, "vector"). The default is
1:n.

It is also possible to represent A by a function denoted af. af must be followed by a
scalar argument n defining the length of the vector argument accepted by af. af can
be a function handle, an inline function, or a string. When af is a string it holds the
name of the function to use.
af is a function of the form y = af (x) where the required return value of af is
determined by the value of sigma. The four possible forms are
A*x

if sigma is not given or is a string other than "sm".

A\x

if sigma is 0 or "sm".

592

GNU Octave

(A - sigma * I) \ x
for the standard eigenvalue problem, where I is the identity matrix of the
same size as A.
(A - sigma * B) \ x
for the general eigenvalue problem.
The return arguments of eigs depend on the number of return arguments requested.
With a single return argument, a vector d of length k is returned containing the k
eigenvalues that have been found. With two return arguments, V is a n-by-k matrix
whose columns are the k eigenvectors corresponding to the returned eigenvalues. The
eigenvalues themselves are returned in d in the form of a n-by-k matrix, where the
elements on the diagonal are the eigenvalues.
Given a third return argument flag, eigs returns the status of the convergence. If
flag is 0 then all eigenvalues have converged. Any other value indicates a failure to
converge.
This function is based on the arpack package, written by R. Lehoucq, K. Maschhoff,
D. Sorensen, and C. Yang. For more information see http://www.caam.rice.edu/
software/ARPACK/.
See also: [eig], page 509, [svds], page 592.

s =
s =
s =
s =
[u,
[u,

svds (A)
svds (A, k)
svds (A, k, sigma)
svds (A, k, sigma, opts)
s, v] = svds ( . . . )
s, v, flag] = svds ( . . . )
Find a few singular values of the matrix A.
The singular values are calculated using
[m, n] = size (A);
s = eigs ([sparse(m, m), A;
A’, sparse(n, n)])
The eigenvalues returned by eigs correspond to the singular values of A. The number
of singular values to calculate is given by k and defaults to 6.
The argument sigma specifies which singular values to find. When sigma is the string
’L’, the default, the largest singular values of A are found. Otherwise, sigma must be
a real scalar and the singular values closest to sigma are found. As a corollary, sigma
= 0 finds the smallest singular values. Note that for relatively small values of sigma,
there is a chance that the requested number of singular values will not be found. In
that case sigma should be increased.
opts is a structure defining options that svds will pass to eigs. The possible fields
of this structure are documented in eigs. By default, svds sets the following three
fields:
tol

The required convergence tolerance for the singular values. The default
value is 1e-10. eigs is passed tol / sqrt(2).

maxit

The maximum number of iterations. The default is 300.

Chapter 22: Sparse Matrices

disp

593

The level of diagnostic printout (0|1|2). If disp is 0 then diagnostics are
disabled. The default value is 0.

If more than one output is requested then svds will return an approximation of the
singular value decomposition of A
A_approx = u*s*v’
where A approx is a matrix of size A but only rank k.
flag returns 0 if the algorithm has succesfully converged, and 1 otherwise. The test
for convergence is
norm (A*v - u*s, 1) <= tol * norm (A, 1)
svds is best for finding only a few singular values from a large sparse matrix. Otherwise, svd (full (A)) will likely be more efficient.
See also: [svd], page 525, [eigs], page 590.

22.3 Iterative Techniques Applied to Sparse Matrices
The left division \ and right division / operators, discussed in the previous section, use
direct solvers to resolve a linear equation of the form x = A \ b or x = b / A. Octave also
includes a number of functions to solve sparse linear equations using iterative techniques.

x = pcg (A, b, tol, maxit, m1, m2, x0, . . . )
[x, flag, relres, iter, resvec, eigest] = pcg ( . . . )
Solve the linear system of equations A * x = b by means of the Preconditioned Conjugate Gradient iterative method.
The input arguments are
• A can be either a square (preferably sparse) matrix or a function handle, inline
function or string containing the name of a function which computes A * x. In
principle, A should be symmetric and positive definite; if pcg finds A not to be
positive definite, a warning is printed and the flag output will be set.
• b is the right-hand side vector.
• tol is the required relative tolerance for the residual error, b - A * x. The iteration
stops if norm (b - A * x) ≤ tol * norm (b). If tol is omitted or empty then a
tolerance of 1e-6 is used.
• maxit is the maximum allowable number of iterations; if maxit is omitted or
empty then a value of 20 is used.
• m = m1 * m2 is the (left) preconditioning matrix, so that the iteration is (theoretically) equivalent to solving by pcg P * x = m \ b, with P = m \ A. Note that
a proper choice of the preconditioner may dramatically improve the overall performance of the method. Instead of matrices m1 and m2, the user may pass two
functions which return the results of applying the inverse of m1 and m2 to a
vector (usually this is the preferred way of using the preconditioner). If m1 is
omitted or empty [] then no preconditioning is applied. If m2 is omitted, m =
m1 will be used as a preconditioner.
• x0 is the initial guess. If x0 is omitted or empty then the function sets x0 to a
zero vector by default.

594

GNU Octave

The arguments which follow x0 are treated as parameters, and passed in a proper
way to any of the functions (A or m) which are passed to pcg. See the examples
below for further details. The output arguments are
• x is the computed approximation to the solution of A * x = b.
• flag reports on the convergence. A value of 0 means the solution converged and
the tolerance criterion given by tol is satisfied. A value of 1 means that the
maxit limit for the iteration count was reached. A value of 3 indicates that the
(preconditioned) matrix was found not to be positive definite.
• relres is the ratio of the final residual to its initial value, measured in the Euclidean norm.
• iter is the actual number of iterations performed.
• resvec describes the convergence history of the method. resvec(i,1) is the
Euclidean norm of the residual, and resvec(i,2) is the preconditioned residual
norm, after the (i-1)-th iteration, i = 1, 2, ..., iter+1. The preconditioned
residual norm is defined as norm (r) ^ 2 = r’ * (m \ r) where r = b - A * x, see
also the description of m. If eigest is not required, only resvec(:,1) is returned.
• eigest returns the estimate for the smallest eigest(1) and largest eigest(2)
eigenvalues of the preconditioned matrix P = m \ A. In particular, if no preconditioning is used, the estimates for the extreme eigenvalues of A are returned. eigest(1) is an overestimate and eigest(2) is an underestimate, so
that eigest(2) / eigest(1) is a lower bound for cond (P, 2), which nevertheless in the limit should theoretically be equal to the actual value of the condition
number. The method which computes eigest works only for symmetric positive
definite A and m, and the user is responsible for verifying this assumption.
Let us consider a trivial problem with a diagonal matrix (we exploit the sparsity of
A)
n = 10;
A = diag (sparse (1:n));
b = rand (n, 1);
[l, u, p] = ilu (A, struct ("droptol", 1.e-3));
Example 1: Simplest use of pcg
x = pcg (A, b)
Example 2: pcg with a function which computes A * x
function y = apply_a (x)
y = [1:N]’ .* x;
endfunction
x = pcg ("apply_a", b)
Example 3: pcg with a preconditioner: l * u
x = pcg (A, b, 1.e-6, 500, l*u)
Example 4: pcg with a preconditioner: l * u. Faster than Example 3 since lower
and upper triangular matrices are easier to invert
x = pcg (A, b, 1.e-6, 500, l, u)

Chapter 22: Sparse Matrices

595

Example 5: Preconditioned iteration, with full diagnostics. The preconditioner
(quite strange, because even the original matrix A is trivial) is defined as a function
function y = apply_m (x)
k = floor (length (x) - 2);
y = x;
y(1:k) = x(1:k) ./ [1:k]’;
endfunction
[x, flag, relres, iter, resvec, eigest] = ...
pcg (A, b, [], [], "apply_m");
semilogy (1:iter+1, resvec);
Example 6: Finally, a preconditioner which depends on a parameter k.
function y = apply_M (x, varargin)
K = varargin{1};
y = x;
y(1:K) = x(1:K) ./ [1:K]’;
endfunction
[x, flag, relres, iter, resvec, eigest] = ...
pcg (A, b, [], [], "apply_m", [], [], 3)
References:
1. C.T. Kelley, Iterative Methods for Linear and Nonlinear Equations, SIAM, 1995.
(the base PCG algorithm)
2. Y. Saad, Iterative Methods for Sparse Linear Systems, PWS 1996. (condition
number estimate from PCG) Revised version of this book is available online at
http://www-users.cs.umn.edu/~saad/books.html
See also: [sparse], page 568, [pcr], page 595.

x = pcr (A, b, tol, maxit, m, x0, . . . )
[x, flag, relres, iter, resvec] = pcr ( . . . )
Solve the linear system of equations A * x = b by means of the Preconditioned Conjugate Residuals iterative method.
The input arguments are
• A can be either a square (preferably sparse) matrix or a function handle, inline
function or string containing the name of a function which computes A * x. In
principle A should be symmetric and non-singular; if pcr finds A to be numerically singular, you will get a warning message and the flag output parameter will
be set.
• b is the right hand side vector.
• tol is the required relative tolerance for the residual error, b - A * x. The iteration stops if norm (b - A * x) <= tol * norm (b - A * x0). If tol is empty or is
omitted, the function sets tol = 1e-6 by default.
• maxit is the maximum allowable number of iterations; if [] is supplied for maxit,
or pcr has less arguments, a default value equal to 20 is used.

596

GNU Octave

• m is the (left) preconditioning matrix, so that the iteration is (theoretically)
equivalent to solving by pcr P * x = m \ b, with P = m \ A. Note that a proper
choice of the preconditioner may dramatically improve the overall performance
of the method. Instead of matrix m, the user may pass a function which returns
the results of applying the inverse of m to a vector (usually this is the preferred
way of using the preconditioner). If [] is supplied for m, or m is omitted, no
preconditioning is applied.
• x0 is the initial guess. If x0 is empty or omitted, the function sets x0 to a zero
vector by default.
The arguments which follow x0 are treated as parameters, and passed in a proper
way to any of the functions (A or m) which are passed to pcr. See the examples
below for further details.
The output arguments are
• x is the computed approximation to the solution of A * x = b.

• flag reports on the convergence. flag = 0 means the solution converged and the
tolerance criterion given by tol is satisfied. flag = 1 means that the maxit limit
for the iteration count was reached. flag = 3 reports a pcr breakdown, see [1]
for details.
• relres is the ratio of the final residual to its initial value, measured in the Euclidean norm.
• iter is the actual number of iterations performed.

• resvec describes the convergence history of the method, so that resvec (i) contains the Euclidean norms of the residual after the (i-1)-th iteration, i = 1,2,
..., iter+1.
Let us consider a trivial problem with a diagonal matrix (we exploit the sparsity of
A)
n = 10;
A = sparse (diag (1:n));
b = rand (N, 1);
Example 1: Simplest use of pcr
x = pcr (A, b)
Example 2: pcr with a function which computes A * x.
function y = apply_a (x)
y = [1:10]’ .* x;
endfunction
x = pcr ("apply_a", b)
Example 3: Preconditioned iteration, with full diagnostics. The preconditioner
(quite strange, because even the original matrix A is trivial) is defined as a function

Chapter 22: Sparse Matrices

597

function y = apply_m (x)
k = floor (length (x) - 2);
y = x;
y(1:k) = x(1:k) ./ [1:k]’;
endfunction
[x, flag, relres, iter, resvec] = ...
pcr (A, b, [], [], "apply_m")
semilogy ([1:iter+1], resvec);
Example 4: Finally, a preconditioner which depends on a parameter k.
function y = apply_m (x, varargin)
k = varargin{1};
y = x;
y(1:k) = x(1:k) ./ [1:k]’;
endfunction
[x, flag, relres, iter, resvec] = ...
pcr (A, b, [], [], "apply_m"’, [], 3)
References:
[1] W. Hackbusch, Iterative Solution of Large Sparse Systems of Equations, section
9.5.4; Springer, 1994
See also: [sparse], page 568, [pcg], page 593.
The speed with which an iterative solver converges to a solution can be accelerated with
the use of a pre-conditioning matrix M. In this case the linear equation M^-1 * x = M^-1 *
A \ b is solved instead. Typical pre-conditioning matrices are partial factorizations of the
original matrix.

L = ichol (A)
L = ichol (A, opts)
Compute the incomplete Cholesky factorization of the sparse square matrix A.
By default, ichol uses only the lower triangle of A and produces a lower triangular
factor L such that L*L’ approximates A.
The factor given by this routine may be useful as a preconditioner for a system
of linear equations being solved by iterative methods such as PCG (Preconditioned
Conjugate Gradient).
The factorization may be modified by passing options in a structure opts. The option
name is a field of the structure and the setting is the value of field. Names and
specifiers are case sensitive.
type

Type of factorization.
"nofill" (default)
Incomplete Cholesky factorization with no fill-in (IC(0)).
"ict"

Incomplete Cholesky factorization with threshold dropping
(ICT).

598

GNU Octave

diagcomp

A non-negative scalar alpha for incomplete Cholesky factorization of A +
alpha * diag (diag (A)) instead of A. This can be useful when A is not
positive definite. The default value is 0.

droptol

A non-negative scalar specifying the drop tolerance for factorization if
performing ICT. The default value is 0 which produces the complete
Cholesky factorization.
Non-diagonal entries of L are set to 0 unless
abs (L(i,j)) >= droptol * norm (A(j:end, j), 1).

michol

Modified incomplete Cholesky factorization:
"off" (default)
Row and column sums are not necessarily preserved.
"on"

The diagonal of L is modified so that row (and column) sums
are preserved even when elements have been dropped during
the factorization. The relationship preserved is: A * e = L *
L’ * e, where e is a vector of ones.

shape
"lower" (default)
Use only the lower triangle of A and return a lower triangular
factor L such that L*L’ approximates A.
"upper"

Use only the upper triangle of A and return an upper triangular factor U such that U’*U approximates A.

EXAMPLES
The following problem demonstrates how to factorize a sample symmetric positive
definite matrix with the full Cholesky decomposition and with the incomplete one.
A = [ 0.37, -0.05, -0.05, -0.07;
-0.05, 0.116, 0.0,
-0.05;
-0.05, 0.0,
0.116, -0.05;
-0.07, -0.05, -0.05,
0.202];
A = sparse (A);
nnz (tril (A))
ans = 9
L = chol (A, "lower");
nnz (L)
ans = 10
norm (A - L * L’, "fro") / norm (A, "fro")
ans = 1.1993e-16
opts.type = "nofill";
L = ichol (A, opts);
nnz (L)
ans = 9
norm (A - L * L’, "fro") / norm (A, "fro")
ans = 0.019736

Chapter 22: Sparse Matrices

599

Another example for decomposition is a finite difference matrix used to solve a boundary value problem on the unit square.
nx = 400; ny = 200;
hx = 1 / (nx + 1); hy = 1 / (ny + 1);
Dxx = spdiags ([ones(nx, 1), -2*ones(nx, 1), ones(nx, 1)],
[-1 0 1 ], nx, nx) / (hx ^ 2);
Dyy = spdiags ([ones(ny, 1), -2*ones(ny, 1), ones(ny, 1)],
[-1 0 1 ], ny, ny) / (hy ^ 2);
A = -kron (Dxx, speye (ny)) - kron (speye (nx), Dyy);
nnz (tril (A))
ans = 239400
opts.type = "nofill";
L = ichol (A, opts);
nnz (tril (A))
ans = 239400
norm (A - L * L’, "fro") / norm (A, "fro")
ans = 0.062327
References for implemented algorithms:
[1] Y. Saad. "Preconditioning Techniques." Iterative Methods for Sparse Linear Systems, PWS Publishing Company, 1996.
[2] M. Jones, P. Plassmann: An Improved Incomplete Cholesky Factorization, 1992.
See also: [chol], page 515, [ilu], page 599, [pcg], page 593.

ilu
ilu
[L,
[L,

(A)
(A, opts)

U] = ilu ( . . . )
U, P] = ilu ( . . . )
Compute the incomplete LU factorization of the sparse square matrix A.
ilu returns a unit lower triangular matrix L, an upper triangular matrix U, and
optionally a permutation matrix P, such that L*U approximates P*A.
The factors given by this routine may be useful as preconditioners for a system of linear
equations being solved by iterative methods such as BICG (BiConjugate Gradients)
or GMRES (Generalized Minimum Residual Method).
The factorization may be modified by passing options in a structure opts. The option
name is a field of the structure and the setting is the value of field. Names and
specifiers are case sensitive.
type

Type of factorization.
"nofill"

ILU factorization with no fill-in (ILU(0)).
Additional supported options: milu.

"crout"

Crout version of ILU factorization (ILUC).
Additional supported options: milu, droptol.

"ilutp" (default)
ILU factorization with threshold and pivoting.

600

GNU Octave

Additional supported options:
thresh.

milu, droptol, udiag,

droptol

A non-negative scalar specifying the drop tolerance for factorization. The
default value is 0 which produces the complete LU factorization.
Non-diagonal entries of U are set to 0 unless
abs (U(i,j)) >= droptol * norm (A(:,j)).
Non-diagonal entries of L are set to 0 unless
abs (L(i,j)) >= droptol * norm (A(:,j))/U(j,j).

milu

Modified incomplete LU factorization:
"row"

Row-sum modified incomplete LU factorization. The factorization preserves row sums: A * e = L * U * e, where e is a
vector of ones.

"col"

Column-sum modified incomplete LU factorization. The factorization preserves column sums: e’ * A = e’ * L * U.

"off" (default)
Row and column sums are not necessarily preserved.
udiag

If true, any zeros on the diagonal of the upper triangular factor are replaced by the local drop tolerance droptol * norm (A(:,j))/U(j,j).
The default is false.

thresh

Pivot threshold for factorization. It can range between 0 (diagonal pivoting) and 1 (default), where the maximum magnitude entry in the column
is chosen to be the pivot.

If ilu is called with just one output, the returned matrix is L + U - speye (size
(A)), where L is unit lower triangular and U is upper triangular.
With two outputs, ilu returns a unit lower triangular matrix L and an upper triangular matrix U. For opts.type == "ilutp", one of the factors is permuted based on
the value of opts.milu. When opts.milu == "row", U is a column permuted upper
triangular factor. Otherwise, L is a row-permuted unit lower triangular factor.
If there are three named outputs and opts.milu != "row", P is returned such that L
and U are incomplete factors of P*A. When opts.milu == "row", P is returned such
that L and U are incomplete factors of A*P.
EXAMPLES
A = gallery ("neumann", 1600) + speye (1600);
opts.type = "nofill";
nnz (A)
ans = 7840
nnz (lu (A))
ans = 126478
nnz (ilu (A, opts))
ans = 7840

Chapter 22: Sparse Matrices

601

This shows that A has 7,840 nonzeros, the complete LU factorization
has 126,478 nonzeros, and the incomplete LU factorization, with 0
level of fill-in, has 7,840 nonzeros, the same amount as A.
Taken from:
http://www.mathworks.com/help/matlab/ref/ilu.html
A = gallery ("wathen", 10, 10);
b = sum (A, 2);
tol = 1e-8;
maxit = 50;
opts.type = "crout";
opts.droptol = 1e-4;
[L, U] = ilu (A, opts);
x = bicg (A, b, tol, maxit, L, U);
norm (A * x - b, inf)
This example uses ILU as preconditioner for a random FEM-Matrix, which has a
large condition number. Without L and U BICG would not converge.
See also: [lu], page 518, [ichol], page 597, [bicg], page 528, [gmres], page 530.

22.4 Real Life Example using Sparse Matrices
A common application for sparse matrices is in the solution of Finite Element Models.
Finite element models allow numerical solution of partial differential equations that do not
have closed form solutions, typically because of the complex shape of the domain.
In order to motivate this application, we consider the boundary value Laplace equation.
This system can model scalar potential fields, such as heat or electrical potential. Given
a medium Ω with boundary ∂Ω. At all points on the ∂Ω the boundary conditions are
known, and we wish to calculate the potential in Ω. Boundary conditions may specify the
potential (Dirichlet boundary condition), its normal derivative across the boundary (Neumann boundary condition), or a weighted sum of the potential and its derivative (Cauchy
boundary condition).
In a thermal model, we want to calculate the temperature in Ω and know the boundary
temperature (Dirichlet condition) or heat flux (from which we can calculate the Neumann
condition by dividing by the thermal conductivity at the boundary). Similarly, in an electrical model, we want to calculate the voltage in Ω and know the boundary voltage (Dirichlet)
or current (Neumann condition after diving by the electrical conductivity). In an electrical
model, it is common for much of the boundary to be electrically isolated; this is a Neumann
boundary condition with the current equal to zero.
The simplest finite element models will divide Ω into simplexes (triangles in 2D, pyramids
in 3D). We take as a 3-D example a cylindrical liquid filled tank with a small non-conductive
ball from the EIDORS project4 . This is model is designed to reflect an application of
electrical impedance tomography, where current patterns are applied to such a tank in order
to image the internal conductivity distribution. In order to describe the FEM geometry, we
have a matrix of vertices nodes and simplices elems.
4

EIDORS - Electrical Impedance Tomography and Diffuse optical Tomography Reconstruction Software
http://eidors3d.sourceforge.net

602

GNU Octave

The following example creates a simple rectangular 2-D electrically conductive medium
with 10 V and 20 V imposed on opposite sides (Dirichlet boundary conditions). All other
edges are electrically isolated.
node_y = [1;1.2;1.5;1.8;2]*ones(1,11);
node_x = ones(5,1)*[1,1.05,1.1,1.2, ...
1.3,1.5,1.7,1.8,1.9,1.95,2];
nodes = [node_x(:), node_y(:)];
[h,w] = size (node_x);
elems = [];
for idx = 1:w-1
widx = (idx-1)*h;
elems = [elems; ...
widx+[(1:h-1);(2:h);h+(1:h-1)]’; ...
widx+[(2:h);h+(2:h);h+(1:h-1)]’ ];
endfor
E = size (elems,1); # No. of simplices
N = size (nodes,1); # No. of vertices
D = size (elems,2); # dimensions+1
This creates a N-by-2 matrix nodes and a E-by-3 matrix elems with values, which define
finite element triangles:
nodes(1:7,:)’
1.00 1.00 1.00 1.00 1.00 1.05 1.05 ...
1.00 1.20 1.50 1.80 2.00 1.00 1.20 ...
elems(1:7,:)’
1
2
3
4
2
3
4 ...
2
3
4
5
7
8
9 ...
6
7
8
9
6
7
8 ...
Using a first order FEM, we approximate the electrical conductivity distribution in Ω
as constant on each simplex (represented by the vector conductivity). Based on the
finite element geometry, we first calculate a system (or stiffness) matrix for each simplex
(represented as 3-by-3 elements on the diagonal of the element-wise system matrix SE).
Based on SE and a N-by-DE connectivity matrix C, representing the connections between
simplices and vertices, the global connectivity matrix S is calculated.
## Element conductivity
conductivity = [1*ones(1,16), ...
2*ones(1,48), 1*ones(1,16)];
## Connectivity matrix
C = sparse ((1:D*E), reshape (elems’, ...
D*E, 1), 1, D*E, N);
## Calculate system matrix
Siidx = floor ([0:D*E-1]’/D) * D * ...

Chapter 22: Sparse Matrices

603

ones(1,D) + ones(D*E,1)*(1:D) ;
Sjidx = [1:D*E]’*ones (1,D);
Sdata = zeros (D*E,D);
dfact = factorial (D-1);
for j = 1:E
a = inv ([ones(D,1), ...
nodes(elems(j,:), :)]);
const = conductivity(j) * 2 / ...
dfact / abs (det (a));
Sdata(D*(j-1)+(1:D),:) = const * ...
a(2:D,:)’ * a(2:D,:);
endfor
## Element-wise system matrix
SE = sparse(Siidx,Sjidx,Sdata);
## Global system matrix
S = C’* SE *C;
The system matrix acts like the conductivity S in Ohm’s law SV = I. Based on the
Dirichlet and Neumann boundary conditions, we are able to solve for the voltages at each
vertex V.
## Dirichlet boundary conditions
D_nodes = [1:5, 51:55];
D_value = [10*ones(1,5), 20*ones(1,5)];
V = zeros (N,1);
V(D_nodes) = D_value;
idx = 1:N; # vertices without Dirichlet
# boundary condns
idx(D_nodes) = [];
## Neumann boundary conditions. Note that
## N_value must be normalized by the
## boundary length and element conductivity
N_nodes = [];
N_value = [];
Q = zeros (N,1);
Q(N_nodes) = N_value;
V(idx) = S(idx,idx) \ ( Q(idx) - ...
S(idx,D_nodes) * V(D_nodes));
Finally, in order to display the solution, we show each solved voltage value in the z-axis
for each simplex vertex. See Figure 22.6.

604

GNU Octave

elemx = elems(:,[1,2,3,1])’;
xelems = reshape (nodes(elemx, 1), 4, E);
yelems = reshape (nodes(elemx, 2), 4, E);
velems = reshape (V(elemx), 4, E);
plot3 (xelems,yelems,velems,"k");
print "grid.eps";

20
18
16
14
12
10
1
1.2
1.4
1.6
1.8

2

1

1.2

1.4

1.6

1.8

2

Figure 22.6: Example finite element model the showing triangular elements. The height
of each vertex corresponds to the solution value.

605

23 Numerical Integration
Octave comes with several built-in functions for computing the integral of a function numerically (termed quadrature). These functions all solve 1-dimensional integration problems.

23.1 Functions of One Variable
Octave supports five different algorithms for computing the integral
Z

b

f (x)dx

a

of a function f over the interval from a to b. These are
quad

Numerical integration based on Gaussian quadrature.

quadv

Numerical integration using an adaptive vectorized Simpson’s rule.

quadl

Numerical integration using an adaptive Lobatto rule.

quadgk

Numerical integration using an adaptive Gauss-Konrod rule.

quadcc

Numerical integration using adaptive Clenshaw-Curtis rules.

trapz, cumtrapz
Numerical integration of data using the trapezoidal method.
The best quadrature algorithm to use depends on the integrand. If you have empirical data,
rather than a function, the choice is trapz or cumtrapz. If you are uncertain about the
characteristics of the integrand, quadcc will be the most robust as it can handle discontinuities, singularities, oscillatory functions, and infinite intervals. When the integrand is
smooth quadgk may be the fastest of the algorithms.
Function
quad
quadv
quadl
quadgk

Characteristics
Low accuracy with nonsmooth integrands
Medium accuracy with smooth integrands
Medium accuracy with smooth integrands. Slower than quadgk.
Medium accuracy (1e-6 – 1e-9) with smooth integrands.
Handles oscillatory functions and infinite bounds
quadcc
Low to High accuracy with nonsmooth/smooth integrands
Handles oscillatory functions, singularities, and infinite bounds
Here is an example of using quad to integrate the function
q

f (x) = x sin(1/x) |1 − x|
from x = 0 to x = 3.
This is a fairly difficult integration (plot the function over the range of integration to see
why).
The first step is to define the function:
function y = f (x)
y = x .* sin (1./x) .* sqrt (abs (1 - x));
endfunction

606

GNU Octave

Note the use of the ‘dot’ forms of the operators. This is not necessary for the quad
integrator, but is required by the other integrators. In any case, it makes it much easier to
generate a set of points for plotting because it is possible to call the function with a vector
argument to produce a vector result.
The second step is to call quad with the limits of integration:
[q, ier, nfun, err] = quad ("f", 0, 3)
⇒ 1.9819
⇒ 1
⇒ 5061
⇒ 1.1522e-07
Although quad returns a nonzero value for ier, the result is reasonably accurate (to see
why, examine what happens to the result if you move the lower bound to 0.1, then 0.01,
then 0.001, etc.).
The function "f" can be the string name of a function, a function handle, or an inline
function. These options make it quite easy to do integration without having to fully define
a function in an m-file. For example:
# Verify integral (x^3) = x^4/4
f = inline ("x.^3");
quadgk (f, 0, 1)
⇒ 0.25000
# Verify gamma function = (n-1)! for n = 4
f = @(x) x.^3 .* exp (-x);
quadcc (f, 0, Inf)
⇒ 6.0000

q =
q =
q =
[q,

quad
quad
quad
ier,

(f, a, b)
(f, a, b, tol)
(f, a, b, tol, sing)

nfun, err] = quad ( . . . )
Numerically evaluate the integral of f from a to b using Fortran routines from
quadpack.
f is a function handle, inline function, or a string containing the name of the function
to evaluate. The function must have the form y = f (x) where y and x are scalars.
a and b are the lower and upper limits of integration. Either or both may be infinite.
The optional argument tol is a vector that specifies the desired accuracy of the result.
The first element of the vector is the desired absolute tolerance, and the second
element is the desired relative tolerance. To choose a relative test only, set the absolute
tolerance to zero. To choose an absolute test only, set the relative tolerance to zero.
Both tolerances default to sqrt (eps) or approximately 1.5e-8.
The optional argument sing is a vector of values at which the integrand is known to
be singular.
The result of the integration is returned in q.
ier contains an integer error code (0 indicates a successful integration).
nfun indicates the number of function evaluations that were made.

Chapter 23: Numerical Integration

607

err contains an estimate of the error in the solution.
The function quad_options can set other optional parameters for quad.
Note: because quad is written in Fortran it cannot be called recursively. This prevents its use in integrating over more than one variable by routines dblquad and
triplequad.
See also: [quad options], page 607, [quadv], page 607, [quadl], page 608, [quadgk],
page 608, [quadcc], page 610, [trapz], page 611, [dblquad], page 613, [triplequad],
page 614.

quad_options ()
val = quad_options (opt)
quad_options (opt, val)
Query or set options for the function quad.
When called with no arguments, the names of all available options and their current
values are displayed.
Given one argument, return the value of the option opt.
When called with two arguments, quad_options sets the option opt to value val.
Options include
"absolute tolerance"
Absolute tolerance; may be zero for pure relative error test.
"relative tolerance"
Non-negative relative tolerance. If the absolute tolerance is zero, the relative tolerance must be greater than or equal to max (50*eps, 0.5e-28).
"single precision absolute tolerance"
Absolute tolerance for single precision; may be zero for pure relative error
test.
"single precision relative tolerance"
Non-negative relative tolerance for single precision. If the absolute tolerance is zero, the relative tolerance must be greater than or equal to
max (50*eps, 0.5e-28).

q =
q =
q =
q =
[q,

quadv
quadv
quadv
quadv
nfun]

(f,
(f,
(f,
(f,

a, b)
a, b, tol)
a, b, tol, trace)
a, b, tol, trace, p1, p2, . . . )
= quadv ( . . . )

Numerically evaluate the integral of f from a to b using an adaptive Simpson’s rule.
f is a function handle, inline function, or string containing the name of the function
to evaluate. quadv is a vectorized version of quad and the function defined by f must
accept a scalar or vector as input and return a scalar, vector, or array as output.
a and b are the lower and upper limits of integration. Both limits must be finite.
The optional argument tol defines the absolute tolerance used to stop the adaptation
procedure. The default value is 1e-6.

608

GNU Octave

The algorithm used by quadv involves recursively subdividing the integration interval
and applying Simpson’s rule on each subinterval. If trace is true then after computing
each of these partial integrals display: (1) the total number of function evaluations, (2)
the left end of the subinterval, (3) the length of the subinterval, (4) the approximation
of the integral over the subinterval.
Additional arguments p1, etc., are passed directly to the function f. To use default
values for tol and trace, one may pass empty matrices ([]).
The result of the integration is returned in q.
The optional output nfun indicates the total number of function evaluations performed.
Note: quadv is written in Octave’s scripting language and can be used recursively in
dblquad and triplequad, unlike the quad function.
See also: [quad], page 606, [quadl], page 608, [quadgk], page 608, [quadcc], page 610,
[trapz], page 611, [dblquad], page 613, [triplequad], page 614.

q =
q =
q =
q =
[q,

quadl
quadl
quadl
quadl
nfun]

(f,
(f,
(f,
(f,

a, b)
a, b, tol)
a, b, tol, trace)
a, b, tol, trace, p1, p2, . . . )
= quadl ( . . . )

Numerically evaluate the integral of f from a to b using an adaptive Lobatto rule.
f is a function handle, inline function, or string containing the name of the function
to evaluate. The function f must be vectorized and return a vector of output values
when given a vector of input values.
a and b are the lower and upper limits of integration. Both limits must be finite.
The optional argument tol defines the absolute tolerance with which to perform the
integration. The default value is 1e-6.
The algorithm used by quadl involves recursively subdividing the integration interval.
If trace is defined then for each subinterval display: (1) the total number of function
evaluations, (2) the left end of the subinterval, (3) the length of the subinterval, (4)
the approximation of the integral over the subinterval.
Additional arguments p1, etc., are passed directly to the function f. To use default
values for tol and trace, one may pass empty matrices ([]).
The result of the integration is returned in q.
The optional output nfun indicates the total number of function evaluations performed.
Reference: W. Gander and W. Gautschi, Adaptive Quadrature - Revisited, BIT Vol.
40, No. 1, March 2000, pp. 84–101. http://www.inf.ethz.ch/personal/gander/
See also: [quad], page 606, [quadv], page 607, [quadgk], page 608, [quadcc], page 610,
[trapz], page 611, [dblquad], page 613, [triplequad], page 614.

q = quadgk (f, a, b)
q = quadgk (f, a, b, abstol)
q = quadgk (f, a, b, abstol, trace)

Chapter 23: Numerical Integration

609

q = quadgk (f, a, b, prop, val, . . . )
[q, err] = quadgk ( . . . )
Numerically evaluate the integral of f from a to b using adaptive Gauss-Konrod
quadrature.
f is a function handle, inline function, or string containing the name of the function
to evaluate. The function f must be vectorized and return a vector of output values
when given a vector of input values.
a and b are the lower and upper limits of integration. Either or both limits may be
infinite or contain weak end singularities. Variable transformation will be used to
treat any infinite intervals and weaken the singularities. For example:
quadgk (@(x) 1 ./ (sqrt (x) .* (x + 1)), 0, Inf)
Note that the formulation of the integrand uses the element-by-element operator ./
and all user functions to quadgk should do the same.
The optional argument tol defines the absolute tolerance used to stop the integration
procedure. The default value is 1e-10.
The algorithm used by quadgk involves subdividing the integration interval and evaluating each subinterval. If trace is true then after computing each of these partial
integrals display: (1) the number of subintervals at this step, (2) the current estimate
of the error err, (3) the current estimate for the integral q.
Alternatively, properties of quadgk can be passed to the function as pairs "prop",
val. Valid properties are
AbsTol

Define the absolute error tolerance for the quadrature. The default absolute tolerance is 1e-10 (1e-5 for single).

RelTol

Define the relative error tolerance for the quadrature. The default relative
tolerance is 1e-6 (1e-4 for single).

MaxIntervalCount
quadgk initially subdivides the interval on which to perform the
quadrature into 10 intervals. Subintervals that have an unacceptable
error are subdivided and re-evaluated. If the number of subintervals
exceeds 650 subintervals at any point then a poor convergence is signaled
and the current estimate of the integral is returned. The property
"MaxIntervalCount" can be used to alter the number of subintervals
that can exist before exiting.
WayPoints
Discontinuities in the first derivative of the function to integrate can be
flagged with the "WayPoints" property. This forces the ends of a subinterval to fall on the breakpoints of the function and can result in significantly improved estimation of the error in the integral, faster computation, or both. For example,
quadgk (@(x) abs (1 - x.^2), 0, 2, "Waypoints", 1)
signals the breakpoint in the integrand at x = 1.
Trace

If logically true quadgk prints information on the convergence of the
quadrature at each iteration.

610

GNU Octave

If any of a, b, or waypoints is complex then the quadrature is treated as a contour
integral along a piecewise continuous path defined by the above. In this case the
integral is assumed to have no edge singularities. For example,
quadgk (@(z) log (z), 1+1i, 1+1i, "WayPoints",
[1-1i, -1,-1i, -1+1i])
integrates log (z) along the square defined by [1+1i, 1-1i, -1-1i, -1+1i].
The result of the integration is returned in q.
err is an approximate bound on the error in the integral abs (q - I), where I is the
exact value of the integral.
Reference: L.F. Shampine, "Vectorized adaptive quadrature in matlab", Journal of
Computational and Applied Mathematics, pp. 131–140, Vol 211, Issue 2, Feb 2008.
See also: [quad], page 606, [quadv], page 607, [quadl], page 608, [quadcc], page 610,
[trapz], page 611, [dblquad], page 613, [triplequad], page 614.

q =
q =
q =
[q,

quadcc (f, a, b)
quadcc (f, a, b, tol)
quadcc (f, a, b, tol, sing)
err, nr_points] = quadcc ( . . . )
Numerically evaluate the integral of f from a to b using doubly-adaptive ClenshawCurtis quadrature.
f is a function handle, inline function, or string containing the name of the function
to evaluate. The function f must be vectorized and must return a vector of output
values if given a vector of input values. For example,
f = @(x) x .* sin (1./x) .* sqrt (abs (1 - x));
which uses the element-by-element “dot” form for all operators.
a and b are the lower and upper limits of integration. Either or both limits may be
infinite. quadcc handles an inifinite limit by substituting the variable of integration
with x = tan (pi/2*u).
The optional argument tol defines the relative tolerance used to stop the integration
procedure. The default value is 1e-6.
The optional argument sing contains a list of points where the integrand has known
singularities, or discontinuities in any of its derivatives, inside the integration interval.
For the example above, which has a discontinuity at x=1, the call to quadcc would
be as follows
int = quadcc (f, a, b, 1.0e-6, [ 1 ]);
The result of the integration is returned in q.
err is an estimate of the absolute integration error.
nr points is the number of points at which the integrand was evaluated.
If the adaptive integration did not converge, the value of err will be larger than the
requested tolerance. Therefore, it is recommended to verify this value for difficult
integrands.
quadcc is capable of dealing with non-numeric values of the integrand such as NaN or
Inf. If the integral diverges, and quadcc detects this, then a warning is issued and
Inf or -Inf is returned.

Chapter 23: Numerical Integration

611

Note: quadcc is a general purpose quadrature algorithm and, as such, may be less
efficient for a smooth or otherwise well-behaved integrand than other methods such
as quadgk.
The algorithm uses Clenshaw-Curtis quadrature rules of increasing degree in each
interval and bisects the interval if either the function does not appear to be smooth
or a rule of maximum degree has been reached. The error estimate is computed from
the L2-norm of the difference between two successive interpolations of the integrand
over the nodes of the respective quadrature rules.
Reference: P. Gonnet, Increasing the Reliability of Adaptive Quadrature Using Explicit Interpolants, ACM Transactions on Mathematical Software, Vol. 37, Issue 3,
Article No. 3, 2010.
See also: [quad], page 606, [quadv], page 607, [quadl], page 608, [quadgk], page 608,
[trapz], page 611, [dblquad], page 613, [triplequad], page 614.
Sometimes one does not have the function, but only the raw (x, y) points from which to
perform an integration. This can occur when collecting data in an experiment. The trapz
function can integrate these values as shown in the following example where "data" has
been collected on the cosine function over the range [0, pi/2).
x = 0:0.1:pi/2;
y = cos (x);
trapz (x, y)
⇒ 0.99666

# Uniformly spaced points

x = 0:0.1:pi/2;
y = sin (x);
trapz (x, y)
⇒ 0.92849

# Uniformly spaced points

The answer is reasonably close to the exact value of 1. Ordinary quadrature is sensitive
to the characteristics of the integrand. Empirical integration depends not just on the
integrand, but also on the particular points chosen to represent the function. Repeating the
example above with the sine function over the range [0, pi/2) produces far inferior results.

However, a slightly different choice of data points can change the result significantly.
The same integration, with the same number of points, but spaced differently produces a
more accurate answer.
x = linspace (0, pi/2, 16);
y = sin (x);
trapz (x, y)
⇒ 0.99909

# Uniformly spaced, but including endpoint

In general there may be no way of knowing the best distribution of points ahead of time.
Or the points may come from an experiment where there is no freedom to select the best
distribution. In any case, one must remain aware of this issue when using trapz.

q = trapz (y)
q = trapz (x, y)
q = trapz ( . . . , dim)
Numerically evaluate the integral of points y using the trapezoidal method.

612

GNU Octave

trapz (y) computes the integral of y along the first non-singleton dimension. When
the argument x is omitted an equally spaced x vector with unit spacing (1) is assumed.
trapz (x, y) evaluates the integral with respect to the spacing in x and the values
in y. This is useful if the points in y have been sampled unevenly.
If the optional dim argument is given, operate along this dimension.
Application Note: If x is not specified then unit spacing will be used. To scale the
integral to the correct value you must multiply by the actual spacing value (deltaX).
As an example, the integral of x3 over the range [0, 1] is x4 /4 or 0.25. The following
code uses trapz to calculate the integral in three different ways.
x = 0:0.1:1;
y = x.^3;
q = trapz (y)
⇒ q = 2.525
q * 0.1
⇒ q = 0.2525
trapz (x, y)
⇒ q = 0.2525

# No scaling
# Approximation to integral by scaling
# Same result by specifying x

See also: [cumtrapz], page 612.

q = cumtrapz (y)
q = cumtrapz (x, y)
q = cumtrapz ( . . . , dim)
Cumulative numerical integration of points y using the trapezoidal method.
cumtrapz (y) computes the cumulative integral of y along the first non-singleton
dimension. Where trapz reports only the overall integral sum, cumtrapz reports the
current partial sum value at each point of y.
When the argument x is omitted an equally spaced x vector with unit spacing (1)
is assumed. cumtrapz (x, y) evaluates the integral with respect to the spacing in x
and the values in y. This is useful if the points in y have been sampled unevenly.
If the optional dim argument is given, operate along this dimension.
Application Note: If x is not specified then unit spacing will be used. To scale the
integral to the correct value you must multiply by the actual spacing value (deltaX).
See also: [trapz], page 611, [cumsum], page 483.

23.2 Orthogonal Collocation
[r, amat, bmat, q] = colloc (n, "left", "right")
Compute derivative and integral weight matrices for orthogonal collocation.
Reference: J. Villadsen, M. L. Michelsen, Solution of Differential Equation Models
by Polynomial Approximation.
Here is an example of using colloc to generate weight matrices for solving the second
order differential equation u0 −αu00 = 0 with the boundary conditions u(0) = 0 and u(1) = 1.

Chapter 23: Numerical Integration

613

First, we can generate the weight matrices for n points (including the endpoints of the
interval), and incorporate the boundary conditions in the right hand side (for a specific
value of α).
n = 7;
alpha = 0.1;
[r, a, b] = colloc (n-2, "left", "right");
at = a(2:n-1,2:n-1);
bt = b(2:n-1,2:n-1);
rhs = alpha * b(2:n-1,n) - a(2:n-1,n);
Then the solution at the roots r is
u = [ 0; (at - alpha * bt) \ rhs; 1]
⇒ [ 0.00; 0.004; 0.01 0.00; 0.12; 0.62; 1.00 ]

23.3 Functions of Multiple Variables
Octave does not have built-in functions for computing the integral of functions of multiple
variables directly. It is possible, however, to compute the integral of a function of multiple
variables using the existing functions for one-dimensional integrals.
To illustrate how the integration can be performed, we will integrate the function
√
f (x, y) = sin(πxy) xy
for x and y between 0 and 1.
The first approach creates a function that integrates f with respect to x, and then
integrates that function with respect to y. Because quad is written in Fortran it cannot
be called recursively. This means that quad cannot integrate a function that calls quad,
and hence cannot be used to perform the double integration. Any of the other integrators,
however, can be used which is what the following code demonstrates.
function q = g(y)
q = ones (size (y));
for i = 1:length (y)
f = @(x) sin (pi*x.*y(i)) .* sqrt (x.*y(i));
q(i) = quadgk (f, 0, 1);
endfor
endfunction
I = quadgk ("g", 0, 1)
⇒ 0.30022
The above process can be simplified with the dblquad and triplequad functions for
integrals over two and three variables. For example:
I = dblquad (@(x, y) sin (pi*x.*y) .* sqrt (x.*y), 0, 1, 0, 1)
⇒ 0.30022

dblquad (f, xa, xb, ya, yb)
dblquad (f, xa, xb, ya, yb, tol)
dblquad (f, xa, xb, ya, yb, tol, quadf)

614

GNU Octave

dblquad (f, xa, xb, ya, yb, tol, quadf, . . . )
Numerically evaluate the double integral of f.
f is a function handle, inline function, or string containing the name of the function
to evaluate. The function f must have the form z = f (x, y) where x is a vector and
y is a scalar. It should return a vector of the same length and orientation as x.
xa, ya and xb, yb are the lower and upper limits of integration for x and y respectively.
The underlying integrator determines whether infinite bounds are accepted.
The optional argument tol defines the absolute tolerance used to integrate each subintegral. The default value is 1e-6.
The optional argument quadf specifies which underlying integrator function to use.
Any choice but quad is available and the default is quadcc.
Additional arguments, are passed directly to f. To use the default value for tol or
quadf one may pass ’:’ or an empty matrix ([]).
See also: [triplequad], page 614, [quad], page 606, [quadv], page 607, [quadl], page 608,
[quadgk], page 608, [quadcc], page 610, [trapz], page 611.
(f, xa, xb, ya, yb, za, zb)
(f, xa, xb, ya, yb, za, zb, tol)
(f, xa, xb, ya, yb, za, zb, tol, quadf)
(f, xa, xb, ya, yb, za, zb, tol, quadf, . . . )
Numerically evaluate the triple integral of f.
f is a function handle, inline function, or string containing the name of the function
to evaluate. The function f must have the form w = f (x, y, z) where either x or y is
a vector and the remaining inputs are scalars. It should return a vector of the same
length and orientation as x or y.
xa, ya, za and xb, yb, zb are the lower and upper limits of integration for x, y,
and z respectively. The underlying integrator determines whether infinite bounds are
accepted.
The optional argument tol defines the absolute tolerance used to integrate each subintegral. The default value is 1e-6.
The optional argument quadf specifies which underlying integrator function to use.
Any choice but quad is available and the default is quadcc.
Additional arguments, are passed directly to f. To use the default value for tol or
quadf one may pass ’:’ or an empty matrix ([]).

triplequad
triplequad
triplequad
triplequad

See also: [dblquad], page 613, [quad], page 606, [quadv], page 607, [quadl], page 608,
[quadgk], page 608, [quadcc], page 610, [trapz], page 611.
The above mentioned approach works, but is fairly slow, and that problem increases
exponentially with the dimensionality of the integral. Another possible solution is to use
Orthogonal Collocation as described in the previous section (see Section 23.2 [Orthogonal
Collocation], page 612). The integral of a function f (x, y) for x and y between 0 and 1 can
be approximated using n points by
Z

0

1

Z

0

1

f (x, y)dxdy ≈

n X
n
X
i=1 j=1

qi qj f (ri , rj ),

615

where q and r is as returned by colloc (n). The generalization to more than two variables
is straight forward. The following code computes the studied integral using n = 8 points.
f = @(x,y) sin (pi*x*y’) .* sqrt (x*y’);
n = 8;
[t, ~, ~, q] = colloc (n);
I = q’*f(t,t)*q;
⇒ 0.30022
It should be noted that the number of points determines the quality of the approximation.
If the integration needs to be performed between a and b, instead of 0 and 1, then a change
of variables is needed.

617

24 Differential Equations
Octave has built-in functions for solving ordinary differential equations, and differentialalgebraic equations. All solvers are based on reliable ODE routines written in Fortran.

24.1 Ordinary Differential Equations
The function lsode can be used to solve ODEs of the form
dx
= f (x, t)
dt
using Hindmarsh’s ODE solver lsode.

[x, istate, msg] = lsode (fcn, x_0, t)
[x, istate, msg] = lsode (fcn, x_0, t, t_crit)
Ordinary Differential Equation (ODE) solver.
The set of differential equations to solve is
dx
= f (x, t)
dt
with
x(t0 ) = x0
The solution is returned in the matrix x, with each row corresponding to an element
of the vector t. The first element of t should be t0 and should correspond to the initial
state of the system x 0, so that the first row of the output is x 0.
The first argument, fcn, is a string, inline, or function handle that names the function
f to call to compute the vector of right hand sides for the set of equations. The
function must have the form
xdot = f (x, t)
in which xdot and x are vectors and t is a scalar.
If fcn is a two-element string array or a two-element cell array of strings, inline
functions, or function handles, the first element names the function f described above,
and the second element names a function to compute the Jacobian of f . The Jacobian
function must have the form
jac = j (x, t)
in which jac is the matrix of partial derivatives



∂f1
∂x1
∂f2
∂x1

∂f1
∂x2
∂f2
∂x2

∂f3
∂x1

∂f3
∂x2


∂fi
J=
=
 ..
∂xj
 .

..
.

···
···
..
.
···

∂f1
∂xN
∂f2
∂xN






.. 
. 

∂f3
∂xN

The second argument specifies the initial state of the system x0 . The third argument
is a vector, t, specifying the time values for which a solution is sought.

618

GNU Octave

The fourth argument is optional, and may be used to specify a set of times that
the ODE solver should not integrate past. It is useful for avoiding difficulties with
singularities and points where there is a discontinuity in the derivative.
After a successful computation, the value of istate will be 2 (consistent with the
Fortran version of lsode).
If the computation is not successful, istate will be something other than 2 and msg
will contain additional information.
You can use the function lsode_options to set optional parameters for lsode.
See also: [daspk], page 624, [dassl], page 628, [dasrt], page 630.

lsode_options ()
val = lsode_options (opt)
lsode_options (opt, val)
Query or set options for the function lsode.
When called with no arguments, the names of all available options and their current
values are displayed.
Given one argument, return the value of the option opt.
When called with two arguments, lsode_options sets the option opt to value val.
Options include
"absolute tolerance"
Absolute tolerance. May be either vector or scalar. If a vector, it must
match the dimension of the state vector.
"relative tolerance"
Relative tolerance parameter. Unlike the absolute tolerance, this parameter may only be a scalar.
The local error test applied at each integration step is
abs (local error in x(i)) <= ...
rtol * abs (y(i)) + atol(i)
"integration method"
A string specifying the method of integration to use to solve the ODE
system. Valid values are
"adams"
"non-stiff"
No Jacobian used (even if it is available).
"bdf"
"stiff"

Use stiff backward differentiation formula (BDF) method. If
a function to compute the Jacobian is not supplied, lsode
will compute a finite difference approximation of the Jacobian
matrix.

"initial step size"
The step size to be attempted on the first step (default is determined
automatically).

Chapter 24: Differential Equations

619

"maximum order"
Restrict the maximum order of the solution method. If using the Adams
method, this option must be between 1 and 12. Otherwise, it must be
between 1 and 5, inclusive.
"maximum step size"
Setting the maximum stepsize will avoid passing over very large regions
(default is not specified).
"minimum step size"
The minimum absolute step size allowed (default is 0).
"step limit"
Maximum number of steps allowed (default is 100000).
Here is an example of solving a set of three differential equations using lsode. Given
the function
## oregonator differential equation
function xdot = f (x, t)
xdot = zeros (3,1);
xdot(1) = 77.27 * (x(2) - x(1)*x(2) + x(1) ...
- 8.375e-06*x(1)^2);
xdot(2) = (x(3) - x(1)*x(2) - x(2)) / 77.27;
xdot(3) = 0.161*(x(1) - x(3));
endfunction
and the initial condition x0 = [ 4; 1.1; 4 ], the set of equations can be integrated using
the command
t = linspace (0, 500, 1000);
y = lsode ("f", x0, t);
If you try this, you will see that the value of the result changes dramatically between t
= 0 and 5, and again around t = 305. A more efficient set of output points might be
t = [0, logspace(-1, log10(303), 150), ...
logspace(log10(304), log10(500), 150)];
See Alan C. Hindmarsh, ODEPACK, A Systematized Collection of ODE Solvers, in
Scientific Computing, R. S. Stepleman, editor, (1983) for more information about the inner
workings of lsode.
An m-file for the differential equation used above is included with the Octave distribution
in the examples directory under the name oregonator.m.

24.1.1 Matlab-compatible solvers
Octave also provides a set of solvers for initial value problems for Ordinary Differential
Equations that have a matlab-compatible interface. The options for this class of methods
are set using the functions.
• odeset

620

GNU Octave

• odeget
Currently implemented solvers are:
• Runge-Kutta methods
• ode45 Integrates a system of non–stiff ordinary differential equations (non–stiff
ODEs and DAEs) using second order Dormand-Prince method. This is a fourth–
order accurate integrator therefore the local error normally expected is O(h5 ). This
solver requires six function evaluations per integration step.
• ode23 Integrates a system of non–stiff ordinary differential equations (non-stiff
ODEs and DAEs) using second order Bogacki-Shampine method. This is a secondorder accurate integrator therefore the local error normally expected is O(h3 ). This
solver requires three function evaluations per integration step.

[t, y] = ode45 (fun, trange, init)
[t, y] = ode45 (fun, trange, init, ode_opt)
[t, y, te, ye, ie] = ode45 ( . . . )
solution = ode45 ( . . . )
ode45 ( . . . )
Solve a set of non-stiff Ordinary Differential Equations (non-stiff ODEs) with the well
known explicit Dormand-Prince method of order 4.
fun is a function handle, inline function, or string containing the name of the function
that defines the ODE: y’ = f(t,y). The function must accept two inputs where the
first is time t and the second is a column vector of unknowns y.
trange specifies the time interval over which the ODE will be evaluated. Typically,
it is a two-element vector specifying the initial and final times ([tinit, tfinal]).
If there are more than two elements then the solution will also be evaluated at these
intermediate time instances.
By default, ode45 uses an adaptive timestep with the integrate_adaptive algorithm. The tolerance for the timestep computation may be changed by using the
options "RelTol" and "AbsTol".
init contains the initial value for the unknowns. If it is a row vector then the solution
y will be a matrix in which each column is the solution for the corresponding initial
value in init.
The optional fourth argument ode opt specifies non-default options to the ODE
solver. It is a structure generated by odeset.
The function typically returns two outputs. Variable t is a column vector and contains
the times where the solution was found. The output y is a matrix in which each
column refers to a different unknown of the problem and each row corresponds to a
time in t.
The output can also be returned as a structure solution which has a field x containing
a row vector of times where the solution was evaluated and a field y containing the
solution matrix such that each column corresponds to a time in x. Use fieldnames
(solution) to see the other fields and additional information returned.
If no output arguments are requested, and no OutputFcn is specified in ode opt, then
the OutputFcn is set to odeplot and the results of the solver are plotted immediately.

Chapter 24: Differential Equations

621

If using the "Events" option then three additional outputs may be returned. te holds
the time when an Event function returned a zero. ye holds the value of the solution
at time te. ie contains an index indicating which Event function was triggered in the
case of multiple Event functions.
Example: Solve the Van der Pol equation
fvdp = @(t,y) [y(2); (1 - y(1)^2) * y(2) - y(1)];
[t,y] = ode45 (fvdp, [0, 20], [2, 0]);
See also: [odeset], page 622, [odeget], page 624, [ode23], page 621.

[t, y] = ode23 (fun, trange, init)
[t, y] = ode23 (fun, trange, init, ode_opt)
[t, y, te, ye, ie] = ode23 ( . . . )
solution = ode23 ( . . . )
ode23 ( . . . )
Solve a set of non-stiff Ordinary Differential Equations (non-stiff ODEs) with the well
known explicit Bogacki-Shampine method of order 3. For the definition of this method
see http://en.wikipedia.org/wiki/List_of_Runge%E2%80%93Kutta_methods.
fun is a function handle, inline function, or string containing the name of the function
that defines the ODE: y’ = f(t,y). The function must accept two inputs where the
first is time t and the second is a column vector of unknowns y.
trange specifies the time interval over which the ODE will be evaluated. Typically,
it is a two-element vector specifying the initial and final times ([tinit, tfinal]).
If there are more than two elements then the solution will also be evaluated at these
intermediate time instances.
By default, ode23 uses an adaptive timestep with the integrate_adaptive algorithm. The tolerance for the timestep computation may be changed by using the
options "RelTol" and "AbsTol".
init contains the initial value for the unknowns. If it is a row vector then the solution
y will be a matrix in which each column is the solution for the corresponding initial
value in init.
The optional fourth argument ode opt specifies non-default options to the ODE
solver. It is a structure generated by odeset.
The function typically returns two outputs. Variable t is a column vector and contains
the times where the solution was found. The output y is a matrix in which each
column refers to a different unknown of the problem and each row corresponds to a
time in t.
The output can also be returned as a structure solution which has a field x containing
a row vector of times where the solution was evaluated and a field y containing the
solution matrix such that each column corresponds to a time in x. Use fieldnames
(solution) to see the other fields and additional information returned.
If no output arguments are requested, and no OutputFcn is specified in ode opt, then
the OutputFcn is set to odeplot and the results of the solver are plotted immediately.
If using the "Events" option then three additional outputs may be returned. te holds
the time when an Event function returned a zero. ye holds the value of the solution

622

GNU Octave

at time te. ie contains an index indicating which Event function was triggered in the
case of multiple Event functions.
Example: Solve the Van der Pol equation
fvdp = @(t,y) [y(2); (1 - y(1)^2) * y(2) - y(1)];
[t,y] = ode23 (fvdp, [0, 20], [2, 0]);
See also: [odeset], page 622, [odeget], page 624, [ode45], page 620.

odestruct
odestruct
odestruct
odestruct
odeset ()

=
=
=
=

odeset
odeset
odeset
odeset

()
("field1", value1, "field2", value2, . . . )
(oldstruct, "field1", value1, "field2", value2, . . . )
(oldstruct, newstruct)

Create or modify an ODE options structure.
When called with no input argument and one output argument, return a new ODE
options structure that contains all possible fields initialized to their default values. If
no output argument is requested, display a list of the common ODE solver options
along with their default value.
If called with name-value input argument pairs "field1", "value1", "field2", "value2",
. . . return a new ODE options structure with all the most common option fields
initialized, and set the values of the fields "field1", "field2", . . . to the values value1,
value2, . . . .
If called with an input structure oldstruct then overwrite the values of the options
"field1", "field2", . . . with new values value1, value2, . . . and return the modified
structure.
When called with two input ODE options structures oldstruct and newstruct overwrite all values from the structure oldstruct with new values from the structure
newstruct. Empty values in newstruct will not overwrite values in oldstruct.
The most commonly used ODE options, which are always assigned a value by odeset,
are the following:
AbsTol

Absolute error tolerance.

BDF

Use BDF formulas in implicit multistep methods. Note: This option is
not yet implemented.

Events

Event function. An event function must have the form [value,
isterminal, direction] = my_events_f (t, y)

InitialSlope
Consistent initial slope vector for DAE solvers.
InitialStep Initial time step size.
Jacobian

Jacobian matrix, specified as a constant matrix or a function of time and
state.

JConstant Specify whether the Jacobian is a constant matrix or depends on the
state.

Chapter 24: Differential Equations

623

JPattern

If the Jacobian matrix is sparse and non-constant but maintains a constant sparsity pattern, specify the sparsity pattern.

Mass

Mass matrix, specified as a constant matrix or a function of time and
state.

MassSingular
Specify whether the mass matrix is singular. Accepted values include
"yes", "no", "maybe".
MaxOrder Maximum order of formula.
MaxStep

Maximum time step value.

MStateDependence
Specify whether the mass matrix depends on the state or only on time.
MvPattern
If the mass matrix is sparse and non-constant but maintains a constant
sparsity pattern, specify the sparsity pattern. Note: This option is not
yet implemented.
NonNegative
Specify elements of the state vector that are expected to remain nonnegative during the simulation.
NormControl
Control error relative to the 2-norm of the solution, rather than its absolute value.
OutputFcn
Function to monitor the state during the simulation. For the form of the
function to use see odeplot.
OutputSel Indices of elements of the state vector to be passed to the output monitoring function.
Refine

Specify whether output should be returned only at the end of each time
step or also at intermediate time instances. The value should be a scalar
indicating the number of equally spaced time points to use within each
timestep at which to return output. Note: This option is not yet implemented.

RelTol

Relative error tolerance.

Stats

Print solver statistics after simulation.

Vectorized Specify whether odefun can be passed multiple values of the state at
once.
Field names that are not in the above list are also accepted and added to the result
structure.
See also: [odeget], page 624.

624

GNU Octave

val = odeget (ode_opt, field)
val = odeget (ode_opt, field, default)
Query the value of the property field in the ODE options structure ode opt.
If called with two input arguments and the first input argument ode opt is an ODE
option structure and the second input argument field is a string specifying an option
name, then return the option value val corresponding to field from ode opt.
If called with an optional third input argument, and field is not set in the structure
ode opt, then return the default value default instead.
See also: [odeset], page 622.

stop_solve = odeplot (t, y, flag)
Open a new figure window and plot the solution of an ode problem at each time step
during the integration.
The types and values of the input parameters t and y depend on the input flag that
is of type string. Valid values of flag are:
"init"

The input t must be a column vector of length 2 with the first and last
time step ([tfirst tlast]. The input y contains the initial conditions
for the ode problem (y0).

""

The input t must be a scalar double specifying the time for which the
solution in input y was calculated.

"done"

The inputs should be empty, but are ignored if they are present.

odeplot always returns false, i.e., don’t stop the ode solver.
Example: solve an anonymous implementation of the "Van der Pol" equation and
display the results while solving.
fvdp = @(t,y) [y(2); (1 - y(1)^2) * y(2) - y(1)];
opt = odeset ("OutputFcn", @odeplot, "RelTol", 1e-6);
sol = ode45 (fvdp, [0 20], [2 0], opt);
Background Information: This function is called by an ode solver function if it was
specified in the "OutputFcn" property of an options structure created with odeset.
The ode solver will initially call the function with the syntax odeplot ([tfirst,
tlast], y0, "init"). The function initializes internal variables, creates a new figure
window, and sets the x limits of the plot. Subsequently, at each time step during the
integration the ode solver calls odeplot (t, y, []). At the end of the solution the
ode solver calls odeplot ([], [], "done") so that odeplot can perform any clean-up
actions required.
See also: [odeset], page 622, [odeget], page 624, [ode23], page 621, [ode45], page 620.

24.2 Differential-Algebraic Equations
The function daspk can be used to solve DAEs of the form
0 = f (ẋ, x, t),
where ẋ =
daspk.

dx
dt

x(t = 0) = x0 , ẋ(t = 0) = ẋ0

is the derivative of x. The equation is solved using Petzold’s DAE solver

Chapter 24: Differential Equations

625

[x, xdot, istate, msg] = daspk (fcn, x_0, xdot_0, t, t_crit)
Solve the set of differential-algebraic equations
0 = f (x, ẋ, t)
with
x(t0 ) = x0 , ẋ(t0 ) = ẋ0
The solution is returned in the matrices x and xdot, with each row in the result
matrices corresponding to one of the elements in the vector t. The first element of t
should be t0 and correspond to the initial state of the system x 0 and its derivative
xdot 0, so that the first row of the output x is x 0 and the first row of the output
xdot is xdot 0.
The first argument, fcn, is a string, inline, or function handle that names the function
f to call to compute the vector of residuals for the set of equations. It must have the
form
res = f (x, xdot, t)
in which x, xdot, and res are vectors, and t is a scalar.
If fcn is a two-element string array or a two-element cell array of strings, inline
functions, or function handles, the first element names the function f described above,
and the second element names a function to compute the modified Jacobian
J=

∂f
∂f
+c
∂x
∂ ẋ

The modified Jacobian function must have the form
jac = j (x, xdot, t, c)
The second and third arguments to daspk specify the initial condition of the states
and their derivatives, and the fourth argument specifies a vector of output times at
which the solution is desired, including the time corresponding to the initial condition.
The set of initial states and derivatives are not strictly required to be consistent.
If they are not consistent, you must use the daspk_options function to provide
additional information so that daspk can compute a consistent starting point.
The fifth argument is optional, and may be used to specify a set of times that the DAE
solver should not integrate past. It is useful for avoiding difficulties with singularities
and points where there is a discontinuity in the derivative.
After a successful computation, the value of istate will be greater than zero (consistent
with the Fortran version of daspk).
If the computation is not successful, the value of istate will be less than zero and msg
will contain additional information.
You can use the function daspk_options to set optional parameters for daspk.
See also: [dassl], page 628.

626

GNU Octave

daspk_options ()
val = daspk_options (opt)
daspk_options (opt, val)
Query or set options for the function daspk.
When called with no arguments, the names of all available options and their current
values are displayed.
Given one argument, return the value of the option opt.
When called with two arguments, daspk_options sets the option opt to value val.
Options include
"absolute tolerance"
Absolute tolerance. May be either vector or scalar. If a vector, it must
match the dimension of the state vector, and the relative tolerance must
also be a vector of the same length.
"relative tolerance"
Relative tolerance. May be either vector or scalar. If a vector, it must
match the dimension of the state vector, and the absolute tolerance must
also be a vector of the same length.
The local error test applied at each integration step is
abs (local error in x(i))
<= rtol(i) * abs (Y(i)) + atol(i)
"compute consistent initial condition"
Denoting the differential variables in the state vector by ‘Y_d’ and the
algebraic variables by ‘Y_a’, ddaspk can solve one of two initialization
problems:
1. Given Y d, calculate Y a and Y’ d
2. Given Y’, calculate Y.
In either case, initial values for the given components are input, and
initial guesses for the unknown components must also be provided as
input. Set this option to 1 to solve the first problem, or 2 to solve the
second (the default is 0, so you must provide a set of initial conditions
that are consistent).
If this option is set to a nonzero value, you must also set the "algebraic
variables" option to declare which variables in the problem are algebraic.
"use initial condition heuristics"
Set to a nonzero value to use the initial condition heuristics options described below.
"initial condition heuristics"
A vector of the following parameters that can be used to control the initial
condition calculation.
MXNIT

Maximum number of Newton iterations (default is 5).

MXNJ

Maximum number of Jacobian evaluations (default is 6).

Chapter 24: Differential Equations

627

MXNH

Maximum number of values of the artificial stepsize
parameter to be tried if the "compute consistent initial
condition" option has been set to 1 (default is 5).
Note that the maximum total number of Newton iterations
allowed is MXNIT*MXNJ*MXNH if the "compute consistent
initial condition" option has been set to 1 and
MXNIT*MXNJ if it is set to 2.

LSOFF

Set to a nonzero value to disable the linesearch algorithm
(default is 0).

STPTOL

Minimum scaled step in linesearch algorithm (default is
eps^(2/3)).

EPINIT

Swing factor in the Newton iteration convergence test. The
test is applied to the residual vector, premultiplied by the
approximate Jacobian. For convergence, the weighted RMS
norm of this vector (scaled by the error weights) must be less
than EPINIT*EPCON, where EPCON = 0.33 is the analogous
test constant used in the time steps. The default is EPINIT
= 0.01.

"print initial condition info"
Set this option to a nonzero value to display detailed information about
the initial condition calculation (default is 0).
"exclude algebraic variables from error test"
Set to a nonzero value to exclude algebraic variables from the error test.
You must also set the "algebraic variables" option to declare which
variables in the problem are algebraic (default is 0).
"algebraic variables"
A vector of the same length as the state vector. A nonzero element
indicates that the corresponding element of the state vector is an algebraic
variable (i.e., its derivative does not appear explicitly in the equation set).
This option is required by the "compute consistent initial
condition" and "exclude algebraic variables from error test"
options.
"enforce inequality constraints"
Set to one of the following values to enforce the inequality constraints
specified by the "inequality constraint types" option (default is 0).
1. To have constraint checking only in the initial condition calculation.
2. To enforce constraint checking during the integration.
3. To enforce both options 1 and 2.
"inequality constraint types"
A vector of the same length as the state specifying the type of inequality
constraint. Each element of the vector corresponds to an element of the
state and should be assigned one of the following codes
-2

Less than zero.

628

GNU Octave

-1

Less than or equal to zero.

0

Not constrained.

1

Greater than or equal to zero.

2

Greater than zero.

This option only has an effect if the "enforce inequality constraints"
option is nonzero.
"initial step size"
Differential-algebraic problems may occasionally suffer from severe scaling
difficulties on the first step. If you know a great deal about the scaling
of your problem, you can help to alleviate this problem by specifying an
initial stepsize (default is computed automatically).
"maximum order"
Restrict the maximum order of the solution method. This option must
be between 1 and 5, inclusive (default is 5).
"maximum step size"
Setting the maximum stepsize will avoid passing over very large regions
(default is not specified).
Octave also includes dassl, an earlier version of daspk, and dasrt, which can be used
to solve DAEs with constraints (stopping conditions).

[x, xdot, istate, msg] = dassl (fcn, x_0, xdot_0, t, t_crit)
Solve the set of differential-algebraic equations
0 = f (x, ẋ, t)
with
x(t0 ) = x0 , ẋ(t0 ) = ẋ0
The solution is returned in the matrices x and xdot, with each row in the result
matrices corresponding to one of the elements in the vector t. The first element of t
should be t0 and correspond to the initial state of the system x 0 and its derivative
xdot 0, so that the first row of the output x is x 0 and the first row of the output
xdot is xdot 0.
The first argument, fcn, is a string, inline, or function handle that names the function
f to call to compute the vector of residuals for the set of equations. It must have the
form
res = f (x, xdot, t)
in which x, xdot, and res are vectors, and t is a scalar.
If fcn is a two-element string array or a two-element cell array of strings, inline
functions, or function handles, the first element names the function f described above,
and the second element names a function to compute the modified Jacobian
J=

∂f
∂f
+c
∂x
∂ ẋ

Chapter 24: Differential Equations

629

The modified Jacobian function must have the form
jac = j (x, xdot, t, c)
The second and third arguments to dassl specify the initial condition of the states
and their derivatives, and the fourth argument specifies a vector of output times at
which the solution is desired, including the time corresponding to the initial condition.
The set of initial states and derivatives are not strictly required to be consistent. In
practice, however, dassl is not very good at determining a consistent set for you, so
it is best if you ensure that the initial values result in the function evaluating to zero.
The fifth argument is optional, and may be used to specify a set of times that the DAE
solver should not integrate past. It is useful for avoiding difficulties with singularities
and points where there is a discontinuity in the derivative.
After a successful computation, the value of istate will be greater than zero (consistent
with the Fortran version of dassl).
If the computation is not successful, the value of istate will be less than zero and msg
will contain additional information.
You can use the function dassl_options to set optional parameters for dassl.
See also: [daspk], page 624, [dasrt], page 630, [lsode], page 617.

dassl_options ()
val = dassl_options (opt)
dassl_options (opt, val)
Query or set options for the function dassl.
When called with no arguments, the names of all available options and their current
values are displayed.
Given one argument, return the value of the option opt.
When called with two arguments, dassl_options sets the option opt to value val.
Options include
"absolute tolerance"
Absolute tolerance. May be either vector or scalar. If a vector, it must
match the dimension of the state vector, and the relative tolerance must
also be a vector of the same length.
"relative tolerance"
Relative tolerance. May be either vector or scalar. If a vector, it must
match the dimension of the state vector, and the absolute tolerance must
also be a vector of the same length.
The local error test applied at each integration step is
abs (local error in x(i))
<= rtol(i) * abs (Y(i)) + atol(i)
"compute consistent initial condition"
If nonzero, dassl will attempt to compute a consistent set of initial conditions. This is generally not reliable, so it is best to provide a consistent
set and leave this option set to zero.

630

GNU Octave

"enforce nonnegativity constraints"
If you know that the solutions to your equations will always be nonnegative, it may help to set this parameter to a nonzero value. However,
it is probably best to try leaving this option set to zero first, and only
setting it to a nonzero value if that doesn’t work very well.
"initial step size"
Differential-algebraic problems may occasionally suffer from severe scaling
difficulties on the first step. If you know a great deal about the scaling
of your problem, you can help to alleviate this problem by specifying an
initial stepsize.
"maximum order"
Restrict the maximum order of the solution method. This option must
be between 1 and 5, inclusive.
"maximum step size"
Setting the maximum stepsize will avoid passing over very large regions
(default is not specified).
"step limit"
Maximum number of integration steps to attempt on a single call to the
underlying Fortran code.

[x,
...
...
...

xdot, t_out,
= dasrt (fcn,
= dasrt (fcn,
= dasrt (fcn,

istat, msg] = dasrt (fcn, g, x_0, xdot_0, t)
g, x_0, xdot_0, t, t_crit)
x_0, xdot_0, t)
x_0, xdot_0, t, t_crit)

Solve the set of differential-algebraic equations
0 = f (x, ẋ, t)
with
x(t0 ) = x0 , ẋ(t0 ) = ẋ0
with functional stopping criteria (root solving).
The solution is returned in the matrices x and xdot, with each row in the result
matrices corresponding to one of the elements in the vector t out. The first element
of t should be t0 and correspond to the initial state of the system x 0 and its derivative
xdot 0, so that the first row of the output x is x 0 and the first row of the output
xdot is xdot 0.
The vector t provides an upper limit on the length of the integration. If the stopping
condition is met, the vector t out will be shorter than t, and the final element of t out
will be the point at which the stopping condition was met, and may not correspond
to any element of the vector t.
The first argument, fcn, is a string, inline, or function handle that names the function
f to call to compute the vector of residuals for the set of equations. It must have the
form
res = f (x, xdot, t)

Chapter 24: Differential Equations

631

in which x, xdot, and res are vectors, and t is a scalar.
If fcn is a two-element string array or a two-element cell array of strings, inline
functions, or function handles, the first element names the function f described above,
and the second element names a function to compute the modified Jacobian
J=

∂f
∂f
+c
∂x
∂ ẋ

The modified Jacobian function must have the form
jac = j (x, xdot, t, c)
The optional second argument names a function that defines the constraint functions
whose roots are desired during the integration. This function must have the form
g_out = g (x, t)
and return a vector of the constraint function values. If the value of any of the
constraint functions changes sign, dasrt will attempt to stop the integration at the
point of the sign change.
If the name of the constraint function is omitted, dasrt solves the same problem as
daspk or dassl.
Note that because of numerical errors in the constraint functions due to round-off
and integration error, dasrt may return false roots, or return the same root at two
or more nearly equal values of T. If such false roots are suspected, the user should
consider smaller error tolerances or higher precision in the evaluation of the constraint
functions.
If a root of some constraint function defines the end of the problem, the input to
dasrt should nevertheless allow integration to a point slightly past that root, so that
dasrt can locate the root by interpolation.
The third and fourth arguments to dasrt specify the initial condition of the states
and their derivatives, and the fourth argument specifies a vector of output times at
which the solution is desired, including the time corresponding to the initial condition.
The set of initial states and derivatives are not strictly required to be consistent. In
practice, however, dassl is not very good at determining a consistent set for you, so
it is best if you ensure that the initial values result in the function evaluating to zero.
The sixth argument is optional, and may be used to specify a set of times that the DAE
solver should not integrate past. It is useful for avoiding difficulties with singularities
and points where there is a discontinuity in the derivative.
After a successful computation, the value of istate will be greater than zero (consistent
with the Fortran version of dassl).
If the computation is not successful, the value of istate will be less than zero and msg
will contain additional information.
You can use the function dasrt_options to set optional parameters for dasrt.
See also: [dasrt options], page 632, [daspk], page 624, [dasrt], page 630, [lsode],
page 617.

632

GNU Octave

dasrt_options ()
val = dasrt_options (opt)
dasrt_options (opt, val)
Query or set options for the function dasrt.
When called with no arguments, the names of all available options and their current
values are displayed.
Given one argument, return the value of the option opt.
When called with two arguments, dasrt_options sets the option opt to value val.
Options include
"absolute tolerance"
Absolute tolerance. May be either vector or scalar. If a vector, it must
match the dimension of the state vector, and the relative tolerance must
also be a vector of the same length.
"relative tolerance"
Relative tolerance. May be either vector or scalar. If a vector, it must
match the dimension of the state vector, and the absolute tolerance must
also be a vector of the same length.
The local error test applied at each integration step is
abs (local error in x(i)) <= ...
rtol(i) * abs (Y(i)) + atol(i)
"initial step size"
Differential-algebraic problems may occasionally suffer from severe scaling
difficulties on the first step. If you know a great deal about the scaling
of your problem, you can help to alleviate this problem by specifying an
initial stepsize.
"maximum order"
Restrict the maximum order of the solution method. This option must
be between 1 and 5, inclusive.
"maximum step size"
Setting the maximum stepsize will avoid passing over very large regions.
"step limit"
Maximum number of integration steps to attempt on a single call to the
underlying Fortran code.
See K. E. Brenan, et al., Numerical Solution of Initial-Value Problems in DifferentialAlgebraic Equations, North-Holland (1989) for more information about the implementation
of dassl.

633

25 Optimization
Octave comes with support for solving various kinds of optimization problems. Specifically
Octave can solve problems in Linear Programming, Quadratic Programming, Nonlinear
Programming, and Linear Least Squares Minimization.

25.1 Linear Programming
Octave can solve Linear Programming problems using the glpk function. That is, Octave
can solve
min cT x
x

subject to the linear constraints Ax = b where x ≥ 0.
The glpk function also supports variations of this problem.

[xopt, fmin, errnum, extra] = glpk (c, A, b, lb, ub, ctype, vartype,
sense, param)
Solve a linear program using the GNU glpk library.
Given three arguments, glpk solves the following standard LP:
min C T x
x

subject to
x≥0

Ax = b
but may also solve problems of the form

[min | max]C T x
x

x

subject to
Ax[= | ≤ | ≥]b

LB ≤ x ≤ U B

Input arguments:
c

A column array containing the objective function coefficients.

A

A matrix containing the constraints coefficients.

b

A column array containing the right-hand side value for each constraint
in the constraint matrix.

lb

An array containing the lower bound on each of the variables. If lb is not
supplied, the default lower bound for the variables is zero.

ub

An array containing the upper bound on each of the variables. If ub is
not supplied, the default upper bound is assumed to be infinite.

ctype

An array of characters containing the sense of each constraint in the
constraint matrix. Each element of the array may be one of the following
values
"F"

A free (unbounded) constraint (the constraint is ignored).

634

GNU Octave

vartype

"U"

An inequality constraint with an upper bound (A(i,:)*x <=
b(i)).

"S"

An equality constraint (A(i,:)*x = b(i)).

"L"

An inequality with a lower bound (A(i,:)*x >= b(i)).

"D"

An inequality constraint with both upper and lower bounds
(A(i,:)*x >= -b(i)) and (A(i,:)*x <= b(i)).

A column array containing the types of the variables.
"C"

A continuous variable.

"I"

An integer variable.

sense

If sense is 1, the problem is a minimization. If sense is -1, the problem is
a maximization. The default value is 1.

param

A structure containing the following parameters used to define the behavior of solver. Missing elements in the structure take on default values,
so you only need to set the elements that you wish to change from the
default.
Integer parameters:
msglev (default: 1)
Level of messages output by solver routines:
0 (GLP_MSG_OFF)
No output.
1 (GLP_MSG_ERR)
Error and warning messages only.
2 (GLP_MSG_ON)
Normal output.
3 (GLP_MSG_ALL)
Full output (includes informational messages).
scale (default: 16)
Scaling option. The values can be combined with the bitwise
OR operator and may be the following:
1 (GLP_SF_GM)
Geometric mean scaling.
16 (GLP_SF_EQ)
Equilibration scaling.
32 (GLP_SF_2N)
Round scale factors to power of two.
64 (GLP_SF_SKIP)
Skip if problem is well scaled.

Chapter 25: Optimization

635

Alternatively, a value of 128 (GLP_SF_AUTO) may be also specified, in which case the routine chooses the scaling options
automatically.
dual (default: 1)
Simplex method option:
1 (GLP_PRIMAL)
Use two-phase primal simplex.
2 (GLP_DUALP)
Use two-phase dual simplex, and if it fails, switch
to the primal simplex.
3 (GLP_DUAL)
Use two-phase dual simplex.
price (default: 34)
Pricing option (for both primal and dual simplex):
17 (GLP_PT_STD)
Textbook pricing.
34 (GLP_PT_PSE)
Steepest edge pricing.
itlim (default: intmax)
Simplex iterations limit. It is decreased by one each time
when one simplex iteration has been performed, and reaching
zero value signals the solver to stop the search.
outfrq (default: 200)
Output frequency, in iterations. This parameter specifies how
frequently the solver sends information about the solution to
the standard output.
branch (default: 4)
Branching technique option (for MIP only):
1 (GLP_BR_FFV)
First fractional variable.
2 (GLP_BR_LFV)
Last fractional variable.
3 (GLP_BR_MFV)
Most fractional variable.
4 (GLP_BR_DTH)
Heuristic by Driebeck and Tomlin.
5 (GLP_BR_PCH)
Hybrid pseudocost heuristic.

636

GNU Octave

btrack (default: 4)
Backtracking technique option (for MIP only):
1 (GLP_BT_DFS)
Depth first search.
2 (GLP_BT_BFS)
Breadth first search.
3 (GLP_BT_BLB)
Best local bound.
4 (GLP_BT_BPH)
Best projection heuristic.
presol (default: 1)
If this flag is set, the simplex solver uses the built-in LP
presolver. Otherwise the LP presolver is not used.
lpsolver (default: 1)
Select which solver to use. If the problem is a MIP problem
this flag will be ignored.
1

Revised simplex method.

2

Interior point method.

rtest (default: 34)
Ratio test technique:
17 (GLP_RT_STD)
Standard ("textbook").
34 (GLP_RT_HAR)
Harris’ two-pass ratio test.
tmlim (default: intmax)
Searching time limit, in milliseconds.
outdly (default: 0)
Output delay, in seconds. This parameter specifies how long
the solver should delay sending information about the solution to the standard output.
save (default: 0)
If this parameter is nonzero, save a copy of the problem in
CPLEX LP format to the file "outpb.lp". There is currently
no way to change the name of the output file.
Real parameters:
tolbnd (default: 1e-7)
Relative tolerance used to check if the current basic solution
is primal feasible. It is not recommended that you change
this parameter unless you have a detailed understanding of
its purpose.

Chapter 25: Optimization

637

toldj (default: 1e-7)
Absolute tolerance used to check if the current basic solution
is dual feasible. It is not recommended that you change this
parameter unless you have a detailed understanding of its
purpose.
tolpiv (default: 1e-10)
Relative tolerance used to choose eligible pivotal elements of
the simplex table. It is not recommended that you change
this parameter unless you have a detailed understanding of
its purpose.
objll (default: -DBL_MAX)
Lower limit of the objective function. If the objective function reaches this limit and continues decreasing, the solver
stops the search. This parameter is used in the dual simplex
method only.
objul (default: +DBL_MAX)
Upper limit of the objective function. If the objective function reaches this limit and continues increasing, the solver
stops the search. This parameter is used in the dual simplex
only.
tolint (default: 1e-5)
Relative tolerance used to check if the current basic solution
is integer feasible. It is not recommended that you change
this parameter unless you have a detailed understanding of
its purpose.
tolobj (default: 1e-7)
Relative tolerance used to check if the value of the objective
function is not better than in the best known integer feasible
solution. It is not recommended that you change this parameter unless you have a detailed understanding of its purpose.
Output values:
xopt

The optimizer (the value of the decision variables at the optimum).

fopt

The optimum value of the objective function.

errnum

Error code.
0

No error.

1 (GLP_EBADB)
Invalid basis.
2 (GLP_ESING)
Singular matrix.
3 (GLP_ECOND)
Ill-conditioned matrix.

638

GNU Octave

4 (GLP_EBOUND)
Invalid bounds.
5 (GLP_EFAIL)
Solver failed.
6 (GLP_EOBJLL)
Objective function lower limit reached.
7 (GLP_EOBJUL)
Objective function upper limit reached.
8 (GLP_EITLIM)
Iterations limit exhausted.
9 (GLP_ETMLIM)
Time limit exhausted.
10 (GLP_ENOPFS)
No primal feasible solution.
11 (GLP_ENODFS)
No dual feasible solution.
12 (GLP_EROOT)
Root LP optimum not provided.
13 (GLP_ESTOP)
Search terminated by application.
14 (GLP_EMIPGAP)
Relative MIP gap tolerance reached.
15 (GLP_ENOFEAS)
No primal/dual feasible solution.
16 (GLP_ENOCVG)
No convergence.
17 (GLP_EINSTAB)
Numerical instability.
18 (GLP_EDATA)
Invalid data.
19 (GLP_ERANGE)
Result out of range.
extra

A data structure containing the following fields:
lambda

Dual variables.

redcosts

Reduced Costs.

time

Time (in seconds) used for solving LP/MIP problem.

status

Status of the optimization.
1 (GLP_UNDEF)
Solution status is undefined.

Chapter 25: Optimization

639

2 (GLP_FEAS)
Solution is feasible.
3 (GLP_INFEAS)
Solution is infeasible.
4 (GLP_NOFEAS)
Problem has no feasible solution.
5 (GLP_OPT)
Solution is optimal.
6 (GLP_UNBND)
Problem has no unbounded solution.
Example:
c = [10, 6, 4]’;
A = [ 1, 1, 1;
10, 4, 5;
2, 2, 6];
b = [100, 600, 300]’;
lb = [0, 0, 0]’;
ub = [];
ctype = "UUU";
vartype = "CCC";
s = -1;
param.msglev = 1;
param.itlim = 100;
[xmin, fmin, status, extra] = ...
glpk (c, A, b, lb, ub, ctype, vartype, s, param);

25.2 Quadratic Programming
Octave can also solve Quadratic Programming problems, this is
1
min xT Hx + xT q
x 2
subject to
lb ≤ x ≤ ub

Ax = b

[x,
[x,
[x,
[x,
[x,
[x,

Alb ≤ Ain x ≤ Aub

(x0, H)
(x0, H, q)
(x0, H, q, A, b)
(x0, H, q, A, b, lb, ub)
(x0, H, q, A, b, lb, ub, A_lb, A_in, A_ub)
( . . . , options)
Solve a quadratic program (QP).

obj,
obj,
obj,
obj,
obj,
obj,

info,
info,
info,
info,
info,
info,

lambda]
lambda]
lambda]
lambda]
lambda]
lambda]

=
=
=
=
=
=

qp
qp
qp
qp
qp
qp

640

GNU Octave

Solve the quadratic program defined by
1
min xT Hx + xT q
x 2
subject to
lb ≤ x ≤ ub

Ax = b

Alb ≤ Ain x ≤ Aub

using a null-space active-set method.
Any bound (A, b, lb, ub, A in, A lb, A ub) may be set to the empty matrix ([]) if
not present. The constraints A and A in are matrices with each row representing a
single constraint. The other bounds are scalars or vectors depending on the number
of constraints. The algorithm is faster if the initial guess is feasible.
options

An optional structure containing the following parameter(s) used to define
the behavior of the solver. Missing elements in the structure take on
default values, so you only need to set the elements that you wish to
change from the default.
MaxIter (default: 200)
Maximum number of iterations.

info

Structure containing run-time information about the algorithm. The following fields are defined:
solveiter
The number of iterations required to find the solution.
info

x =
x =
[x,
[x,
[x,
[x,

An integer indicating the status of the solution.
0

The problem is feasible and convex. Global solution found.

1

The problem is not convex. Local solution found.

2

The problem is not convex and unbounded.

3

Maximum number of iterations reached.

6

The problem is infeasible.

pqpnonneg (c, d)
pqpnonneg (c, d, x0)
minval] = pqpnonneg ( . . . )
minval, exitflag] = pqpnonneg ( . . . )
minval, exitflag, output] = pqpnonneg ( . . . )
minval, exitflag, output, lambda] = pqpnonneg ( . . . )
Minimize 1/2*x’*c*x + d’*x subject to x >= 0.
c and d must be real, and c must be symmetric and positive definite.
x0 is an optional initial guess for x.
Outputs:
• minval
The minimum attained model value, 1/2*xmin’*c*xmin + d’*xmin

Chapter 25: Optimization

641

• exitflag

An indicator of convergence. 0 indicates that the iteration count was exceeded,
and therefore convergence was not reached; >0 indicates that the algorithm converged. (The algorithm is stable and will converge given enough iterations.)

• output

A structure with two fields:
• "algorithm": The algorithm used ("nnls")

• "iterations": The number of iterations taken.

• lambda

Not implemented.

See also: [optimset], page 645, [lsqnonneg], page 644, [qp], page 639.

25.3 Nonlinear Programming
Octave can also perform general nonlinear minimization using a successive quadratic programming solver.

[x, obj, info, iter, nf, lambda] = sqp (x0, phi)
[...] = sqp (x0, phi, g)
[...] = sqp (x0, phi, g, h)
[...] = sqp (x0, phi, g, h, lb, ub)
[...] = sqp (x0, phi, g, h, lb, ub, maxiter)
[...] = sqp (x0, phi, g, h, lb, ub, maxiter, tol)
Minimize an objective function using sequential quadratic programming (SQP).
Solve the nonlinear program
min φ(x)
x

subject to
g(x) = 0

h(x) ≥ 0

lb ≤ x ≤ ub

using a sequential quadratic programming method.
The first argument is the initial guess for the vector x0.
The second argument is a function handle pointing to the objective function phi. The
objective function must accept one vector argument and return a scalar.
The second argument may also be a 2- or 3-element cell array of function handles.
The first element should point to the objective function, the second should point
to a function that computes the gradient of the objective function, and the third
should point to a function that computes the Hessian of the objective function. If the
gradient function is not supplied, the gradient is computed by finite differences. If
the Hessian function is not supplied, a BFGS update formula is used to approximate
the Hessian.
When supplied, the gradient function phi{2} must accept one vector argument and
return a vector. When supplied, the Hessian function phi{3} must accept one vector
argument and return a matrix.

642

GNU Octave

The third and fourth arguments g and h are function handles pointing to functions
that compute the equality constraints and the inequality constraints, respectively. If
the problem does not have equality (or inequality) constraints, then use an empty
matrix ([]) for g (or h). When supplied, these equality and inequality constraint
functions must accept one vector argument and return a vector.
The third and fourth arguments may also be 2-element cell arrays of function handles.
The first element should point to the constraint function and the second should point
to a function that computes the gradient of the constraint function:
∂f (x) ∂f (x)
∂f (x)
,
,...,
∂x1
∂x2
∂xN

!T

The fifth and sixth arguments, lb and ub, contain lower and upper bounds on x.
These must be consistent with the equality and inequality constraints g and h. If the
arguments are vectors then x(i) is bound by lb(i) and ub(i). A bound can also be a
scalar in which case all elements of x will share the same bound. If only one bound
(lb, ub) is specified then the other will default to (-realmax, +realmax).
The seventh argument maxiter specifies the maximum number of iterations. The
default value is 100.
The eighth argument tol specifies the tolerance for the stopping criteria. The default
value is sqrt (eps).
The value returned in info may be one of the following:
101

The algorithm terminated normally. All constraints meet the specified
tolerance.

102

The BFGS update failed.

103

The maximum number of iterations was reached.

104

The stepsize has become too small, i.e., Δx, is less than tol * norm (x).

An example of calling sqp:
function r = g (x)
r = [ sumsq(x)-10;
x(2)*x(3)-5*x(4)*x(5);
x(1)^3+x(2)^3+1 ];
endfunction
function obj = phi (x)
obj = exp (prod (x)) - 0.5*(x(1)^3+x(2)^3+1)^2;
endfunction
x0 = [-1.8; 1.7; 1.9; -0.8; -0.8];
[x, obj, info, iter, nf, lambda] = sqp (x0, @phi, @g, [])
x =

Chapter 25: Optimization

643

-1.71714
1.59571
1.82725
-0.76364
-0.76364
obj = 0.053950
info = 101
iter = 8
nf = 10
lambda =
-0.0401627
0.0379578
-0.0052227
See also: [qp], page 639.

25.4 Linear Least Squares
Octave also supports linear least squares minimization. That is, Octave can find the parameter b such that the model y = xb fits data (x, y) as well as possible, assuming zero-mean
Gaussian noise. If the noise is assumed to be isotropic the problem can be solved using the
‘\’ or ‘/’ operators, or the ols function. In the general case where the noise is assumed to
be anisotropic the gls is needed.

[beta, sigma, r] = ols (y, x)
Ordinary least squares (OLS) estimation.
OLS applies to the multivariate model y = x b + e where y is a t × p matrix, x is a
t × k matrix, b is a k × p matrix, and e is a t × p matrix.
Each row of y is a p-variate observation in which each column represents a variable.
Likewise, the rows of x represent k-variate observations or possibly designed values.
Furthermore, the collection of observations x must be of adequate rank, k, otherwise
b cannot be uniquely estimated.

The observation errors, e, are assumed to originate from an underlying p-variate distribution with zero mean and p-by-p covariance matrix S, both constant conditioned
on x. Furthermore, the matrix S is constant with respect to each observation such
that ē = 0 and cov(vec(e)) = kron(s,I). (For cases that don’t meet this criteria,
such as autocorrelated errors, see generalized least squares, gls, for more efficient
estimations.)
The return values beta, sigma, and r are defined as follows.
beta

The OLS estimator for matrix b. beta is calculated directly via
(x T x)−1 x T y if the matrix x T x is of full rank. Otherwise, beta = pinv
(x) * y where pinv (x) denotes the pseudoinverse of x.

sigma

The OLS estimator for the matrix s,
sigma = (y-x*beta)’ * (y-x*beta) / (t-rank(x))

644

GNU Octave

r

The matrix of OLS residuals, r = y - x*beta.

See also: [gls], page 644, [pinv], page 514.

[beta, v, r] = gls (y, x, o)
Generalized least squares (GLS) model.
Perform a generalized least squares estimation for the multivariate model y = x b + e
where y is a t × p matrix, x is a t × k matrix, b is a k × p matrix and e is a t × p
matrix.
Each row of y is a p-variate observation in which each column represents a variable.
Likewise, the rows of x represent k-variate observations or possibly designed values.
Furthermore, the collection of observations x must be of adequate rank, k, otherwise
b cannot be uniquely estimated.
The observation errors, e, are assumed to originate from an underlying p-variate
distribution with zero mean but possibly heteroscedastic observations. That is, in
general, ē = 0 and cov(vec(e)) = s2 o in which s is a scalar and o is a t p × t p matrix.
The return values beta, v, and r are defined as follows.
beta

The GLS estimator for matrix b.

v

The GLS estimator for scalar s2 .

r

The matrix of GLS residuals, r = y − x ∗ beta.

See also: [ols], page 643.

x =
x =
x =
[x,
[x,
[x,
[x,
[x,

lsqnonneg (c, d)
lsqnonneg (c, d, x0)
lsqnonneg (c, d, x0, options)
resnorm] = lsqnonneg ( . . . )
resnorm, residual] = lsqnonneg ( . . . )
resnorm, residual, exitflag] = lsqnonneg ( . . . )
resnorm, residual, exitflag, output] = lsqnonneg ( . . . )
resnorm, residual, exitflag, output, lambda] = lsqnonneg ( . . . )
Minimize norm (c*x - d) subject to x >= 0.
c and d must be real.
x0 is an optional initial guess for x.
Currently, lsqnonneg recognizes these options: "MaxIter", "TolX". For a description
of these options, see [optimset], page 645.
Outputs:
• resnorm
The squared 2-norm of the residual: norm (c*x-d)^2
• residual
The residual: d-c*x
• exitflag
An indicator of convergence. 0 indicates that the iteration count was exceeded,
and therefore convergence was not reached; >0 indicates that the algorithm converged. (The algorithm is stable and will converge given enough iterations.)

Chapter 25: Optimization

645

• output
A structure with two fields:
• "algorithm": The algorithm used ("nnls")
• "iterations": The number of iterations taken.
• lambda
Not implemented.
See also: [optimset], page 645, [pqpnonneg], page 640, [lscov], page 645.

x =
x =
x =
[x,

lscov
lscov
lscov
stdx,

(A, b)
(A, b, V)
(A, b, V, alg)

mse, S] = lscov ( . . . )
Compute a generalized linear least squares fit.
Estimate x under the model b = Ax + w, where the noise w is assumed to follow a
normal distribution with covariance matrix σ 2 V .
If the size of the coefficient matrix A is n-by-p, the size of the vector/array of constant
terms b must be n-by-k.
The optional input argument V may be a n-by-1 vector of positive weights (inverse
variances), or a n-by-n symmetric positive semidefinite matrix representing the covariance of b. If V is not supplied, the ordinary least squares solution is returned.
The alg input argument, a guidance on solution method to use, is currently ignored.
Besides the least-squares estimate matrix x (p-by-k), the function also returns stdx
(p-by-k), the error standard deviation of estimated x; mse (k-by-1), the estimated
data error covariance scale factors (σ 2 ); and S (p-by-p, or p-by-p-by-k if k > 1), the
error covariance of x.
Reference: Golub and Van Loan (1996), Matrix Computations (3rd Ed.), Johns Hopkins, Section 5.6.3
See also: [ols], page 643, [gls], page 644, [lsqnonneg], page 644.

optimset ()
options = optimset
options = optimset
options = optimset
options = optimset

()
(par, val, . . . )
(old, par, val, . . . )
(old, new)
Create options structure for optimization functions.
When called without any input or output arguments, optimset prints a list of all
valid optimization parameters.
When called with one output and no inputs, return an options structure with all valid
option parameters initialized to [].
When called with a list of parameter/value pairs, return an options structure with
only the named parameters initialized.
When the first input is an existing options structure old, the values are updated from
either the par/val list or from the options structure new.

646

GNU Octave

Valid parameters are:
AutoScaling
ComplexEqn
Display
Request verbose display of results from optimizations. Values are:
"off" [default]
No display.
"iter"

Display intermediate results for every loop iteration.

"final"

Display the result of the final loop iteration.

"notify"

Display the result of the final loop iteration if the function
has failed to converge.

FinDiffType
FunValCheck
When enabled, display an error if the objective function returns an invalid
value (a complex number, NaN, or Inf). Must be set to "on" or "off"
[default]. Note: the functions fzero and fminbnd correctly handle Inf
values and only complex values or NaN will cause an error in this case.
GradObj

When set to "on", the function to be minimized must return a second
argument which is the gradient, or first derivative, of the function at the
point x. If set to "off" [default], the gradient is computed via finite
differences.

Jacobian

When set to "on", the function to be minimized must return a second
argument which is the Jacobian, or first derivative, of the function at the
point x. If set to "off" [default], the Jacobian is computed via finite
differences.

MaxFunEvals
Maximum number of function evaluations before optimization stops.
Must be a positive integer.
MaxIter

Maximum number of algorithm iterations before optimization stops.
Must be a positive integer.

OutputFcn
A user-defined function executed once per algorithm iteration.
TolFun

Termination criterion for the function output. If the difference in the
calculated objective function between one algorithm iteration and the
next is less than TolFun the optimization stops. Must be a positive
scalar.

TolX

Termination criterion for the function input. If the difference in x, the
current search point, between one algorithm iteration and the next is less
than TolX the optimization stops. Must be a positive scalar.

TypicalX
Updating
See also: [optimget], page 647.

Chapter 25: Optimization

647

optimget (options, parname)
optimget (options, parname, default)
Return the specific option parname from the optimization options structure options
created by optimset.
If parname is not defined then return default if supplied, otherwise return an empty
matrix.
See also: [optimset], page 645.

649

26 Statistics
Octave has support for various statistical methods. This includes basic descriptive statistics,
probability distributions, statistical tests, random number generation, and much more.
The functions that analyze data all assume that multi-dimensional data is arranged in a
matrix where each row is an observation, and each column is a variable. Thus, the matrix
defined by
a = [ 0.9, 0.7;
0.1, 0.1;
0.5, 0.4 ];
contains three observations from a two-dimensional distribution. While this is the default
data arrangement, most functions support different arrangements.
It should be noted that the statistics functions don’t test for data containing NaN, NA,
or Inf. These values need to be detected and dealt with explicitly. See [isnan], page 444,
[isna], page 43, [isinf], page 444, [isfinite], page 445.

26.1 Descriptive Statistics
One principal goal of descriptive statistics is to represent the essence of a large data set
concisely. Octave provides the mean, median, and mode functions which all summarize a
data set with just a single number corresponding to the central tendency of the data.

mean
mean
mean
mean

(x)
(x, dim)
(x, opt)
(x, dim, opt)
Compute the mean of the elements of the vector x.
The mean is defined as
mean(x) = x̄ =

N
1 X
xi
N i=1

where N is the number of elements of x.
If x is a matrix, compute the mean for each column and return them in a row vector.
If the optional argument dim is given, operate along this dimension.
The optional argument opt selects the type of mean to compute. The following options
are recognized:
"a"

Compute the (ordinary) arithmetic mean. [default]

"g"

Compute the geometric mean.

"h"

Compute the harmonic mean.

Both dim and opt are optional. If both are supplied, either may appear first.
See also: [median], page 650, [mode], page 650.

650

GNU Octave

median (x)
median (x, dim)
Compute the median value of the elements of the vector x.
When the elements of x are sorted, say s = sort (x), the median is defined as
median(x) =



s(dN/2e),
N odd;
(s(N/2) + s(N/2 + 1))/2, N even.

where N is the number of elements of x.
If x is of a discrete type such as integer or logical, then the case of even N rounds up
(or toward true).
If x is a matrix, compute the median value for each column and return them in a row
vector.
If the optional dim argument is given, operate along this dimension.
See also: [mean], page 649, [mode], page 650.

mode (x)
mode (x, dim)
[m, f, c] = mode ( . . . )
Compute the most frequently occurring value in a dataset (mode).
mode determines the frequency of values along the first non-singleton dimension and
returns the value with the highest frequency. If two, or more, values have the same
frequency mode returns the smallest.
If the optional argument dim is given, operate along this dimension.
The return variable f is the number of occurrences of the mode in the dataset.
The cell array c contains all of the elements with the maximum frequency.
See also: [mean], page 649, [median], page 650.
Using just one number, such as the mean, to represent an entire data set may not give
an accurate picture of the data. One way to characterize the fit is to measure the dispersion
of the data. Octave provides several functions for measuring dispersion.

range (x)
range (x, dim)
Return the range, i.e., the difference between the maximum and the minimum of the
input data.
If x is a vector, the range is calculated over the elements of x. If x is a matrix, the
range is calculated over each column of x.
If the optional argument dim is given, operate along this dimension.
The range is a quickly computed measure of the dispersion of a data set, but is less
accurate than iqr if there are outlying data points.
See also: [iqr], page 651, [std], page 651.

Chapter 26: Statistics

651

iqr (x)
iqr (x, dim)
Return the interquartile range, i.e., the difference between the upper and lower quartile of the input data.
If x is a matrix, do the above for first non-singleton dimension of x.
If the optional argument dim is given, operate along this dimension.
As a measure of dispersion, the interquartile range is less affected by outliers than
either range or std.
See also: [range], page 650, [std], page 651.

meansq (x)
meansq (x, dim)
Compute the mean square of the elements of the vector x.
The mean square is defined as
meansq(x) =

PN

i=1

xi 2

N

where N is the number of elements of x.
If x is a matrix, return a row vector containing the mean square of each column.
If the optional argument dim is given, operate along this dimension.
See also: [var], page 652, [std], page 651, [moment], page 653.

std (x)
std (x, opt)
std (x, opt, dim)
Compute the standard deviation of the elements of the vector x.
The standard deviation is defined as
std(x) = σ =

sP

N
i=1 (xi

− x̄)2
N −1

where x̄ is the mean value of x and N is the number of elements of x.
If x is a matrix, compute the standard deviation for each column and return them in
a row vector.
The argument opt determines the type of normalization to use. Valid values are
0:

normalize with N − 1, provides the square root of the best unbiased
estimator of the variance [default]

1:

normalize with N , this provides the square root of the second moment
around the mean

If the optional argument dim is given, operate along this dimension.
See also: [var], page 652, [range], page 650, [iqr], page 651, [mean], page 649, [median],
page 650.

652

GNU Octave

In addition to knowing the size of a dispersion it is useful to know the shape of the data
set. For example, are data points massed to the left or right of the mean? Octave provides
several common measures to describe the shape of the data set. Octave can also calculate
moments allowing arbitrary shape measures to be developed.

var (x)
var (x, opt)
var (x, opt, dim)
Compute the variance of the elements of the vector x.
The variance is defined as
PN

− x̄)2
N −1
where x̄ is the mean value of x and N is the number of elements of x.
If x is a matrix, compute the variance for each column and return them in a row
vector.
The argument opt determines the type of normalization to use. Valid values are
2

var(x) = σ =

i=1 (xi

0:

normalize with N −1, provides the best unbiased estimator of the variance
[default]

1:

normalizes with N , this provides the second moment around the mean

If N is equal to 1 the value of opt is ignored and normalization by N is used.
If the optional argument dim is given, operate along this dimension.
See also: [cov], page 660, [std], page 651, [skewness], page 652, [kurtosis], page 653,
[moment], page 653.

skewness (x)
skewness (x, flag)
skewness (x, flag, dim)
Compute the sample skewness of the elements of x.
The sample skewness is defined as
skewness(x) =

1
N

PN

i=1 (xi
σ3

− x̄)3

,

where N is the length of x, x̄ its mean and σ its (uncorrected) standard deviation.
The optional argument flag controls which normalization is used. If flag is equal to
1 (default value, used when flag is omitted or empty), return the sample skewness as
defined above. If flag is equal to 0, return the adjusted skewness coefficient instead:
p

PN

1
(xi − x̄)3
N (N − 1)
skewness(x) =
× N i=1 3
N −2
σ
The adjusted skewness coefficient is obtained by replacing the sample second and
third central moments by their bias-corrected versions.
If x is a matrix, or more generally a multi-dimensional array, return the skewness along
the first non-singleton dimension. If the optional dim argument is given, operate along
this dimension.

See also: [var], page 652, [kurtosis], page 653, [moment], page 653.

Chapter 26: Statistics

653

kurtosis (x)
kurtosis (x, flag)
kurtosis (x, flag, dim)
Compute the sample kurtosis of the elements of x.
The sample kurtosis is defined as
κ1 =

1
N

PN

i=1 (xi
σ4

− x̄)4

,

where N is the length of x, x̄ its mean, and σ its (uncorrected) standard deviation.
The optional argument flag controls which normalization is used. If flag is equal to
1 (default value, used when flag is omitted or empty), return the sample kurtosis as
defined above. If flag is equal to 0, return the "bias-corrected" kurtosis coefficient
instead:
N −1
κ0 = 3 +
((N + 1) κ1 − 3(N − 1))
(N −2)(N −3)

The bias-corrected kurtosis coefficient is obtained by replacing the sample second and
fourth central moments by their unbiased versions. It is an unbiased estimate of the
population kurtosis for normal populations.
If x is a matrix, or more generally a multi-dimensional array, return the kurtosis along
the first non-singleton dimension. If the optional dim argument is given, operate along
this dimension.
See also: [var], page 652, [skewness], page 652, [moment], page 653.
(x, p)
(x, p, type)
(x, p, dim)
(x, p, type, dim)
(x, p, dim, type)
Compute the p-th central moment of the vector x:

moment
moment
moment
moment
moment

PN

i=1 (xi

− x̄)p

N
where x̄ is the mean value of x and N is the number of elements of x.
If x is a matrix, return the row vector containing the p-th central moment of each
column.
If the optional argument dim is given, operate along this dimension.
The optional string type specifies the type of moment to be computed. Valid options
are:
"c"
"a"
"ac"

Central Moment (default).
Absolute Central Moment. The moment about the mean ignoring sign
defined as
PN
p
i=1 |xi − x̄|
N

654

GNU Octave

"r"

Raw Moment. The moment about zero defined as
moment(x) =

"ar"

PN

i=1

xi p

N

Absolute Raw Moment. The moment about zero ignoring sign defined as
PN

|xi |
N

p

i=1

If both type and dim are given they may appear in any order.
See also: [var], page 652, [skewness], page 652, [kurtosis], page 653.

q
q
q
q

=
=
=
=

(x)
(x, p)
(x, p, dim)
(x, p, dim, method)
For a sample, x, calculate the quantiles, q, corresponding to the cumulative probability
values in p. All non-numeric values (NaNs) of x are ignored.

quantile
quantile
quantile
quantile

If x is a matrix, compute the quantiles for each column and return them in a matrix,
such that the i-th row of q contains the p(i)th quantiles of each column of x.
If p is unspecified, return the quantiles for [0.00 0.25 0.50 0.75 1.00]. The optional argument dim determines the dimension along which the quantiles are calculated. If dim is omitted it defaults to the first non-singleton dimension.
The methods available to calculate sample quantiles are the nine methods used by R
(http://www.r-project.org/). The default value is method = 5.
Discontinuous sample quantile methods 1, 2, and 3
1. Method 1: Inverse of empirical distribution function.
2. Method 2: Similar to method 1 but with averaging at discontinuities.
3. Method 3: SAS definition: nearest even order statistic.
Continuous sample quantile methods 4 through 9, where p(k) is the linear interpolation function respecting each method’s representative cdf.
4. Method 4: p(k) = k/N . That is, linear interpolation of the empirical cdf, where
N is the length of P.
5. Method 5: p(k) = (k − 0.5)/N . That is, a piecewise linear function where the
knots are the values midway through the steps of the empirical cdf.
6. Method 6: p(k) = k/(N + 1).
7. Method 7: p(k) = (k − 1)/(N − 1).

8. Method 8: p(k) = (k − 1/3)/(N + 1/3). The resulting quantile estimates are
approximately median-unbiased regardless of the distribution of x.
9. Method 9: p(k) = (k − 3/8)/(N + 1/4). The resulting quantile estimates are
approximately unbiased for the expected order statistics if x is normally distributed.

Chapter 26: Statistics

655

Hyndman and Fan (1996) recommend method 8. Maxima, S, and R (versions prior to
2.0.0) use 7 as their default. Minitab and SPSS use method 6. matlab uses method
5.
References:
• Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language.
Wadsworth & Brooks/Cole.
• Hyndman, R. J. and Fan, Y. (1996) Sample quantiles in statistical packages,
American Statistician, 50, 361–365.
• R: A Language and Environment for Statistical Computing; http: / / cran .
r-project.org/doc/manuals/fullrefman.pdf.
Examples:
x = randi (1000, [10, 1]); # Create empirical data in range 1-1000
q = quantile (x, [0, 1]);
# Return minimum, maximum of distribution
q = quantile (x, [0.25 0.5 0.75]); # Return quartiles of distribution

See also: [prctile], page 655.

q = prctile (x)
q = prctile (x, p)
q = prctile (x, p, dim)
For a sample x, compute the quantiles, q, corresponding to the cumulative probability
values, p, in percent.
If x is a matrix, compute the percentiles for each column and return them in a matrix,
such that the i-th row of q contains the p(i)th percentiles of each column of x.
If p is unspecified, return the quantiles for [0 25 50 75 100].
The optional argument dim determines the dimension along which the percentiles are
calculated. If dim is omitted it defaults to the first non-singleton dimension.
Programming Note: All non-numeric values (NaNs) of x are ignored.
See also: [quantile], page 654.
A summary view of a data set can be generated quickly with the statistics function.

statistics (x)
statistics (x, dim)
Return a vector with the minimum, first quartile, median, third quartile, maximum,
mean, standard deviation, skewness, and kurtosis of the elements of the vector x.
If x is a matrix, calculate statistics over the first non-singleton dimension.
If the optional argument dim is given, operate along this dimension.
See also: [min], page 485, [max], page 485, [median], page 650, [mean], page 649,
[std], page 651, [skewness], page 652, [kurtosis], page 653.

656

GNU Octave

26.2 Basic Statistical Functions
Octave supports various helpful statistical functions. Many are useful as initial steps to
prepare a data set for further analysis. Others provide different measures from those of the
basic descriptive statistics.

center (x)
center (x, dim)
Center data by subtracting its mean.
If x is a vector, subtract its mean.
If x is a matrix, do the above for each column.
If the optional argument dim is given, operate along this dimension.
Programming Note: center has obvious application for normalizing statistical data.
It is also useful for improving the precision of general numerical calculations. Whenever there is a large value that is common to a batch of data, the mean can be
subtracted off, the calculation performed, and then the mean added back to obtain
the final answer.
See also: [zscore], page 656.

z =
z =
z =
[z,

zscore (x)
zscore (x, opt)
zscore (x, opt, dim)
mu, sigma] = zscore ( . . . )
Compute the Z score of x
If x is a vector, subtract its mean and divide by its standard deviation. If the standard
deviation is zero, divide by 1 instead.
The optional parameter opt determines the normalization to use when computing the
standard deviation and has the same definition as the corresponding parameter for
std.
If x is a matrix, calculate along the first non-singleton dimension. If the third optional
argument dim is given, operate along this dimension.
The optional outputs mu and sigma contain the mean and standard deviation.
See also: [mean], page 649, [std], page 651, [center], page 656.

n = histc (x, edges)
n = histc (x, edges, dim)
[n, idx] = histc ( . . . )
Compute histogram counts.
When x is a vector, the function counts the number of elements of x that fall in the
histogram bins defined by edges. This must be a vector of monotonically increasing
values that define the edges of the histogram bins. n(k) contains the number of
elements in x for which edges(k) <= x < edges(k + 1). The final element of n
contains the number of elements of x exactly equal to the last element of edges.
When x is an N -dimensional array, the computation is carried out along dimension
dim. If not specified dim defaults to the first non-singleton dimension.

Chapter 26: Statistics

657

When a second output argument is requested an index matrix is also returned. The
idx matrix has the same size as x. Each element of idx contains the index of the
histogram bin in which the corresponding element of x was counted.
See also: [hist], page 295.
unique function documented at [unique], page 685, is often useful for statistics.

c = nchoosek (n, k)
c = nchoosek (set, k)
Compute the binomial coefficient of n or list all possible combinations of a set of
items.
If n is a scalar then calculate the binomial coefficient of n and k which is defined as
n
k

!

=

n(n − 1)(n − 2) · · · (n − k + 1)
n!
=
k!
k!(n − k)!

This is the number of combinations of n items taken in groups of size k.
If the first argument is a vector, set, then generate all combinations of the elements
of set, taken k at a time, with one row per combination. The result c has k columns
and nchoosek (length (set), k) rows.
For example:
How many ways can three items be grouped into pairs?
nchoosek (3, 2)
⇒ 3
What are the possible pairs?
nchoosek (1:3, 2)
⇒ 1
2
1
3
2
3
Programming Note: When calculating the binomial coefficient nchoosek works only
for non-negative, integer arguments. Use bincoeff for non-integer and negative scalar
arguments, or for computing many binomial coefficients at once with vector inputs
for n or k.
See also: [bincoeff], page 494, [perms], page 657.

perms (v)
Generate all permutations of v with one row per permutation.
The result has size factorial (n) * n, where n is the length of v.
Example
perms ([1, 2, 3])
⇒
1
2
3
2
1
3
1
3
2
2
3
1
3
1
2
3
2
1

658

GNU Octave

Programming Note: The maximum length of v should be less than or equal to 10 to
limit memory consumption.
See also: [permute], page 449, [randperm], page 464, [nchoosek], page 657.

ranks (x, dim)
Return the ranks of x along the first non-singleton dimension adjusted for ties.
If the optional argument dim is given, operate along this dimension.
See also: [spearman], page 660, [kendall], page 661.

run_count (x, n)
run_count (x, n, dim)
Count the upward runs along the first non-singleton dimension of x of length 1, 2,
. . . , n-1 and greater than or equal to n.
If the optional argument dim is given then operate along this dimension.
See also: [runlength], page 658.

count = runlength (x)
[count, value] = runlength (x)
Find the lengths of all sequences of common values.
count is a vector with the lengths of each repeated value.
The optional output value contains the value that was repeated in the sequence.
runlength ([2, 2, 0, 4, 4, 4, 0, 1, 1, 1, 1])
⇒ [2, 1, 3, 1, 4]
See also: [run count], page 658.

probit (p)
Return the probit (the quantile of the standard normal distribution) for each element
of p.
See also: [logit], page 658.

logit (p)
Compute the logit for each value of p
The logit is defined as
 p 
logit(p) = log
1−p
See also: [probit], page 658, [logistic cdf], page 666.

cloglog (x)
Return the complementary log-log function of x.
The complementary log-log function is defined as
cloglog(x) = − log(− log(x))

[t, l_x] = table (x)
[t, l_x, l_y] = table (x, y)
Create a contingency table t from data vectors.
The l x and l y vectors are the corresponding levels.
Currently, only 1- and 2-dimensional tables are supported.

Chapter 26: Statistics

659

26.3 Statistical Plots
Octave can create Quantile Plots (QQ-Plots), and Probability Plots (PP-Plots). These are
simple graphical tests for determining if a data set comes from a certain distribution.
Note that Octave can also show histograms of data using the hist function as described
in Section 15.2.1 [Two-Dimensional Plots], page 287.

[q, s]
[q, s]
[q, s]
[q, s]
qqplot

= qqplot
= qqplot
= qqplot
= qqplot
(. . . )

(x)
(x, y)
(x, dist)
(x, y, params)

Perform a QQ-plot (quantile plot).
If F is the CDF of the distribution dist with parameters params and G its inverse,
and x a sample vector of length n, the QQ-plot graphs ordinate s(i) = i-th largest
element of x versus abscissa q(if) = G((i - 0.5)/n).
If the sample comes from F, except for a transformation of location and scale, the
pairs will approximately follow a straight line.
If the second argument is a vector y the empirical CDF of y is used as dist.
The default for dist is the standard normal distribution. The optional argument
params contains a list of parameters of dist. For example, for a quantile plot of the
uniform distribution on [2,4] and x, use
qqplot (x, "unif", 2, 4)
dist can be any string for which a function distinv or dist inv exists that calculates
the inverse CDF of distribution dist.
If no output arguments are given, the data are plotted directly.

[p, y] = ppplot (x, dist, params)
Perform a PP-plot (probability plot).
If F is the CDF of the distribution dist with parameters params and x a sample vector
of length n, the PP-plot graphs ordinate y(i) = F (i-th largest element of x) versus
abscissa p(i) = (i - 0.5)/n. If the sample comes from F, the pairs will approximately
follow a straight line.
The default for dist is the standard normal distribution.
The optional argument params contains a list of parameters of dist.
For example, for a probability plot of the uniform distribution on [2,4] and x, use
ppplot (x, "uniform", 2, 4)
dist can be any string for which a function dist cdf that calculates the CDF of
distribution dist exists.
If no output is requested then the data are plotted immediately.

660

GNU Octave

26.4 Correlation and Regression Analysis
cov
cov
cov
cov

(x)
(x, opt)
(x, y)
(x, y, opt)
Compute the covariance matrix.
If each row of x and y is an observation, and each column is a variable, then the
(i, j)-th entry of cov (x, y) is the covariance between the i-th variable in x and the
j-th variable in y.
N
1 X
σij =
(xi − x̄)(yi − ȳ)
N − 1 i=1
where x̄ and ȳ are the mean values of x and y.
If called with one argument, compute cov (x, x), the covariance between the columns
of x.
The argument opt determines the type of normalization to use. Valid values are
0:

normalize with N − 1, provides the best unbiased estimator of the covariance [default]

1:

normalize with N , this provides the second moment around the mean

Compatibility Note:: Octave always treats rows of x and y as multivariate random
variables. For two inputs, however, matlab treats x and y as two univariate distributions regardless of their shapes, and will calculate cov ([x(:), y(:)]) whenever
the number of elements in x and y are equal. This will result in a 2x2 matrix. Code
relying on matlab’s definition will need to be changed when running in Octave.
See also: [corr], page 660.

corr (x)
corr (x, y)
Compute matrix of correlation coefficients.
If each row of x and y is an observation and each column is a variable, then the
(i, j)-th entry of corr (x, y) is the correlation between the i-th variable in x and the
j-th variable in y.
cov(x, y)
corr(x, y) =
std(x) std(y)
If called with one argument, compute corr (x, x), the correlation between the
columns of x.
See also: [cov], page 660.

spearman (x)
spearman (x, y)
Compute Spearman’s rank correlation coefficient ρ.
For two data vectors x and y, Spearman’s ρ is the correlation coefficient of the ranks
of x and y.

Chapter 26: Statistics

661

If x and y are drawn from independent distributions, ρ has zero mean and variance
1/(N − 1), where N is the length of the x and y vectors, and is asymptotically
normally distributed.
spearman (x) is equivalent to spearman (x, x).
See also: [ranks], page 658, [kendall], page 661.

kendall (x)
kendall (x, y)
Compute Kendall’s τ .
For two data vectors x, y of common length N , Kendall’s τ is the correlation of the
signs of all rank differences of x and y; i.e., if both x and y have distinct entries, then
τ=

X
1
sign(qi − qj ) sign(ri − rj )
N (N − 1) i,j

in which the qi and ri are the ranks of x and y, respectively.
If x and y are drawn from independent distributions, Kendall’s τ is asymptotically
2(2N +5)
normal with mean 0 and variance 9N
.
(N −1)
kendall (x) is equivalent to kendall (x, x).

See also: [ranks], page 658, [spearman], page 660.

[theta, beta, dev, dl, d2l, p] = logistic_regression (y, x, print,
theta, beta)
Perform ordinal logistic regression.
Suppose y takes values in k ordered categories, and let gamma_i (x) be the cumulative
probability that y falls in one of the first i categories given the covariate x. Then
[theta, beta] = logistic_regression (y, x)
fits the model
logit (gamma_i (x)) = theta_i - beta’ * x,

i = 1 ... k-1

The number of ordinal categories, k, is taken to be the number of distinct values of
round (y). If k equals 2, y is binary and the model is ordinary logistic regression.
The matrix x is assumed to have full column rank.
Given y only, theta = logistic_regression (y) fits the model with baseline logit
odds only.
The full form is
[theta, beta, dev, dl, d2l, gamma]
= logistic_regression (y, x, print, theta, beta)
in which all output arguments and all input arguments except y are optional.
Setting print to 1 requests summary information about the fitted model to be displayed. Setting print to 2 requests information about convergence at each iteration.
Other values request no information to be displayed. The input arguments theta and
beta give initial estimates for theta and beta.
The returned value dev holds minus twice the log-likelihood.

662

GNU Octave

The returned values dl and d2l are the vector of first and the matrix of second
derivatives of the log-likelihood with respect to theta and beta.
p holds estimates for the conditional distribution of y given x.

26.5 Distributions
Octave has functions for computing the Probability Density Function (PDF), the Cumulative Distribution function (CDF), and the quantile (the inverse of the CDF) for a large
number of distributions.
The following table summarizes the supported distributions (in alphabetical order).
Distribution
Beta
Binomial
Cauchy
Chi-Square
Univariate Discrete
Empirical
Exponential
F
Gamma
Geometric
Hypergeometric
Kolmogorov Smirnov

PDF
betapdf
binopdf
cauchy pdf
chi2pdf
discrete pdf
empirical pdf
exppdf
fpdf
gampdf
geopdf
hygepdf
Not Available

Laplace
Logistic
Log-Normal
Univariate Normal
Pascal
Poisson
Standard Normal
t (Student)
Uniform Discrete
Uniform
Weibull

laplace pdf
logistic pdf
lognpdf
normpdf
nbinpdf
poisspdf
stdnormal pdf
tpdf
unidpdf
unifpdf
wblpdf

CDF
betacdf
binocdf
cauchy cdf
chi2cdf
discrete cdf
empirical cdf
expcdf
fcdf
gamcdf
geocdf
hygecdf
kolmogorov
smirnov cdf
laplace cdf
logistic cdf
logncdf
normcdf
nbincdf
poisscdf
stdnormal cdf
tcdf
unidcdf
unifcdf
wblcdf

Quantile
betainv
binoinv
cauchy inv
chi2inv
discrete inv
empirical inv
expinv
finv
gaminv
geoinv
hygeinv
Not Available
laplace inv
logistic inv
logninv
norminv
nbininv
poissinv
stdnormal inv
tinv
unidinv
unifinv
wblinv

betapdf (x, a, b)
For each element of x, compute the probability density function (PDF) at x of the
Beta distribution with parameters a and b.

betacdf (x, a, b)
For each element of x, compute the cumulative distribution function (CDF) at x of
the Beta distribution with parameters a and b.

Chapter 26: Statistics

663

betainv (x, a, b)
For each element of x, compute the quantile (the inverse of the CDF) at x of the Beta
distribution with parameters a and b.

binopdf (x, n, p)
For each element of x, compute the probability density function (PDF) at x of the
binomial distribution with parameters n and p, where n is the number of trials and
p is the probability of success.

binocdf (x, n, p)
For each element of x, compute the cumulative distribution function (CDF) at x of
the binomial distribution with parameters n and p, where n is the number of trials
and p is the probability of success.

binoinv (x, n, p)
For each element of x, compute the quantile (the inverse of the CDF) at x of the
binomial distribution with parameters n and p, where n is the number of trials and
p is the probability of success.

cauchy_pdf (x)
cauchy_pdf (x, location, scale)
For each element of x, compute the probability density function (PDF) at x of the
Cauchy distribution with location parameter location and scale parameter scale > 0.
Default values are location = 0, scale = 1.

cauchy_cdf (x)
cauchy_cdf (x, location, scale)
For each element of x, compute the cumulative distribution function (CDF) at x of
the Cauchy distribution with location parameter location and scale parameter scale.
Default values are location = 0, scale = 1.

cauchy_inv (x)
cauchy_inv (x, location, scale)
For each element of x, compute the quantile (the inverse of the CDF) at x of the
Cauchy distribution with location parameter location and scale parameter scale.
Default values are location = 0, scale = 1.

chi2pdf (x, n)
For each element of x, compute the probability density function (PDF) at x of the
chi-square distribution with n degrees of freedom.

chi2cdf (x, n)
For each element of x, compute the cumulative distribution function (CDF) at x of
the chi-square distribution with n degrees of freedom.

chi2inv (x, n)
For each element of x, compute the quantile (the inverse of the CDF) at x of the
chi-square distribution with n degrees of freedom.

664

GNU Octave

discrete_pdf (x, v, p)
For each element of x, compute the probability density function (PDF) at x of a
univariate discrete distribution which assumes the values in v with probabilities p.

discrete_cdf (x, v, p)
For each element of x, compute the cumulative distribution function (CDF) at x of a
univariate discrete distribution which assumes the values in v with probabilities p.

discrete_inv (x, v, p)
For each element of x, compute the quantile (the inverse of the CDF) at x of the
univariate distribution which assumes the values in v with probabilities p.

empirical_pdf (x, data)
For each element of x, compute the probability density function (PDF) at x of the
empirical distribution obtained from the univariate sample data.

empirical_cdf (x, data)
For each element of x, compute the cumulative distribution function (CDF) at x of
the empirical distribution obtained from the univariate sample data.

empirical_inv (x, data)
For each element of x, compute the quantile (the inverse of the CDF) at x of the
empirical distribution obtained from the univariate sample data.

exppdf (x, lambda)
For each element of x, compute the probability density function (PDF) at x of the
exponential distribution with mean lambda.

expcdf (x, lambda)
For each element of x, compute the cumulative distribution function (CDF) at x of
the exponential distribution with mean lambda.
The arguments can be of common size or scalars.

expinv (x, lambda)
For each element of x, compute the quantile (the inverse of the CDF) at x of the
exponential distribution with mean lambda.

fpdf (x, m, n)
For each element of x, compute the probability density function (PDF) at x of the F
distribution with m and n degrees of freedom.

fcdf (x, m, n)
For each element of x, compute the cumulative distribution function (CDF) at x of
the F distribution with m and n degrees of freedom.

finv (x, m, n)
For each element of x, compute the quantile (the inverse of the CDF) at x of the F
distribution with m and n degrees of freedom.

gampdf (x, a, b)
For each element of x, return the probability density function (PDF) at x of the
Gamma distribution with shape parameter a and scale b.

Chapter 26: Statistics

665

gamcdf (x, a, b)
For each element of x, compute the cumulative distribution function (CDF) at x of
the Gamma distribution with shape parameter a and scale b.

gaminv (x, a, b)
For each element of x, compute the quantile (the inverse of the CDF) at x of the
Gamma distribution with shape parameter a and scale b.

geopdf (x, p)
For each element of x, compute the probability density function (PDF) at x of the
geometric distribution with parameter p.
The geometric distribution models the number of failures (x-1) of a Bernoulli trial
with probability p before the first success (x).

geocdf (x, p)
For each element of x, compute the cumulative distribution function (CDF) at x of
the geometric distribution with parameter p.
The geometric distribution models the number of failures (x-1) of a Bernoulli trial
with probability p before the first success (x).

geoinv (x, p)
For each element of x, compute the quantile (the inverse of the CDF) at x of the
geometric distribution with parameter p.
The geometric distribution models the number of failures (x-1) of a Bernoulli trial
with probability p before the first success (x).

hygepdf (x, t, m, n)
Compute the probability density function (PDF) at x of the hypergeometric distribution with parameters t, m, and n.
This is the probability of obtaining x marked items when randomly drawing a sample
of size n without replacement from a population of total size t containing m marked
items.
The parameters t, m, and n must be positive integers with m and n not greater than
t.

hygecdf (x, t, m, n)
Compute the cumulative distribution function (CDF) at x of the hypergeometric
distribution with parameters t, m, and n.
This is the probability of obtaining not more than x marked items when randomly
drawing a sample of size n without replacement from a population of total size t
containing m marked items.
The parameters t, m, and n must be positive integers with m and n not greater than
t.

hygeinv (x, t, m, n)
For each element of x, compute the quantile (the inverse of the CDF) at x of the
hypergeometric distribution with parameters t, m, and n.

666

GNU Octave

This is the probability of obtaining x marked items when randomly drawing a sample
of size n without replacement from a population of total size t containing m marked
items.
The parameters t, m, and n must be positive integers with m and n not greater than
t.

kolmogorov_smirnov_cdf (x, tol)
Return the cumulative distribution function (CDF) at x of the Kolmogorov-Smirnov
distribution.
This is defined as
Q(x) =

∞
X

(−1)k exp(−2k 2 x2 )

k=−∞

for x > 0.
The optional parameter tol specifies the precision up to which the series should be
evaluated; the default is tol = eps.

laplace_pdf (x)
For each element of x, compute the probability density function (PDF) at x of the
Laplace distribution.

laplace_cdf (x)
For each element of x, compute the cumulative distribution function (CDF) at x of
the Laplace distribution.

laplace_inv (x)
For each element of x, compute the quantile (the inverse of the CDF) at x of the
Laplace distribution.

logistic_pdf (x)
For each element of x, compute the PDF at x of the logistic distribution.

logistic_cdf (x)
For each element of x, compute the cumulative distribution function (CDF) at x of
the logistic distribution.

logistic_inv (x)
For each element of x, compute the quantile (the inverse of the CDF) at x of the
logistic distribution.

lognpdf (x)
lognpdf (x, mu, sigma)
For each element of x, compute the probability density function (PDF) at x of the
lognormal distribution with parameters mu and sigma.
If a random variable follows this distribution, its logarithm is normally distributed
with mean mu and standard deviation sigma.
Default values are mu = 0, sigma = 1.

Chapter 26: Statistics

667

logncdf (x)
logncdf (x, mu, sigma)
For each element of x, compute the cumulative distribution function (CDF) at x of
the lognormal distribution with parameters mu and sigma.
If a random variable follows this distribution, its logarithm is normally distributed
with mean mu and standard deviation sigma.
Default values are mu = 0, sigma = 1.

logninv (x)
logninv (x, mu, sigma)
For each element of x, compute the quantile (the inverse of the CDF) at x of the
lognormal distribution with parameters mu and sigma.
If a random variable follows this distribution, its logarithm is normally distributed
with mean mu and standard deviation sigma.
Default values are mu = 0, sigma = 1.

nbinpdf (x, n, p)
For each element of x, compute the probability density function (PDF) at x of the
negative binomial distribution with parameters n and p.
When n is integer this is the Pascal distribution. When n is extended to real numbers
this is the Polya distribution.
The number of failures in a Bernoulli experiment with success probability p before
the n-th success follows this distribution.

nbincdf (x, n, p)
For each element of x, compute the cumulative distribution function (CDF) at x of
the negative binomial distribution with parameters n and p.
When n is integer this is the Pascal distribution. When n is extended to real numbers
this is the Polya distribution.
The number of failures in a Bernoulli experiment with success probability p before
the n-th success follows this distribution.

nbininv (x, n, p)
For each element of x, compute the quantile (the inverse of the CDF) at x of the
negative binomial distribution with parameters n and p.
When n is integer this is the Pascal distribution. When n is extended to real numbers
this is the Polya distribution.
The number of failures in a Bernoulli experiment with success probability p before
the n-th success follows this distribution.

normpdf (x)
normpdf (x, mu, sigma)
For each element of x, compute the probability density function (PDF) at x of the
normal distribution with mean mu and standard deviation sigma.
Default values are mu = 0, sigma = 1.

668

GNU Octave

normcdf (x)
normcdf (x, mu, sigma)
For each element of x, compute the cumulative distribution function (CDF) at x of
the normal distribution with mean mu and standard deviation sigma.
Default values are mu = 0, sigma = 1.

norminv (x)
norminv (x, mu, sigma)
For each element of x, compute the quantile (the inverse of the CDF) at x of the
normal distribution with mean mu and standard deviation sigma.
Default values are mu = 0, sigma = 1.

poisspdf (x, lambda)
For each element of x, compute the probability density function (PDF) at x of the
Poisson distribution with parameter lambda.

poisscdf (x, lambda)
For each element of x, compute the cumulative distribution function (CDF) at x of
the Poisson distribution with parameter lambda.

poissinv (x, lambda)
For each element of x, compute the quantile (the inverse of the CDF) at x of the
Poisson distribution with parameter lambda.

stdnormal_pdf (x)
For each element of x, compute the probability density function (PDF) at x of the
standard normal distribution (mean = 0, standard deviation = 1).

stdnormal_cdf (x)
For each element of x, compute the cumulative distribution function (CDF) at x of
the standard normal distribution (mean = 0, standard deviation = 1).

stdnormal_inv (x)
For each element of x, compute the quantile (the inverse of the CDF) at x of the
standard normal distribution (mean = 0, standard deviation = 1).

tpdf (x, n)
For each element of x, compute the probability density function (PDF) at x of the t
(Student) distribution with n degrees of freedom.

tcdf (x, n)
For each element of x, compute the cumulative distribution function (CDF) at x of
the t (Student) distribution with n degrees of freedom.

tinv (x, n)
For each element of x, compute the quantile (the inverse of the CDF) at x of the t
(Student) distribution with n degrees of freedom.
This function is analogous to looking in a table for the t-value of a single-tailed
distribution.

Chapter 26: Statistics

669

unidpdf (x, n)
For each element of x, compute the probability density function (PDF) at x of a
discrete uniform distribution which assumes the integer values 1–n with equal probability.
Warning: The underlying implementation uses the double class and will only be
accurate for n < flintmax (253 on IEEE 754 compatible systems).

unidcdf (x, n)
For each element of x, compute the cumulative distribution function (CDF) at x
of a discrete uniform distribution which assumes the integer values 1–n with equal
probability.

unidinv (x, n)
For each element of x, compute the quantile (the inverse of the CDF) at x of the
discrete uniform distribution which assumes the integer values 1–n with equal probability.

unifpdf (x)
unifpdf (x, a, b)
For each element of x, compute the probability density function (PDF) at x of the
uniform distribution on the interval [a, b].
Default values are a = 0, b = 1.

unifcdf (x)
unifcdf (x, a, b)
For each element of x, compute the cumulative distribution function (CDF) at x of
the uniform distribution on the interval [a, b].
Default values are a = 0, b = 1.

unifinv (x)
unifinv (x, a, b)
For each element of x, compute the quantile (the inverse of the CDF) at x of the
uniform distribution on the interval [a, b].
Default values are a = 0, b = 1.

wblpdf (x)
wblpdf (x, scale)
wblpdf (x, scale, shape)
Compute the probability density function (PDF) at x of the Weibull distribution with
scale parameter scale and shape parameter shape.
This is given by
shape
shape
x
· xshape−1 · e−( scale )
shape
scale

for x ≥ 0.

Default values are scale = 1, shape = 1.

670

GNU Octave

wblcdf (x)
wblcdf (x, scale)
wblcdf (x, scale, shape)
Compute the cumulative distribution function (CDF) at x of the Weibull distribution
with scale parameter scale and shape parameter shape.
This is defined as
x

shape

1 − e−( scale )
for x ≥ 0.

wblinv (x)
wblinv (x, scale)
wblinv (x, scale, shape)
Compute the quantile (the inverse of the CDF) at x of the Weibull distribution with
scale parameter scale and shape parameter shape.
Default values are scale = 1, shape = 1.

26.6 Tests
Octave can perform many different statistical tests. The following table summarizes the
available tests.
Hypothesis
Equal mean values
Equal medians
Equal variances
Equal distributions
Equal marginal frequencies
Equal success probabilities
Independent observations
Uncorrelated observations
Given mean value
Observations from distribution
Regression

Test Functions
anova, hotelling test2, t test 2,
welch test, wilcoxon test, z test 2
kruskal wallis test, sign test
bartlett test, manova, var test
chisquare test homogeneity,
kolmogorov smirnov test 2, u test
mcnemar test
prop test 2
chisquare test independence,
run test
cor test
hotelling test, t test, z test
kolmogorov smirnov test
f test regression, t test regression

The tests return a p-value that describes the outcome of the test. Assuming that the
test hypothesis is true, the p-value is the probability of obtaining a worse result than the
observed one. So large p-values corresponds to a successful test. Usually a test hypothesis
is accepted if the p-value exceeds 0.05.

[pval, f, df_b, df_w] = anova (y, g)
Perform a one-way analysis of variance (ANOVA).
The goal is to test whether the population means of data taken from k different groups
are all equal.

Chapter 26: Statistics

671

Data may be given in a single vector y with groups specified by a corresponding
vector of group labels g (e.g., numbers from 1 to k). This is the general form which
does not impose any restriction on the number of data in each group or the group
labels.
If y is a matrix and g is omitted, each column of y is treated as a group. This form
is only appropriate for balanced ANOVA in which the numbers of samples from each
group are all equal.
Under the null of constant means, the statistic f follows an F distribution with df b
and df w degrees of freedom.
The p-value (1 minus the CDF of this distribution at f ) is returned in pval.
If no output argument is given, the standard one-way ANOVA table is printed.
See also: [manova], page 674.

[pval, chisq, df] = bartlett_test (x1, . . . )
Perform a Bartlett test for the homogeneity of variances in the data vectors x1, x2,
. . . , xk, where k > 1.
Under the null of equal variances, the test statistic chisq approximately follows a
chi-square distribution with df degrees of freedom.
The p-value (1 minus the CDF of this distribution at chisq) is returned in pval.
If no output argument is given, the p-value is displayed.

[pval, chisq, df] = chisquare_test_homogeneity (x, y, c)
Given two samples x and y, perform a chisquare test for homogeneity of the null
hypothesis that x and y come from the same distribution, based on the partition
induced by the (strictly increasing) entries of c.
For large samples, the test statistic chisq approximately follows a chisquare distribution with df = length (c) degrees of freedom.
The p-value (1 minus the CDF of this distribution at chisq) is returned in pval.
If no output argument is given, the p-value is displayed.

[pval, chisq, df] = chisquare_test_independence (x)
Perform a chi-square test for independence based on the contingency table x.
Under the null hypothesis of independence, chisq approximately has a chi-square
distribution with df degrees of freedom.
The p-value (1 minus the CDF of this distribution at chisq) of the test is returned in
pval.
If no output argument is given, the p-value is displayed.

cor_test (x, y, alt, method)
Test whether two samples x and y come from uncorrelated populations.
The optional argument string alt describes the alternative hypothesis, and can be
"!=" or "<>" (nonzero), ">" (greater than 0), or "<" (less than 0). The default is the
two-sided case.
The optional argument string method specifies which correlation coefficient to use
for testing. If method is "pearson" (default), the (usual) Pearson’s product moment

672

GNU Octave

correlation coefficient is used. In this case, the data should come from a bivariate
normal distribution. Otherwise, the other two methods offer nonparametric alternatives. If method is "kendall", then Kendall’s rank correlation tau is used. If method
is "spearman", then Spearman’s rank correlation rho is used. Only the first character
is necessary.
The output is a structure with the following elements:
pval

The p-value of the test.

stat

The value of the test statistic.

dist

The distribution of the test statistic.

params

The parameters of the null distribution of the test statistic.

alternative
The alternative hypothesis.
method

The method used for testing.

If no output argument is given, the p-value is displayed.

[pval, f, df_num, df_den] = f_test_regression (y, x, rr, r)
Perform an F test for the null hypothesis rr * b = r in a classical normal regression
model y = X * b + e.
Under the null, the test statistic f follows an F distribution with df num and df den
degrees of freedom.
The p-value (1 minus the CDF of this distribution at f ) is returned in pval.
If not given explicitly, r = 0.
If no output argument is given, the p-value is displayed.

[pval, tsq] = hotelling_test (x, m)
For a sample x from a multivariate normal distribution with unknown mean and
covariance matrix, test the null hypothesis that mean (x) == m.
Hotelling’s T 2 is returned in tsq. Under the null, (n − p)T 2 /(p(n − 1)) has an F
distribution with p and n − p degrees of freedom, where n and p are the numbers of
samples and variables, respectively.
The p-value of the test is returned in pval.
If no output argument is given, the p-value of the test is displayed.

[pval, tsq] = hotelling_test_2 (x, y)
For two samples x from multivariate normal distributions with the same number of
variables (columns), unknown means and unknown equal covariance matrices, test
the null hypothesis mean (x) == mean (y).
Hotelling’s two-sample T 2 is returned in tsq. Under the null,
(nx + ny − p − 1)T 2
p(nx + ny − 2)
has an F distribution with p and nx + ny − p − 1 degrees of freedom, where nx and
ny are the sample sizes and p is the number of variables.
The p-value of the test is returned in pval.
If no output argument is given, the p-value of the test is displayed.

Chapter 26: Statistics

673

[pval, ks] = kolmogorov_smirnov_test (x, dist, params, alt)
Perform a Kolmogorov-Smirnov test of the null hypothesis that the sample x comes
from the (continuous) distribution dist.
if F and G are the CDFs corresponding to the sample and dist, respectively, then the
null is that F == G.
The optional argument params contains a list of parameters of dist. For example, to
test whether a sample x comes from a uniform distribution on [2,4], use
kolmogorov_smirnov_test (x, "unif", 2, 4)
dist can be any string for which a function distcdf that calculates the CDF of distribution dist exists.
With the optional argument string alt, the alternative of interest can be selected. If
alt is "!=" or "<>", the null is tested against the two-sided alternative F != G. In
this case, the test statistic ks follows a two-sided Kolmogorov-Smirnov distribution.
If alt is ">", the one-sided alternative F > G is considered. Similarly for "<", the
one-sided alternative F > G is considered. In this case, the test statistic ks has a
one-sided Kolmogorov-Smirnov distribution. The default is the two-sided case.
The p-value of the test is returned in pval.
If no output argument is given, the p-value is displayed.

[pval, ks, d] = kolmogorov_smirnov_test_2 (x, y, alt)
Perform a 2-sample Kolmogorov-Smirnov test of the null hypothesis that the samples
x and y come from the same (continuous) distribution.
If F and G are the CDFs corresponding to the x and y samples, respectively, then
the null is that F == G.
With the optional argument string alt, the alternative of interest can be selected. If
alt is "!=" or "<>", the null is tested against the two-sided alternative F != G. In
this case, the test statistic ks follows a two-sided Kolmogorov-Smirnov distribution.
If alt is ">", the one-sided alternative F > G is considered. Similarly for "<", the
one-sided alternative F < G is considered. In this case, the test statistic ks has a
one-sided Kolmogorov-Smirnov distribution. The default is the two-sided case.
The p-value of the test is returned in pval.
The third returned value, d, is the test statistic, the maximum vertical distance
between the two cumulative distribution functions.
If no output argument is given, the p-value is displayed.

[pval, k, df] = kruskal_wallis_test (x1, . . . )
Perform a Kruskal-Wallis one-factor analysis of variance.
Suppose a variable is observed for k > 1 different groups, and let x1, . . . , xk be the
corresponding data vectors.
Under the null hypothesis that the ranks in the pooled sample are not affected by the
group memberships, the test statistic k is approximately chi-square with df = k - 1
degrees of freedom.
If the data contains ties (some value appears more than once) k is divided by
1 - sum ties / (n^3 - n)

674

GNU Octave

where sum ties is the sum of t^2 - t over each group of ties where t is the number
of ties in the group and n is the total number of values in the input data. For more
info on this adjustment see William H. Kruskal and W. Allen Wallis, Use of Ranks
in One-Criterion Variance Analysis, Journal of the American Statistical Association,
Vol. 47, No. 260 (Dec 1952).
The p-value (1 minus the CDF of this distribution at k) is returned in pval.
If no output argument is given, the p-value is displayed.

manova (x, g)
Perform a one-way multivariate analysis of variance (MANOVA).
The goal is to test whether the p-dimensional population means of data taken from
k different groups are all equal. All data are assumed drawn independently from
p-dimensional normal distributions with the same covariance matrix.
The data matrix is given by x. As usual, rows are observations and columns are
variables. The vector g specifies the corresponding group labels (e.g., numbers from
1 to k).
The LR test statistic (Wilks’ Lambda) and approximate p-values are computed and
displayed.
See also: [anova], page 670.

[pval, chisq, df] = mcnemar_test (x)
For a square contingency table x of data cross-classified on the row and column
variables, McNemar’s test can be used for testing the null hypothesis of symmetry of
the classification probabilities.
Under the null, chisq is approximately distributed as chisquare with df degrees of
freedom.
The p-value (1 minus the CDF of this distribution at chisq) is returned in pval.
If no output argument is given, the p-value of the test is displayed.

[pval, z] = prop_test_2 (x1, n1, x2, n2, alt)
If x1 and n1 are the counts of successes and trials in one sample, and x2 and n2 those
in a second one, test the null hypothesis that the success probabilities p1 and p2 are
the same.
Under the null, the test statistic z approximately follows a standard normal distribution.
With the optional argument string alt, the alternative of interest can be selected. If
alt is "!=" or "<>", the null is tested against the two-sided alternative p1 != p2. If
alt is ">", the one-sided alternative p1 > p2 is used. Similarly for "<", the one-sided
alternative p1 < p2 is used. The default is the two-sided case.
The p-value of the test is returned in pval.
If no output argument is given, the p-value of the test is displayed.

[pval, chisq] = run_test (x)
Perform a chi-square test with 6 degrees of freedom based on the upward runs in the
columns of x.

Chapter 26: Statistics

675

run_test can be used to decide whether x contains independent data.
The p-value of the test is returned in pval.
If no output argument is given, the p-value is displayed.

[pval, b, n] = sign_test (x, y, alt)
For two matched-pair samples x and y, perform a sign test of the null hypothesis
PROB (x > y) == PROB (x < y) == 1/2.
Under the null, the test statistic b roughly follows a binomial distribution with parameters n = sum (x != y) and p = 1/2.
With the optional argument alt, the alternative of interest can be selected. If alt is
"!=" or "<>", the null hypothesis is tested against the two-sided alternative PROB
(x < y) != 1/2. If alt is ">", the one-sided alternative PROB (x > y) > 1/2 ("x
is stochastically greater than y") is considered. Similarly for "<", the one-sided
alternative PROB (x > y) < 1/2 ("x is stochastically less than y") is considered. The
default is the two-sided case.
The p-value of the test is returned in pval.
If no output argument is given, the p-value of the test is displayed.

[pval, t, df] = t_test (x, m, alt)
For a sample x from a normal distribution with unknown mean and variance, perform
a t-test of the null hypothesis mean (x) == m.
Under the null, the test statistic t follows a Student distribution with df = length
(x) - 1 degrees of freedom.
With the optional argument string alt, the alternative of interest can be selected. If
alt is "!=" or "<>", the null is tested against the two-sided alternative mean (x) != m.
If alt is ">", the one-sided alternative mean (x) > m is considered. Similarly for "<",
the one-sided alternative mean (x) < m is considered. The default is the two-sided
case.
The p-value of the test is returned in pval.
If no output argument is given, the p-value of the test is displayed.

[pval, t, df] = t_test_2 (x, y, alt)
For two samples x and y from normal distributions with unknown means and unknown
equal variances, perform a two-sample t-test of the null hypothesis of equal means.
Under the null, the test statistic t follows a Student distribution with df degrees of
freedom.
With the optional argument string alt, the alternative of interest can be selected.
If alt is "!=" or "<>", the null is tested against the two-sided alternative mean (x)
!= mean (y). If alt is ">", the one-sided alternative mean (x) > mean (y) is used.
Similarly for "<", the one-sided alternative mean (x) < mean (y) is used. The default
is the two-sided case.
The p-value of the test is returned in pval.
If no output argument is given, the p-value of the test is displayed.

676

GNU Octave

[pval, t, df] = t_test_regression (y, x, rr, r, alt)
Perform a t test for the null hypothesis rr * b = r in a classical normal regression
model y = x * b + e.
Under the null, the test statistic t follows a t distribution with df degrees of freedom.
If r is omitted, a value of 0 is assumed.
With the optional argument string alt, the alternative of interest can be selected.
If alt is "!=" or "<>", the null is tested against the two-sided alternative rr * b !=
r. If alt is ">", the one-sided alternative rr * b > r is used. Similarly for "<", the
one-sided alternative rr * b < r is used. The default is the two-sided case.
The p-value of the test is returned in pval.
If no output argument is given, the p-value of the test is displayed.

[pval, z] = u_test (x, y, alt)
For two samples x and y, perform a Mann-Whitney U-test of the null hypothesis
PROB (x > y) == 1/2 == PROB (x < y).
Under the null, the test statistic z approximately follows a standard normal distribution. Note that this test is equivalent to the Wilcoxon rank-sum test.
With the optional argument string alt, the alternative of interest can be selected. If
alt is "!=" or "<>", the null is tested against the two-sided alternative PROB (x >
y) != 1/2. If alt is ">", the one-sided alternative PROB (x > y) > 1/2 is considered.
Similarly for "<", the one-sided alternative PROB (x > y) < 1/2 is considered. The
default is the two-sided case.
The p-value of the test is returned in pval.
If no output argument is given, the p-value of the test is displayed.

[pval, f, df_num, df_den] = var_test (x, y, alt)
For two samples x and y from normal distributions with unknown means and unknown
variances, perform an F-test of the null hypothesis of equal variances.
Under the null, the test statistic f follows an F-distribution with df num and df den
degrees of freedom.
With the optional argument string alt, the alternative of interest can be selected. If
alt is "!=" or "<>", the null is tested against the two-sided alternative var (x) != var
(y). If alt is ">", the one-sided alternative var (x) > var (y) is used. Similarly for
"<", the one-sided alternative var (x) > var (y) is used. The default is the two-sided
case.
The p-value of the test is returned in pval.
If no output argument is given, the p-value of the test is displayed.

[pval, t, df] = welch_test (x, y, alt)
For two samples x and y from normal distributions with unknown means and unknown
and not necessarily equal variances, perform a Welch test of the null hypothesis of
equal means.
Under the null, the test statistic t approximately follows a Student distribution with
df degrees of freedom.

Chapter 26: Statistics

677

With the optional argument string alt, the alternative of interest can be selected. If
alt is "!=" or "<>", the null is tested against the two-sided alternative mean (x) != m.
If alt is ">", the one-sided alternative mean(x) > m is considered. Similarly for "<",
the one-sided alternative mean(x) < m is considered. The default is the two-sided
case.
The p-value of the test is returned in pval.
If no output argument is given, the p-value of the test is displayed.

[pval, z] = wilcoxon_test (x, y, alt)
For two matched-pair sample vectors x and y, perform a Wilcoxon signed-rank test
of the null hypothesis PROB (x > y) == 1/2.
Under the null, the test statistic z approximately follows a standard normal distribution when n > 25.
Caution: This function assumes a normal distribution for z and thus is invalid for n
≤ 25.
With the optional argument string alt, the alternative of interest can be selected. If
alt is "!=" or "<>", the null is tested against the two-sided alternative PROB (x >
y) != 1/2. If alt is ">", the one-sided alternative PROB (x > y) > 1/2 is considered.
Similarly for "<", the one-sided alternative PROB (x > y) < 1/2 is considered. The
default is the two-sided case.
The p-value of the test is returned in pval.
If no output argument is given, the p-value of the test is displayed.

[pval, z] = z_test (x, m, v, alt)
Perform a Z-test of the null hypothesis mean (x) == m for a sample x from a normal
distribution with unknown mean and known variance v.
Under the null, the test statistic z follows a standard normal distribution.
With the optional argument string alt, the alternative of interest can be selected. If
alt is "!=" or "<>", the null is tested against the two-sided alternative mean (x) != m.
If alt is ">", the one-sided alternative mean (x) > m is considered. Similarly for "<",
the one-sided alternative mean (x) < m is considered. The default is the two-sided
case.
The p-value of the test is returned in pval.
If no output argument is given, the p-value of the test is displayed along with some
information.

[pval, z] = z_test_2 (x, y, v_x, v_y, alt)
For two samples x and y from normal distributions with unknown means and known
variances v x and v y, perform a Z-test of the hypothesis of equal means.
Under the null, the test statistic z follows a standard normal distribution.
With the optional argument string alt, the alternative of interest can be selected.
If alt is "!=" or "<>", the null is tested against the two-sided alternative mean (x)
!= mean (y). If alt is ">", the one-sided alternative mean (x) > mean (y) is used.
Similarly for "<", the one-sided alternative mean (x) < mean (y) is used. The default
is the two-sided case.

678

GNU Octave

The p-value of the test is returned in pval.
If no output argument is given, the p-value of the test is displayed along with some
information.

26.7 Random Number Generation
Octave can generate random numbers from a large number of distributions. The random
number generators are based on the random number generators described in Section 16.3
[Special Utility Matrices], page 457.
The following table summarizes the available random number generators (in alphabetical
order).
Distribution
Beta Distribution
Binomial Distribution
Cauchy Distribution
Chi-Square Distribution
Univariate Discrete Distribution
Empirical Distribution
Exponential Distribution
F Distribution
Gamma Distribution
Geometric Distribution
Hypergeometric Distribution
Laplace Distribution
Logistic Distribution
Log-Normal Distribution
Pascal Distribution
Univariate Normal Distribution
Poisson Distribution
Standard Normal Distribution
t (Student) Distribution
Univariate Discrete Distribution
Uniform Distribution
Weibull Distribution
Wiener Process
(a,
(a,
(a,
(a,
Return
and b.

betarnd
betarnd
betarnd
betarnd

Function
betarnd
binornd
cauchy rnd
chi2rnd
discrete rnd
empirical rnd
exprnd
frnd
gamrnd
geornd
hygernd
laplace rnd
logistic rnd
lognrnd
nbinrnd
normrnd
poissrnd
stdnormal rnd
trnd
unidrnd
unifrnd
wblrnd
wienrnd

b)
b, r)
b, r, c, . . . )
b, [sz])
a matrix of random samples from the Beta distribution with parameters a

When called with a single size argument, return a square matrix with the dimension
specified. When called with more than one scalar argument the first two arguments are

Chapter 26: Statistics

679

taken as the number of rows and columns and any further arguments specify additional
matrix dimensions. The size may also be specified with a vector of dimensions sz.
If no size arguments are given then the result matrix is the common size of a and b.
(n, p)
(n, p, r)
(n, p, r, c, . . . )
(n, p, [sz])
Return a matrix of random samples from the binomial distribution with parameters
n and p, where n is the number of trials and p is the probability of success.
When called with a single size argument, return a square matrix with the dimension
specified. When called with more than one scalar argument the first two arguments are
taken as the number of rows and columns and any further arguments specify additional
matrix dimensions. The size may also be specified with a vector of dimensions sz.
If no size arguments are given then the result matrix is the common size of n and p.

binornd
binornd
binornd
binornd

(location, scale)
(location, scale, r)
(location, scale, r, c, . . . )
(location, scale, [sz])
Return a matrix of random samples from the Cauchy distribution with parameters
location and scale.
When called with a single size argument, return a square matrix with the dimension
specified. When called with more than one scalar argument the first two arguments are
taken as the number of rows and columns and any further arguments specify additional
matrix dimensions. The size may also be specified with a vector of dimensions sz.
If no size arguments are given then the result matrix is the common size of location
and scale.

cauchy_rnd
cauchy_rnd
cauchy_rnd
cauchy_rnd

(n)
(n, r)
(n, r, c, . . . )
(n, [sz])
Return a matrix of random samples from the chi-square distribution with n degrees
of freedom.
When called with a single size argument, return a square matrix with the dimension
specified. When called with more than one scalar argument the first two arguments are
taken as the number of rows and columns and any further arguments specify additional
matrix dimensions. The size may also be specified with a vector of dimensions sz.
If no size arguments are given then the result matrix is the size of n.

chi2rnd
chi2rnd
chi2rnd
chi2rnd

(v, p)
(v, p, r)
(v, p, r, c, . . . )
(v, p, [sz])
Return a matrix of random samples from the univariate distribution which assumes
the values in v with probabilities p.

discrete_rnd
discrete_rnd
discrete_rnd
discrete_rnd

680

GNU Octave

When called with a single size argument, return a square matrix with the dimension
specified. When called with more than one scalar argument the first two arguments are
taken as the number of rows and columns and any further arguments specify additional
matrix dimensions. The size may also be specified with a vector of dimensions sz.
If no size arguments are given then the result matrix is the common size of v and p.
(data)
(data, r)
(data, r, c, . . . )
(data, [sz])
Return a matrix of random samples from the empirical distribution obtained from
the univariate sample data.

empirical_rnd
empirical_rnd
empirical_rnd
empirical_rnd

When called with a single size argument, return a square matrix with the dimension
specified. When called with more than one scalar argument the first two arguments are
taken as the number of rows and columns and any further arguments specify additional
matrix dimensions. The size may also be specified with a vector of dimensions sz.
If no size arguments are given then the result matrix is a random ordering of the
sample data.
(lambda)
(lambda, r)
(lambda, r, c, . . . )
(lambda, [sz])
Return a matrix of random samples from the exponential distribution with mean
lambda.

exprnd
exprnd
exprnd
exprnd

When called with a single size argument, return a square matrix with the dimension
specified. When called with more than one scalar argument the first two arguments are
taken as the number of rows and columns and any further arguments specify additional
matrix dimensions. The size may also be specified with a vector of dimensions sz.
If no size arguments are given then the result matrix is the size of lambda.

frnd
frnd
frnd
frnd

(m, n)
(m, n, r)
(m, n, r, c, . . . )
(m, n, [sz])
Return a matrix of random samples from the F distribution with m and n degrees of
freedom.
When called with a single size argument, return a square matrix with the dimension
specified. When called with more than one scalar argument the first two arguments are
taken as the number of rows and columns and any further arguments specify additional
matrix dimensions. The size may also be specified with a vector of dimensions sz.
If no size arguments are given then the result matrix is the common size of m and n.

gamrnd (a, b)
gamrnd (a, b, r)
gamrnd (a, b, r, c, . . . )

Chapter 26: Statistics

681

gamrnd (a, b, [sz])
Return a matrix of random samples from the Gamma distribution with shape parameter a and scale b.
When called with a single size argument, return a square matrix with the dimension
specified. When called with more than one scalar argument the first two arguments are
taken as the number of rows and columns and any further arguments specify additional
matrix dimensions. The size may also be specified with a vector of dimensions sz.
If no size arguments are given then the result matrix is the common size of a and b.
(p)
(p, r)
(p, r, c, . . . )
(p, [sz])
Return a matrix of random samples from the geometric distribution with parameter
p.
When called with a single size argument, return a square matrix with the dimension
specified. When called with more than one scalar argument the first two arguments are
taken as the number of rows and columns and any further arguments specify additional
matrix dimensions. The size may also be specified with a vector of dimensions sz.
If no size arguments are given then the result matrix is the size of p.
The geometric distribution models the number of failures (x-1) of a Bernoulli trial
with probability p before the first success (x).

geornd
geornd
geornd
geornd

(t, m, n)
(t, m, n, r)
(t, m, n, r, c, . . . )
(t, m, n, [sz])
Return a matrix of random samples from the hypergeometric distribution with parameters t, m, and n.
The parameters t, m, and n must be positive integers with m and n not greater than
t.
When called with a single size argument, return a square matrix with the dimension
specified. When called with more than one scalar argument the first two arguments are
taken as the number of rows and columns and any further arguments specify additional
matrix dimensions. The size may also be specified with a vector of dimensions sz.
If no size arguments are given then the result matrix is the common size of t, m, and
n.

hygernd
hygernd
hygernd
hygernd

laplace_rnd (r)
laplace_rnd (r, c, . . . )
laplace_rnd ([sz])
Return a matrix of random samples from the Laplace distribution.
When called with a single size argument, return a square matrix with the dimension
specified. When called with more than one scalar argument the first two arguments are
taken as the number of rows and columns and any further arguments specify additional
matrix dimensions. The size may also be specified with a vector of dimensions sz.

682

GNU Octave

logistic_rnd (r)
logistic_rnd (r, c, . . . )
logistic_rnd ([sz])
Return a matrix of random samples from the logistic distribution.
When called with a single size argument, return a square matrix with the dimension
specified. When called with more than one scalar argument the first two arguments are
taken as the number of rows and columns and any further arguments specify additional
matrix dimensions. The size may also be specified with a vector of dimensions sz.
(mu, sigma)
(mu, sigma, r)
(mu, sigma, r, c, . . . )
(mu, sigma, [sz])
Return a matrix of random samples from the lognormal distribution with parameters
mu and sigma.

lognrnd
lognrnd
lognrnd
lognrnd

When called with a single size argument, return a square matrix with the dimension
specified. When called with more than one scalar argument the first two arguments are
taken as the number of rows and columns and any further arguments specify additional
matrix dimensions. The size may also be specified with a vector of dimensions sz.
If no size arguments are given then the result matrix is the common size of mu and
sigma.
(n, p)
(n, p, r)
(n, p, r, c, . . . )
(n, p, [sz])
Return a matrix of random samples from the negative binomial distribution with
parameters n and p.

nbinrnd
nbinrnd
nbinrnd
nbinrnd

When called with a single size argument, return a square matrix with the dimension
specified. When called with more than one scalar argument the first two arguments are
taken as the number of rows and columns and any further arguments specify additional
matrix dimensions. The size may also be specified with a vector of dimensions sz.
If no size arguments are given then the result matrix is the common size of n and p.
(mu, sigma)
(mu, sigma, r)
(mu, sigma, r, c, . . . )
(mu, sigma, [sz])
Return a matrix of random samples from the normal distribution with parameters
mean mu and standard deviation sigma.

normrnd
normrnd
normrnd
normrnd

When called with a single size argument, return a square matrix with the dimension
specified. When called with more than one scalar argument the first two arguments are
taken as the number of rows and columns and any further arguments specify additional
matrix dimensions. The size may also be specified with a vector of dimensions sz.
If no size arguments are given then the result matrix is the common size of mu and
sigma.

Chapter 26: Statistics

683

(lambda)
(lambda, r)
(lambda, r, c, . . . )
(lambda, [sz])
Return a matrix of random samples from the Poisson distribution with parameter
lambda.
When called with a single size argument, return a square matrix with the dimension
specified. When called with more than one scalar argument the first two arguments are
taken as the number of rows and columns and any further arguments specify additional
matrix dimensions. The size may also be specified with a vector of dimensions sz.
If no size arguments are given then the result matrix is the size of lambda.

poissrnd
poissrnd
poissrnd
poissrnd

stdnormal_rnd (r)
stdnormal_rnd (r, c, . . . )
stdnormal_rnd ([sz])
Return a matrix of random samples from the standard normal distribution (mean =
0, standard deviation = 1).
When called with a single size argument, return a square matrix with the dimension
specified. When called with more than one scalar argument the first two arguments are
taken as the number of rows and columns and any further arguments specify additional
matrix dimensions. The size may also be specified with a vector of dimensions sz.

trnd
trnd
trnd
trnd

(n)
(n, r)
(n, r, c, . . . )
(n, [sz])
Return a matrix of random samples from the t (Student) distribution with n degrees
of freedom.
When called with a single size argument, return a square matrix with the dimension
specified. When called with more than one scalar argument the first two arguments are
taken as the number of rows and columns and any further arguments specify additional
matrix dimensions. The size may also be specified with a vector of dimensions sz.
If no size arguments are given then the result matrix is the size of n.
(n)
(n, r)
(n, r, c, . . . )
(n, [sz])
Return a matrix of random samples from the discrete uniform distribution which
assumes the integer values 1–n with equal probability.
n may be a scalar or a multi-dimensional array.
When called with a single size argument, return a square matrix with the dimension
specified. When called with more than one scalar argument the first two arguments are
taken as the number of rows and columns and any further arguments specify additional
matrix dimensions. The size may also be specified with a vector of dimensions sz.
If no size arguments are given then the result matrix is the size of n.

unidrnd
unidrnd
unidrnd
unidrnd

684

GNU Octave

(a, b)
(a, b, r)
(a, b, r, c, . . . )
(a, b, [sz])
Return a matrix of random samples from the uniform distribution on [a, b].
When called with a single size argument, return a square matrix with the dimension
specified. When called with more than one scalar argument the first two arguments are
taken as the number of rows and columns and any further arguments specify additional
matrix dimensions. The size may also be specified with a vector of dimensions sz.
If no size arguments are given then the result matrix is the common size of a and b.

unifrnd
unifrnd
unifrnd
unifrnd

(scale, shape)
(scale, shape, r)
(scale, shape, r, c, . . . )
(scale, shape, [sz])
Return a matrix of random samples from the Weibull distribution with parameters
scale and shape.
When called with a single size argument, return a square matrix with the dimension
specified. When called with more than one scalar argument the first two arguments are
taken as the number of rows and columns and any further arguments specify additional
matrix dimensions. The size may also be specified with a vector of dimensions sz.
If no size arguments are given then the result matrix is the common size of scale and
shape.

wblrnd
wblrnd
wblrnd
wblrnd

wienrnd (t, d, n)
Return a simulated realization of the d-dimensional Wiener Process on the interval
[0, t].
If d is omitted, d = 1 is used. The first column of the return matrix contains time,
the remaining columns contain the Wiener process.
The optional parameter n defines the number of summands used for simulating the
process over an interval of length 1. If n is omitted, n = 1000 is used.

685

27 Sets
Octave has a number of functions for managing sets of data. A set is defined as a collection
of unique elements and is typically represented by a vector of numbers sorted in ascending
order. Any vector or matrix can be converted to a set by removing duplicates through the
use of the unique function. However, it isn’t necessary to explicitly create a set as all of
the functions which operate on sets will convert their input to a set before proceeding.

unique
unique
[y, i,
[y, i,
[y, i,

(x)
(x, "rows")

j] = unique ( . . . )
j] = unique ( . . . , "first")
j] = unique ( . . . , "last")

Return the unique elements of x sorted in ascending order.
If the input x is a column vector then return a column vector; Otherwise, return a
row vector. x may also be a cell array of strings.
If the optional argument "rows" is given then return the unique rows of x sorted in
ascending order. The input must be a 2-D matrix to use this option.
If requested, return index vectors i and j such that y = x(i) and x = y(j).
Additionally, if i is a requested output then one of "first" or "last" may be given
as an input. If "last" is specified, return the highest possible indices in i, otherwise,
if "first" is specified, return the lowest. The default is "last".
See also: [union], page 686, [intersect], page 685, [setdiff], page 686, [setxor], page 686,
[ismember], page 686.

27.1 Set Operations
Octave supports several basic set operations. Octave can compute the union, intersection,
and difference of two sets. Octave also supports the Exclusive Or set operation.
The functions for set operations all work in the same way by accepting two input sets
and returning a third set. As an example, assume that a and b contains two sets, then
union (a, b)
computes the union of the two sets.
Finally, determining whether elements belong to a set can be done with the ismember
function. Because sets are ordered this operation is very efficient and is of order O(log2(n))
which is preferable to the find function which is of order O(n).

c = intersect (a, b)
c = intersect (a, b, "rows")
[c, ia, ib] = intersect ( . . . )
Return the unique elements common to both a and b sorted in ascending order.
If a and b are both row vectors then return a row vector; Otherwise, return a column
vector. The inputs may also be cell arrays of strings.
If the optional input "rows" is given then return the common rows of a and b. The
inputs must be 2-D matrices to use this option.
If requested, return index vectors ia and ib such that c = a(ia) and c = b(ib).

686

GNU Octave

See also: [unique], page 685, [union], page 686, [setdiff], page 686, [setxor], page 686,
[ismember], page 686.

c = union (a, b)
c = union (a, b, "rows")
[c, ia, ib] = union ( . . . )
Return the unique elements that are in either a or b sorted in ascending order.
If a and b are both row vectors then return a row vector; Otherwise, return a column
vector. The inputs may also be cell arrays of strings.
If the optional input "rows" is given then return rows that are in either a or b. The
inputs must be 2-D matrices to use this option.
The optional outputs ia and ib are index vectors such that a(ia) and b(ib) are
disjoint sets whose union is c.
See also: [unique], page 685, [intersect], page 685, [setdiff], page 686, [setxor],
page 686, [ismember], page 686.

c = setdiff (a, b)
c = setdiff (a, b, "rows")
[c, ia] = setdiff ( . . . )
Return the unique elements in a that are not in b sorted in ascending order.
If a is a row vector return a column vector; Otherwise, return a column vector. The
inputs may also be cell arrays of strings.
If the optional input "rows" is given then return the rows in a that are not in b. The
inputs must be 2-D matrices to use this option.
If requested, return the index vector ia such that c = a(ia).
See also: [unique], page 685, [union], page 686, [intersect], page 685, [setxor], page 686,
[ismember], page 686.

c = setxor (a, b)
c = setxor (a, b, "rows")
[c, ia, ib] = setxor ( . . . )
Return the unique elements exclusive to sets a or b sorted in ascending order.
If a and b are both row vectors then return a row vector; Otherwise, return a column
vector. The inputs may also be cell arrays of strings.
If the optional input "rows" is given then return the rows exclusive to sets a and b.
The inputs must be 2-D matrices to use this option.
If requested, return index vectors ia and ib such that a(ia) and b(ib) are disjoint
sets whose union is c.
See also: [unique], page 685, [union], page 686, [intersect], page 685, [setdiff], page 686,
[ismember], page 686.

tf = ismember (a, s)
tf = ismember (a, s, "rows")
[tf, s_idx] = ismember ( . . . )
Return a logical matrix tf with the same shape as a which is true (1) if the element
in a is found in s and false (0) if it is not.

Chapter 27: Sets

687

If a second output argument is requested then the index into s of each matching
element is also returned.
a = [3, 10, 1];
s = [0:9];
[tf, s_idx] = ismember (a, s)
⇒ tf = [1, 0, 1]
⇒ s_idx = [4, 0, 2]
The inputs a and s may also be cell arrays.
a = {"abc"};
s = {"abc", "def"};
[tf, s_idx] = ismember (a, s)
⇒ tf = [1, 0]
⇒ s_idx = [1, 0]
If the optional third argument "rows" is given then compare rows in a with rows in
s. The inputs must be 2-D matrices with the same number of columns to use this
option.
a = [1:3; 5:7; 4:6];
s = [0:2; 1:3; 2:4; 3:5; 4:6];
[tf, s_idx] = ismember (a, s, "rows")
⇒ tf = logical ([1; 0; 1])
⇒ s_idx = [2; 0; 5];

See also: [lookup], page 446, [unique], page 685, [union], page 686, [intersect],
page 685, [setdiff], page 686, [setxor], page 686.

powerset (a)
powerset (a, "rows")
Compute the powerset (all subsets) of the set a.
The set a must be a numerical matrix or a cell array of strings. The output will
always be a cell array of either vectors or strings.
With the optional argument "rows", each row of the set a is considered one element
of the set. The input must be a 2-D numeric matrix to use this argument.
See also: [unique], page 685, [union], page 686, [intersect], page 685, [setdiff], page 686,
[setxor], page 686, [ismember], page 686.

689

28 Polynomial Manipulations
In Octave, a polynomial is represented by its coefficients (arranged in descending order).
For example, a vector c of length N + 1 corresponds to the following polynomial of order N
p(x) = c1 xN + . . . + cN x + cN +1 .

28.1 Evaluating Polynomials
The value of a polynomial represented by the vector c can be evaluated at the point x very
easily, as the following example shows:
N = length (c) - 1;
val = dot (x.^(N:-1:0), c);
While the above example shows how easy it is to compute the value of a polynomial, it isn’t
the most stable algorithm. With larger polynomials you should use more elegant algorithms,
such as Horner’s Method, which is exactly what the Octave function polyval does.
In the case where x is a square matrix, the polynomial given by c is still well-defined.
As when x is a scalar the obvious implementation is easily expressed in Octave, but also in
this case more elegant algorithms perform better. The polyvalm function provides such an
algorithm.

y =
y =
[y,
[y,

polyval (p, x)
polyval (p, x, [], mu)
dy] = polyval (p, x, s)
dy] = polyval (p, x, s, mu)
Evaluate the polynomial p at the specified values of x.
If x is a vector or matrix, the polynomial is evaluated for each of the elements of x.
When mu is present, evaluate the polynomial for (x-mu(1))/mu(2).
In addition to evaluating the polynomial, the second output represents the prediction
interval, y +/- dy, which contains at least 50% of the future predictions. To calculate
the prediction interval, the structured variable s, originating from polyfit, must be
supplied.
See also: [polyvalm], page 689, [polyaffine], page 695, [polyfit], page 695, [roots],
page 690, [poly], page 704.

polyvalm (c, x)
Evaluate a polynomial in the matrix sense.
polyvalm (c, x) will evaluate the polynomial in the matrix sense, i.e., matrix multiplication is used instead of element by element multiplication as used in polyval.
The argument x must be a square matrix.
See also: [polyval], page 689, [roots], page 690, [poly], page 704.

690

GNU Octave

28.2 Finding Roots
Octave can find the roots of a given polynomial. This is done by computing the companion
matrix of the polynomial (see the compan function for a definition), and then finding its
eigenvalues.

roots (c)
Compute the roots of the polynomial c.
For a vector c with N components, return the roots of the polynomial
c1 xN −1 + · · · + cN −1 x + cN .
As an example, the following code finds the roots of the quadratic polynomial
p(x) = x2 − 5.
c = [1, 0, -5];
roots (c)
⇒ 2.2361
⇒ -2.2361
√
Note that the true result is ± 5 which is roughly ±2.2361.

See also: [poly], page 704, [compan], page 690, [fzero], page 551.

z = polyeig (C0, C1, . . . , Cl)
[v, z] = polyeig (C0, C1, . . . , Cl)
Solve the polynomial eigenvalue problem of degree l.
Given an n*n matrix polynomial
C(s) = C0 + C1 s + ... + Cl s^l
polyeig solves the eigenvalue problem
(C0 + C1 + ... + Cl)v = 0.
Note that the eigenvalues z are the zeros of the matrix polynomial. z is a row vector
with n*l elements. v is a matrix (n x n*l) with columns that correspond to the
eigenvectors.
See also: [eig], page 509, [eigs], page 590, [compan], page 690.

compan (c)
Compute the companion matrix corresponding to polynomial coefficient vector c.
The companion matrix is


−c2 /c1

1


A= 0

..

.
0

−c3 /c1
0
1
..
.
0

···
···
···
..
.
···

−cN /c1
0
0
..
.
1



−cN +1 /c1

0


0
.

..

.
0

The eigenvalues of the companion matrix are equal to the roots of the polynomial.
See also: [roots], page 690, [poly], page 704, [eig], page 509.

Chapter 28: Polynomial Manipulations

691

[multp, idxp] = mpoles (p)
[multp, idxp] = mpoles (p, tol)
[multp, idxp] = mpoles (p, tol, reorder)
Identify unique poles in p and their associated multiplicity.
The output is ordered from largest pole to smallest pole.
If the relative difference of two poles is less than tol then they are considered to be
multiples. The default value for tol is 0.001.
If the optional parameter reorder is zero, poles are not sorted.
The output multp is a vector specifying the multiplicity of the poles. multp(n) refers
to the multiplicity of the Nth pole p(idxp(n)).
For example:
p = [2 3 1 1 2];
[m, n] = mpoles (p)
⇒ m = [1; 1; 2; 1; 2]
⇒ n = [2; 5; 1; 4; 3]
⇒ p(n) = [3, 2, 2, 1, 1]
See also: [residue], page 693, [poly], page 704, [roots], page 690, [conv], page 691,
[deconv], page 692.

28.3 Products of Polynomials
conv (a, b)
conv (a, b, shape)
Convolve two vectors a and b.
The output convolution is a vector with length equal to length (a) + length (b)
- 1. When a and b are the coefficient vectors of two polynomials, the convolution
represents the coefficient vector of the product polynomial.
The optional shape argument may be
shape = "full"
Return the full convolution. (default)
shape = "same"
Return the central part of the convolution with the same size as a.
See also: [deconv], page 692, [conv2], page 692, [convn], page 691, [fftconv], page 736.

C = convn (A, B)
C = convn (A, B, shape)
Return the n-D convolution of A and B.
The size of the result is determined by the optional shape argument which takes the
following values
shape = "full"
Return the full convolution. (default)

692

GNU Octave

shape = "same"
Return central part of the convolution with the same size as A. The
central part of the convolution begins at the indices floor ([size(B)/2]
+ 1).
shape = "valid"
Return only the parts which do not include zero-padded edges. The size
of the result is max (size (A) - size (B) + 1, 0).
See also: [conv2], page 692, [conv], page 691.

deconv (y, a)
Deconvolve two vectors.
[b, r] = deconv (y, a) solves for b and r such that y = conv (a, b) + r.
If y and a are polynomial coefficient vectors, b will contain the coefficients of the
polynomial quotient and r will be a remainder polynomial of lowest order.
See also: [conv], page 691, [residue], page 693.

conv2 (A, B)
conv2 (v1, v2, m)
conv2 ( . . . , shape)
Return the 2-D convolution of A and B.
The size of the result is determined by the optional shape argument which takes the
following values
shape = "full"
Return the full convolution. (default)
shape = "same"
Return the central part of the convolution with the same size as A. The
central part of the convolution begins at the indices floor ([size(B)/2]
+ 1).
shape = "valid"
Return only the parts which do not include zero-padded edges. The size
of the result is max (size (A) - size (B) + 1, 0).
When the third argument is a matrix, return the convolution of the matrix m by the
vector v1 in the column direction and by the vector v2 in the row direction.
See also: [conv], page 691, [convn], page 691.

q = polygcd (b, a)
q = polygcd (b, a, tol)
Find the greatest common divisor of two polynomials.
This is equivalent to the polynomial found by multiplying together all the common
roots. Together with deconv, you can reduce a ratio of two polynomials.
The tolerance tol defaults to sqrt (eps).
Caution: This is a numerically unstable algorithm and should not be used on large
polynomials.

Chapter 28: Polynomial Manipulations

693

Example code:
polygcd (poly (1:8), poly (3:12)) - poly (3:8)
⇒ [ 0, 0, 0, 0, 0, 0, 0 ]
deconv (poly (1:8), polygcd (poly (1:8), poly (3:12))) - poly (1:2)
⇒ [ 0, 0, 0 ]
See also: [poly], page 704, [roots], page 690, [conv], page 691, [deconv], page 692,
[residue], page 693.

[r, p, k, e] = residue (b, a)
[b, a] = residue (r, p, k)
[b, a] = residue (r, p, k, e)
The first calling form computes the partial fraction expansion for the quotient of the
polynomials, b and a.
The quotient is defined as
M
N
X
X
B(s)
rm
=
+
ki sN −i .
A(s) m=1 (s − pm )em i=1

where M is the number of poles (the length of the r, p, and e), the k vector is a
polynomial of order N − 1 representing the direct contribution, and the e vector
specifies the multiplicity of the m-th residue’s pole.
For example,
b = [1, 1, 1];
a = [1, -5, 8, -4];
[r, p, k, e] = residue (b, a)
⇒ r = [-2; 7; 3]
⇒ p = [2; 2; 1]
⇒ k = [](0x0)
⇒ e = [1; 2; 1]
which represents the following partial fraction expansion
−2
7
3
s2 + s + 1
=
+
+
3
2
2
s − 5s + 8s − 4
s − 2 (s − 2)
s−1
The second calling form performs the inverse operation and computes the reconstituted quotient of polynomials, b(s)/a(s), from the partial fraction expansion; represented by the residues, poles, and a direct polynomial specified by r, p and k, and the
pole multiplicity e.
If the multiplicity, e, is not explicitly specified the multiplicity is determined by the
function mpoles.
For example:

694

GNU Octave

r =
p =
k =
[b,

[-2; 7; 3];
[2; 2; 1];
[1, 0];
a] = residue (r, p, k)
⇒ b = [1, -5, 9, -3, 1]
⇒ a = [1, -5, 8, -4]

where mpoles is used to determine e = [1; 2; 1]
Alternatively the multiplicity may be defined explicitly, for example,
r = [7; 3; -2];
p = [2; 1; 2];
k = [1, 0];
e = [2; 1; 1];
[b, a] = residue (r, p, k, e)
⇒ b = [1, -5, 9, -3, 1]
⇒ a = [1, -5, 8, -4]
which represents the following partial fraction expansion
−2
7
3
s4 − 5s3 + 9s2 − 3s + 1
+
+
+
s
=
s − 2 (s − 2)2 s − 1
s3 − 5s2 + 8s − 4
See also: [mpoles], page 691, [poly], page 704, [roots], page 690, [conv], page 691,
[deconv], page 692.

28.4 Derivatives / Integrals / Transforms
Octave comes with functions for computing the derivative and the integral of a polynomial.
The functions polyder and polyint both return new polynomials describing the result. As
an example we’ll compute the definite integral of p(x) = x2 + 1 from 0 to 3.
c = [1, 0, 1];
integral = polyint (c);
area = polyval (integral, 3) - polyval (integral, 0)
⇒ 12

polyder (p)
[k] = polyder (a, b)
[q, d] = polyder (b, a)
Return the coefficients of the derivative of the polynomial whose coefficients are given
by the vector p.
If a pair of polynomials is given, return the derivative of the product a ∗ b.
If two inputs and two outputs are given, return the derivative of the polynomial
quotient b/a. The quotient numerator is in q and the denominator in d.
See also: [polyint], page 694, [polyval], page 689, [polyreduce], page 705.

polyint (p)
polyint (p, k)
Return the coefficients of the integral of the polynomial whose coefficients are represented by the vector p.

Chapter 28: Polynomial Manipulations

695

The variable k is the constant of integration, which by default is set to zero.
See also: [polyder], page 694, [polyval], page 689.

polyaffine (f, mu)
Return the coefficients of the polynomial vector f after an affine transformation.
If f is the vector representing the polynomial f(x), then g = polyaffine (f, mu) is
the vector representing:
g(x) = f( (x - mu(1)) / mu(2) )
See also: [polyval], page 689, [polyfit], page 695.

28.5 Polynomial Interpolation
Octave comes with good support for various kinds of interpolation, most of which are described in Chapter 29 [Interpolation], page 707. One simple alternative to the functions
described in the aforementioned chapter, is to fit a single polynomial, or a piecewise polynomial (spline) to some given data points. To avoid a highly fluctuating polynomial, one
most often wants to fit a low-order polynomial to data. This usually means that it is necessary to fit the polynomial in a least-squares sense, which just is what the polyfit function
does.

p = polyfit (x, y, n)
[p, s] = polyfit (x, y, n)
[p, s, mu] = polyfit (x, y, n)
Return the coefficients of a polynomial p(x) of degree n that minimizes the leastsquares-error of the fit to the points [x, y].
If n is a logical vector, it is used as a mask to selectively force the corresponding
polynomial coefficients to be used or ignored.
The polynomial coefficients are returned in a row vector.
The optional output s is a structure containing the following fields:
‘R’

Triangular factor R from the QR decomposition.

‘X’

The Vandermonde matrix used to compute the polynomial coefficients.

‘C’

The unscaled covariance matrix, formally equal to the inverse of x’*x,
but computed in a way minimizing roundoff error propagation.

‘df’

The degrees of freedom.

‘normr’

The norm of the residuals.

‘yf’

The values of the polynomial for each value of x.

The second output may be used by polyval to calculate the statistical error limits of
the predicted values. In particular, the standard deviation of p coefficients is given
by
sqrt (diag (s.C)/s.df)*s.normr.
When the third output, mu, is present the coefficients, p, are associated with a polynomial in

696

GNU Octave

xhat = (x - mu(1)) / mu(2)
where mu(1) = mean (x), and mu(2) = std (x).
This linear transformation of x improves the numerical stability of the fit.
See also: [polyval], page 689, [polyaffine], page 695, [roots], page 690, [vander],
page 472, [zscore], page 656.
In situations where a single polynomial isn’t good enough, a solution is to use several
polynomials pieced together. The function splinefit fits a piecewise polynomial (spline)
to a set of data.

pp
pp
pp
pp
pp
pp
pp

=
=
=
=
=
=
=

(x, y, breaks)
(x, y, p)
( . . . , "periodic", periodic)
( . . . , "robust", robust)
( . . . , "beta", beta)
( . . . , "order", order)
( . . . , "constraints", constraints)
Fit a piecewise cubic spline with breaks (knots) breaks to the noisy data, x and y.

splinefit
splinefit
splinefit
splinefit
splinefit
splinefit
splinefit

x is a vector, and y is a vector or N-D array. If y is an N-D array, then x(j) is matched
to y(:,. . . ,:,j).
p is a positive integer defining the number of intervals along x, and p+1 is the number
of breaks. The number of points in each interval differ by no more than 1.
The optional property periodic is a logical value which specifies whether a periodic
boundary condition is applied to the spline. The length of the period is max (breaks)
- min (breaks). The default value is false.
The optional property robust is a logical value which specifies if robust fitting is to be
applied to reduce the influence of outlying data points. Three iterations of weighted
least squares are performed. Weights are computed from previous residuals. The
sensitivity of outlier identification is controlled by the property beta. The value of
beta is restricted to the range, 0 < beta < 1. The default value is beta = 1/2. Values
close to 0 give all data equal weighting. Increasing values of beta reduce the influence
of outlying data. Values close to unity may cause instability or rank deficiency.
The fitted spline is returned as a piecewise polynomial, pp, and may be evaluated
using ppval.
The splines are constructed of polynomials with degree order. The default is a cubic,
order=3. A spline with P pieces has P+order degrees of freedom. With periodic
boundary conditions the degrees of freedom are reduced to P.
The optional property, constaints, is a structure specifying linear constraints on the
fit. The structure has three fields, "xc", "yc", and "cc".
"xc"

Vector of the x-locations of the constraints.

"yc"

Constraining values at the locations xc. The default is an array of zeros.

"cc"

Coefficients (matrix). The default is an array of ones. The number of
rows is limited to the order of the piecewise polynomials, order.

Chapter 28: Polynomial Manipulations

697

Constraints are linear combinations of derivatives of order 0 to order-1 according to

cc(1, j) · y(xc(j)) + cc(2, j) · y0(xc(j)) + ... = yc(:, . . . , :, j).

See also: [interp1], page 707, [unmkpp], page 703, [ppval], page 704, [spline], page 711,
[pchip], page 742, [ppder], page 704, [ppint], page 704, [ppjumps], page 704.

The number of breaks (or knots) used to construct the piecewise polynomial is a significant factor in suppressing the noise present in the input data, x and y. This is demonstrated
by the example below.

x = 2 * pi * rand (1, 200);
y = sin (x) + sin (2 * x) + 0.2 * randn (size (x));
## Uniform breaks
breaks = linspace (0, 2 * pi, 41); % 41 breaks, 40 pieces
pp1 = splinefit (x, y, breaks);
## Breaks interpolated from data
pp2 = splinefit (x, y, 10); % 11 breaks, 10 pieces
## Plot
xx = linspace (0, 2 * pi, 400);
y1 = ppval (pp1, xx);
y2 = ppval (pp2, xx);
plot (x, y, ".", xx, [y1; y2])
axis tight
ylim auto
legend ({"data", "41 breaks, 40 pieces", "11 breaks, 10 pieces"})

The result of which can be seen in Figure 28.1.

698

GNU Octave

data
41 breaks, 40 pieces
11 breaks, 10 pieces

2

1

0

-1

-2

0

1

2

3

4

5

6

Figure 28.1: Comparison of a fitting a piecewise polynomial with 41 breaks to one with
11 breaks. The fit with the large number of breaks exhibits a fast ripple that is not present
in the underlying function.
The piecewise polynomial fit, provided by splinefit, has continuous derivatives up to
the order-1. For example, a cubic fit has continuous first and second derivatives. This is
demonstrated by the code
## Data (200 points)
x = 2 * pi * rand (1, 200);
y = sin (x) + sin (2 * x) + 0.1 * randn (size (x));
## Piecewise constant
pp1 = splinefit (x, y, 8, "order", 0);
## Piecewise linear
pp2 = splinefit (x, y, 8, "order", 1);
## Piecewise quadratic
pp3 = splinefit (x, y, 8, "order", 2);
## Piecewise cubic
pp4 = splinefit (x, y, 8, "order", 3);
## Piecewise quartic
pp5 = splinefit (x, y, 8, "order", 4);
## Plot
xx = linspace (0, 2 * pi, 400);
y1 = ppval (pp1, xx);
y2 = ppval (pp2, xx);
y3 = ppval (pp3, xx);
y4 = ppval (pp4, xx);
y5 = ppval (pp5, xx);
plot (x, y, ".", xx, [y1; y2; y3; y4; y5])

Chapter 28: Polynomial Manipulations

699

axis tight
ylim auto
legend ({"data", "order 0", "order 1", "order 2", "order 3", "order 4"})
The result of which can be seen in Figure 28.2.

data
order 0
order 1
order 2
order 3
order 4

2

1

0

-1

-2

0

1

2

3

4

5

6

Figure 28.2: Comparison of a piecewise constant, linear, quadratic, cubic, and quartic polynomials with 8 breaks to noisy data. The higher order solutions more accurately
represent the underlying function, but come with the expense of computational complexity.
When the underlying function to provide a fit to is periodic, splinefit is able to apply
the boundary conditions needed to manifest a periodic fit. This is demonstrated by the
code below.
## Data (100 points)
x = 2 * pi * [0, (rand (1, 98)), 1];
y = sin (x) - cos (2 * x) + 0.2 * randn (size (x));
## No constraints
pp1 = splinefit (x, y, 10, "order", 5);
## Periodic boundaries
pp2 = splinefit (x, y, 10, "order", 5, "periodic", true);
## Plot
xx = linspace (0, 2 * pi, 400);
y1 = ppval (pp1, xx);
y2 = ppval (pp2, xx);
plot (x, y, ".", xx, [y1; y2])
axis tight
ylim auto
legend ({"data", "no constraints", "periodic"})
The result of which can be seen in Figure 28.3.

700

GNU Octave

3
data
no constraints
periodic
2

1

0

-1

-2

0

1

2

3

4

5

6

Figure 28.3: Comparison of piecewise polynomial fits to a noisy periodic function with,
and without, periodic boundary conditions.
More complex constraints may be added as well. For example, the code below illustrates
a periodic fit with values that have been clamped at the endpoints, and a second periodic
fit which is hinged at the endpoints.
## Data (200 points)
x = 2 * pi * rand (1, 200);
y = sin (2 * x) + 0.1 * randn (size (x));
## Breaks
breaks = linspace (0, 2 * pi, 10);
## Clamped endpoints, y = y’ = 0
xc = [0, 0, 2*pi, 2*pi];
cc = [(eye (2)), (eye (2))];
con = struct ("xc", xc, "cc", cc);
pp1 = splinefit (x, y, breaks, "constraints", con);
## Hinged periodic endpoints, y = 0
con = struct ("xc", 0);
pp2 = splinefit (x, y, breaks, "constraints", con, "periodic", true);
## Plot
xx = linspace (0, 2 * pi, 400);
y1 = ppval (pp1, xx);
y2 = ppval (pp2, xx);
plot (x, y, ".", xx, [y1; y2])
axis tight
ylim auto
legend ({"data", "clamped", "hinged periodic"})
The result of which can be seen in Figure 28.4.

Chapter 28: Polynomial Manipulations

701

1.5
data
clamped
hinged periodic
1

0.5

0

-0.5

-1

-1.5

0

1

2

3

4

5

6

Figure 28.4: Comparison of two periodic piecewise cubic fits to a noisy periodic signal.
One fit has its endpoints clamped and the second has its endpoints hinged.
The splinefit function also provides the convenience of a robust fitting, where the effect
of outlying data is reduced. In the example below, three different fits are provided. Two
with differing levels of outlier suppression and a third illustrating the non-robust solution.
## Data
x = linspace (0, 2*pi, 200);
y = sin (x) + sin (2 * x) + 0.05 * randn (size (x));
## Add outliers
x = [x, linspace(0,2*pi,60)];
y = [y, -ones(1,60)];
## Fit splines with hinged conditions
con = struct ("xc", [0, 2*pi]);
## Robust fitting, beta = 0.25
pp1 = splinefit (x, y, 8, "constraints", con, "beta", 0.25);
## Robust fitting, beta = 0.75
pp2 = splinefit (x, y, 8, "constraints", con, "beta", 0.75);
## No robust fitting
pp3 = splinefit (x, y, 8, "constraints", con);
## Plot
xx = linspace (0, 2*pi, 400);
y1 = ppval (pp1, xx);
y2 = ppval (pp2, xx);
y3 = ppval (pp3, xx);
plot (x, y, ".", xx, [y1; y2; y3])
legend ({"data with outliers","robust, beta = 0.25", ...
"robust, beta = 0.75", "no robust fitting"})

702

GNU Octave

axis tight
ylim auto
The result of which can be seen in Figure 28.5.

2
data with outliers
robust, beta = 0.25
robust, beta = 0.75
no robust fitting

1

0

-1

-2
0

1

2

3

4

5

6

Figure 28.5: Comparison of two different levels of robust fitting (beta = 0.25 and 0.75)
to noisy data combined with outlying data. A conventional fit, without robust fitting (beta
= 0) is also included.
A very specific form of polynomial interpretation is the Padé approximant. For control
systems, a continuous-time delay can be modeled very simply with the approximant.

[num, den] = padecoef (T)
[num, den] = padecoef (T, N)
Compute the N th-order Padé approximant of the continuous-time delay T in transfer
function form.
The Padé approximant of e−sT is defined by the following equation
e−sT ≈

Pn (s)
Qn (s)

where both Pn (s) and Qn (s) are N th -order rational functions defined by the following
expressions
N
X
(2N − k)!N !
(−sT )k
Pn (s) =
(2N
)!k!(N
−
k)!
k=0
Qn (s) = Pn (−s)
The inputs T and N must be non-negative numeric scalars. If N is unspecified it
defaults to 1.

Chapter 28: Polynomial Manipulations

703

The output row vectors num and den contain the numerator and denominator coefficients in descending powers of s. Both are N th-order polynomials.
For example:
t = 0.1;
n = 4;
[num, den] = padecoef (t, n)
⇒ num =
1.0000e-04

-2.0000e-02

1.8000e+00

-8.4000e+01

1.6800e+03

2.0000e-02

1.8000e+00

8.4000e+01

1.6800e+03

⇒ den =
1.0000e-04

The function, ppval, evaluates the piecewise polynomials, created by mkpp or other
means, and unmkpp returns detailed information about the piecewise polynomial.
The following example shows how to combine two linear functions and a quadratic into
one function. Each of these functions is expressed on adjoined intervals.
x = [-2, -1, 1, 2];
p = [ 0, 1, 0;
1, -2, 1;
0, -1, 1 ];
pp = mkpp (x, p);
xi = linspace (-2, 2, 50);
yi = ppval (pp, xi);
plot (xi, yi);

pp = mkpp (breaks, coefs)
pp = mkpp (breaks, coefs, d)
Construct a piecewise polynomial (pp) structure from sample points breaks and coefficients coefs.
breaks must be a vector of strictly increasing values. The number of intervals is given
by ni = length (breaks) - 1.
When m is the polynomial order coefs must be of size: ni-by-(m + 1).
The i-th row of coefs, coefs (i,:), contains the coefficients for the polynomial over
the i-th interval, ordered from highest (m) to lowest (0).
coefs may also be a multi-dimensional array, specifying a vector-valued or array-valued
polynomial. In that case the polynomial order m is defined by the length of the last
dimension of coefs. The size of first dimension(s) are given by the scalar or vector d.
If d is not given it is set to 1. In any case coefs is reshaped to a 2-D matrix of size
[ni*prod(d) m].
See also: [unmkpp], page 703, [ppval], page 704, [spline], page 711, [pchip], page 742,
[ppder], page 704, [ppint], page 704, [ppjumps], page 704.

[x, p, n, k, d] = unmkpp (pp)
Extract the components of a piecewise polynomial structure pp.
The components are:
x

Sample points.

704

GNU Octave

p

Polynomial coefficients for points in sample interval. p (i, :) contains
the coefficients for the polynomial over interval i ordered from highest to
lowest. If d > 1, p (r, i, :) contains the coefficients for the r-th polynomial defined on interval i.

n

Number of polynomial pieces.

k

Order of the polynomial plus 1.

d

Number of polynomials defined for each interval.

See also: [mkpp], page 703, [ppval], page 704, [spline], page 711, [pchip], page 742.

yi = ppval (pp, xi)
Evaluate the piecewise polynomial structure pp at the points xi.
If pp describes a scalar polynomial function, the result is an array of the same shape
as xi. Otherwise, the size of the result is [pp.dim, length(xi)] if xi is a vector, or
[pp.dim, size(xi)] if it is a multi-dimensional array.
See also: [mkpp], page 703, [unmkpp], page 703, [spline], page 711, [pchip], page 742.

ppd = ppder (pp)
ppd = ppder (pp, m)
Compute the piecewise m-th derivative of a piecewise polynomial struct pp.
If m is omitted the first derivative is calculated.
See also: [mkpp], page 703, [ppval], page 704, [ppint], page 704.

ppi = ppint (pp)
ppi = ppint (pp, c)
Compute the integral of the piecewise polynomial struct pp.
c, if given, is the constant of integration.
See also: [mkpp], page 703, [ppval], page 704, [ppder], page 704.

jumps = ppjumps (pp)
Evaluate the boundary jumps of a piecewise polynomial.
If there are n intervals, and the dimensionality of pp is d, the resulting array has
dimensions [d, n-1].
See also: [mkpp], page 703.

28.6 Miscellaneous Functions
poly (A)
poly (x)
If A is a square N -by-N matrix, poly (A) is the row vector of the coefficients of det
(z * eye (N) - A), the characteristic polynomial of A.
For example, the following code finds the eigenvalues of A which are the roots of poly
(A).
roots (poly (eye (3)))
⇒ 1.00001 + 0.00001i
1.00001 - 0.00001i
0.99999 + 0.00000i

Chapter 28: Polynomial Manipulations

705

In fact, all three eigenvalues are exactly 1 which emphasizes that for numerical performance the eig function should be used to compute eigenvalues.
If x is a vector, poly (x) is a vector of the coefficients of the polynomial whose roots
are the elements of x. That is, if c is a polynomial, then the elements of d = roots
(poly (c)) are contained in c. The vectors c and d are not identical, however, due
to sorting and numerical errors.
See also: [roots], page 690, [eig], page 509.

polyout (c)
polyout (c, x)
str = polyout ( . . . )
Display a formatted version of the polynomial c.
The formatted polynomial
c(x) = c1 xn + . . . + cn x + cn+1
is returned as a string or written to the screen if nargout is zero.
The second argument x specifies the variable name to use for each term and defaults
to the string "s".
See also: [polyreduce], page 705.

polyreduce (c)
Reduce a polynomial coefficient vector to a minimum number of terms by stripping
off any leading zeros.
See also: [polyout], page 705.

707

29 Interpolation
29.1 One-dimensional Interpolation
Octave supports several methods for one-dimensional interpolation, most of which are described in this section. Section 28.5 [Polynomial Interpolation], page 695, and Section 30.4
[Interpolation on Scattered Data], page 730, describe additional methods.

yi
yi
yi
yi
yi
yi
pp

=
=
=
=
=
=
=

(x, y, xi)
(y, xi)
( . . . , method)
( . . . , extrap)
( . . . , "left")
( . . . , "right")
( . . . , "pp")
One-dimensional interpolation.
Interpolate input data to determine the value of yi at the points xi. If not specified, x
is taken to be the indices of y (1:length (y)). If y is a matrix or an N-dimensional
array, the interpolation is performed on each column of y.
The interpolation method is one of:

interp1
interp1
interp1
interp1
interp1
interp1
interp1

"nearest"
Return the nearest neighbor.
"previous"
Return the previous neighbor.
"next"

Return the next neighbor.

"linear" (default)
Linear interpolation from nearest neighbors.
"pchip"

Piecewise cubic Hermite interpolating polynomial—shape-preserving interpolation with smooth first derivative.

"cubic"

Cubic interpolation (same as "pchip").

"spline"

Cubic spline interpolation—smooth first and second derivatives throughout the curve.

Adding ’*’ to the start of any method above forces interp1 to assume that x is
uniformly spaced, and only x(1) and x(2) are referenced. This is usually faster, and
is never slower. The default method is "linear".
If extrap is the string "extrap", then extrapolate values beyond the endpoints using
the current method. If extrap is a number, then replace values beyond the endpoints
with that number. When unspecified, extrap defaults to NA.
If the string argument "pp" is specified, then xi should not be supplied and interp1
returns a piecewise polynomial object. This object can later be used with ppval to
evaluate the interpolation. There is an equivalence, such that ppval (interp1 (x,
y, method, "pp"), xi) == interp1 (x, y, xi, method, "extrap").

708

GNU Octave

Duplicate points in x specify a discontinuous interpolant. There may be at most 2
consecutive points with the same value. If x is increasing, the default discontinuous interpolant is right-continuous. If x is decreasing, the default discontinuous interpolant
is left-continuous. The continuity condition of the interpolant may be specified by
using the options "left" or "right" to select a left-continuous or right-continuous
interpolant, respectively. Discontinuous interpolation is only allowed for "nearest"
and "linear" methods; in all other cases, the x-values must be unique.
An example of the use of interp1 is
xf = [0:0.05:10];
yf = sin (2*pi*xf/5);
xp = [0:10];
yp = sin (2*pi*xp/5);
lin = interp1 (xp, yp, xf);
near = interp1 (xp, yp, xf, "nearest");
pch = interp1 (xp, yp, xf, "pchip");
spl = interp1 (xp, yp, xf, "spline");
plot (xf,yf,"r", xf,near,"g", xf,lin,"b", xf,pch,"c", xf,spl,"m",
xp,yp,"r*");
legend ("original", "nearest", "linear", "pchip", "spline");
See also: [pchip], page 742, [spline], page 711, [interpft], page 709, [interp2], page 711,
[interp3], page 712, [interpn], page 713.
There are some important differences between the various interpolation methods. The
"spline" method enforces that both the first and second derivatives of the interpolated
values have a continuous derivative, whereas the other methods do not. This means that the
results of the "spline" method are generally smoother. If the function to be interpolated
is in fact smooth, then "spline" will give excellent results. However, if the function to be
evaluated is in some manner discontinuous, then "pchip" interpolation might give better
results.
This can be demonstrated by the code
t = -2:2;
dt = 1;
ti =-2:0.025:2;
dti = 0.025;
y = sign (t);
ys = interp1 (t,y,ti,"spline");
yp = interp1 (t,y,ti,"pchip");
ddys = diff (diff (ys)./dti) ./ dti;
ddyp = diff (diff (yp)./dti) ./ dti;
figure (1);
plot (ti,ys,"r-", ti,yp,"g-");
legend ("spline", "pchip", 4);
figure (2);
plot (ti,ddys,"r+", ti,ddyp,"g*");
legend ("spline", "pchip");
The result of which can be seen in Figure 29.1 and Figure 29.2.

Chapter 29: Interpolation

709

1.5

1

0.5

0

-0.5

-1

spline
pchip
-1.5
-2

-1

0

1

2

Figure 29.1: Comparison of "pchip" and "spline" interpolation methods for a step
function

4
spline
pchip

2

0

-2

-4
-2

-1

0

1

2

Figure 29.2: Comparison of the second derivative of the "pchip" and "spline" interpolation methods for a step function

Fourier interpolation, is a resampling technique where a signal is converted to the frequency domain, padded with zeros and then reconverted to the time domain.

710

GNU Octave

interpft (x, n)
interpft (x, n, dim)
Fourier interpolation.
If x is a vector then x is resampled with n points. The data in x is assumed to be
equispaced. If x is a matrix or an N-dimensional array, the interpolation is performed
on each column of x.
If dim is specified, then interpolate along the dimension dim.
interpft assumes that the interpolated function is periodic, and so assumptions are
made about the endpoints of the interpolation.
See also: [interp1], page 707.
There are two significant limitations on Fourier interpolation. First, the function signal is
assumed to be periodic, and so non-periodic signals will be poorly represented at the edges.
Second, both the signal and its interpolation are required to be sampled at equispaced
points. An example of the use of interpft is
t = 0 : 0.3 : pi; dt = t(2)-t(1);
n = length (t); k = 100;
ti = t(1) + [0 : k-1]*dt*n/k;
y = sin (4*t + 0.3) .* cos (3*t - 0.1);
yp = sin (4*ti + 0.3) .* cos (3*ti - 0.1);
plot (ti, yp, "g", ti, interp1 (t, y, ti, "spline"), "b", ...
ti, interpft (y, k), "c", t, y, "r+");
legend ("sin(4t+0.3)cos(3t-0.1)", "spline", "interpft", "data");
which demonstrates the poor behavior of Fourier interpolation for non-periodic functions,
as can be seen in Figure 29.3.

1.5
sin(4t+0.3)cos(3t-0.1)
spline
interpft
data
1

0.5

0

-0.5

-1
0

0.5

1

1.5

2

2.5

3

3.5

Figure 29.3: Comparison of interp1 and interpft for non-periodic data

Chapter 29: Interpolation

711

In addition, the support functions spline and lookup that underlie the interp1 function
can be called directly.

pp = spline (x, y)
yi = spline (x, y, xi)
Return the cubic spline interpolant of points x and y.
When called with two arguments, return the piecewise polynomial pp that may be
used with ppval to evaluate the polynomial at specific points.
When called with a third input argument, spline evaluates the spline at the points
xi. The third calling form spline (x, y, xi) is equivalent to ppval (spline (x,
y), xi).
The variable x must be a vector of length n.
y can be either a vector or array. If y is a vector it must have a length of either n or
n + 2. If the length of y is n, then the "not-a-knot" end condition is used. If the
length of y is n + 2, then the first and last values of the vector y are the values of the
first derivative of the cubic spline at the endpoints.
If y is an array, then the size of y must have the form
[s1 , s2 , · · · , sk , n]
or
[s1 , s2 , · · · , sk , n + 2].
The array is reshaped internally to a matrix where the leading dimension is given by
s1 s2 · · · sk
and each row of this matrix is then treated separately. Note that this is exactly the
opposite of interp1 but is done for matlab compatibility.
See also: [pchip], page 742, [ppval], page 704, [mkpp], page 703, [unmkpp], page 703.

29.2 Multi-dimensional Interpolation
There are three multi-dimensional interpolation functions in Octave, with similar capabilities. Methods using Delaunay tessellation are described in Section 30.4 [Interpolation on
Scattered Data], page 730.

zi
zi
zi
zi
zi
zi

=
=
=
=
=
=

(x, y, z, xi, yi)
(z, xi, yi)
(z, n)
(z)
( . . . , method)
( . . . , method, extrap)
Two-dimensional interpolation.
Interpolate reference data x, y, z to determine zi at the coordinates xi, yi. The
reference data x, y can be matrices, as returned by meshgrid, in which case the sizes
of x, y, and z must be equal. If x, y are vectors describing a grid then length (x)

interp2
interp2
interp2
interp2
interp2
interp2

712

GNU Octave

== columns (z) and length (y) == rows (z). In either case the input data must be
strictly monotonic.
If called without x, y, and just a single reference data matrix z, the 2-D region x =
1:columns (z), y = 1:rows (z) is assumed. This saves memory if the grid is regular
and the distance between points is not important.
If called with a single reference data matrix z and a refinement value n, then perform
interpolation over a grid where each original interval has been recursively subdivided
n times. This results in 2^n-1 additional points for every interval in the original grid.
If n is omitted a value of 1 is used. As an example, the interval [0,1] with n==2 results
in a refined interval with points at [0, 1/4, 1/2, 3/4, 1].
The interpolation method is one of:
"nearest"
Return the nearest neighbor.
"linear" (default)
Linear interpolation from nearest neighbors.
"pchip"

Piecewise cubic Hermite interpolating polynomial—shape-preserving interpolation with smooth first derivative.

"cubic"

Cubic interpolation (same as "pchip").

"spline"

Cubic spline interpolation—smooth first and second derivatives throughout the curve.

extrap is a scalar number. It replaces values beyond the endpoints with extrap. Note
that if extrap is used, method must be specified as well. If extrap is omitted and
the method is "spline", then the extrapolated values of the "spline" are used.
Otherwise the default extrap value for any other method is "NA".
See also: [interp1], page 707, [interp3], page 712, [interpn], page 713, [meshgrid],
page 339.

vi
vi
vi
vi
vi
vi

=
=
=
=
=
=

(x, y, z, v, xi, yi, zi)
(v, xi, yi, zi)
(v, n)
(v)
( . . . , method)
( . . . , method, extrapval)
Three-dimensional interpolation.

interp3
interp3
interp3
interp3
interp3
interp3

Interpolate reference data x, y, z, v to determine vi at the coordinates xi, yi, zi. The
reference data x, y, z can be matrices, as returned by meshgrid, in which case the
sizes of x, y, z, and v must be equal. If x, y, z are vectors describing a cubic grid then
length (x) == columns (v), length (y) == rows (v), and length (z) == size (v,
3). In either case the input data must be strictly monotonic.
If called without x, y, z, and just a single reference data matrix v, the 3-D region x =
1:columns (v), y = 1:rows (v), z = 1:size (v, 3) is assumed. This saves memory
if the grid is regular and the distance between points is not important.

Chapter 29: Interpolation

713

If called with a single reference data matrix v and a refinement value n, then perform interpolation over a 3-D grid where each original interval has been recursively
subdivided n times. This results in 2^n-1 additional points for every interval in the
original grid. If n is omitted a value of 1 is used. As an example, the interval [0,1]
with n==2 results in a refined interval with points at [0, 1/4, 1/2, 3/4, 1].
The interpolation method is one of:
"nearest"
Return the nearest neighbor.
"linear" (default)
Linear interpolation from nearest neighbors.
"cubic"

Piecewise cubic Hermite interpolating polynomial—shape-preserving interpolation with smooth first derivative (not implemented yet).

"spline"

Cubic spline interpolation—smooth first and second derivatives throughout the curve.

extrapval is a scalar number. It replaces values beyond the endpoints with extrapval.
Note that if extrapval is used, method must be specified as well. If extrapval is
omitted and the method is "spline", then the extrapolated values of the "spline"
are used. Otherwise the default extrapval value for any other method is "NA".
See also: [interp1], page 707, [interp2], page 711, [interpn], page 713, [meshgrid],
page 339.

vi
vi
vi
vi
vi
vi

=
=
=
=
=
=

(x1, x2, . . . , v, y1, y2, . . . )
(v, y1, y2, . . . )
(v, m)
(v)
( . . . , method)
( . . . , method, extrapval)
Perform n-dimensional interpolation, where n is at least two.

interpn
interpn
interpn
interpn
interpn
interpn

Each element of the n-dimensional array v represents a value at a location given
by the parameters x1, x2, . . . , xn. The parameters x1, x2, . . . , xn are either ndimensional arrays of the same size as the array v in the "ndgrid" format or vectors.
The parameters y1, etc. respect a similar format to x1, etc., and they represent the
points at which the array vi is interpolated.
If x1, . . . , xn are omitted, they are assumed to be x1 = 1 : size (v, 1), etc. If
m is specified, then the interpolation adds a point half way between each of the
interpolation points. This process is performed m times. If only v is specified, then
m is assumed to be 1.
The interpolation method is one of:
"nearest"
Return the nearest neighbor.
"linear" (default)
Linear interpolation from nearest neighbors.

714

GNU Octave

"pchip"

Piecewise cubic Hermite interpolating polynomial—shape-preserving interpolation with smooth first derivative (not implemented yet).

"cubic"

Cubic interpolation (same as "pchip" [not implemented yet]).

"spline"

Cubic spline interpolation—smooth first and second derivatives throughout the curve.

The default method is "linear".
extrapval is a scalar number. It replaces values beyond the endpoints with extrapval.
Note that if extrapval is used, method must be specified as well. If extrapval is
omitted and the method is "spline", then the extrapolated values of the "spline"
are used. Otherwise the default extrapval value for any other method is "NA".
See also: [interp1], page 707, [interp2], page 711, [interp3], page 712, [spline], page 711,
[ndgrid], page 339.

A significant difference between interpn and the other two multi-dimensional interpolation functions is the fashion in which the dimensions are treated. For interp2 and interp3,
the y-axis is considered to be the columns of the matrix, whereas the x-axis corresponds to
the rows of the array. As Octave indexes arrays in column major order, the first dimension
of any array is the columns, and so interpn effectively reverses the ’x’ and ’y’ dimensions.
Consider the example,
x = y = z = -1:1;
f = @(x,y,z) x.^2 - y - z.^2;
[xx, yy, zz] = meshgrid (x, y, z);
v = f (xx,yy,zz);
xi = yi = zi = -1:0.1:1;
[xxi, yyi, zzi] = meshgrid (xi, yi, zi);
vi = interp3 (x, y, z, v, xxi, yyi, zzi, "spline");
[xxi, yyi, zzi] = ndgrid (xi, yi, zi);
vi2 = interpn (x, y, z, v, xxi, yyi, zzi, "spline");
mesh (zi, yi, squeeze (vi2(1,:,:)));
where vi and vi2 are identical. The reversal of the dimensions is treated in the meshgrid
and ndgrid functions respectively. The result of this code can be seen in Figure 29.4.

715

2

1.5

1

0.5

0
1
0.5

1
0.5

0
0
-0.5

-0.5
-1

-1

Figure 29.4: Demonstration of the use of interpn

717

30 Geometry
Much of the geometry code in Octave is based on the Qhull library1 . Some of the documentation for Qhull, particularly for the options that can be passed to delaunay, voronoi
and convhull, etc., is relevant to Octave users.

30.1 Delaunay Triangulation
The Delaunay triangulation is constructed from a set of circum-circles. These circum-circles
are chosen so that there are at least three of the points in the set to triangulation on the
circumference of the circum-circle. None of the points in the set of points falls within any
of the circum-circles.
In general there are only three points on the circumference of any circum-circle. However,
in some cases, and in particular for the case of a regular grid, 4 or more points can be on a
single circum-circle. In this case the Delaunay triangulation is not unique.

tri = delaunay (x, y)
tetr = delaunay (x, y, z)
tri = delaunay (x)
tri = delaunay ( . . . , options)
Compute the Delaunay triangulation for a 2-D or 3-D set of points.
For 2-D sets, the return value tri is a set of triangles which satisfies the Delaunay
circum-circle criterion, i.e., no data point from [x, y] is within the circum-circle of the
defining triangle. The set of triangles tri is a matrix of size [n, 3]. Each row defines
a triangle and the three columns are the three vertices of the triangle. The value
of tri(i,j) is an index into x and y for the location of the j-th vertex of the i-th
triangle.
For 3-D sets, the return value tetr is a set of tetrahedrons which satisfies the Delaunay
circum-circle criterion, i.e., no data point from [x, y, z] is within the circum-circle of
the defining tetrahedron. The set of tetrahedrons is a matrix of size [n, 4]. Each row
defines a tetrahedron and the four columns are the four vertices of the tetrahedron.
The value of tetr(i,j) is an index into x, y, z for the location of the j-th vertex of
the i-th tetrahedron.
The input x may also be a matrix with two or three columns where the first column
contains x-data, the second y-data, and the optional third column contains z-data.
The optional last argument, which must be a string or cell array of strings, contains
options passed to the underlying qhull command. See the documentation for the
Qhull library for details http://www.qhull.org/html/qh-quick.htm#options.
The default options are {"Qt", "Qbb", "Qc", "Qz"}.
If options is not present or [] then the default arguments are used. Otherwise,
options replaces the default argument list. To append user options to the defaults it
is necessary to repeat the default arguments in options. Use a null string to pass no
arguments.
1

Barber, C.B., Dobkin, D.P., and Huhdanpaa, H.T., The Quickhull Algorithm for Convex Hulls, ACM
Trans. on Mathematical Software, 22(4):469–483, Dec 1996, http://www.qhull.org

718

GNU Octave

x = rand (1, 10);
y = rand (1, 10);
tri = delaunay (x, y);
triplot (tri, x, y);
hold on;
plot (x, y, "r*");
axis ([0,1,0,1]);
See also: [delaunayn], page 718, [convhull], page 728, [voronoi], page 724, [triplot],
page 719, [trimesh], page 719, [tetramesh], page 721, [trisurf], page 720.
For 3-D inputs delaunay returns a set of tetrahedra that satisfy the Delaunay circumcircle criteria. Similarly, delaunayn returns the N-dimensional simplex satisfying the Delaunay circum-circle criteria. The N-dimensional extension of a triangulation is called a
tessellation.

T = delaunayn (pts)
T = delaunayn (pts, options)
Compute the Delaunay triangulation for an N-dimensional set of points.
The Delaunay triangulation is a tessellation of the convex hull of a set of points such
that no N-sphere defined by the N-triangles contains any other points from the set.
The input matrix pts of size [n, dim] contains n points in a space of dimension dim.
The return matrix T has size [m, dim+1]. Each row of T contains a set of indices
back into the original set of points pts which describes a simplex of dimension dim.
For example, a 2-D simplex is a triangle and 3-D simplex is a tetrahedron.
An optional second argument, which must be a string or cell array of strings, contains
options passed to the underlying qhull command. See the documentation for the
Qhull library for details http://www.qhull.org/html/qh-quick.htm#options.
The default options depend on the dimension of the input:
• 2-D and 3-D: options = {"Qt", "Qbb", "Qc", "Qz"}
• 4-D and higher: options = {"Qt", "Qbb", "Qc", "Qx"}

If options is not present or [] then the default arguments are used. Otherwise,
options replaces the default argument list. To append user options to the defaults it
is necessary to repeat the default arguments in options. Use a null string to pass no
arguments.
See also: [delaunay], page 717, [convhulln], page 728, [voronoin], page 725, [trimesh],
page 719, [tetramesh], page 721.

An example of a Delaunay triangulation of a set of points is
rand ("state", 2);
x = rand (10, 1);
y = rand (10, 1);
T = delaunay (x, y);
X = [ x(T(:,1)); x(T(:,2)); x(T(:,3)); x(T(:,1)) ];
Y = [ y(T(:,1)); y(T(:,2)); y(T(:,3)); y(T(:,1)) ];
axis ([0, 1, 0, 1]);
plot (X, Y, "b", x, y, "r*");

Chapter 30: Geometry

719

The result of which can be seen in Figure 30.1.

1

0.8

0.6

0.4

0.2

0
0

0.2

0.4

0.6

0.8

1

Figure 30.1: Delaunay triangulation of a random set of points

30.1.1 Plotting the Triangulation
Octave has the functions triplot, trimesh, and trisurf to plot the Delaunay triangulation
of a 2-dimensional set of points. tetramesh will plot the triangulation of a 3-dimensional
set of points.

triplot (tri, x, y)
triplot (tri, x, y, linespec)
h = triplot ( . . . )
Plot a 2-D triangular mesh.
tri is typically the output of a Delaunay triangulation over the grid of x, y. Every
row of tri represents one triangle and contains three indices into [x, y] which are the
vertices of the triangles in the x-y plane.
The linestyle to use for the plot can be defined with the argument linespec of the
same format as the plot command.
The optional return value h is a graphics handle to the created patch object.
See also: [plot], page 288, [trimesh], page 719, [trisurf], page 720, [delaunay],
page 717.

trimesh (tri, x, y, z, c)
trimesh (tri, x, y, z)
trimesh (tri, x, y)
trimesh ( . . . , prop, val, . . . )
h = trimesh ( . . . )
Plot a 3-D triangular wireframe mesh.

720

GNU Octave

In contrast to mesh, which plots a mesh using rectangles, trimesh plots the mesh
using triangles.
tri is typically the output of a Delaunay triangulation over the grid of x, y. Every
row of tri represents one triangle and contains three indices into [x, y] which are the
vertices of the triangles in the x-y plane. z determines the height above the plane of
each vertex. If no z input is given then the triangles are plotted as a 2-D figure.
The color of the trimesh is computed by linearly scaling the z values to fit the range
of the current colormap. Use caxis and/or change the colormap to control the
appearance.
Optionally, the color of the mesh can be specified independently of z by supplying c,
which is a vector for colormap data, or a matrix with three columns for RGB data.
The number of colors specified in c must either equal the number of vertices in z or
the number of triangles in tri.
Any property/value pairs are passed directly to the underlying patch object.
The optional return value h is a graphics handle to the created patch object.
See also: [mesh], page 323, [tetramesh], page 721, [triplot], page 719, [trisurf],
page 720, [delaunay], page 717, [patch], page 379, [hidden], page 325.

trisurf (tri, x, y, z, c)
trisurf (tri, x, y, z)
trisurf ( . . . , prop, val, . . . )
h = trisurf ( . . . )
Plot a 3-D triangular surface.
In contrast to surf, which plots a surface mesh using rectangles, trisurf plots the
mesh using triangles.
tri is typically the output of a Delaunay triangulation over the grid of x, y. Every
row of tri represents one triangle and contains three indices into [x, y] which are the
vertices of the triangles in the x-y plane. z determines the height above the plane of
each vertex.
The color of the trisurf is computed by linearly scaling the z values to fit the range
of the current colormap. Use caxis and/or change the colormap to control the
appearance.
Optionally, the color of the mesh can be specified independently of z by supplying c,
which is a vector for colormap data, or a matrix with three columns for RGB data.
The number of colors specified in c must either equal the number of vertices in z or
the number of triangles in tri. When specifying the color at each vertex the triangle
will be colored according to the color of the first vertex only (see patch documentation
and the "FaceColor" property when set to "flat").
Any property/value pairs are passed directly to the underlying patch object.
The optional return value h is a graphics handle to the created patch object.
See also: [surf], page 325, [triplot], page 719, [trimesh], page 719, [delaunay], page 717,
[patch], page 379, [shading], page 342.

Chapter 30: Geometry

721

tetramesh (T, X)
tetramesh (T, X, C)
tetramesh ( . . . , property, val, . . . )
h = tetramesh ( . . . )
Display the tetrahedrons defined in the m-by-4 matrix T as 3-D patches.
T is typically the output of a Delaunay triangulation of a 3-D set of points. Every row
of T contains four indices into the n-by-3 matrix X of the vertices of a tetrahedron.
Every row in X represents one point in 3-D space.
The vector C specifies the color of each tetrahedron as an index into the current
colormap. The default value is 1:m where m is the number of tetrahedrons; the indices
are scaled to map to the full range of the colormap. If there are more tetrahedrons
than colors in the colormap then the values in C are cyclically repeated.
Calling tetramesh (..., "property", "value", ...) passes all property/value
pairs directly to the patch function as additional arguments.
The optional return value h is a vector of patch handles where each handle represents
one tetrahedron in the order given by T. A typical use case for h is to turn the
respective patch "visible" property "on" or "off".
Type demo tetramesh to see examples on using tetramesh.
See also: [trimesh], page 719, [delaunay], page 717, [delaunayn], page 718, [patch],
page 379.

The difference between triplot, and trimesh or trisurf, is that the former only plots
the 2-dimensional triangulation itself, whereas the second two plot the value of a function
f (x, y). An example of the use of the triplot function is
rand ("state", 2)
x = rand (20, 1);
y = rand (20, 1);
tri = delaunay (x, y);
triplot (tri, x, y);
which plots the Delaunay triangulation of a set of random points in 2-dimensions. The
output of the above can be seen in Figure 30.2.

722

GNU Octave

1

0.8

0.6

0.4

0.2

0
0

0.2

0.4

0.6

0.8

1

Figure 30.2: Delaunay triangulation of a random set of points

30.1.2 Identifying Points in Triangulation
It is often necessary to identify whether a particular point in the N-dimensional space
is within the Delaunay tessellation of a set of points in this N-dimensional space, and if
so which N-simplex contains the point and which point in the tessellation is closest to the
desired point. The functions tsearch and dsearch perform this function in a triangulation,
and tsearchn and dsearchn in an N-dimensional tessellation.
To identify whether a particular point represented by a vector p falls within one of the
simplices of an N-simplex, we can write the Cartesian coordinates of the point in a parametric form with respect to the N-simplex. This parametric form is called the Barycentric
Coordinates of the point. If the points defining the N-simplex are given by N + 1 vectors
t(i,:), then the Barycentric coordinates defining the point p are given by
p = beta * t
where beta contains N + 1 values that together as a vector represent the Barycentric coordinates of the point p. To ensure a unique solution for the values of beta an additional
criteria of
sum (beta) == 1
is imposed, and we can therefore write the above as
p - t(end, :) = beta(1:end-1) * (t(1:end-1, :)
- ones (N, 1) * t(end, :)
Solving for beta we can then write
beta(1:end-1) = (p - t(end, :)) /
(t(1:end-1, :) - ones (N, 1) * t(end, :))
beta(end) = sum (beta(1:end-1))

Chapter 30: Geometry

723

which gives the formula for the conversion of the Cartesian coordinates of the point p to
the Barycentric coordinates beta. An important property of the Barycentric coordinates is
that for all points in the N-simplex
0 <= beta(i) <= 1
Therefore, the test in tsearch and tsearchn essentially only needs to express each point in
terms of the Barycentric coordinates of each of the simplices of the N-simplex and test the
values of beta. This is exactly the implementation used in tsearchn. tsearch is optimized
for 2-dimensions and the Barycentric coordinates are not explicitly formed.

idx = tsearch (x, y, t, xi, yi)
Search for the enclosing Delaunay convex hull.
For t = delaunay (x, y), finds the index in t containing the points (xi, yi). For
points outside the convex hull, idx is NaN.
See also: [delaunay], page 717, [delaunayn], page 718.

idx = tsearchn (x, t, xi)
[idx, p] = tsearchn (x, t, xi)
Search for the enclosing Delaunay convex hull.
For t = delaunayn (x), finds the index in t containing the points xi. For points
outside the convex hull, idx is NaN.
If requested tsearchn also returns the Barycentric coordinates p of the enclosing
triangles.
See also: [delaunay], page 717, [delaunayn], page 718.
An example of the use of tsearch can be seen with the simple triangulation
x = [-1; -1; 1; 1];
y = [-1; 1; -1; 1];
tri = [1, 2, 3; 2, 3, 4];
consisting of two triangles defined by tri. We can then identify which triangle a point falls
in like
tsearch (x, y, tri, -0.5, -0.5)
⇒ 1
tsearch (x, y, tri, 0.5, 0.5)
⇒ 2
and we can confirm that a point doesn’t lie within one of the triangles like
tsearch (x, y, tri, 2, 2)
⇒ NaN
The dsearch and dsearchn find the closest point in a tessellation to the desired point.
The desired point does not necessarily have to be in the tessellation, and even if it the
returned point of the tessellation does not have to be one of the vertexes of the N-simplex
within which the desired point is found.

idx = dsearch (x, y, tri, xi, yi)
idx = dsearch (x, y, tri, xi, yi, s)
Return the index idx of the closest point in x, y to the elements [xi(:), yi(:)].

724

GNU Octave

The variable s is accepted for compatibility but is ignored.
See also: [dsearchn], page 724, [tsearch], page 723.

idx =
idx =
idx =
[idx,

dsearchn (x, tri, xi)
dsearchn (x, tri, xi, outval)
dsearchn (x, xi)
d] = dsearchn ( . . . )

Return the index idx of the closest point in x to the elements xi.
If outval is supplied, then the values of xi that are not contained within one of the
simplices tri are set to outval. Generally, tri is returned from delaunayn (x).
See also: [dsearch], page 723, [tsearch], page 723.
An example of the use of dsearch, using the above values of x, y and tri is
dsearch (x, y, tri, -2, -2)
⇒ 1
If you wish the points that are outside the tessellation to be flagged, then dsearchn can
be used as
dsearchn ([x, y], tri, [-2, -2], NaN)
⇒ NaN
dsearchn ([x, y], tri, [-0.5, -0.5], NaN)
⇒ 1
where the point outside the tessellation are then flagged with NaN.

30.2 Voronoi Diagrams
A Voronoi diagram or Voronoi tessellation of a set of points s in an N-dimensional space,
is the tessellation of the N-dimensional space such that all points in v(p), a partitions of
the tessellation where p is a member of s, are closer to p than any other point in s. The
Voronoi diagram is related to the Delaunay triangulation of a set of points, in that the
vertexes of the Voronoi tessellation are the centers of the circum-circles of the simplices of
the Delaunay tessellation.

voronoi (x, y)
voronoi (x, y, options)
voronoi ( . . . , "linespec")
voronoi (hax, . . . )
h = voronoi ( . . . )
[vx, vy] = voronoi ( . . . )
Plot the Voronoi diagram of points (x, y).
The Voronoi facets with points at infinity are not drawn.
The options argument, which must be a string or cell array of strings, contains options
passed to the underlying qhull command. See the documentation for the Qhull library
for details http://www.qhull.org/html/qh-quick.htm#options.
If "linespec" is given it is used to set the color and line style of the plot.
If an axes graphics handle hax is supplied then the Voronoi diagram is drawn on the
specified axes rather than in a new figure.

Chapter 30: Geometry

725

If a single output argument is requested then the Voronoi diagram will be plotted and
a graphics handle h to the plot is returned.
[vx, vy] = voronoi (. . . ) returns the Voronoi vertices instead of plotting the diagram.
x = rand (10, 1);
y = rand (size (x));
h = convhull (x, y);
[vx, vy] = voronoi (x, y);
plot (vx, vy, "-b", x, y, "o", x(h), y(h), "-g");
legend ("", "points", "hull");
See also: [voronoin], page 725, [delaunay], page 717, [convhull], page 728.

[C, F] = voronoin (pts)
[C, F] = voronoin (pts, options)
Compute N-dimensional Voronoi facets.
The input matrix pts of size [n, dim] contains n points in a space of dimension dim.
C contains the points of the Voronoi facets. The list F contains, for each facet, the
indices of the Voronoi points.
An optional second argument, which must be a string or cell array of strings, contains
options passed to the underlying qhull command. See the documentation for the Qhull
library for details http://www.qhull.org/html/qh-quick.htm#options.
The default options depend on the dimension of the input:
• 2-D and 3-D: options = {"Qbb"}

• 4-D and higher: options = {"Qbb", "Qx"}
If options is not present or [] then the default arguments are used. Otherwise,
options replaces the default argument list. To append user options to the defaults it
is necessary to repeat the default arguments in options. Use a null string to pass no
arguments.
See also: [voronoi], page 724, [convhulln], page 728, [delaunayn], page 718.
An example of the use of voronoi is
rand ("state",9);
x = rand (10,1);
y = rand (10,1);
tri = delaunay (x, y);
[vx, vy] = voronoi (x, y, tri);
triplot (tri, x, y, "b");
hold on;
plot (vx, vy, "r");
The result of which can be seen in Figure 30.3. Note that the circum-circle of one of the
triangles has been added to this figure, to make the relationship between the Delaunay
tessellation and the Voronoi diagram clearer.

726

GNU Octave

1
Delaunay Triangulation
Voronoi Diagram

0.8

0.6

0.4

0.2

0
0

0.2

0.4

0.6

0.8

1

Figure 30.3: Delaunay triangulation and Voronoi diagram of a random set of points
Additional information about the size of the facets of a Voronoi diagram, and which
points of a set of points is in a polygon can be had with the polyarea and inpolygon
functions respectively.

polyarea (x, y)
polyarea (x, y, dim)
Determine area of a polygon by triangle method.
The variables x and y define the vertex pairs, and must therefore have the same
shape. They can be either vectors or arrays. If they are arrays then the columns of
x and y are treated separately and an area returned for each.
If the optional dim argument is given, then polyarea works along this dimension of
the arrays x and y.
An example of the use of polyarea might be
rand ("state", 2);
x = rand (10, 1);
y = rand (10, 1);
[c, f] = voronoin ([x, y]);
af = zeros (size (f));
for i = 1 : length (f)
af(i) = polyarea (c (f {i, :}, 1), c (f {i, :}, 2));
endfor
Facets of the Voronoi diagram with a vertex at infinity have infinity area. A simplified
version of polyarea for rectangles is available with rectint

area = rectint (a, b)
Compute area or volume of intersection of rectangles or N-D boxes.

Chapter 30: Geometry

727

Compute the area of intersection of rectangles in a and rectangles in b. N-dimensional
boxes are supported in which case the volume, or hypervolume is computed according
to the number of dimensions.
2-dimensional rectangles are defined as [xpos ypos width height] where xpos and
ypos are the position of the bottom left corner. Higher dimensions are supported
where the coordinates for the minimum value of each dimension follow the length
of the box in that dimension, e.g., [xpos ypos zpos kpos ... width height depth
k_length ...].
Each row of a and b define a rectangle, and if both define multiple rectangles, then
the output, area, is a matrix where the i-th row corresponds to the i-th row of a and
the j-th column corresponds to the j-th row of b.
See also: [polyarea], page 726.

in = inpolygon (x, y, xv, yv)
[in, on] = inpolygon (x, y, xv, yv)
For a polygon defined by vertex points (xv, yv), return true if the points (x, y) are
inside (or on the boundary) of the polygon; Otherwise, return false.
The input variables x and y, must have the same dimension.
The optional output on returns true if the points are exactly on the polygon edge,
and false otherwise.
See also: [delaunay], page 717.

An example of the use of inpolygon might be
randn ("state", 2);
x = randn (100, 1);
y = randn (100, 1);
vx = cos (pi * [-1 : 0.1: 1]);
vy = sin (pi * [-1 : 0.1 : 1]);
in = inpolygon (x, y, vx, vy);
plot (vx, vy, x(in), y(in), "r+", x(!in), y(!in), "bo");
axis ([-2, 2, -2, 2]);
The result of which can be seen in Figure 30.4.

728

GNU Octave

2

1

0

-1

-2
-2

-1

0

1

2

Figure 30.4: Demonstration of the inpolygon function to determine the points inside a
polygon

30.3 Convex Hull
The convex hull of a set of points is the minimum convex envelope containing all of the
points. Octave has the functions convhull and convhulln to calculate the convex hull of
2-dimensional and N-dimensional sets of points.

H = convhull (x, y)
H = convhull (x, y, options)
Compute the convex hull of the set of points defined by the arrays x and y. The
hull H is an index vector into the set of points and specifies which points form the
enclosing hull.
An optional third argument, which must be a string or cell array of strings, contains
options passed to the underlying qhull command. See the documentation for the
Qhull library for details http://www.qhull.org/html/qh-quick.htm#options.
The default option is {"Qt"}.
If options is not present or [] then the default arguments are used. Otherwise,
options replaces the default argument list. To append user options to the defaults it
is necessary to repeat the default arguments in options. Use a null string to pass no
arguments.
See also: [convhulln], page 728, [delaunay], page 717, [voronoi], page 724.

h = convhulln (pts)
h = convhulln (pts, options)
[h, v] = convhulln ( . . . )
Compute the convex hull of the set of points pts.

Chapter 30: Geometry

729

pts is a matrix of size [n, dim] containing n points in a space of dimension dim.
The hull h is an index vector into the set of points and specifies which points form
the enclosing hull.
An optional second argument, which must be a string or cell array of strings, contains
options passed to the underlying qhull command. See the documentation for the
Qhull library for details http://www.qhull.org/html/qh-quick.htm#options.
The default options depend on the dimension of the input:
• 2D, 3D, 4D: options = {"Qt"}

• 5D and higher: options = {"Qt", "Qx"}
If options is not present or [] then the default arguments are used. Otherwise,
options replaces the default argument list. To append user options to the defaults it
is necessary to repeat the default arguments in options. Use a null string to pass no
arguments.
If the second output v is requested the volume of the enclosing convex hull is calculated.
See also: [convhull], page 728, [delaunayn], page 718, [voronoin], page 725.
An example of the use of convhull is
x = -3:0.05:3;
y = abs (sin (x));
k = convhull (x, y);
plot (x(k), y(k), "r-", x, y, "b+");
axis ([-3.05, 3.05, -0.05, 1.05]);
The output of the above can be seen in Figure 30.5.

1

0.8

0.6

0.4

0.2

0
-3

-2

-1

0

1

Figure 30.5: The convex hull of a simple set of points

2

3

730

GNU Octave

30.4 Interpolation on Scattered Data
An important use of the Delaunay tessellation is that it can be used to interpolate from
scattered data to an arbitrary set of points. To do this the N-simplex of the known set of
points is calculated with delaunay or delaunayn. Then the simplices in to which the desired
points are found are identified. Finally the vertices of the simplices are used to interpolate to
the desired points. The functions that perform this interpolation are griddata, griddata3
and griddatan.

zi = griddata (x, y, z, xi, yi)
zi = griddata (x, y, z, xi, yi, method)
[xi, yi, zi] = griddata ( . . . )
Generate a regular mesh from irregular data using interpolation.
The function is defined by z = f (x, y). Inputs x, y, z are vectors of the same length
or x, y are vectors and z is matrix.
The interpolation points are all (xi, yi). If xi, yi are vectors then they are made
into a 2-D mesh.
The interpolation method can be "nearest", "cubic" or "linear". If method is
omitted it defaults to "linear".
See also: [griddata3], page 730, [griddatan], page 730, [delaunay], page 717.

vi = griddata3 (x, y, z, v, xi, yi, zi)
vi = griddata3 (x, y, z, v, xi, yi, zi, method)
vi = griddata3 (x, y, z, v, xi, yi, zi, method, options)
Generate a regular mesh from irregular data using interpolation.
The function is defined by v = f (x, y, z). The interpolation points are specified by
xi, yi, zi.
The interpolation method can be "nearest" or "linear". If method is omitted it
defaults to "linear".
The optional argument options is passed directly to Qhull when computing the Delaunay triangulation used for interpolation. See delaunayn for information on the
defaults and how to pass different values.
See also: [griddata], page 730, [griddatan], page 730, [delaunayn], page 718.

yi = griddatan (x, y, xi)
yi = griddatan (x, y, xi, method)
yi = griddatan (x, y, xi, method, options)
Generate a regular mesh from irregular data using interpolation.
The function is defined by y = f (x). The interpolation points are all xi.
The interpolation method can be "nearest" or "linear". If method is omitted it
defaults to "linear".
The optional argument options is passed directly to Qhull when computing the Delaunay triangulation used for interpolation. See delaunayn for information on the
defaults and how to pass different values.
See also: [griddata], page 730, [griddata3], page 730, [delaunayn], page 718.

Chapter 30: Geometry

731

An example of the use of the griddata function is
rand ("state", 1);
x = 2*rand (1000,1) - 1;
y = 2*rand (size (x)) - 1;
z = sin (2*(x.^2+y.^2));
[xx,yy] = meshgrid (linspace (-1,1,32));
zz = griddata (x, y, z, xx, yy);
mesh (xx, yy, zz);
that interpolates from a random scattering of points, to a uniform grid. The output of the
above can be seen in Figure 30.6.

1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
1
0.5

1
0.5

0
0
-0.5

-0.5
-1

-1

Figure 30.6: Interpolation from a scattered data to a regular grid

733

31 Signal Processing
This chapter describes the signal processing and fast Fourier transform functions available
in Octave. Fast Fourier transforms are computed with the fftw or fftpack libraries
depending on how Octave is built.

fft (x)
fft (x, n)
fft (x, n, dim)
Compute the discrete Fourier transform of x using a Fast Fourier Transform (FFT)
algorithm.
The FFT is calculated along the first non-singleton dimension of the array. Thus if x
is a matrix, fft (x) computes the FFT for each column of x.
If called with two arguments, n is expected to be an integer specifying the number of
elements of x to use, or an empty matrix to specify that its value should be ignored.
If n is larger than the dimension along which the FFT is calculated, then x is resized
and padded with zeros. Otherwise, if n is smaller than the dimension along which the
FFT is calculated, then x is truncated.
If called with three arguments, dim is an integer specifying the dimension of the
matrix along which the FFT is performed.
See also: [ifft], page 733, [fft2], page 733, [fftn], page 734, [fftw], page 734.

ifft (x)
ifft (x, n)
ifft (x, n, dim)
Compute the inverse discrete Fourier transform of x using a Fast Fourier Transform
(FFT) algorithm.
The inverse FFT is calculated along the first non-singleton dimension of the array.
Thus if x is a matrix, fft (x) computes the inverse FFT for each column of x.
If called with two arguments, n is expected to be an integer specifying the number of
elements of x to use, or an empty matrix to specify that its value should be ignored.
If n is larger than the dimension along which the inverse FFT is calculated, then x is
resized and padded with zeros. Otherwise, if n is smaller than the dimension along
which the inverse FFT is calculated, then x is truncated.
If called with three arguments, dim is an integer specifying the dimension of the
matrix along which the inverse FFT is performed.
See also: [fft], page 733, [ifft2], page 734, [ifftn], page 734, [fftw], page 734.

fft2 (A)
fft2 (A, m, n)
Compute the two-dimensional discrete Fourier transform of A using a Fast Fourier
Transform (FFT) algorithm.
The optional arguments m and n may be used specify the number of rows and columns
of A to use. If either of these is larger than the size of A, A is resized and padded
with zeros.

734

GNU Octave

If A is a multi-dimensional matrix, each two-dimensional sub-matrix of A is treated
separately.
See also: [ifft2], page 734, [fft], page 733, [fftn], page 734, [fftw], page 734.

ifft2 (A)
ifft2 (A, m, n)
Compute the inverse two-dimensional discrete Fourier transform of A using a Fast
Fourier Transform (FFT) algorithm.
The optional arguments m and n may be used specify the number of rows and columns
of A to use. If either of these is larger than the size of A, A is resized and padded
with zeros.
If A is a multi-dimensional matrix, each two-dimensional sub-matrix of A is treated
separately.
See also: [fft2], page 733, [ifft], page 733, [ifftn], page 734, [fftw], page 734.

fftn (A)
fftn (A, size)
Compute the N-dimensional discrete Fourier transform of A using a Fast Fourier
Transform (FFT) algorithm.
The optional vector argument size may be used specify the dimensions of the array
to be used. If an element of size is smaller than the corresponding dimension of A,
then the dimension of A is truncated prior to performing the FFT. Otherwise, if
an element of size is larger than the corresponding dimension then A is resized and
padded with zeros.
See also: [ifftn], page 734, [fft], page 733, [fft2], page 733, [fftw], page 734.

ifftn (A)
ifftn (A, size)
Compute the inverse N-dimensional discrete Fourier transform of A using a Fast
Fourier Transform (FFT) algorithm.
The optional vector argument size may be used specify the dimensions of the array to
be used. If an element of size is smaller than the corresponding dimension of A, then
the dimension of A is truncated prior to performing the inverse FFT. Otherwise, if
an element of size is larger than the corresponding dimension then A is resized and
padded with zeros.
See also: [fftn], page 734, [ifft], page 733, [ifft2], page 734, [fftw], page 734.
Octave uses the fftw libraries to perform FFT computations. When Octave starts up
and initializes the fftw libraries, they read a system wide file (on a Unix system, it is typically /etc/fftw/wisdom) that contains information useful to speed up FFT computations.
This information is called the wisdom. The system-wide file allows wisdom to be shared
between all applications using the fftw libraries.
Use the fftw function to generate and save wisdom. Using the utilities provided together with the fftw libraries (fftw-wisdom on Unix systems), you can even add wisdom
generated by Octave to the system-wide wisdom file.

Chapter 31: Signal Processing

735

method = fftw ("planner")
fftw ("planner", method)
wisdom = fftw ("dwisdom")
fftw ("dwisdom", wisdom)
fftw ("threads", nthreads)
nthreads = fftw ("threads")
Manage fftw wisdom data.
Wisdom data can be used to significantly accelerate the calculation of the FFTs, but
implies an initial cost in its calculation. When the fftw libraries are initialized, they
read a system wide wisdom file (typically in /etc/fftw/wisdom), allowing wisdom to
be shared between applications other than Octave. Alternatively, the fftw function
can be used to import wisdom. For example,
wisdom = fftw ("dwisdom")
will save the existing wisdom used by Octave to the string wisdom. This string can
then be saved to a file and restored using the save and load commands respectively.
This existing wisdom can be re-imported as follows
fftw ("dwisdom", wisdom)
If wisdom is an empty string, then the wisdom used is cleared.
During the calculation of Fourier transforms further wisdom is generated. The fashion
in which this wisdom is generated is also controlled by the fftw function. There are
five different manners in which the wisdom can be treated:
"estimate"
Specifies that no run-time measurement of the optimal means of calculating a particular is performed, and a simple heuristic is used to pick a
(probably sub-optimal) plan. The advantage of this method is that there
is little or no overhead in the generation of the plan, which is appropriate
for a Fourier transform that will be calculated once.
"measure"
In this case a range of algorithms to perform the transform is considered
and the best is selected based on their execution time.
"patient"
Similar to "measure", but a wider range of algorithms is considered.
"exhaustive"
Like "measure", but all possible algorithms that may be used to treat
the transform are considered.
"hybrid"

As run-time measurement of the algorithm can be expensive, this is a
compromise where "measure" is used for transforms up to the size of
8192 and beyond that the "estimate" method is used.

The default method is "estimate". The current method can be queried with
method = fftw ("planner")
or set by using
fftw ("planner", method)

736

GNU Octave

Note that calculated wisdom will be lost when restarting Octave. However, the wisdom data can be reloaded if it is saved to a file as described above. Saved wisdom
files should not be used on different platforms since they will not be efficient and the
point of calculating the wisdom is lost.
The number of threads used for computing the plans and executing the transforms
can be set with
fftw ("threads", NTHREADS)
Note that octave must be compiled with multi-threaded fftw support for this feature.
The number of processors available to the current process is used per default.
See also: [fft], page 733, [ifft], page 733, [fft2], page 733, [ifft2], page 734, [fftn],
page 734, [ifftn], page 734.

fftconv (x, y)
fftconv (x, y, n)
Convolve two vectors using the FFT for computation.
c = fftconv (x, y) returns a vector of length equal to length (x) + length (y) 1. If x and y are the coefficient vectors of two polynomials, the returned value is the
coefficient vector of the product polynomial.
The computation uses the FFT by calling the function fftfilt. If the optional
argument n is specified, an N-point FFT is used.
See also: [deconv], page 692, [conv], page 691, [conv2], page 692.

fftfilt (b, x)
fftfilt (b, x, n)
Filter x with the FIR filter b using the FFT.
If x is a matrix, filter each column of the matrix.
Given the optional third argument, n, fftfilt uses the overlap-add method to filter
x with b using an N-point FFT. The FFT size must be an even power of 2 and must
be greater than or equal to the length of b. If the specified n does not meet these
criteria, it is automatically adjusted to the nearest value that does.
See also: [filter], page 736, [filter2], page 737.

y =
[y,
[y,
[y,

filter (b, a, x)
sf] = filter (b, a, x, si)
sf] = filter (b, a, x, [], dim)
sf] = filter (b, a, x, si, dim)
Apply a 1-D digital filter to the data x.
filter returns the solution to the following linear, time-invariant difference equation:
N
X

k=0

ak+1 yn−k =

M
X

k=0

bk+1 xn−k ,

1≤n≤P

where a ∈  k, [1, y(t-1), ..., y(t-k)] is the t-th row of the result.
The resulting matrix may be used as a regressor matrix in autoregressions.

bartlett (m)
Return the filter coefficients of a Bartlett (triangular) window of length m.
For a definition of the Bartlett window see, e.g., A.V. Oppenheim & R. W. Schafer,
Discrete-Time Signal Processing.

blackman (m)
blackman (m, "periodic")
blackman (m, "symmetric")
Return the filter coefficients of a Blackman window of length m.
If the optional argument "periodic" is given, the periodic form of the window is
returned. This is equivalent to the window of length m+1 with the last coefficient
removed. The optional argument "symmetric" is equivalent to not specifying a second
argument.
For a definition of the Blackman window, see, e.g., A.V. Oppenheim & R. W. Schafer,
Discrete-Time Signal Processing.

detrend (x, p)
If x is a vector, detrend (x, p) removes the best fit of a polynomial of order p from
the data x.
If x is a matrix, detrend (x, p) does the same for each column in x.
The second argument p is optional. If it is not specified, a value of 1 is assumed. This
corresponds to removing a linear trend.
The order of the polynomial can also be given as a string, in which case p must be
either "constant" (corresponds to p=0) or "linear" (corresponds to p=1).
See also: [polyfit], page 695.

[d, dd] = diffpara (x, a, b)
Return the estimator d for the differencing parameter of an integrated time series.
The frequencies from [2 ∗ pi ∗ a/t, 2 ∗ pi ∗ b/T ] are used for the estimation. If b is
omitted, the interval [2 ∗ pi/T, 2 ∗ pi ∗ a/T ] is used. If both b and a are omitted then
a = 0.5 ∗ sqrt(T ) and b = 1.5 ∗ sqrt(T ) is used, where T is the sample size. If x is a
matrix, the differencing parameter of each column is estimated.
The estimators for all frequencies in the intervals described above is returned in dd.
The value of d is simply the mean of dd.
Reference: P.J. Brockwell & R.A. Davis. Time Series: Theory and Methods. Springer
1987.

Chapter 31: Signal Processing

741

durbinlevinson (c, oldphi, oldv)
Perform one step of the Durbin-Levinson algorithm.
The vector c specifies the autocovariances [gamma_0, ..., gamma_t] from lag 0 to t,
oldphi specifies the coefficients based on c(t-1) and oldv specifies the corresponding
error.
If oldphi and oldv are omitted, all steps from 1 to t of the algorithm are performed.

fftshift (x)
fftshift (x, dim)
Perform a shift of the vector x, for use with the fft and ifft functions, in order to
move the frequency 0 to the center of the vector or matrix.
If x is a vector of N elements corresponding to N time samples spaced by dt, then
fftshift (fft (x)) corresponds to frequencies
f = [ -(ceil((N-1)/2):-1:1), 0, (1:floor((N-1)/2)) ] * df
where df = 1/(N ∗ dt).
If x is a matrix, the same holds for rows and columns. If x is an array, then the same
holds along each dimension.
The optional dim argument can be used to limit the dimension along which the
permutation occurs.
See also: [ifftshift], page 741.

ifftshift (x)
ifftshift (x, dim)
Undo the action of the fftshift function.
For even length x, fftshift is its own inverse, but odd lengths differ slightly.
See also: [fftshift], page 741.

fractdiff (x, d)
Compute the fractional differences (1 − L)d x where L denotes the lag-operator and d
is greater than -1.

hamming (m)
hamming (m, "periodic")
hamming (m, "symmetric")
Return the filter coefficients of a Hamming window of length m.
If the optional argument "periodic" is given, the periodic form of the window is
returned. This is equivalent to the window of length m+1 with the last coefficient
removed. The optional argument "symmetric" is equivalent to not specifying a second
argument.
For a definition of the Hamming window see, e.g., A.V. Oppenheim & R. W. Schafer,
Discrete-Time Signal Processing.

hanning (m)
hanning (m, "periodic")
hanning (m, "symmetric")
Return the filter coefficients of a Hanning window of length m.

742

GNU Octave

If the optional argument "periodic" is given, the periodic form of the window is
returned. This is equivalent to the window of length m+1 with the last coefficient
removed. The optional argument "symmetric" is equivalent to not specifying a second
argument.
For a definition of the Hanning window see, e.g., A.V. Oppenheim & R. W. Schafer,
Discrete-Time Signal Processing.

hurst (x)
Estimate the Hurst parameter of sample x via the rescaled range statistic.
If x is a matrix, the parameter is estimated for every column.

pp = pchip (x, y)
yi = pchip (x, y, xi)
Return the Piecewise Cubic Hermite Interpolating Polynomial (pchip) of points x
and y.
If called with two arguments, return the piecewise polynomial pp that may be used
with ppval to evaluate the polynomial at specific points.
When called with a third input argument, pchip evaluates the pchip polynomial at
the points xi. The third calling form is equivalent to ppval (pchip (x, y), xi).
The variable x must be a strictly monotonic vector (either increasing or decreasing)
of length n.
y can be either a vector or array. If y is a vector then it must be the same length n
as x. If y is an array then the size of y must have the form
[s1 , s2 , · · · , sk , n]
The array is reshaped internally to a matrix where the leading dimension is given by
s1 s2 · · · sk
and each row of this matrix is then treated separately. Note that this is exactly
opposite to interp1 but is done for matlab compatibility.
See also: [spline], page 711, [ppval], page 704, [mkpp], page 703, [unmkpp], page 703.

[Pxx, w] = periodogram
[Pxx, w] = periodogram
[Pxx, w] = periodogram
[Pxx, f] = periodogram
[Pxx, f] = periodogram
periodogram ( . . . )

(x)
(x, win)
(x, win, nfft)
(x, win, nfft, Fs)
( . . . , "range")

Return the periodogram (Power Spectral Density) of x.
The possible inputs are:
x
data vector. If x is real-valued a one-sided spectrum is estimated. If x
is complex-valued, or "range" specifies "twosided", the full spectrum is
estimated.

Chapter 31: Signal Processing

743

win

window weight data. If window is empty or unspecified a default rectangular window is used. Otherwise, the window is applied to the signal (x
.* win) before computing the periodogram. The window data must be a
vector of the same length as x.

nfft

number of frequency bins. The default is 256 or the next higher power of
2 greater than the length of x (max (256, 2.^nextpow2 (length (x)))).
If nfft is greater than the length of the input then x will be zero-padded
to the length of nfft.

Fs

sampling rate. The default is 1.

range

range of spectrum. "onesided" computes spectrum from [0:nfft/2+1].
"twosided" computes spectrum from [0:nfft-1].

The optional second output w are the normalized angular frequencies. For a one-sided
calculation w is in the range [0, pi] if nfft is even and [0, pi) if nfft is odd. Similarly,
for a two-sided calculation w is in the range [0, 2*pi] or [0, 2*pi) depending on nfft.
If a sampling frequency is specified, Fs, then the output frequencies f will be in the
range [0, Fs/2] or [0, Fs/2) for one-sided calculations. For two-sided calculations the
range will be [0, Fs).
When called with no outputs the periodogram is immediately plotted in the current
figure window.
See also: [fft], page 733.

sinetone (freq, rate, sec, ampl)
Return a sinetone of frequency freq with a length of sec seconds at sampling rate rate
and with amplitude ampl.
The arguments freq and ampl may be vectors of common size.
The defaults are rate = 8000, sec = 1, and ampl = 64.
See also: [sinewave], page 743.

sinewave (m, n, d)
Return an m-element vector with i-th element given by sin (2 * pi * (i+d-1) / n).
The default value for d is 0 and the default value for n is m.
See also: [sinetone], page 743.

spectral_adf (c)
spectral_adf (c, win)
spectral_adf (c, win, b)
Return the spectral density estimator given a vector of autocovariances c, window
name win, and bandwidth, b.
The window name, e.g., "triangle" or "rectangle" is used to search for a function
called win_lw.
If win is omitted, the triangle window is used.
If b is omitted, 1 / sqrt (length (x)) is used.
See also: [spectral xdf], page 744.

744

GNU Octave

spectral_xdf (x)
spectral_xdf (x, win)
spectral_xdf (x, win, b)
Return the spectral density estimator given a data vector x, window name win, and
bandwidth, b.
The window name, e.g., "triangle" or "rectangle" is used to search for a function
called win_sw.
If win is omitted, the triangle window is used.
If b is omitted, 1 / sqrt (length (x)) is used.
See also: [spectral adf], page 743.

spencer (x)
Return Spencer’s 15 point moving average of each column of x.

y =
y =
y =
y =
y =
[y,

(x)
(x, win_size)
(x, win_size, inc)
(x, win_size, inc, num_coef)
(x, win_size, inc, num_coef, win_type)
stft ( . . . )
Compute the short-time Fourier transform of the vector x with num coef coefficients
by applying a window of win size data points and an increment of inc points.
Before computing the Fourier transform, one of the following windows is applied:

stft
stft
stft
stft
stft
c] =

"hanning"
win type = 1
"hamming"
win type = 2
"rectangle"
win type = 3
The window names can be passed as strings or by the win type number.
The following defaults are used for unspecified arguments: win size = 80, inc = 24,
num coef = 64, and win type = 1.
y = stft (x, ...) returns the absolute values of the Fourier coefficients according to
the num coef positive frequencies.
[y, c] = stft (x, ...) returns the entire STFT-matrix y and a 3-element vector
c containing the window size, increment, and window type, which is needed by the
synthesis function.
See also: [synthesis], page 744.

x = synthesis (y, c)
Compute a signal from its short-time Fourier transform y and a 3-element vector c
specifying window size, increment, and window type.
The values y and c can be derived by
[y, c] = stft (x , ...)
See also: [stft], page 744.

745

[a, v] = yulewalker (c)
Fit an AR (p)-model with Yule-Walker estimates given a vector c of autocovariances
[gamma_0, ..., gamma_p].
Returns the AR coefficients, a, and the variance of white noise, v.

747

32 Image Processing
Since an image is basically a matrix, Octave is a very powerful environment for processing
and analyzing images. To illustrate how easy it is to do image processing in Octave, the
following example will load an image, smooth it by a 5-by-5 averaging filter, and compute
the gradient of the smoothed image.
I = imread ("myimage.jpg");
S = conv2 (I, ones (5, 5) / 25, "same");
[Dx, Dy] = gradient (S);
In this example S contains the smoothed image, and Dx and Dy contains the partial spatial
derivatives of the image.

32.1 Loading and Saving Images
The first step in most image processing tasks is to load an image into Octave which is done
with the imread function. The imwrite function is the corresponding function for writing
images to the disk.
In summary, most image processing code will follow the structure of this code
I = imread ("my_input_image.img");
J = process_my_image (I);
imwrite (J, "my_output_image.img");

[img,
[...]
[...]
[...]
[...]

map, alpha] = imread (filename)
= imread (url)
= imread ( . . . , ext)
= imread ( . . . , idx)
= imread ( . . . , param1, value1, . . . )

Read images from various file formats.
Read an image as a matrix from the file filename or from the online resource url. If
neither is given, but ext was specified, look for a file with the extension ext.
The size and class of the output depends on the format of the image. A color image
is returned as an MxNx3 matrix. Grayscale and black-and-white images are of size
MxN. Multipage images will have an additional 4th dimension.
The bit depth of the image determines the class of the output: "uint8", "uint16",
or "single" for grayscale and color, and "logical" for black-and-white. Note that
indexed images always return the indexes for a colormap, independent of whether
map is a requested output. To obtain the actual RGB image, use ind2rgb. When
more than one indexed image is being read, map is obtained from the first. In some
rare cases this may be incorrect and imfinfo can be used to obtain the colormap of
each image.
See the Octave manual for more information in representing images.
Some file formats, such as TIFF and GIF, are able to store multiple images in a
single file. idx can be a scalar or vector specifying the index of the images to read.
By default, Octave will read only the first page.

748

GNU Octave

Depending on the file format, it is possible to configure the reading of images with
parameter, value pairs. The following options are supported:
"Frames" or "Index"
This is an alternative method to specify idx. When specifying it in this
way, its value can also be the string "all".
"Info"

This option exists for matlab compatibility, but has no effect. For maximum performance when reading multiple images from a single file, use
the "Index" option.

"PixelRegion"
Controls the image region that is read. The value must be a cell array with
two arrays of 3 elements {[rows], [cols]}. The elements in the array
are the start, increment, and end pixel to be read. If the increment value
is omitted it defaults to 1. For example, the following are all equivalent:
imread (filename, "PixelRegion", {[200 600], [300 700]});
imread (filename, "PixelRegion", {[200 1 600], [300 1 700]});
imread (filename)(200:600, 300:700);
See also: [imwrite], page 748, [imfinfo], page 750, [imformats], page 752.
(img, filename)
(img, filename, ext)
(img, map, filename)
( . . . , param1, val1, . . . )
Write images in various file formats.

imwrite
imwrite
imwrite
imwrite

The image img can be a binary, grayscale, RGB, or multi-dimensional image. The
size and class of img should be the same as what should be expected when reading it
with imread: the 3rd and 4th dimensions reserved for color space, and multiple pages
respectively. If it’s an indexed image, the colormap map must also be specified.
If ext is not supplied, the file extension of filename is used to determine the format.
The actual supported formats are dependent on options made during the build of
Octave. Use imformats to check the support of the different image formats.
Depending on the file format, it is possible to configure the writing of images with
param, val pairs. The following options are supported:
‘Alpha’

Alpha (transparency) channel for the image. This must be a matrix
with same class, and number of rows and columns of img. In case of a
multipage image, the size of the 4th dimension must also match and the
third dimension must be a singleton. By default, image will be completely
opaque.

‘Compression’
Compression to use one the image. Can be one of the following: "none"
(default), "bzip", "fax3", "fax4", "jpeg", "lzw", "rle", or "deflate". Note
that not all compression types are available for all image formats in which
it defaults to your Magick library.

Chapter 32: Image Processing

749

‘DelayTime’
For formats that accept animations (such as GIF), controls for how long
a frame is displayed until it moves to the next one. The value must be
scalar (which will applied to all frames in img), or a vector of length equal
to the number of frames in im. The value is in seconds, must be between
0 and 655.35, and defaults to 0.5.
‘DisposalMethod’
For formats that accept animations (such as GIF), controls what happens
to a frame before drawing the next one. Its value can be one of the following strings: "doNotSpecify" (default); "leaveInPlace"; "restoreBG";
and "restorePrevious", or a cell array of those string with length equal
to the number of frames in img.
‘LoopCount’
For formats that accept animations (such as GIF), controls how many
times the sequence is repeated. A value of Inf means an infinite loop
(default), a value of 0 or 1 that the sequence is played only once (loops zero
times), while a value of 2 or above loops that number of times (looping
twice means it plays the complete sequence 3 times). This option is
ignored when there is only a single image at the end of writing the file.
‘Quality’

Set the quality of the compression. The value should be an integer between 0 and 100, with larger values indicating higher visual quality and
lower compression. Defaults to 75.

‘WriteMode’
Some file formats, such as TIFF and GIF, are able to store multiple
images in a single file. This option specifies if img should be appended
to the file (if it exists) or if a new file should be created for it (possibly
overwriting an existing file). The value should be the string "Overwrite"
(default), or "Append".
Despite this option, the most efficient method of writing a multipage
image is to pass a 4 dimensional img to imwrite, the same matrix that
could be expected when using imread with the option "Index" set to
"all".
See also: [imread], page 747, [imfinfo], page 750, [imformats], page 752.

val = IMAGE_PATH ()
old_val = IMAGE_PATH (new_val)
IMAGE_PATH (new_val, "local")
Query or set the internal variable that specifies a colon separated list of directories in
which to search for image files.
When called from inside a function with the "local" option, the variable is changed
locally for the function and any subroutines it calls. The original variable value is
restored when exiting the function.
See also: [EXEC PATH], page 849, [OCTAVE HOME], page 859.

750

GNU Octave

It is possible to get information about an image file on disk, without actually reading it
into Octave. This is done using the imfinfo function which provides read access to many
of the parameters stored in the header of the image file.

info = imfinfo (filename)
info = imfinfo (url)
info = imfinfo ( . . . , ext)
Read image information from a file.
imfinfo returns a structure containing information about the image stored in the
file filename. If there is no file filename, and ext was specified, it will look for a file
named filename and extension ext, i.e., a file named filename.ext.
The output structure info contains the following fields:
‘Filename’
The full name of the image file.
‘FileModDate’
Date of last modification to the file.
‘FileSize’
Number of bytes of the image on disk
‘Format’

Image format (e.g., "jpeg").

‘Height’

Image height in pixels.

‘Width’

Image Width in pixels.

‘BitDepth’
Number of bits per channel per pixel.
‘ColorType’
Image type. Value is "grayscale", "indexed", "truecolor", "CMYK",
or "undefined".
‘XResolution’
X resolution of the image.
‘YResolution’
Y resolution of the image.
‘ResolutionUnit’
Units of image resolution.
"undefined".

Value is "Inch", "Centimeter", or

‘DelayTime’
Time in 1/100ths of a second (0 to 65535) which must expire before
displaying the next image in an animated sequence.
‘LoopCount’
Number of iterations to loop an animation.
‘ByteOrder’
Endian option for formats that support it. Value is "little-endian",
"big-endian", or "undefined".

Chapter 32: Image Processing

751

‘Gamma’

Gamma level of the image. The same color image displayed on two different workstations may look different due to differences in the display
monitor.

‘Quality’

JPEG/MIFF/PNG compression level. Value is an integer in the range [0
100].

‘DisposalMethod’
Only valid for GIF images, control how successive frames are rendered
(how the preceding frame is disposed of) when creating a GIF animation. Values can be "doNotSpecify", "leaveInPlace", "restoreBG",
or "restorePrevious". For non-GIF files, value is an empty string.
‘Chromaticities’
Value is a 1x8 Matrix with the x,y chromaticity values for white, red,
green, and blue points, in that order.
‘Comment’

Image comment.

‘Compression’
Compression type. Value can be "none", "bzip", "fax3", "fax4",
"jpeg", "lzw", "rle", "deflate", "lzma", "jpeg2000", "jbig2",
"jbig2", or "undefined".
‘Colormap’
Colormap for each image.
‘Orientation’
The orientation of the image with respect to the rows and columns. Value
is an integer between 1 and 8 as defined in the TIFF 6 specifications, and
for matlab compatibility.
‘Software’
Name and version of the software or firmware of the camera or image
input device used to generate the image.
‘Make’

The manufacturer of the recording equipment. This is the manufacture
of the DSC, scanner, video digitizer or other equipment that generated
the image.

‘Model’

The model name or model number of the recording equipment as mentioned on the field "Make".

‘DateTime’
The date and time of image creation as defined by the Exif standard, i.e.,
it is the date and time the file was changed.
‘ImageDescription’
The title of the image as defined by the Exif standard.
‘Artist’

Name of the camera owner, photographer or image creator.

‘Copyright’
Copyright notice of the person or organization claiming rights to the
image.

752

GNU Octave

‘DigitalCamera’
A struct with information retrieved from the Exif tag.
‘GPSInfo’

A struct with geotagging information retrieved from the Exif tag.

See also: [imread], page 747, [imwrite], page 748, [imshow], page 753, [imformats],
page 752.
By default, Octave’s image IO functions (imread, imwrite, and imfinfo) use the
GraphicsMagick library for their operations. This means a vast number of image formats is
supported but considering the large amount of image formats in science and its commonly
closed nature, it is impossible to have a library capable of reading them all. Because of
this, the function imformats keeps a configurable list of available formats, their extensions,
and what functions should the image IO functions use. This allows one to expand Octave’s
image IO capabilities by creating functions aimed at acting on specific file formats.
While it would be possible to call the extra functions directly, properly configuring
Octave with imformats allows one to keep a consistent code that is abstracted from file
formats.
It is important to note that a file format is not actually defined by its file extension and
that GraphicsMagick is capable to read and write more file formats than the ones listed by
imformats. What this means is that even with an incorrect or missing extension the image
may still be read correctly, and that even unlisted formats are not necessarily unsupported.

imformats
formats =
formats =
formats =
formats =
formats =
formats =

()

(ext)
(format)
("add", format)
("remove", ext)
("update", ext, format)
("factory")
Manage supported image formats.
formats is a structure with information about each supported file format, or from a
specific format ext, the value displayed on the field ext. It contains the following
fields:

imformats
imformats
imformats
imformats
imformats
imformats

ext

The name of the file format. This may match the file extension but
Octave will automatically detect the file format.

description
A long description of the file format.
isa

A function handle to confirm if a file is of the specified format.

write

A function handle to write if a file is of the specified format.

read

A function handle to open files the specified format.

info

A function handle to obtain image information of the specified format.

alpha

Logical value if format supports alpha channel (transparency or matte).

multipage Logical value if format supports multipage (multiple images per file).

Chapter 32: Image Processing

753

It is possible to change the way Octave manages file formats with the options "add",
"remove", and "update", and supplying a structure format with the required fields.
The option "factory" resets the configuration to the default.
This can be used by Octave packages to extend the image reading capabilities Octave,
through use of the PKG ADD and PKG DEL commands.
See also: [imfinfo], page 750, [imread], page 747, [imwrite], page 748.

32.2 Displaying Images
A natural part of image processing is visualization of an image. The most basic function
for this is the imshow function that shows the image given in the first input argument.

imshow (im)
imshow (im, limits)
imshow (im, map)
imshow (rgb, . . . )
imshow (filename)
imshow ( . . . , string_param1, value1, . . . )
h = imshow ( . . . )
Display the image im, where im can be a 2-dimensional (grayscale image) or a 3dimensional (RGB image) matrix.
If limits is a 2-element vector [low, high], the image is shown using a display range
between low and high. If an empty matrix is passed for limits, the display range is
computed as the range between the minimal and the maximal value in the image.
If map is a valid color map, the image will be shown as an indexed image using the
supplied color map.
If a filename is given instead of an image, the file will be read and shown.
If given, the parameter string param1 has value value1. string param1 can be any
of the following:
"displayrange"
value1 is the display range as described above.
"colormap"
value1 is the colormap to use when displaying an indexed image.
"xdata"

If value1 is a two element vector, it must contain horizontal axis limits in
the form [xmin xmax]; Otherwise value1 must be a vector and only the
first and last elements will be used for xmin and xmax respectively.

"ydata"

If value1 is a two element vector, it must contain vertical axis limits in
the form [ymin ymax]; Otherwise value1 must be a vector and only the
first and last elements will be used for ymin and ymax respectively.

The optional return value h is a graphics handle to the image.
See also: [image], page 754, [imagesc], page 754, [colormap], page 757, [gray2ind],
page 756, [rgb2ind], page 756.

754

GNU Octave

image (img)
image (x, y, img)
image ( . . . , "prop", val, . . . )
image ("prop1", val1, . . . )
h = image ( . . . )
Display a matrix as an indexed color image.
The elements of img are indices into the current colormap.
x and y are optional 2-element vectors, [min, max], which specify the range for the
axis labels. If a range is specified as [max, min] then the image will be reversed along
that axis. For convenience, x and y may be specified as N-element vectors matching
the length of the data in img. However, only the first and last elements will be used
to determine the axis limits.
Multiple property/value pairs may be specified for the image object, but they must
appear in pairs.
The optional return value h is a graphics handle to the image.
Implementation Note: The origin (0, 0) for images is located in the upper left. For
ordinary plots, the origin is located in the lower left. Octave handles this inversion by
plotting the data normally, and then reversing the direction of the y-axis by setting
the ydir property to "reverse". This has implications whenever an image and an
ordinary plot need to be overlaid. The recommended solution is to display the image
and then plot the reversed ydata using, for example, flipud (ydata).
Calling Forms: The image function can be called in two forms: High-Level and LowLevel. When invoked with normal options, the High-Level form is used which first
calls newplot to prepare the graphic figure and axes. When the only inputs to image
are property/value pairs the Low-Level form is used which creates a new instance of
an image object and inserts it in the current axes.
See also: [imshow], page 753, [imagesc], page 754, [colormap], page 757.

imagesc (img)
imagesc (x, y, img)
imagesc ( . . . , climits)
imagesc ( . . . , "prop", val, . . . )
imagesc ("prop1", val1, . . . )
imagesc (hax, . . . )
h = imagesc ( . . . )
Display a scaled version of the matrix img as a color image.
The colormap is scaled so that the entries of the matrix occupy the entire colormap.
If climits = [lo, hi] is given, then that range is set to the "clim" of the current
axes.
The axis values corresponding to the matrix elements are specified in x and y, either
as pairs giving the minimum and maximum values for the respective axes, or as values
for each row and column of the matrix img.
The optional return value h is a graphics handle to the image.
Calling Forms: The imagesc function can be called in two forms: High-Level and
Low-Level. When invoked with normal options, the High-Level form is used which

Chapter 32: Image Processing

755

first calls newplot to prepare the graphic figure and axes. When the only inputs
to image are property/value pairs the Low-Level form is used which creates a new
instance of an image object and inserts it in the current axes.
See also: [image], page 754, [imshow], page 753, [caxis], page 317.

32.3 Representing Images
In general Octave supports four different kinds of images, grayscale images, RGB images,
binary images, and indexed images. A grayscale image is represented with an M-by-N
matrix in which each element corresponds to the intensity of a pixel. An RGB image is
represented with an M-by-N-by-3 array where each 3-vector corresponds to the red, green,
and blue intensities of each pixel.
The actual meaning of the value of a pixel in a grayscale or RGB image depends on the
class of the matrix. If the matrix is of class double pixel intensities are between 0 and 1, if
it is of class uint8 intensities are between 0 and 255, and if it is of class uint16 intensities
are between 0 and 65535.
A binary image is an M-by-N matrix of class logical. A pixel in a binary image is black
if it is false and white if it is true.
An indexed image consists of an M-by-N matrix of integers and a C-by-3 color map.
Each integer corresponds to an index in the color map, and each row in the color map
corresponds to an RGB color. The color map must be of class double with values between
0 and 1.

im2double (img)
im2double (img, "indexed")
Convert image to double precision.
The conversion of img to double precision, is dependent on the type of input image.
The following input classes are supported:
‘uint8, uint16, and int16’
The range of values from the class is scaled to the interval [0 1].
‘logical’

True and false values are assigned a value of 0 and 1 respectively.

‘single’

Values are cast to double.

‘double’

Returns the same image.

If img is an indexed image, then the second argument should be the string "indexed".
If so, then img must either be of floating point class, or unsigned integer class and it
will simply be cast to double. If it is an integer class, a +1 offset is applied.
See also: [double], page 47.

iscolormap (cmap)
Return true if cmap is a colormap.
A colormap is a real matrix, of class single or double, with 3 columns. Each row
represents a single color. The 3 columns contain red, green, and blue intensities
respectively.

756

GNU Octave

All values in a colormap should be in the [0 1] range but this is not enforced. Each
function must decide what to do for values outside this range.
See also: [colormap], page 757, [rgbplot], page 758.

img =
img =
img =
img =
[img,

gray2ind (I)
gray2ind (I, n)
gray2ind (BW)
gray2ind (BW, n)
map] = gray2ind ( . . . )

Convert a grayscale or binary intensity image to an indexed image.
The indexed image will consist of n different intensity values. If not given n defaults
to 64 for grayscale images or 2 for binary black and white images.
The output img is of class uint8 if n is less than or equal to 256; Otherwise the return
class is uint16.
See also: [ind2gray], page 756, [rgb2ind], page 756.

I = ind2gray (x, map)
Convert a color indexed image to a grayscale intensity image.
The image x must be an indexed image which will be converted using the colormap
map. If map does not contain enough colors for the image, pixels in x outside the
range are mapped to the last color in the map before conversion to grayscale.
The output I is of the same class as the input x and may be one of uint8, uint16,
single, or double.
Implementation Note: There are several ways of converting colors to grayscale intensities. This functions uses the luminance value obtained from rgb2ntsc which is I =
0.299*R + 0.587*G + 0.114*B. Other possibilities include the value component from
rgb2hsv or using a single color channel from ind2rgb.
See also: [gray2ind], page 756, [ind2rgb], page 757.

[x, map] = rgb2ind (rgb)
[x, map] = rgb2ind (R, G, B)
Convert an image in red-green-blue (RGB) color space to an indexed image.
The input image rgb can be specified as a single matrix of size MxNx3, or as three
separate variables, R, G, and B, its three color channels, red, green, and blue.
It outputs an indexed image x and a colormap map to interpret an image exactly
the same as the input. No dithering or other form of color quantization is performed.
The output class of the indexed image x can be uint8, uint16 or double, whichever is
required to specify the number of unique colors in the image (which will be equal to
the number of rows in map) in order.
Multi-dimensional indexed images (of size MxNx3xK) are also supported, both via a
single input (rgb) or its three color channels as separate variables.
See also: [ind2rgb], page 757, [rgb2hsv], page 765, [rgb2ntsc], page 766.

Chapter 32: Image Processing

757

rgb = ind2rgb (x, map)
[R, G, B] = ind2rgb (x, map)
Convert an indexed image to red, green, and blue color components.
The image x must be an indexed image which will be converted using the colormap
map. If map does not contain enough colors for the image, pixels in x outside the
range are mapped to the last color in the map.
The output may be a single RGB image (MxNx3 matrix where M and N are the
original image x dimensions, one for each of the red, green and blue channels). Alternatively, the individual red, green, and blue color matrices of size MxN may be
returned.
Multi-dimensional indexed images (of size MxNx1xK) are also supported.
See also: [rgb2ind], page 756, [ind2gray], page 756, [hsv2rgb], page 765, [ntsc2rgb],
page 766.

[x, map] = frame2im (f)
Convert movie frame to indexed image.
A movie frame is simply a struct with the fields "cdata" and "colormap".
Support for N-dimensional images or movies is given when f is a struct array. In such
cases, x will be a MxNx1xK or MxNx3xK for indexed and RGB movies respectively,
with each frame concatenated along the 4th dimension.
See also: [im2frame], page 757.

im2frame (rgb)
im2frame (x, map)
Convert image to movie frame.
A movie frame is simply a struct with the fields "cdata" and "colormap".
Support for N-dimensional images is given when each image projection, matrix sizes
of MxN and MxNx3 for RGB images, is concatenated along the fourth dimension. In
such cases, the returned value is a struct array.
See also: [frame2im], page 757.

cmap = colormap ()
cmap = colormap (map)
cmap = colormap ("default")
cmap = colormap (map_name)
cmap = colormap (hax, . . . )
colormap map_name
Query or set the current colormap.
With no input arguments, colormap returns the current color map.
colormap (map) sets the current colormap to map. The colormap should be an n row
by 3 column matrix. The columns contain red, green, and blue intensities respectively.
All entries must be between 0 and 1 inclusive. The new colormap is returned.
colormap ("default") restores the default colormap (the viridis map with 64 entries). The default colormap is returned.

758

GNU Octave

The map may also be specified by a string, map name, which is the name of a function
that returns a colormap.
If the first argument hax is an axes handle, then the colormap for the parent figure
of hax is queried or set.
For convenience, it is also possible to use this function with the command form,
colormap map_name.
The list of built-in colormaps is:
Map
viridis
jet
cubehelix
hsv
rainbow
————hot
cool
spring
summer
autumn
winter
————gray
bone
copper
pink
ocean
————colorcube
flag
lines
prism
————white

Description
default
colormap traversing blue, cyan, green, yellow, red.
colormap traversing black, blue, green, red, white with increasing intensity.
cyclic colormap traversing Hue, Saturation, Value space.
colormap traversing red, yellow, blue, green, violet.
———————————————————————————————
colormap traversing black, red, orange, yellow, white.
colormap traversing cyan, purple, magenta.
colormap traversing magenta to yellow.
colormap traversing green to yellow.
colormap traversing red, orange, yellow.
colormap traversing blue to green.
———————————————————————————————
colormap traversing black to white in shades of gray.
colormap traversing black, gray-blue, white.
colormap traversing black to light copper.
colormap traversing black, gray-pink, white.
colormap traversing black, dark-blue, white.
———————————————————————————————
equally spaced colors in RGB color space.
cyclic 4-color map of red, white, blue, black.
cyclic colormap with colors from axes "ColorOrder" property.
cyclic 6-color map of red, orange, yellow, green, blue, violet.
———————————————————————————————
all white colormap (no colors).

See also: [viridis], page 762, [jet], page 761, [cubehelix], page 760, [hsv], page 761,
[rainbow], page 762, [hot], page 760, [cool], page 759, [spring], page 762, [summer],
page 762, [autumn], page 759, [winter], page 763, [gray], page 760, [bone], page 759,
[copper], page 760, [pink], page 761, [ocean], page 761, [colorcube], page 759, [flag],
page 760, [lines], page 761, [prism], page 762, [white], page 762.

rgbplot (cmap)
rgbplot (cmap, style)
h = rgbplot ( . . . )
Plot the components of a colormap.
Two different styles are available for displaying the cmap:

Chapter 32: Image Processing

759

profile (default)
Plot the RGB line profile of the colormap for each of the channels (red,
green and blue) with the plot lines colored appropriately. Each line represents the intensity of an RGB component across the colormap.
composite Draw the colormap across the X-axis so that the actual index colors are
visible rather than the individual color components.
The optional return value h is a graphics handle to the created plot.
Run demo rgbplot to see an example of rgbplot and each style option.
See also: [colormap], page 757.

map = autumn ()
map = autumn (n)
Create color colormap. This colormap ranges from red through orange to yellow.
The argument n must be a scalar. If unspecified, the length of the current colormap,
or 64, is used.
See also: [colormap], page 757.

map = bone ()
map = bone (n)
Create color colormap. This colormap varies from black to white with gray-blue
shades.
The argument n must be a scalar. If unspecified, the length of the current colormap,
or 64, is used.
See also: [colormap], page 757.

map = colorcube ()
map = colorcube (n)
Create color colormap. This colormap is composed of as many equally spaced colors
(not grays) in the RGB color space as possible.
If there are not a perfect number n of regularly spaced colors then the remaining
entries in the colormap are gradients of pure red, green, blue, and gray.
The argument n must be a scalar. If unspecified, the length of the current colormap,
or 64, is used.
See also: [colormap], page 757.

map = cool ()
map = cool (n)
Create color colormap. The colormap varies from cyan to magenta.
The argument n must be a scalar. If unspecified, the length of the current colormap,
or 64, is used.
See also: [colormap], page 757.

760

GNU Octave

map = copper ()
map = copper (n)
Create color colormap. This colormap varies from black to a light copper tone.
The argument n must be a scalar. If unspecified, the length of the current colormap,
or 64, is used.
See also: [colormap], page 757.

map = cubehelix ()
map = cubehelix (n)
Create cubehelix colormap.
This colormap varies from black to white going though blue, green, and red tones while
maintaining a monotonically increasing perception of intensity. This is achieved by
traversing a color cube from black to white through a helix, hence the name cubehelix,
while taking into account the perceived brightness of each channel according to the
NTSC specifications from 1953.
rgbplot (cubehelix (256))
The argument n must be a scalar. If unspecified, the length of the current colormap,
or 64, is used.
Reference: Green, D. A., 2011, "A colour scheme for the display of astronomical
intensity images", Bulletin of the Astronomical Society of India, 39, 289.
See also: [colormap], page 757.

map = flag ()
map = flag (n)
Create color colormap. This colormap cycles through red, white, blue, and black with
each index change.
The argument n must be a scalar. If unspecified, the length of the current colormap,
or 64, is used.
See also: [colormap], page 757.

map = gray ()
map = gray (n)
Create gray colormap. This colormap varies from black to white with shades of gray.
The argument n must be a scalar. If unspecified, the length of the current colormap,
or 64, is used.
See also: [colormap], page 757.

map = hot ()
map = hot (n)
Create color colormap. This colormap ranges from black through dark red, red,
orange, yellow, to white.
The argument n must be a scalar. If unspecified, the length of the current colormap,
or 64, is used.
See also: [colormap], page 757.

Chapter 32: Image Processing

761

hsv (n)
Create color colormap. This colormap begins with red, changes through yellow, green,
cyan, blue, and magenta, before returning to red.
It is useful for displaying periodic functions. The map is obtained by linearly varying
the hue through all possible values while keeping constant maximum saturation and
value. The equivalent code is hsv2rgb ([(0:N-1)’/N, ones(N,2)]).
The argument n must be a scalar. If unspecified, the length of the current colormap,
or 64, is used.
See also: [colormap], page 757.

map = jet ()
map = jet (n)
Create color colormap. This colormap ranges from dark blue through blue, cyan,
green, yellow, red, to dark red.
The argument n must be a scalar. If unspecified, the length of the current colormap,
or 64, is used.
See also: [colormap], page 757.

map = lines ()
map = lines (n)
Create color colormap. This colormap is composed of the list of colors in the current
axes "ColorOrder" property. The default is blue, orange, yellow, purple, green, light
blue, and dark red.
The argument n must be a scalar. If unspecified, the length of the current colormap,
or 64, is used.
See also: [colormap], page 757.

map = ocean ()
map = ocean (n)
Create color colormap. This colormap varies from black to white with shades of blue.
The argument n must be a scalar. If unspecified, the length of the current colormap,
or 64, is used.
See also: [colormap], page 757.

map = pink ()
map = pink (n)
Create color colormap. This colormap varies from black to white with shades of
gray-pink.
This colormap gives a sepia tone when used on grayscale images.
The argument n must be a scalar. If unspecified, the length of the current colormap,
or 64, is used.
See also: [colormap], page 757.

762

GNU Octave

map = prism ()
map = prism (n)
Create color colormap. This colormap cycles through red, orange, yellow, green, blue
and violet with each index change.
The argument n must be a scalar. If unspecified, the length of the current colormap,
or 64, is used.
See also: [colormap], page 757.

map = rainbow ()
map = rainbow (n)
Create color colormap. This colormap ranges from red through orange, yellow, green,
blue, to violet.
The argument n must be a scalar. If unspecified, the length of the current colormap,
or 64, is used.
See also: [colormap], page 757.

map = spring ()
map = spring (n)
Create color colormap. This colormap varies from magenta to yellow.
The argument n must be a scalar. If unspecified, the length of the current colormap,
or 64, is used.
See also: [colormap], page 757.

map = summer ()
map = summer (n)
Create color colormap. This colormap varies from green to yellow.
The argument n must be a scalar. If unspecified, the length of the current colormap,
or 64, is used.
See also: [colormap], page 757.

map = viridis ()
map = viridis (n)
Create color colormap. This colormap ranges from dark purplish-blue through blue,
green, to yellow.
The argument n must be a scalar. If unspecified, the length of the current colormap,
or 64, is used.
See also: [colormap], page 757.

map = white ()
map = white (n)
Create color colormap. This colormap is completely white.
The argument n must be a scalar. If unspecified, the length of the current colormap,
or 64, is used.
See also: [colormap], page 757.

Chapter 32: Image Processing

763

map = winter ()
map = winter (n)
Create color colormap. This colormap varies from blue to green.
The argument n must be a scalar. If unspecified, the length of the current colormap,
or 64, is used.
See also: [colormap], page 757.

cmap = contrast (x)
cmap = contrast (x, n)
Return a gray colormap that maximizes the contrast in an image.
The returned colormap will have n rows. If n is not defined then the size of the current
colormap is used.
See also: [colormap], page 757, [brighten], page 763.
The following three functions modify the existing colormap rather than replace it.

map_out = brighten (beta)
map_out = brighten (map, beta)
map_out = brighten (h, beta)
brighten ( . . . )
Brighten or darken a colormap.
The argument beta must be a scalar between -1 and 1, where a negative value darkens
and a positive value brightens the colormap.
If the map argument is omitted, the function is applied to the current colormap.
The first argument can also be a valid graphics handle h, in which case brighten is
applied to the colormap associated with this handle.
If no output is specified then the result is written to the current colormap.
See also: [colormap], page 757, [contrast], page 763.
()
(t)
(t, inc)
("inf")
Cycle the colormap for t seconds with a color increment of inc.
Both parameters are optional. The default cycle time is 5 seconds and the default
increment is 2. If the option "inf" is given then cycle continuously until Control-C
is pressed.
When rotating, the original color 1 becomes color 2, color 2 becomes color 3, etc. A
positive or negative increment is allowed and a higher value of inc will cause faster
cycling through the colormap.

spinmap
spinmap
spinmap
spinmap

See also: [colormap], page 757.

whitebg ()
whitebg (color)
whitebg ("none")

764

GNU Octave

whitebg (hfig, . . . )
Invert the colors in the current color scheme.
The root properties are also inverted such that all subsequent plot use the new color
scheme.
If the optional argument color is present then the background color is set to color
rather than inverted. color may be a string representing one of the eight known
colors or an RGB triplet. The special string argument "none" restores the plot to
the default colors.
If the first argument hfig is a figure handle, then operate on this figure rather than
the current figure returned by gcf. The root properties will not be changed.
See also: [reset], page 424, [get], page 383, [set], page 383.
The following functions can be used to manipulate colormaps.

[Y, newmap] = cmunique (X, map)
[Y, newmap] = cmunique (RGB)
[Y, newmap] = cmunique (I)
Convert an input image X to an ouput indexed image Y which uses the smallest
colormap possible newmap.
When the input is an indexed image (X with colormap map) the output is a colormap
newmap from which any repeated rows have been eliminated. The output image, Y, is
the original input image with the indices adjusted to match the new, possibly smaller,
colormap.
When the input is an RGB image (an MxNx3 array), the output colormap will contain
one entry for every unique color in the original image. In the worst case the new map
could have as many rows as the number of pixels in the original image.
When the input is a grayscale image I, the output colormap will contain one entry
for every unique intensity value in the original image. In the worst case the new map
could have as many rows as the number of pixels in the original image.
Implementation Details:
newmap is always an Mx3 matrix, even if the input image is an intensity grayscale
image I (all three RGB planes are assigned the same value).
The output image is of class uint8 if the size of the new colormap is less than or equal
to 256. Otherwise, the output image is of class double.
See also: [rgb2ind], page 756, [gray2ind], page 756.

[Y, newmap] = cmpermute (X, map)
[Y, newmap] = cmpermute (X, map, index)
Reorder colors in a colormap.
When called with only two arguments, cmpermute randomly rearranges the colormap
map and returns a new colormap newmap. It also returns the indexed image Y which
is the equivalent of the original input image X when displayed using newmap.
When called with an optional third argument the order of colors in the new colormap
is defined by index.
Caution: index should not have repeated elements or the function will fail.

Chapter 32: Image Processing

765

32.4 Plotting on top of Images
If gnuplot is being used to display images it is possible to plot on top of images. Since an
image is a matrix it is indexed by row and column values. The plotting system is, however,
based on the traditional (x, y) system. To minimize the difference between the two systems
Octave places the origin of the coordinate system in the point corresponding to the pixel
at (1, 1). So, to plot points given by row and column values on top of an image, one should
simply call plot with the column values as the first argument and the row values as the
second. As an example the following code generates an image with random intensities
between 0 and 1, and shows the image with red circles over pixels with an intensity above
0.99.
I = rand (100, 100);
[row, col] = find (I > 0.99);
hold ("on");
imshow (I);
plot (col, row, "ro");
hold ("off");

32.5 Color Conversion
Octave supports conversion from the RGB color system to NTSC and HSV and vice versa.

hsv_map = rgb2hsv (rgb_map)
hsv_img = rgb2hsv (rgb_img)
Transform a colormap or image from RGB to HSV color space.
A color in the RGB space consists of red, green, and blue intensities.
A color in HSV space is represented by hue, saturation and value (brightness) levels
in a cylindrical coordinate system. Hue is the azimuth and describes the dominant
color. Saturation is the radial distance and gives the amount of hue mixed into the
color. Value is the height and is the amount of light in the color.
Output class and size will be the same as input.
See also: [hsv2rgb], page 765, [rgb2ind], page 756, [rgb2ntsc], page 766.

rgb_map = hsv2rgb (hsv_map)
rgb_img = hsv2rgb (hsv_img)
Transform a colormap or image from HSV to RGB color space.
A color in HSV space is represented by hue, saturation and value (brightness) levels
in a cylindrical coordinate system. Hue is the azimuth and describes the dominant
color. Saturation is the radial distance and gives the amount of hue mixed into the
color. Value is the height and is the amount of light in the color.
The input can be both a colormap or RGB image. In the case of floating point input,
values are expected to be on the [0 1] range. In the case of hue (azimuth), since the
value corresponds to an angle, mod (h, 1) is used.

766

GNU Octave

>> hsv2rgb ([0.5 1 1])
⇒ ans = 0 1 1
>> hsv2rgb ([2.5 1 1])
⇒ ans = 0 1 1
>> hsv2rgb ([3.5 1 1])
⇒ ans = 0 1 1
Output class and size will be the same as input.
See also: [rgb2hsv], page 765, [ind2rgb], page 757, [ntsc2rgb], page 766.

yiq_map = rgb2ntsc (rgb_map)
yiq_img = rgb2ntsc (rgb_img)
Transform a colormap or image from red-green-blue (RGB) color space to luminancechrominance (NTSC) space. The input may be of class uint8, uint16, single, or double.
The output is of class double.
Implementation Note: The reference matrix for the transformation is
/Y\
0.299 0.587 0.114 /R\
|I| = 0.596 -0.274 -0.322 |G|
\Q/
0.211 -0.523 0.312 \B/
as documented in http://en.wikipedia.org/wiki/YIQ and truncated to 3 significant figures. Note: The FCC version of NTSC uses only 2 significant digits and is
slightly different.
See also: [ntsc2rgb], page 766, [rgb2hsv], page 765, [rgb2ind], page 756.

rgb_map = ntsc2rgb (yiq_map)
rgb_img = ntsc2rgb (yiq_img)
Transform a colormap or image from luminance-chrominance (NTSC) space to redgreen-blue (RGB) color space.
Implementation Note: The conversion matrix is chosen to be the inverse of the matrix
used for rgb2ntsc such that
x == ntsc2rgb (rgb2ntsc (x))
matlab uses a slightly different matrix where rounding means the equality above
does not hold.
See also: [rgb2ntsc], page 766, [hsv2rgb], page 765, [ind2rgb], page 757.

767

33 Audio Processing
33.1 Audio File Utilities
The following functions allow you to read, write and retrieve information about audio files.
Various formats are supported including wav, flac and ogg vorbis.

info = audioinfo (filename)
Return information about an audio file specified by filename.

[y,
[y,
[y,
[y,

(filename)
(filename, samples)
(filename, datatype)
(filename, samples, datatype)
Read the audio file filename and return the audio data y and sampling rate fs.

fs]
fs]
fs]
fs]

=
=
=
=

audioread
audioread
audioread
audioread

The audio data is stored as matrix with rows corresponding to audio frames and
columns corresponding to channels.
The optional two-element vector argument samples specifies starting and ending
frames.
The optional argument datatype specifies the datatype to return. If it is "native",
then the type of data depends on how the data is stored in the audio file.

audiowrite (filename, y, fs)
audiowrite (filename, y, fs, name, value, . . . )
Write audio data from the matrix y to filename at sampling rate fs with the file
format determined by the file extension.
Additional name/value argument pairs may be used to specify the following options:
‘BitsPerSample’
Number of bits per sample. Valid values are 8, 16, 24, and 32. Default is
16.
‘BitRate’

Valid argument name, but ignored. Left for compatibility with matlab.

‘Quality’

Quality setting for the Ogg Vorbis compressor. Values can range between
0 and 100 with 100 being the highest quality setting. Default is 75.

‘Title’

Title for the audio file.

‘Artist’

Artist name.

‘Comment’

Comment.

audioformats ()
audioformats (format)
Display information about all supported audio formats.
If the optional argument format is given, then display only formats with names that
start with format.

768

GNU Octave

33.2 Audio Device Information
devinfo = audiodevinfo ()
devs = audiodevinfo (io)
name = audiodevinfo (io, id)
id = audiodevinfo (io, name)
id = audiodevinfo (io, rate, bits, chans)
supports = audiodevinfo (io, id, rate, bits, chans)
Return a structure describing the available audio input and output devices.
The devinfo structure has two fields "input" and "output". The value of each field
is a structure array with fields "Name", "DriverVersion" and "ID" describing an audio
device.
If the optional argument io is 1, return information about input devices only. If it is
0, return information about output devices only.
If the optional argument id is provided, return information about the corresponding
device.
If the optional argument name is provided, return the id of the named device.
Given a sampling rate, bits per sample, and number of channels for an input or output
device, return the ID of the first device that supports playback or recording using the
specified parameters.
If also given a device ID, return true if the device supports playback or recording
using those parameters.

33.3 Audio Player
The following methods are used to create and use audioplayer objects. These objects can be
used to play back audio data stored in Octave matrices and arrays. The audioplayer object
supports playback from various devices available to the system, blocking and non-blocking
playback, convenient pausing and resuming and much more.
(y, fs)
(y, fs, nbits)
(y, fs, nbits, id)
(recorder)
(recorder, id)
Create an audioplayer object that will play back data y at sample rate fs.

player
player
player
player
player

=
=
=
=
=

audioplayer
audioplayer
audioplayer
audioplayer
audioplayer

The optional arguments nbits, and id specify the bit depth and player device id,
respectively. Device IDs may be found using the audiodevinfo function. Given an
audioplayer object, use the data from the object to initialize the player.
The signal y can be a vector or a two-dimensional array.
The following example will create an audioplayer object that will play back one second
of white noise at 44100 sample rate using 8 bits per sample.
y = 0.25 * randn (2, 44100);
player = audioplayer (y, 44100, 8);
play (player);

Chapter 33: Audio Processing

769

33.3.1 Playback
The following methods are used to control player playback.

play (player)
play (player, start)
play (player, limits)
Play audio stored in the audioplayer object player without blocking.
Given optional argument start, begin playing at start samples in the recording. Given
a two-element vector limits, begin and end playing at the number of samples specified
by the elements of the vector.

playblocking (player)
playblocking (player, start)
playblocking (player, limits)
Play audio stored in the audioplayer object player with blocking.
Given optional argument start, begin playing at start samples in the recording. Given
a two-element vector limits, begin and end playing at the number of samples specified
by the elements of the vector.

pause (player)
Pause the audioplayer player.

resume (player)
Resume playback for the paused audioplayer object player.

stop (player)
Stop the playback for the audioplayer player and reset the relevant variables to their
starting values.

isplaying (player)
Return true if the audioplayer object player is currently playing back audio and false
otherwise.

33.3.2 Properties
The remaining couple of methods are used to get and set various properties of the audioplayer object.

value = get (player, name)
values = get (player)
Return the value of the property identified by name.
If name is a cell array return the values of the properties identified by the elements
of the cell array. Given only the player object, return a scalar structure with values
of all properties of player. The field names of the structure correspond to property
names.

set (player, name, value)
set (player, properties)
properties = set (player)
Set the value of property specified by name to a given value.

770

GNU Octave

If name and value are cell arrays, set each property to the corresponding value. Given
a structure of properties with fields corresponding to property names, set the value
of those properties to the field values. Given only the audioplayer object, return a
structure of settable properties.

33.4 Audio Recorder
The following methods are used to create and use audiorecorder objects. These objects can
be used to record audio data from various devices available to the system. You can use
convenient methods to retrieve that data or audioplayer objects created from that data.
Methods for blocking and non-blocking recording, pausing and resuming recording and
much more is available.

recorder = audiorecorder ()
recorder = audiorecorder (fs, nbits, channels)
recorder = audiorecorder (fs, nbits, channels, id)
Create an audiorecorder object recording 8 bit mono audio at 8000 Hz sample rate.
The optional arguments fs, nbits, channels, and id specify the sample rate, bit depth,
number of channels and recording device id, respectively. Device IDs may be found
using the audiodevinfo function.

33.4.1 Recording
The following methods control the recording process.

record (recorder)
record (recorder, length)
Record audio without blocking using the audiorecorder object recorder until stopped
or paused by the stop or pause method.
Given the optional argument length, record for length seconds.

recordblocking (recorder, length)
Record audio with blocking (synchronous I/O).
The length of the recording in seconds (length) must be specified.

pause (recorder)
Pause recording with audiorecorder object recorder.

resume (recorder)
Resume recording with the paused audiorecorder object recorder.

stop (recorder)
Stop the audiorecorder object recorder and clean up any audio streams.

isrecording (recorder)
Return true if the audiorecorder object recorder is currently recording audio and false
otherwise.

Chapter 33: Audio Processing

771

33.4.2 Data Retrieval
The following methods allow you to retrieve recorded audio data in various ways.

data = getaudiodata (recorder)
data = getaudiodata (recorder, datatype)
Return recorder audio data as a matrix with values between -1.0 and 1.0 and with as
many columns as there are channels in the recorder.
Given the optional argument datatype, convert the recorded data to the specified
type, which may be one of "double", "single", "int16", "int8" or "uint8".

player = getplayer (recorder)
Return an audioplayer object with data recorded by the audiorecorder object recorder.

player = play (recorder)
player = play (recorder, start)
player = play (recorder, [start, end])
Play the audio recorded in recorder and return a corresponding audioplayer object.
If the optional argument start is provided, begin playing start seconds in to the
recording.
If the optional argument end is provided, stop playing at end seconds in the recording.

33.4.3 Properties
The remaining two methods allow you to read or alter the properties of audiorecorder
objects.

value = get (recorder, name)
values = get (recorder)
Return the value of the property identified by name.
If name is a cell array, return the values of the properties corresponding to the elements of the cell array. Given only the recorder object, return a scalar structure with
values of all properties of recorder. The field names of the structure correspond to
property names.

set (recorder, name, value)
set (recorder, properties)
properties = set (recorder)
Set the value of property specified by name to a given value.
If name and value are cell arrays of the same size, set each property to a corresponding
value. Given a structure with fields corresponding to property names, set the value
of those properties to the corresponding field values. Given only the recorder object,
return a structure of settable properties.

33.5 Audio Data Processing
Octave provides a few functions for dealing with audio data. An audio ‘sample’ is a single
output value from an A/D converter, i.e., a small integer number (usually 8 or 16 bits), and
audio data is just a series of such samples. It can be characterized by three parameters: the

772

GNU Octave

sampling rate (measured in samples per second or Hz, e.g., 8000 or 44100), the number of
bits per sample (e.g., 8 or 16), and the number of channels (1 for mono, 2 for stereo, etc.).
There are many different formats for representing such data. Currently, only the two
most popular, linear encoding and mu-law encoding, are supported by Octave. There is an
excellent FAQ on audio formats by Guido van Rossum guido@cwi.nl which can be found at
any FAQ ftp site, in particular in the directory /pub/usenet/news.answers/audio-fmts
of the archive site rtfm.mit.edu.
Octave simply treats audio data as vectors of samples (non-mono data are not supported
yet). It is assumed that audio files using linear encoding have one of the extensions lin or
raw, and that files holding data in mu-law encoding end in au, mu, or snd.

y = lin2mu (x, n)
Convert audio data from linear to mu-law.
Mu-law values use 8-bit unsigned integers. Linear values use n-bit signed integers or
floating point values in the range -1 ≤ x ≤ 1 if n is 0.
If n is not specified it defaults to 0, 8, or 16 depending on the range of values in x.
See also: [mu2lin], page 772.

y = mu2lin (x, n)
Convert audio data from mu-law to linear.
Mu-law values are 8-bit unsigned integers. Linear values use n-bit signed integers or
floating point values in the range -1 ≤ y ≤ 1 if n is 0.
If n is not specified it defaults to 0.
See also: [lin2mu], page 772.

record (sec)
record (sec, fs)
Record sec seconds of audio from the system’s default audio input at a sampling rate
of 8000 samples per second.
If the optional argument fs is given, it specifies the sampling rate for recording.
For more control over audio recording, use the audiorecorder class.
See also: [sound], page 772, [soundsc], page 773.

sound (y)
sound (y, fs)
sound (y, fs, nbits)
Play audio data y at sample rate fs to the default audio device.
The audio signal y can be a vector or a two-column array, representing mono or stereo
audio, respectively.
If fs is not given, a default sample rate of 8000 samples per second is used.
The optional argument nbits specifies the bit depth to play to the audio device and
defaults to 8 bits.
For more control over audio playback, use the audioplayer class.
See also: [soundsc], page 773, [record], page 772.

Chapter 33: Audio Processing

773

(y)
(y, fs)
(y, fs, nbits)
( . . . , [ymin, ymax])
Scale the audio data y and play it at sample rate fs to the default audio device.
The audio signal y can be a vector or a two-column array, representing mono or stereo
audio, respectively.
If fs is not given, a default sample rate of 8000 samples per second is used.
The optional argument nbits specifies the bit depth to play to the audio device and
defaults to 8 bits.
By default, y is automatically normalized to the range [-1, 1]. If the range [ymin,
ymax] is given, then elements of y that fall within the range ymin ≤ y ≤ ymax are
scaled to the range [-1, 1] instead.
For more control over audio playback, use the audioplayer class.

soundsc
soundsc
soundsc
soundsc

See also: [sound], page 772, [record], page 772.

775

34 Object Oriented Programming
Octave has the ability to create user-defined classes—including the capabilities of operator
and function overloading. Classes can protect internal properties so that they may not be
altered accidentally which facilitates data encapsulation. In addition, rules can be created
to address the issue of class precedence in mixed class operations.
This chapter discusses the means of constructing a user class, how to query and set the
properties of a class, and how to overload operators and functions. Throughout this chapter
real code examples are given using a class designed for polynomials.

34.1 Creating a Class
This chapter illustrates user-defined classes and object oriented programming through a
custom class designed for polynomials. This class was chosen for its simplicity which does
not distract unnecessarily from the discussion of the programming features of Octave. Even
so, a bit of background on the goals of the polynomial class is necessary before the syntax
and techniques of Octave object oriented programming are introduced.
The polynomial class is used to represent polynomials of the form
a0 + a1 x + a2 x2 + . . . an xn
where a0 , a1 , etc. are elements of <. Thus the polynomial can be represented by a vector
a = [a0, a1, a2, ..., an];
This is a sufficient specification to begin writing the constructor for the polynomial class.
All object oriented classes in Octave must be located in a directory that is the name of the
class prepended with the ‘@’ symbol. For example, the polynomial class will have all of its
methods defined in the @polynomial directory.
The constructor for the class must be the name of the class itself; in this example
the constructor resides in the file @polynomial/polynomial.m. Ideally, even when the
constructor is called with no arguments it should return a valid object. A constructor for
the polynomial class might look like
## -*- texinfo -*## @deftypefn {} {} polynomial ()
## @deftypefnx {} {} polynomial (@var{a})
## Create a polynomial object representing the polynomial
##
## @example
## a0 + a1 * x + a2 * x^2 + @dots{} + an * x^n
## @end example
##
## @noindent
## from a vector of coefficients [a0 a1 a2 @dots{} an].
## @end deftypefn
function p = polynomial (a)

776

GNU Octave

if (nargin > 1)
print_usage ();
endif
if (nargin == 0)
p.poly = [0];
p = class (p, "polynomial");
else
if (strcmp (class (a), "polynomial"))
p = a;
elseif (isreal (a) && isvector (a))
p.poly = a(:).’; # force row vector
p = class (p, "polynomial");
else
error ("@polynomial: A must be a real vector");
endif
endif
endfunction
Note that the return value of the constructor must be the output of the class function.
The first argument to the class function is a structure and the second is the name of the
class itself. An example of calling the class constructor to create an instance is
p = polynomial ([1, 0, 1]);
Methods are defined by m-files in the class directory and can have embedded documentation the same as any other m-file. The help for the constructor can be obtained by
using the constructor name alone, that is, for the polynomial constructor help polynomial
will return the help string. Help can be restricted to a particular class by using the class
directory name followed by the method. For example, help @polynomial/polynomial is
another way of displaying the help string for the polynomial constructor. This second means
is the only way to obtain help for the overloaded methods and functions of a class.
The same specification mechanism can be used wherever Octave expects a function name.
For example type @polynomial/display will print the code of the display method of the
polynomial class to the screen, and dbstop @polynomial/display will set a breakpoint at
the first executable line of the display method of the polynomial class.
To check whether a variable belongs to a user class, the isobject and isa functions can
be used. For example:
p = polynomial ([1, 0, 1]);
isobject (p)
⇒ 1
isa (p, "polynomial")
⇒ 1

isobject (x)
Return true if x is a class object.
See also: [class], page 39, [typeinfo], page 39, [isa], page 39, [ismethod], page 777,
[isprop], page 377.

Chapter 34: Object Oriented Programming

777

The available methods of a class can be displayed with the methods function.

methods (obj)
methods ("classname")
mtds = methods ( . . . )
List the names of the public methods for the object obj or the named class classname.
obj may be an Octave class object or a Java object. classname may be the name of
an Octave class or a Java class.
When called with no output arguments, methods prints the list of method names to
the screen. Otherwise, the output argument mtds contains the list in a cell array of
strings.
See also: [fieldnames], page 109.
To inquire whether a particular method exists for a user class, the ismethod function can
be used.

ismethod (obj, method)
ismethod (clsname, method)
Return true if the string method is a valid method of the object obj or of the class
clsname.
See also: [isprop], page 377, [isobject], page 776.
For example:
p = polynomial ([1, 0, 1]);
ismethod (p, "roots")
⇒ 1

34.2 Class Methods
There are a number of basic class methods that can (and should) be defined to allow the
contents of the classes to be queried and set. The most basic of these is the display method.
The display method is used by Octave whenever a class should be displayed on the screen.
Usually this is the result of an Octave expression that doesn’t end with a semicolon. If this
method is not defined, then Octave won’t print anything when displaying the contents of a
class which can be confusing.

display (obj)
Display the contents of the object obj.
The Octave interpreter calls the display function whenever it needs to present a class
on-screen. Typically, this would be a statement which does not end in a semicolon to
suppress output. For example:
myobj = myclass (...)
User-defined classes should overload the display method so that something useful is
printed for a class object. Otherwise, Octave will report only that the object is an
instance of its class.
myobj = myclass (...)
⇒ myobj = 
See also: [class], page 39, [subsref], page 781, [subsasgn], page 782.

778

GNU Octave

An example of a display method for the polynomial class might be
function display (p)
printf ("%s =", inputname (1));
a = p.poly;
first = true;
for i = 1 : length (a);
if (a(i) != 0)
if (first)
first = false;
elseif (a(i) > 0 || isnan (a(i)))
printf (" +");
endif
if (a(i) < 0)
printf (" -");
endif
if (i == 1)
printf (" %.5g", abs (a(i)));
elseif (abs (a(i)) != 1)
printf (" %.5g *", abs (a(i)));
endif
if (i > 1)
printf (" X");
endif
if (i > 2)
printf (" ^ %d", i - 1);
endif
endif
endfor
if (first)
printf (" 0");
endif
printf ("\n");
endfunction
Note that in the display method it makes sense to start the method with the line
printf ("%s =", inputname (1)) to be consistent with the rest of Octave which prints
the variable name to be displayed followed by the value.
To be consistent with the Octave graphic handle classes, a class should also define the
get and set methods. The get method accepts one or two arguments. The first argument
is an object of the appropriate class. If no second argument is given then the method should
return a structure with all the properties of the class. If the optional second argument is
given it should be a property name and the specified property should be retrieved.
function val = get (p, prop)

Chapter 34: Object Oriented Programming

779

if (nargin < 1 || nargin > 2)
print_usage ();
endif
if (nargin == 1)
val.poly = p.poly;
else
if (! ischar (prop))
error ("@polynomial/get: PROPERTY must be a string");
endif
switch (prop)
case "poly"
val = p.poly;
otherwise
error (’@polynomial/get: invalid PROPERTY "%s"’, prop);
endswitch
endif
endfunction
Similarly, the first argument to the set method should be an object and any additional
arguments should be property/value pairs.
function pout = set (p, varargin)
if (numel (varargin) < 2 || rem (numel (varargin), 2) != 0)
error ("@polynomial/set: expecting PROPERTY/VALUE pairs");
endif
pout = p;
while (numel (varargin) > 1)
prop = varargin{1};
val = varargin{2};
varargin(1:2) = [];
if (! ischar (prop) || ! strcmp (prop, "poly"))
error ("@polynomial/set: invalid PROPERTY for polynomial class");
elseif (! (isreal (val) && isvector (val)))
error ("@polynomial/set: VALUE must be a real vector");
endif
pout.poly = val(:).’;
endwhile

# force row vector

endfunction
Note that Octave does not implement pass by reference; Therefore, to modify an object
requires an assignment statement using the return value from the set method.

780

GNU Octave

p = set (p, "poly", [1, 0, 0, 0, 1]);
The set method makes use of the subsasgn method of the class, and therefore this method
must also be defined. The subsasgn method is discussed more thoroughly in the next
section (see Section 34.3 [Indexing Objects], page 781).
Finally, user classes can be considered to be a special type of a structure, and they can
be saved to a file in the same manner as a structure. For example:
p = polynomial ([1, 0, 1]);
save userclass.mat p
clear p
load userclass.mat
All of the file formats supported by save and load are supported. In certain circumstances
a user class might contain a field that it doesn’t make sense to save, or a field that needs
to be initialized before it is saved. This can be done with the saveobj method of the class.

b = saveobj (a)
Method of a class to manipulate an object prior to saving it to a file.
The function saveobj is called when the object a is saved using the save function.
An example of the use of saveobj might be to remove fields of the object that don’t
make sense to be saved or it might be used to ensure that certain fields of the object
are initialized before the object is saved. For example:
function b = saveobj (a)
b = a;
if (isempty (b.field))
b.field = initfield (b);
endif
endfunction
See also: [loadobj], page 780, [class], page 39.
saveobj is called just prior to saving the class to a file. Similarly, the loadobj method is
called just after a class is loaded from a file, and can be used to ensure that any removed
fields are reinserted into the user object.

b = loadobj (a)
Method of a class to manipulate an object after loading it from a file.
The function loadobj is called when the object a is loaded using the load function.
An example of the use of saveobj might be to add fields to an object that don’t make
sense to be saved. For example:
function b = loadobj (a)
b = a;
b.addmissingfield = addfield (b);
endfunction
See also: [saveobj], page 780, [class], page 39.

Chapter 34: Object Oriented Programming

781

34.3 Indexing Objects
34.3.1 Defining Indexing And Indexed Assignment
Objects can be indexed with parentheses or braces, either like obj(idx) or like obj{idx},
or even like obj(idx).field. However, it is up to the programmer to decide what this
indexing actually means. In the case of the polynomial class p(n) might mean either
the coefficient of the n-th power of the polynomial, or it might be the evaluation of the
polynomial at n. The meaning of this subscripted referencing is determined by the subsref
method.

subsref (val, idx)
Perform the subscripted element selection operation on val according to the subscript
specified by idx.
The subscript idx must be a structure array with fields ‘type’ and ‘subs’. Valid
values for ‘type’ are "()", "{}", and ".". The ‘subs’ field may be either ":" or a
cell array of index values.
The following example shows how to extract the first two columns of a matrix
val = magic (3)
⇒ val = [ 8
1
6
3
5
7
4
9
2 ]
idx.type = "()";
idx.subs = {":", 1:2};
subsref (val, idx)
⇒ [ 8
1
3
5
4
9 ]
Note that this is the same as writing val(:, 1:2).
If idx is an empty structure array with fields ‘type’ and ‘subs’, return val.
See also: [subsasgn], page 782, [substruct], page 113.
For example, this class uses the convention that indexing with "()" evaluates the polynomial and indexing with "{}" returns the n-th coefficient (of the n-th power). The code
for the subsref method looks like
function r = subsref (p, s)
if (isempty (s))
error ("@polynomial/subsref: missing index");
endif
switch (s(1).type)
case "()"
idx = s(1).subs;
if (numel (idx) != 1)

782

GNU Octave

error ("@polynomial/subsref: need exactly one index");
endif
r = polyval (fliplr (p.poly), idx{1});
case "{}"
idx = s(1).subs;
if (numel (idx) != 1)
error ("@polynomial/subsref: need exactly one index");
endif
if (isnumeric (idx{1}))
r = p.poly(idx{1}+1);
else
r = p.poly(idx{1});
endif
case "."
fld = s.subs;
if (! strcmp (fld, "poly"))
error (’@polynomial/subsref: invalid property "%s"’, fld);
endif
r = p.poly;
otherwise
error ("@polynomial/subsref: invalid subscript type");
endswitch
if (numel (s) > 1)
r = subsref (r, s(2:end));
endif
endfunction
The equivalent functionality for subscripted assignments uses the subsasgn method.

subsasgn (val, idx, rhs)
Perform the subscripted assignment operation according to the subscript specified by
idx.
The subscript idx must be a structure array with fields ‘type’ and ‘subs’. Valid
values for ‘type’ are "()", "{}", and ".". The ‘subs’ field may be either ":" or a
cell array of index values.
The following example shows how to set the two first columns of a 3-by-3 matrix to
zero.

Chapter 34: Object Oriented Programming

783

val = magic (3);
idx.type = "()";
idx.subs = {":", 1:2};
subsasgn (val, idx, 0)
⇒ [ 0
0
6
0
0
7
0
0
2 ]
Note that this is the same as writing val(:, 1:2) = 0.
If idx is an empty structure array with fields ‘type’ and ‘subs’, return rhs.
See also:
page 783.

[subsref], page 781, [substruct], page 113, [optimize subsasgn calls],

val = optimize_subsasgn_calls ()
old_val = optimize_subsasgn_calls (new_val)
optimize_subsasgn_calls (new_val, "local")
Query or set the internal flag for subsasgn method call optimizations.
If true, Octave will attempt to eliminate the redundant copying when calling the
subsasgn method of a user-defined class.
When called from inside a function with the "local" option, the variable is changed
locally for the function and any subroutines it calls. The original variable value is
restored when exiting the function.
See also: [subsasgn], page 782.
Note that the subsref and subsasgn methods always receive the whole index chain,
while they usually handle only the first element. It is the responsibility of these methods
to handle the rest of the chain (if needed), usually by forwarding it again to subsref or
subsasgn.
If you wish to use the end keyword in subscripted expressions of an object, then there
must be an end method defined. For example, the end method for the polynomial class
might look like
function r = end (obj, index_pos, num_indices)
if (num_indices != 1)
error ("polynomial object may only have one index");
endif
r = length (obj.poly) - 1;
endfunction
which is a fairly generic end method that has a behavior similar to the end keyword for
Octave Array classes. An example using the polynomial class is then
p = polynomial ([1,2,3,4]);
p{end-1}
⇒ 3
Objects can also be used themselves as the index in a subscripted expression and this is
controlled by the subsindex function.

784

GNU Octave

idx = subsindex (obj)
Convert an object to an index vector.
When obj is a class object defined with a class constructor, then subsindex is the
overloading method that allows the conversion of this class object to a valid indexing
vector. It is important to note that subsindex must return a zero-based real integer
vector of the class "double". For example, if the class constructor were
function obj = myclass (a)
obj = class (struct ("a", a), "myclass");
endfunction
then the subsindex function
function idx = subsindex (obj)
idx = double (obj.a) - 1.0;
endfunction
could be used as follows
a = myclass (1:4);
b = 1:10;
b(a)
⇒ 1 2 3 4
See also: [class], page 39, [subsref], page 781, [subsasgn], page 782.

Finally, objects can be used like ranges by providing a colon method.

r = colon (base, limit)
r = colon (base, increment, limit)
Return the result of the colon expression corresponding to base, limit, and optionally,
increment.
This function is equivalent to the operator syntax base : limit or
base : increment : limit.
See also: [linspace], page 458.

34.3.2 Indexed Assignment Optimization
Octave’s ubiquitous lazily-copied pass-by-value semantics implies a problem for performance
of user-defined subsasgn methods. Imagine the following call to subsasgn
ss = substruct ("()", {1});
x = subsasgn (x, ss, 1);
where the corresponding method looking like this:
function x = subsasgn (x, ss, val)
...
x.myfield (ss.subs{1}) = val;
endfunction
The problem is that on entry to the subsasgn method, x is still referenced from the
caller’s scope, which means that the method will first need to unshare (copy) x and
x.myfield before performing the assignment. Upon completing the call, unless an error
occurs, the result is immediately assigned to x in the caller’s scope, so that the previous

Chapter 34: Object Oriented Programming

785

value of x.myfield is forgotten. Hence, the Octave language implies a copy of N elements
(N being the size of x.myfield), where modifying just a single element would actually
suffice. In other words, a constant-time operation is degraded to linear-time one. This
may be a real problem for user classes that intrinsically store large arrays.
To partially solve the problem Octave uses a special optimization for user-defined
subsasgn methods coded as m-files. When the method gets called as a result of the
built-in assignment syntax (not a direct subsasgn call as shown above), i.e., x(1) = 1,
AND if the subsasgn method is declared with identical input and output arguments,
as in the example above, then Octave will ignore the copy of x inside the caller’s scope;
therefore, any changes made to x during the method execution will directly affect the
caller’s copy as well. This allows, for instance, defining a polynomial class where modifying
a single element takes constant time.
It is important to understand the implications that this optimization brings. Since no
extra copy of x will exist in the caller’s scope, it is solely the callee’s responsibility to not
leave x in an invalid state if an error occurs during the execution. Also, if the method
partially changes x and then errors out, the changes will affect x in the caller’s scope.
Deleting or completely replacing x inside subsasgn will not do anything, however, only
indexed assignments matter.
Since this optimization may change the way code works (especially if badly written), a
built-in variable optimize_subsasgn_calls is provided to control it. It is on by default.
Another way to avoid the optimization is to declare subsasgn methods with different output
and input arguments like this:
function y = subsasgn (x, ss, val)
...
endfunction

34.4 Overloading Objects
34.4.1 Function Overloading
Any Octave function can be overloaded, and this allows an object-specific version of a
function to be called as needed. A pertinent example for the polynomial class might be to
overload the polyval function.
function [y, dy] = polyval (p, varargin)
if (nargout > 1)
[y, dy] = polyval (fliplr (p.poly), varargin{:});
else
y = polyval (fliplr (p.poly), varargin{:});
endif
endfunction
This function just hands off the work to the normal Octave polyval function. Another
interesting example of an overloaded function for the polynomial class is the plot function.

786

GNU Octave

function h = plot (p, varargin)
n = 128;
rmax = max (abs (roots
x = [0 : (n - 1)] / (n
if (nargout > 0)
h = plot (x, polyval
else
plot (x, polyval (p,
endif

(p.poly)));
- 1) * 2.2 * rmax - 1.1 * rmax;
(p, x), varargin{:});
x), varargin{:});

endfunction

which allows polynomials to be plotted in the domain near the region of the roots of the
polynomial.

Functions that are of particular interest for overloading are the class conversion functions
such as double. Overloading these functions allows the cast function to work with a user
class. It can also aid in the use of a class object with methods and functions from other
classes since the object can be transformed to the requisite input form for the new function.
An example double function for the polynomial class might look like

function a = double (p)
a = p.poly;
endfunction

34.4.2 Operator Overloading
The following table shows, for each built-in numerical operation, the corresponding function
name to use when providing an overloaded method for a user class.

Chapter 34: Object Oriented Programming

Operation
a+b
a-b
+a
-a
a .* b
a*b
a ./ b
a/b
a .\ b
a\b
a .^ b
a^b
ab
a >= b
a == b
a != b
a&b
a|b
!a
a’
a.’
a:b
a:b:c
[a, b]
[a; b]
a(s1 ,...,sn )
a(s1 ,...,sn ) = b
b(a)
display

Method
plus (a, b)
minus (a, b)
uplus (a)
uminus (a)
times (a, b)
mtimes (a, b)
rdivide (a, b)
mrdivide (a, b)
ldivide (a, b)
mldivide (a, b)
power (a, b)
mpower (a, b)
lt (a, b)
le (a, b)
gt (a, b)
ge (a, b)
eq (a, b)
ne (a, b)
and (a, b)
or (a, b)
not (a)
ctranspose (a)
transpose (a)
colon (a, b)
colon (a, b, c)
horzcat (a, b)
vertcat (a, b)
subsref (a, s)
subsasgn (a, s, b)
subsindex (a)
display (a)

787

Description
Binary addition
Binary subtraction
Unary addition
Unary subtraction
Element-wise multiplication
Matrix multiplication
Element-wise right division
Matrix right division
Element-wise left division
Matrix left division
Element-wise power
Matrix power
Less than
Less than or equal to
Greater than
Greater than or equal to
Equal to
Not equal to
Logical and
Logical or
Logical not
Complex conjugate transpose
Transpose
Two element range
Three element range
Horizontal concatenation
Vertical concatenation
Subscripted reference
Subscripted assignment
Convert object to index
Object display

Table 34.1: Available overloaded operators and their corresponding class method
An example mtimes method for the polynomial class might look like
function p = mtimes (a, b)
p = polynomial (conv (double (a), double (b)));
endfunction

34.4.3 Precedence of Objects
Many functions and operators take two or more arguments and the situation can easily arise
where these functions are called with objects of different classes. It is therefore necessary
to determine the precedence of which method from which class to call when there are
mixed objects given to a function or operator. To do this the superiorto and inferiorto
functions can be used

788

GNU Octave

superiorto (class_name, . . . )
When called from a class constructor, mark the object currently constructed as having
a higher precedence than class name.
More that one such class can be specified in a single call. This function may only be
called from a class constructor.
See also: [inferiorto], page 788.

inferiorto (class_name, . . . )
When called from a class constructor, mark the object currently constructed as having
a lower precedence than class name.
More that one such class can be specified in a single call. This function may only be
called from a class constructor.
See also: [superiorto], page 787.
With the polynomial class, consider the case
2 * polynomial ([1, 0, 1]);
that mixes an object of the class "double" with an object of the class "polynomial". In
this case the return type should be "polynomial" and so the superiorto function is used
in the class constructor. In particular the polynomial class constructor would be modified
to
## -*- texinfo -*## @deftypefn {} {} polynomial ()
## @deftypefnx {} {} polynomial (@var{a})
## Create a polynomial object representing the polynomial
##
## @example
## a0 + a1 * x + a2 * x^2 + @dots{} + an * x^n
## @end example
##
## @noindent
## from a vector of coefficients [a0 a1 a2 @dots{} an].
## @end deftypefn
function p = polynomial (a)
if (nargin > 1)
print_usage ();
endif
if (nargin == 0)
p.poly = [0];
p = class (p, "polynomial");
else
if (strcmp (class (a), "polynomial"))
p = a;
elseif (isreal (a) && isvector (a))

Chapter 34: Object Oriented Programming

789

p.poly = a(:).’; # force row vector
p = class (p, "polynomial");
else
error ("@polynomial: A must be a real vector");
endif
endif
superiorto ("double");
endfunction
Note that user classes always have higher precedence than built-in Octave types. Thus,
marking the polynomial class higher than the "double" class is not actually necessary.
When confronted with two objects of equal precedence, Octave will use the method of
the object that appears first in the list of arguments.

34.5 Inheritance and Aggregation
Using classes to build new classes is supported by Octave through the use of both inheritance
and aggregation.
Class inheritance is provided by Octave using the class function in the class constructor.
As in the case of the polynomial class, the Octave programmer will create a structure that
contains the data fields required by the class, and then call the class function to indicate
that an object is to be created from the structure. Creating a child of an existing object
is done by creating an object of the parent class and providing that object as the third
argument of the class function.
This is most easily demonstrated by example. Suppose the programmer needs a FIR
filter, i.e., a filter with a numerator polynomial but a denominator of 1. In traditional
Octave programming this would be performed as follows.
octave:1> x = [some data vector];
octave:2> n = [some coefficient vector];
octave:3> y = filter (n, 1, x);
The equivalent behavior can be implemented as a class @FIRfilter. The constructor for
this class is the file FIRfilter.m in the class directory @FIRfilter.
## -*- texinfo -*## @deftypefn {} {} FIRfilter ()
## @deftypefnx {} {} FIRfilter (@var{p})
## Create a FIR filter with polynomial @var{p} as coefficient vector.
## @end deftypefn
function f = FIRfilter (p)
if (nargin > 1)
print_usage ();
endif
if (nargin == 0)

790

GNU Octave

p = @polynomial ([1]);
elseif (! isa (p, "polynomial"))
error ("@FIRfilter: P must be a polynomial object");
endif
f.polynomial = [];
f = class (f, "FIRfilter", p);
endfunction
As before, the leading comments provide documentation for the class constructor. This
constructor is very similar to the polynomial class constructor, except that a polynomial object is passed as the third argument to the class function, telling Octave that the FIRfilter
class will be derived from the polynomial class. The FIR filter class itself does not have any
data fields, but it must provide a struct to the class function. Given that the @polynomial
constructor will add an element named polynomial to the object struct, the @FIRfilter just
initializes a struct with a dummy field polynomial which will later be overwritten.
Note that the sample code always provides for the case in which no arguments are
supplied. This is important because Octave will call a constructor with no arguments when
loading objects from saved files in order to determine the inheritance structure.
A class may be a child of more than one class (see [class], page 39), and inheritance may
be nested. There is no limitation to the number of parents or the level of nesting other than
memory or other physical issues.
As before, a class requires a display method. A simple example might be
function display (f)
printf ("%s.polynomial", inputname (1));
display (f.polynomial);
endfunction
Note that the FIRfilter’s display method relies on the display method from the polynomial class to actually display the filter coefficients.
Once a constructor and display method exist, it is possible to create an instance of the
class. It is also possible to check the class type and examine the underlying structure.
octave:1> f = FIRfilter (polynomial ([1 1 1]/3))
f.polynomial = 0.33333 + 0.33333 * X + 0.33333 * X ^ 2
octave:2> class (f)
ans = FIRfilter
octave:3> isa (f, "FIRfilter")
ans = 1
octave:4> isa (f, "polynomial")
ans = 1
octave:5> struct (f)
ans =
scalar structure containing the fields:
polynomial = 0.33333 + 0.33333 * X + 0.33333 * X ^ 2

Chapter 34: Object Oriented Programming

791

The only thing remaining to make this class usable is a method for processing data. But
before that, it is usually desirable to also have a way of changing the data stored in a class.
Since the fields in the underlying struct are private by default, it is necessary to provide a
mechanism to access the fields. The subsref method may be used for both tasks.
function r = subsref (f, x)
switch (x.type)
case "()"
n = f.polynomial;
r = filter (n.poly, 1, x.subs{1});
case "."
fld = x.subs;
if (! strcmp (fld, "polynomial"))
error (’@FIRfilter/subsref: invalid property "%s"’, fld);
endif
r = f.polynomial;
otherwise
error ("@FIRfilter/subsref: invalid subscript type for FIR filter");
endswitch
endfunction
The "()" case allows us to filter data using the polynomial provided to the constructor.
octave:2> f = FIRfilter (polynomial ([1 1 1]/3));
octave:3> x = ones (5,1);
octave:4> y = f(x)
y =
0.33333
0.66667
1.00000
1.00000
1.00000
The "." case allows us to view the contents of the polynomial field.
octave:1> f = FIRfilter (polynomial ([1 1 1]/3));
octave:2> f.polynomial
ans = 0.33333 + 0.33333 * X + 0.33333 * X ^ 2
In order to change the contents of the object a subsasgn method is needed. For example,
the following code makes the polynomial field publicly writable

792

GNU Octave

function fout = subsasgn (f, index, val)
switch (index.type)
case "."
fld = index.subs;
if (! strcmp (fld, "polynomial"))
error (’@FIRfilter/subsasgn: invalid property "%s"’, fld);
endif
fout = f;
fout.polynomial = val;
otherwise
error ("@FIRfilter/subsasgn: Invalid index type")
endswitch
endfunction
so that
octave:1> f = FIRfilter ();
octave:2> f.polynomial = polynomial ([1 2 3])
f.polynomial = 1 + 2 * X + 3 * X ^ 2
Defining the FIRfilter class as a child of the polynomial class implies that a FIRfilter
object may be used any place that a polynomial object may be used. This is not a normal
use of a filter. It may be a more sensible design approach to use aggregation rather than
inheritance. In this case, the polynomial is simply a field in the class structure. A class
constructor for the aggregation case might be
##
##
##
##
##

-*- texinfo -*@deftypefn {} {} FIRfilter ()
@deftypefnx {} {} FIRfilter (@var{p})
Create a FIR filter with polynomial @var{p} as coefficient vector.
@end deftypefn

function f = FIRfilter (p)
if (nargin > 1)
print_usage ();
endif
if (nargin == 0)
f.polynomial = @polynomial ([1]);
else
if (! isa (p, "polynomial"))
error ("@FIRfilter: P must be a polynomial object");
endif
f.polynomial = p;
endif

Chapter 34: Object Oriented Programming

793

f = class (f, "FIRfilter");
endfunction
For this example only the constructor needs changing, and all other class methods stay
the same.

34.6 classdef Classes
Since version 4.0, Octave has limited support for classdef classes. In contrast to the
aforementioned classes, called old style classes in this section, classdef classes can be
defined within a single m-file. Other innovations of classdef classes are:
• access rights for properties and methods,
• static methods, i.e., methods that are independent of an object, and
• the distinction between value and handle classes.
Several features have to be added in future versions of Octave to be fully compatible
to matlab. An overview of what is missing can be found at http://wiki.octave.org/
Classdef.

34.6.1 Creating a classdef Class
A very basic classdef value class (see Section 34.6.5 [Value Classes vs. Handle Classes],
page 797) is defined by:
classdef some_class
properties
endproperties
methods
endmethods
endclassdef
In contrast to old style classes, the properties-endproperties block as well as the
methods-endmethods block can be used to define properties and methods of the class.
Because both blocks are empty, they can be omitted in this particular case.
For simplicity, a more advanced implementation of a classdef class is shown using the
polynomial example again (see Section 34.1 [Creating a Class], page 775):
classdef polynomial2
properties
poly = 0;
endproperties
methods
function p = polynomial2 (a)
if (nargin > 1)
print_usage ();
endif

794

GNU Octave

if (nargin == 1)
if (isa (a, "polynomial2"))
p.poly = a.poly;
elseif (isreal (a) && isvector (a))
p.poly = a(:).’; # force row vector
else
error ("polynomial2: A must be a real vector");
endif
endif
endfunction
function disp (p)
a = p.poly;
first = true;
for i = 1 : length (a);
if (a(i) != 0)
if (first)
first = false;
elseif (a(i) > 0 || isnan (a(i)))
printf (" +");
endif
if (a(i) < 0)
printf (" -");
endif
if (i == 1)
printf (" %.5g", abs (a(i)));
elseif (abs (a(i)) != 1)
printf (" %.5g *", abs (a(i)));
endif
if (i > 1)
printf (" X");
endif
if (i > 2)
printf (" ^ %d", i - 1);
endif
endif
endfor
if (first)
printf (" 0");
endif
printf ("\n");
endfunction
endmethods
endclassdef
An object of class polynomial2 is created by calling the class constructor:

Chapter 34: Object Oriented Programming

795

>> p = polynomial2 ([1, 0, 1])
⇒ p =
1 + X ^ 2

34.6.2 Properties
All class properties must be defined within properties blocks. The definition of a default
value for a property is optional and can be omitted. The default initial value for each class
property is [].
A properties block can have additional attributes to specify access rights or to define
constants:
classdef some_class
properties (Access = mode)
prop1
endproperties
properties (SetAccess = mode, GetAccess = mode)
prop2
endproperties
properties (Constant = true)
prop3 = pi ()
endproperties
properties
prop4 = 1337
endproperties
endclassdef
where mode can be one of:
public

The properties can be accessed from everywhere.

private

The properties can only be accessed from class methods. Subclasses of that
class cannot access them.

protected
The properties can only be accessed from class methods and from subclasses of
that class.
When creating an object of some_class, prop1 has the default value [] and reading
from and writing to prop1 is defined by a single mode. For prop2 the read and write access
can be set differently. Finally, prop3 is a constant property which can only be initialized
once within the properties block.
By default, in the example prop4, properties are not constant and have public read and
write access.

34.6.3 Methods
All class methods must be defined within methods blocks. An exception to this rule is
described at the end of this subsection. Those methods blocks can have additional attributes

796

GNU Octave

specifying the access rights or whether the methods are static, i.e., methods that can be
called without creating an object of that class.
classdef some_class
methods
function obj = some_class ()
disp ("New instance created.");
endfunction
function disp (obj)
disp ("Here is some_class.");
endfunction
endmethods
methods (Access = mode)
function r = func (obj, r)
r = 2 * r;
endfunction
endmethods
methods (Static = true)
function c = circumference (radius)
c = 2 * pi () .* radius;
endfunction
endmethods
endclassdef
The constructor of the class is declared in the methods block and must have the same
name as the class and exactly one output argument which is an object of its class.
It is also possible to overload built-in or inherited methods, like the disp function in
the example above to tell Octave how objects of some_class should be displayed (see
Section 34.2 [Class Methods], page 777).
In general, the first argument in a method definition is always the object that it is called
from. Class methods can either be called by passing the object as the first argument to
that method or by calling the object followed by a dot (".") and the method’s name with
subsequent arguments:
>> obj = some_class ();
New instance created.
>> disp (obj);
# both are
>> obj.disp (); # equal
In some_class, the method func is defined within a methods block setting the Access
attribute to mode, which is one of:
public

The methods can be accessed from everywhere.

private

The methods can only be accessed from other class methods. Subclasses of that
class cannot access them.

Chapter 34: Object Oriented Programming

797

protected
The methods can only be accessed from other class methods and from subclasses
of that class.
The default access for methods is public.
Finally, the method circumference is defined in a static methods block and can be used
without creating an object of some_class. This is useful for methods, that do not depend
on any class properties. The class name and the name of the static method, separated by
a dot ("."), call this static method. In contrast to non-static methods, the object is not
passed as first argument even if called using an object of some_class.
>> some_class.circumference (3)
⇒ ans = 18.850
>> obj = some_class ();
New instance created.
>> obj.circumference (3)
⇒ ans = 18.850
Additionally, class methods can be defined as functions in a folder of the same name
as the class prepended with the ‘@’ symbol (see Section 34.1 [Creating a Class], page 775).
The main classdef file has to be stored in this class folder as well.

34.6.4 Inheritance
Classes can inherit from other classes. In this case all properties and methods of the
superclass are inherited to the subclass, considering their access rights. Use this syntax to
inherit from superclass:
classdef subclass < superclass
...
endclassdef

34.6.5 Value Classes vs. Handle Classes
There are two intrinsically different types of classdef classes, whose major difference is
the behavior regarding variable assignment. The first type are value classes:
classdef value_class
properties
prop1
endproperties
methods
function obj = set_prop1 (obj, val)
obj.prop1 = val;
endfunction
endmethods
endclassdef
Assigning an object of that class to another variable essentially creates a new object:

798

GNU Octave

>>
>>
>>
>>
>>
⇒
>>
⇒

a = value_class ();
a.prop1 = 1;
b = a;
b.prop1 = 2;
b.prop1
ans = 2
a.prop1
ans = 1

But that also means that you might have to assign the output of a method that changes
properties back to the object manually:
>>
>>
>>
⇒

a = value_class ();
a.prop1 = 1;
a.set_prop1 (3);
ans =

1 2
3 3
Octave Manual

Navigation menu

Versions of this User Manual:

Views

Navigation
