Eureka_The_Solver_Owners_Handbook_1987 Eureka The Solver Owners Handbook 1987

Eureka_The_Solver_Owners_Handbook_1987 Eureka_The_Solver_Owners_Handbook_1987

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Eureka: The

Solver~

Owner's Handbook

BORLAND INTERNATIONAL, INC.
4585 SCOTTS VALLEY DRIVE
SCOTTS VALLEY, CALIFORNIA 95066
Copyright © 1987
All Rights Reserved, First Printing, 1987
Printed in U.s.A.

10 9 8 7 6 5 4 3

Eureka: The Solver Owner's Handbook

Table of Contents

Introduction ............................................................................................................. 1
What Can You Do With Eureka? ............................................................................ 1
Structure of This Manual .......................................................................................... 3
Typography .................................................... ;............................................................ 4
The Distribution Disk ............................................................................................... 4
Hardware and Software Requirements ................................................................... 5
Acknowledgements .................................................................................................... 5
CHAPTER 1: Overview ........................................................................................ 7
Command Menus and Windows .............................................................................. 8
The Editor .................................................................................................................. 9
Equation File Syntax ................................................................................................. 9
Variables ................................................................................................................. 9
Initializations ....................................................................................................... 10
Special Symbols .................................................................................................. 10
Reserved Words .................................................................................................. 10
Solving Equations ..................................................................................................... 11
How Eureka Calculates Solutions ..................................................................... 11
Evaluating Equations .............................................................................................. 12
Reporting .................................................................................................................. l2
CHAPTER 2: Getting Started ............................................................................ 13
Setting Up the Program .......................................................................................... 14

Copying to a Floppy Disk .................................................................................. 14
One Floppy Disk Drive .................................................................................. 14
Two Floppy Disk Drives ................................................................................ 15
Copying to a Hard Disk ...................................................................................... 15
Copying to a RAM Disk ..................................................................................... 15
Tutorial Part 1: Using the Menus and the Editor .................................................16
Starting Eureka .................................................................................................... 16
What to Do If You Want Out of Eureka ........................................................... 17
Making a Selection From the Main Menu ...................................................... 20
Creating a File ...........'.......................................................................................... 21
Using the Eureka Editor ............................................................................... 22
Saving A File ....................................................................................................... 23
Tutorial Part 2: Creating and Solving an Equation File ..................................... 24
Creating the Equation File ................................................................................ 24
Writing an Equation ...................................................................................... 25
Writing a Comment ...................................................................................... 25
Solving the Equation File .................................................................................. 26
Modifying the Equation File ............................................................................. 27
Plotting a Graph ................................................................................................. 28
Printing a Report ................................................................................................ 30
Tutorial Part 3: Using Special Features ............................................................... 30
Writing a Directive ." .......................................................................................... 31
Changing Default Settings ................................................................................. 31
Selecting the Variables Option and Finding Other Roots ............................. 32
Evaluating the Solution ...................................................................................... 33
Using the Window Menu .................................................................................. 33
Saving the Equation File ................................................................................... 34
Leaving the Program .......................................................................................... 34
Wrap Up ................................................................................................................... 35
Eureka Editor Summary ......................................................................................... 35
General Editing Techniques .............................................................................. 35
Basic Features of the Editor .............................................................................. 36
More About the Editor Commands .................................................................. 37
CHAPTER 3: Modeling Tips and Techniques ................................................. 39
How Eureka Works ................................................................................................. 39
Eureka's Problem-Solving Methods ................................................................. 40
Solving Techniques .................................................................................................. 40
Maximizing and Minimizing ............................................................................. 40
Continuing a Search ............................................................................................ 41
Modifying Constants and Variables ................................................................... 41
Finding Alternate Solutions .............................................................................. 42
Running Problems from Batch Files ................................................................ 42
Interface with Data Files: Sample BASIC Input File ................................... 43
Eureka: The Solver Owner's Handbook

Modeling Tips for Successful Problem-Solving .................................................... 44
Helping Eureka Find Correct Solutions .......................................................... 44
About Error Messages ........................................................................................ 46
CHAPTER 4: Menu Commands and Settings ................................................. 47
Main Menu ............................................................................................................... 48
The File Commands ................................................................................................ 51
File ....................................................................................................................... 51
Load ..................................................................................................................... 51
New ...................................................................................................................... 52
Save ...................................................................................................................... 52
Write to ................................................................................................................ 53
Directory .............................................................................................................. 53
Change dir ........................................................................................................... 54
Rename ................................................................................................................ 54
OS shell ............................................................................................................... 55
Quit ...................................................................................................................... 55
The Edit Command ................................................................................................. 56
Edit ....................................................................................................................... 56
The Solve Command ............................................................................................... 57
Solve ..................................................................................................................... 57
The Commands Commands .................................................................................... 59
Commands ........................................................................................................... 59
Verif}r .................................................................................................................... 59
Calculator ............................................................................................................. 60
Find other ............................................................................................................ 61
Iterate .................................................................................................................. 61
The Report Commands ........................................................................................... 63
Report ................................................................................................................... 63
Go ......................................................................................................................... 64
Output .................................................................................................................. 64
Formatted ............................................................................................................ 65
Capture ................................................................................................................ 65
Log file name ...................................................................................................... 65
The Graph Commands ............................................................................................ 66
Graph .................... :.............................................................................................. 66
Plot ....................................................................................................................... 66
Output .................................................................................................................. 67
List ....................................................................................................................... 68
Function ............................................................................................................... 69
The Options Commands ......................................................................................... 70
Options ................................................................................................................. 70
Variables ............................................................................................................... 70
Settings ................................................................................................................ 72

Table of Contents

iii

Colors .................................................................................................................... 74
Directories ............................................................................................................ 74
Load setup ........................................................................................................... 75
Write setup ......................................................................................................... 75
The Window Commands ........................................................................................ 76
Window ............................................................................................................... 76
Open .................................................................................................................... 76
Close .................................................................................................................... 77
Next ...................................................................................................................... 77
Zoom .................................................................................................................... 77
Tile ....................................................................................................................... 77
Stack ..................................................................................................................... 78
Goto ...................................................................................................................... 78
CHAPTER 5: The Equation File: Syntax, Directives, and Functions ......... 79
Contents of the Equation File ................................................................................ 79
Equations ............................................................................................................. 80
Initialization Values ............................................................................................. 81
User-defined Functions ...................................................................................... 81
Directives ............................................................................................................ 82
Comments ........................................................................................................... 82
Notes On Equation File Syntax ............................................................................. 82
Variable Names ................................................................................................... 82
Values and Precision .......................................................................................... 83
Mathematical Operators and Order of Evaluation .......................................... 83
Syntax Rules ........................................................................................................ 84
Special Symbols .................................................................................................. 85
Directives ................................................................................................................. 86
accuracy ............................................................................................................... 88
casefold ................................................................................................................ 88
complex (numbers) ............................................................................................. 88
digits .................................................................................................................... 89
end ........................................................................................................................ 90
finanmode ............................................................................................................ 90
finansmooth ......................................................................................................... 90
include .................................................................................................................. 91
initval .................................................................................................................... 91
listdefault ............................................................................................................. 92
max(imize) ........................................................................................................... 92
maxtime ............................................................................................................... 93
min(imize) ........................................................................................................... 93
penalty ................................................................................................................. 93
plotdefault ............................................................................................................ 94
radius ................................................................................................................... 94
iv

Eureka: The Solver Owner's Handbook

rootsign ................................................................................................................ 94
settings ................................................................................................................. 95
solve ...................................................................................................................... 95
substlevel ............................................................................................................. 96
syntax ................................................................................................................... 96
units (conversion) ................................................................................................ 97
Eureka Functions ..................................................................................................... 99
Financial Functions ............................................................................................ 99
Variables ........................................................................................................ 100
Functions ...................................................................................................... 100
Example ........................................................................................................ 100
User-defined Functions ..................................................................................... 101
Relationship of Variables and Functions in Eureka .................................. 101
Writing User-defined Functions ................................................................ 102
Error Messages ............................................................................................ 103
CHAPTER 6: Worked Examples ..................................................................... 105
Problem 1 A Quick Demonstration of Eureka's Mathematical Capabilities 107
Statement of the Problem ................................................................................ 108
Classic Identities .......................................................................................... 108
Derivatives and Integrals ........................................................................... 108
Maximizations and Minimizations ............................................................. 108
Complex Variables ....................................................................................... 109
Equations ........................................................................................................... 109
Equation Files and Solutions ............................................................................ 111
Graphing the Functions .................................................................................... III
Generating Reports ............................................................................................ III
Problem 2 Loan Payoff ....................................................................................... 123
Statement of the Problem ................................................................................ 123
Equations ........................................................................................................... 124
Equation File and Solution .............................................................................. 124
Problem 3 Ladders in an Alley: Solving Simultaneous Nonlinear Equations 127
Statement of the Problem ................................................................................ 127
Equations ........................................................................................................... 127
Equation File and Solution .............................................................................. 127
Problem 4 Distribution of Line Charges .......................................................... 130
Statement of the Problem ................................................................................ 130
Equations ............................................................................................................ 131
Equation File and Solution .............................................................................. 132
Problem 5 Radioactive Decay ............................................................................ 134
Statement of the Problem ................................................................................ 134
Equations ............................................................................................................ 134
Equation File and Solution .............................................................................. 135
Problem 6 Least-Squares Fit ............................................................................. 137
Table of Contents

Statement of the Problem ................................................................................. 137
Equations ............................................................................................................ 137
Additional Techniques: Inputting Data From Another Program ................. 139
Equation File and Solution ............................................................................... 139
Problem 7 Motion of a Projectile in a Gravitational Field
with Air Resistance .............................................................................................. 141
Statement of the Problem ................................................................................. 141
Equations ............................................................................................................. 141
Equation File and Solution ............................................................................... 142
Problem 8 Charged Masses in a Gravitational Field ...................................... ,144
Statement of the Problem ................................................................................. 144
Equations ............................................................................................................ 145
Equation File and Solution ............................................................................... 146
Problem 9 Using the Built-in Polynomial Function to Find Real
and Complex Roots ............................................................................................... 148
Statement of the Problem ................................................................................. 148
Equations ............................................................................................................ 148
Graphing the Functions .................................................................................... 149
Equation File and Solutions ............................................................................. 149
Problem 10 Maximization: Designing a Maximum-Square-Footage
House Plan While Satisfying Several Constraints ............................................ 152
Statement of the Problem ................................................................................. 152
Equations ............................................................................................................. 152
Equation File and Solution ............................................................................... 154
Problem 11 Maximization: Linear Programming in Manufacturing .............. 158
Statement of the Problem ................................................................................. 158
Equations ............................................................................................................ 158
Equation File and Solution ............................................................................... 159
Problem l2 Equity in a House: the paymt, pval and fyal functions .............. 161
Statement of the Problem ................................................................................. 161
Equations ............................................................................................................ 161
Equation File and Solution ............................................................................... 162
Problem 13 Probability in a Card Game ........................................................... 165
Statement of the Problem ................................................................................. 165
Equations ............................................................................................................ 165
Equation File and Solution ............................................................................... 166
Problem 14 Income Distribution: The integ Function .................................... 169
Statement of the Problem ................................................................................. 169
Equations ............................................................................................................ 169
Equation File and Solution ............................................................................... 170
Problem 15 Chemical Calculations .................................................................... 172
Statement of the Problem ................................................................................. 172
Equations ............................................................................................................ 173
Equation File and Solution .; ...................... ;...................................................... 175
vi

Eureka: The Solver Owner's Handbook

Using the $ include Directive ................................................................. ,.. 175
Statement of the Problem ................................................................................ 178
Equations .................................................................. ;........................................ 179
Equation File and Solution .............................................................................. 179
Equations ............................................................................................................ 181
Equation File and Solution .............................................................................. 182
Equations ........................................................................................................... 184
Equation File and Solution .............................................................................. 185
APPENDIX A: Eureka Quick Reference Table

............................................. 187

APPENDIX B: Eureka Editor Commands ...................................................... 191
Cursor Movement Commands ............................................................................. 192
Insert and Delete Commands .............................................................................. 193
Block Commands ................................................................................................... 194
Miscellaneous Editing Commands ...................................................................... 195
APPENDIX C: Built-in Functions ................................................................... 197
abs ........................................................................................................................... 198
atan2 ........................................................................................................................ 198
cos ............................................................................................................................ 198
cosh .......................................................................................................................... 199
deriv ......................................................................................................................... 199
exp ........................................................................................................................... 200
fact ........................................................................................................................... 200
Hoor ......................................................................................................................... 200
frac ........................................................................................................................... 201
fyal ........................................................................................................................... 201
im .............................................................................................................................. 201
integ ........................................................................................................................ 201
In .............................................................................................................................. 202
log10 ........................................................................................................................ 202
msqrt ....................................................................................................................... 203
ncum ........................................................................................................................ 203
paymt ....................................................................................................................... 204
pi .............................................................................................................................. 204
polar ........................................................................................................................ 204
poly .......................................................................................................................... 205
pos ........................................................................................................................... 205
pval .......................................................................................................................... 206
re .............................................................................................................................. 206
sgn ........................................................................................................................... 206
sin .................................................................................. ;......................................... 206
sinh .......................................................................................................................... 207
Table of Contents

vii

sqrt .......................................................................................................................... 207
sum .......................................................................................................................... 207
tan ........................................................................................................................... 208
tanh ......................................................................................................................... 208
APPENDIX D: A DOS Primer ........................................................................ 209
What Is DOS? ........................................................................................................ 209
How to Load a Program ........................................................................................ 210
Directories ............................................................................................................... 211
Subdirectories ......................................................................................................... 212
Where Am I? The $p $g Prompt .......................................................................... 212
The AUTOEXEC.BAT File ................................................................................... 213
Changing Directories ............................................................................................. 214
APPENDIX E: Error Messages ........................................................................ 217
Assignment Must Be to a Variable or Function .................................................. 217
Bad Function Call .................................................................................................. 217
Cannot ComplexifY Floor or Frac ......................................................................... 218
Cannot Have Complex Derivatives or Integrals ................................................. 218
Circular Function Definition ................................................................................ 218
Comma Expected ................................. ""''',, ......................................................... 218
Command Not Understood .............................·...................................................... 219
Constant Expected ................................................................................................. 219
Dynamic Memory Depleted ................................................................................. 219
Equations are Inconsistent .................................................................................... 219
File Already Exists ................................................................................................. 219
File Not Found ...................................................................................................... 220
File Too Large ........................................................................................................ 220
Function is Multiply Defined .............................................................................. 220
Function is Undefined .......................................................................................... 220
Identifier Expected ................................................................................................ 221
Identifier Too Long ................................................................................................ 221
Incorrect Number of Function Arguments .......................................................... 221
Invalid Function Argument ................................................................................... 221
Invalid Function Definition ................................................................................. 222
Invalid Number ..................................................................................................... 222
Memory Heap Overflow ....................................................................................... 222
Must Differentiate With Respect to a Variable ................................................. 222
Numeral or Operator Expected ........................................................................... 222
Polynomial is Invalid ............................................................................................. 223
Printer Not Ready ................................................................................................. 223
Quote Mark Expected .......................................................................................... 223
Relational Operator Expected .............................................................................. 223
Second Argument Must Be a Variable ......................... ;...................................... 223
viii

Eureka: The Solver Owner's Handbook

String Space Overflow .......................................................................................... 224
Sum is Invalid ........................................................................................................ 224
Sum is Too Large ................................................................................................... 224
Syntax Error for Unit Conversion ........................................................................ 224
Too Few Arguments to Poly ................................................................................. 224
Too Many Active Variables ................................................................................... 225
Too Many Constants .............................................................................................. 225
Too Many Formulas ............................................................................................... 225
Too Many Functions .............................................................................................. 225
Too Many Unit Conversions ................................................................................. 225
Too Many Variables ............................................................................................... 226
Unable to Create File ........................................................................................... 226
Unexpected End of File ........................................................................................ 226
Unexpected $ End Directive ............................................................................... 226
Unmatched Parentheses ....................................................................................... 226
Unrecognized Character ....................................................................................... 227
Unrecognized Setting Value ................................................................................. 227
Use of Uninitialized Variable ............................................................................... 227
Variable Expected .................................................................................................. 227
Variable Has Not Been Initialized ....................................................................... 227
APPENDIX F: Customizing Eureka ............................................................... 229
Running EINST ..................................................................................................... 229
The Eureka Directory Option ......................................................................... 230
The Text Editor Command Option ................................................................. 230
The Default Edit Mode Option ...................................................................... 231
The Screen Mode Option ................................................................................ 232
Default Display Mode ................................................................................. 232
Color Display Mode .................................................................................... 232
Black and White Display Mode ................................................................. 232
Monochrome Display Mode ....................................................................... 233
Quitting the Program ............................................................................................ 233
GLOSSARY
INDEX

.......................................................................................................... 235

.................................................................................................................. 240

Table of Contents

Eureka: The Solver Owner's Handbook

Introduction

Welcome to the world of streamlined problem solving. Eureka: The Solver is an
equation solver with a difference; this program solves systems of equations (linear,
nonlinear, and transcendental), and it also:
• Solves high-order polynomials
• Evaluates derivatives and definite integrals
• Minimizes and maximizes functions with or without constraints
Eureka makes it easy for you to solve your real-world mathematical problems
very quickly, without having to first learn numerical approximation techniques.
You don't have to be a computer specialist to use Eureka. It was designed for
people like you - scientists, engineers, financial analysts, and all other professionals and students - people who need to solve equations and who would rather
use a computer instead of a calculator to do the solving.

What Can You Do With Eureka?
Eureka is a versatile tool designed for solving any problem that can be expressed as
a system of linear or nonlinear equations. Its capabilities range from solving the
simple to the very complex.

Without any special training, you can use Eureka to solve:
• Simple linear equations in one variable
• Systems of linear equations
• Nonlinear equations or systems of equations
• Maximizations or minimizations of a variable
• High-order polynomials (real and complex)
• Derivatives and definite integrals
Eureka also:
• Works with inequalities
• Plots and prints the graphs of functions
• Converts units automatically
• Generates reports
Eureka contains several built-in functions, including:
• The familiar trigonometric functions
• The logarithmic and exponential functions
• Several useful statistical and financial functions
The program also includes a Calculator Mode, which has all the features of a
powerful hand-held calculator, that can also access Eureka's built-in functions.
Eureka is very easy to work with. You don't need programming knowledge or
experience to use it. You simply type in an equation file containing the equation(s)
to be solved, then use the Solve command to solve for the variables in your equations. The equations you enter use standard mathematical representations, very
similar to those you enter in a calculator when solving the same problems the hard
way.

If you get stuck while in Eureka, you can ask the program for on-screen help by
simply pressing a key.
The Eureka text editor, which you use to enter the equation file, is virtually
identical to Borland's SideKick and Turbo Pascal editors. If you are familiar with
one of these editors, Eureka's editor will be no problem for you. Or, if you have
ever used WordStar or a WordStar-like editor, you are already familiar with the
editing syntax and commands. For those who do not have experience with a similar
ASCII text editor, we have included several useful tables, a tutorial, and an appendix covering the Eureka editor.
Eureka displays solutions on the screen in their own windows, along with information about how confident it is that the solution is correct.

Eureka: The Solver Owner's Handbook

After Eureka solves the equation file, you can plot graphs of the functions·in the
file by simply specifying the function and the range you want plotted. Eureka will
display a graph of the function, with the coordinate axes scaled appropriately. You
can also print out the equation file, the solutions, and the graphs in the form of a
written report.

Structure of This Manual
This chapter, "Introduction," tells you the basics about Eureka: what you can use
the program for, how to use the manual, and what hardware and software you need
to run the program.
Chapter 1, "Overview," gives an overall picture of the different parts of Eureka
and how they fit together. Experienced software users can probably read this chapter and get started right away with Eureka.
Chapter 2, "Getting Started," shows you how to load Eureka, then gives a progressive tutorial designed to familiarize you with using the editor, creating and
solving problems, and incorporating some special techniques into your files.
Chapter 3, "Modeling Tips and Techniques," summarizes the tasks that Eureka
performs, describes the program's advanced features, and gives some tips on modeling.
'
Chapter 4, "Menu Commands and Settings," describes each of the commands in
the main menu and the submenus, and explains the settings that you can change
with the Settings command.
Chapter 5, "The Equation File: Syntax, Directives, and Functions" describes
what you put into an equation file, shows you how to formulate the entries, tells
what directives are and shows how to use them in the file, introduces Eureka's
built-in functions, provides detailed explanations of the built-in financial functions,
and explains what user-defined functions are and how to write them.
Chapter 6, "Worked Examples," is a collection of solved equation files taken from
typical scientific, engineering, and financial problems.
Appendix A, "Quick Reference Table," briefly explains many of Eureka's functions.
Appendix B, "Eureka Editor Commands," summarizes the Eureka editor commands.
Appendix C, "Built-in Functions," describes Eureka's built-in functions.
Appendix D, "A DOS Primer; contains basic information about DOS, with a
focus on creating and using directories.

Introduction

Appendix E, "Error Messages," lists all Eureka error messages and their meanings.
Appendix F, "Customizing Eureka," tells you how to use the EINST program to
custom-install Eureka.
The Glossary gives definitions of key terms used in this manual.
The Index lists page references for program features, commands, and windows.

Typography
The body of this manual is set in this typeface - Roman. Special typefaces are used
for the following purposes:
Alternate Type

This typeface is used to show something as it appears on the
screen or to show anything you must type.

< >

Angle brackets indicate data that depends on your system and
should not be typed verbatim.

Italics

Italics are used to introduce a new term; all new terms are
defined in the Giossary.

Mathematical expressions are set in Roman type according to standard textbook
notation except when they represent what you actually type into a Eureka equation
file; these are set in alternate type, and some mathematical operators appear differently. For instance, the multiplier symbol, x, is replaced with * in alternate type,
exponents (e.g.l) are represented by a caret (xA2), and s and ~. are represented
by < = and> =.

The Distribution Disk
The Eureka distribution disk contains:
• EUREKA.EXE, the program itsel£
• PROB} through PROB12, sample equation files.
• READ ME, a message you should read before loading Eureka.
• README.COM, a program to assist you in reading README.
• EINST.COM, a program to customize Eureka keystrokes, set a path to your
files, set the default edit mode, and set the screen mode.
• EUREKA.MAC, SuperKey macros for mathematical symbols (SuperKey is a
keyboard enhancer program from Borland International).
4

Eureka: The Solver Owner's Handbook

• HELP.EKA, Eureka's on-line help text.
• NDP.COM, a program that determines if an 8087 math coprocessor chip is
present in your computer.
• Sample equation files corresponding to the worked problems in Chapter 6
(various file names ending in .EKA)
The file README contains important information about using Eureka. Before
loading and using EUREKA.EXE, you should look over the README file.

Hardware and Software Requirements
Eureka runs on the IBM PC family of computers, including the XT and AT, along
with most IBM compatibles.
Eureka requires:
• DOS 2.0 or higher
• At least 384K of RAM
Eureka will run on any 80-column monitor. A math co-processor chip (an 8087)
significantly enhances performance, but is not required. Eureka will display plots
in graphics mode if you have a eGA, EGA, or Hercules graphics card but a
graphics card is not required.
The Eureka program is in a file called EUREKA.EXE. It is not copy-protected,
so you can easily transfer it to a hard disk or RAM disk. However, you should
read Borland's No-Nonsense Licep.se Agreement at the front of this manual for an
explanation of your responsibilities with respect to copying Eureka, and then sign
it and mail to us .

Ackoowledgements
In this manual, references are made to several products:
• SideKick, Turbo Basic, and Turbo Pascal are trademarks of Borland International, Inc.
• WordS tar is a trademark of MicroPro, Inc.
• IBM PC, XT, and AT are trademarks of International Business Machines, Inc.

Introduction

Eureka: The Solver Owner's Handbook

Overview

This chapter takes a brief look at Eureka: The Solver. If you are an experienced
software user, this chapter may be all you need to get going with Eureka. Check
Appendix A, "Quick Reference Table," for more information.

If you are new to computers and software, you'll probably want to read this
chapter and then turn to Chapter 2 for the step-by-step tutorials.
This chapter summarizes the four typical steps that you and Eureka go through
to solve a problem, and explains features of each step:
• Edit (create an equation file for a problem)
• Solve (find a solution)
• Verify (verify the solution's accuracy)
• Report (print the results)
All Eureka problems begin with the preparation of an equation file using the
Eureka editor. Chapter 5 details the contents, syntax, and format of an equation
file. Chapter 2 gives tutorials designed to familiarize you with preparing an equation file.

Cammand Menus and Windows
Eureka uses pull-down command menus and windows. If you have not used pulldown menus before, you will find the following few comments helpful.
The main menu runs across the top of the screen. When the menu is active,
there is always one highlighted item. You can select an item by moving to it with
the arrow keys, then pressing Enter, or simply press the key corresponding to the
first letter in the item. For instance, to select Solve, you can simply press S. Each
time you select a menu item, Eureka either opens up a new menu or executes a
command.

If the main menu is not active, it is because some window or submenu is active.
Generally, when a window is active, all keyboard actions affect only that window.
In most cases, you can move around in the windows with the cursor keys. The most
important windows are:
• Edit
• Progress
• Solution
• Verify
" Plot
• List
• Report
• Help
• Error message
These windows are fully described in Chapters 2 and 4. The following keys
affect windows:

Esc

Either closes the active window or moves it to the background,
making another window or menu active.

Arrow Keys

With Scroll-Lock on, move windows. With Scroll-Lock on and Shift or
Num-Lock on, resize windows.

Calls up a Help window that gives context-sensitive help.
Wherever you are in Eureka, you can press FI and a help message
appears on the screen.

Eureka: The Solver Owner's Handbook

The Editor
Eureka features an ASCII editor for editing problem files. You can create and modify problem files with this editor while in Eureka, or with the ASCII editor of your
choice. The command syntax of the Eureka editor is described in Chapter 4.
To call up the editor and create a new file, select Edit from the main menu or
select New from the File menu.
To call up the editor and modify an existing file, select the File menu from the
main menu, select Load, and specify an existing equation file name. When you
press Enter, Eureka brings the named file into the Edit window.
To save a file for later modification or examination, select Save from the File
menu.

Equation File Syntax
An equation file consists of a collection of formulas, typically one per line, written

in standard mathematical notation. Eureka recognizes the follOwing symbols:
Relational operators
Arithmetic operators
Functions

= < >

+ _ */A()
exp, In, sin, cos, abs

as well as variable names, numerical constants, and some special operators, functions, and directives. The caret symbol is the exponentiation operator, as in

xA2 = x * x or 5A3 = 5 * 5 * 5 = 125

Variables
A variable name may be any sequence of letters, digits, and periods, starting with a
letter. Uppercase and lowercase are distinguished in variable names, but not in
built-in functions or file names.

Overview

I nitializations
If you know an approximate solution in advance, we recommend that you enter the
approximate values as initializations. The notation
x := 12.75

means that Eureka should initially assign the value 12.75 to the variable x. Eureka
will use its own default initialization (x : = 1) if you give no value.
More information on the equation :file syntax can be found in Chapter 5.

Special Symbols
The symbols that follow have special meanings within the Eureka program. The
symbols and their meanings are listed, along with reference to where you can find a
detailed explanation of each.
Symbol
$
#

{}

Where to Find an Explanation
Initialization or user-defined function, Chapter 5
Directives, Chapters 4 and 5
Commands/Calculator, Chapter 4
Comments, Chapter 5
Comments, Chapter 5

Reserved Words
Eureka's reserved words call up built-in functions. Look in Chapter 5 for directives, and in Appendix C for most built-in functions. Refer to Chapter 5 for definitions of the financial functions, marked here with F. Directives are marked here
with a dollar sign ($).

Eureka: The Solver Owner's Handbook

$
$
$

$
$

Eureka Reserved Words
im
abs
$ include
accuracy
atan2
$ initval
integ
casefold
$ listdefault
complex
In
cos
log 10
cosh
digits
$ max
deriv
$ maxtime
$ min
end
ncum
exp
fact
F paymt
finanmode
$ penalty
pi
finansmooth
floor
$ plotdefault
polar
frac
fval
poly

pos
F pval
$ radius
re
$ rootsign
$ settings
sgn
sin
sinh
$ solve
sqrt
$ substlevel
sum
$ syntax
tan
tanh
$ units

Solving Equations
After preparing the equation file, you select Solve from the main menu. Eureka
solves the problem and displays the solutions to the equation file's functions and
equations in the Solution window.

How Eureka Calculates Solutions
When Eureka solves a problem, it searches for a solution and then displays resultant values for all of the relevant variables in the Solution window. If there are too
many variables to fit into the window, you can scroll through the Solution window
with PgUp and PgDn.
To determine the accuracy of a minimization or maximization solution, Eureka
computes a confidence level between 0 and 100. This is a rough indicator of how
well Eureka has performed. 100 is perfect. A confidence level between 93 and 100
means that Eureka successfully found a solution.
In some cases a solution may not exist or may not be unique. Read Chapters 3
and 5 for techniques that will help you avoid giving such problems to Eureka.

Overview

Evaluating Equations
You use Verify from the Commands menu to evaluate expressions in the equation
file, using data from the Solution window. Use Verify to verify the accuracy of a
previously computed result.
When you select Verify, Eureka evaluates all expressions in the equation file.
The program ignores directives and initializations, solves no equations, and
changes no variable values. Eureka displays the results of the evaluation in the
Verify window.
When an equation or inequality appears in the equation file, Eureka evaluates
the left- and right-hand sides using data from the Solution window, then calculates
the difference between the two sides of the equation (or inequality). If the difference indicates that the equation or inequality is invalid, the program gives an error
message.

Reporting
When you need a hard copy of the solution and the equation file, select Report from
the main menu. From the Report menu, select the output device and appropriate
format, then select Co. The report contains a copy of the equation file, the Solution
window information, the last plot or table graphed, and the Verify window information.

Eureka: The Solver Owner's Handbook

Getting Started

This chapter helps you get going with Eureka: The Solver through a progressive
tutorial. It also contains tables that summarize Eureka's main menu, special function keys, and the editor commands.

If you are a novice personal computer user or are unfamiliar with menu-driven
software, we recommend that you take a few minutes to sit down at your computer
with this manual and follow through the tutorial. When you complete it, you will
know:
• How to load (and leave) Eureka
• How to create, solve, modify, and print out a problem
• How to work with some of the program's special features
In the tutorial, you will first set up (load) the program so you can use it when and
how you want. Then you will read the text and follow the instructions given in the
tutorial, which gives you hands-on experience with using Eureka to solve a real
problem.
The tutorial first introduces you to Eureka's menus and text editor, takes you
step-by-step through creating and solving a problem, then shows you how to implement some of Eureka's special features.
Note: Any time you need help while in Eureka, just press FI to bring up contextsensitive help screens.

Setting Up the Program
Before you begin working with Eureka, you should make a copy of the distribution
disk. Use the copy as your working disk, and store the original in a safe place in
case anything should happen to the copy.
You don't need to know much about your computer to use Eureka: All you need
to know is:
• How to turn the system on
• How to handle and format a floppy disk
• How to start DOS
Appendix D, "A DOS Primer," contains useful information about DOS.

Copying to a Floppy Disk

One Floppy Disk Drive
To copy the files from the distribution disk to another floppy disk using a system
with one disk drive:

1. Insert your DOS disk in the disk drive. Type
DISKCOPY Enter

Your system will respond with
Insert Source diskette in Drive A and press Enter when ready

2. Take the DOS disk out of the disk drive and insert the Eureka disk, then
press Enter.
Mter reading the contents of the Eureka disk into memory, your system will
prompt you to
Insert Target diskette into Drive A and press any key when ready

3. Insert a blank, formatted floppy disk into the drive and press any key. When
your system finishes copying the contents of the Eureka disk from memory to
the new disk, it will ask if you want to
Copy another?

Type
N Enter

Eureka: The Solver Owner's Handbook

Two Floppy Disk Drives
To copy the files from the Eureka disk to another floppy disk using a system with
two disk drives:
1. Insert the Eureka disk in Drive A and a blank, formatted disk in Drive B.
2. Type
COPY A:*.* B: Enter

Copying to a Hard Disk
To copy the files from the distribution disk to a hard disk:
1. Insert the Eureka disk in Drive A. If you use directories, note the full path
name of the directory in which you want the program to reside.

2. Type
COPY A:*.* C: Enter

(assuming your hard disk is Drive C).

Copying to a RAM Disk
If your system has a RAM disk, you can use it to speed up the program's access
time. This means you must copy the Eureka program to the RAM disk each and
every time you boot the computer, unless you use an AUTOEXEC.BAT file. (See
the next paragraph for how to modify the AUTOEXEC.BAT file.) Type
COPY A:EUREKA.EXE 0: Enter

(assuming that your RAM disk is on Drive D) each time you want to load Eureka.

If you already have an AUTOEXEC.BAT file, you can automate this process by
putting a reference to EUREKA.EXE in the AUTOEXEC.BAT file. For instance,
assuming your hard disk is Drive C,
1. Use your word processor to bring the AUTOEXEC.BAT file to the screen. (It
should be in the root directory.)

2. Find the line that copies programs from their home directories to Drive D.

Getting Started

3. Add
C:\EUREKA.EXE

to this line. For example, if EUREKA.EXE is in a directory called
EUREKA, the line will look like this:
COpy C:\EUREKA\EUREKA.EXE 0:

4. Save the updated AUTOEXEC.BAT file.
You can also create an AUTOEXEC.BAT file if you don't already have one. For
more information, see Appendix D.

Tutorial Part 1: Using the Menus and the Editor
This first part of the tutorial explains how to use Eureka's menus and submenus,
and how to create and save a file. It tells you how to:
• Start Eureka
• Understand Eureka's program screens
• Write a new file
• Save the file

Starting Eureka
Eureka requires no installation. Once the program is in your computer (either on a
floppy disk in the disk drive, or on your hard disk), all you need to do is call it up.
For a floppy-disk system, make sure your working disk (the floppy disk with the
copied version of Eureka on it) is in the desired drive (typically Drive A, which we
will refer to in this manual).
Then to start the program, log on to the drive or directory that contains the
working disk and type
EUREKA Enter

Eureka: The Solver Owner's Handbook

After a few moments, the Eureka opening screen appears (Figure 2-1).

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Opening Screen

What to Do If You Want Out of Eureka
Before you begin pressing keys on the keyboard and getting into Eureka, it's a good
idea to know how to get out.
The Escape (Esc) key always returns you to Eureka's previous active state. Since
you are starting at the opening screen, pressing Esc from within one of Eureka's
program screens will eventually bring you back to the opening screen. The main
menu bar is always visible; you can't escape past it (unless you exit the program by
pressing F Q.)
As Figure 2-1 shows, the opening screen is made up of two features: the main
menu and the status line.Once the opening screen appears, you can use any Eureka
feature: select an item from the main menu and then make a selection from a
submenu.
Look over the screen on your display and compare it to Figure 2-1. Note that the
main menu remains on the screen virtually all the time you use Eureka; so does the
status line. What appears on the status line varies from time to time; this is
explained later.

Getting Started

A menu presents a list of options from which you can choose. When you select an
item from the main menu, that item's pull-down menu extends from the main
menu. Each pull-down menu contains a group of related functions. The File menu,
for instance, lists operations concerned with files - making them, saving them, and
so on.
The status line lists keys that perform special functions in Eureka. For instance,
• FI displays a window of context-sensitive help text.
• F5 "zooms" the active window to fill the screen.
Different keys are active under different circumstances, so the keys displayed on
the status line change from time to time. You'll see this in the tutorials and in the
worked examples in Chapter 6.
Table 2-1 summarizes Eureka's main menu options, and Table 2-2 lists the special function keys.

Eureka: The Solver Owner's Handbook

Table 2-1

Main Menu Summary

Entry

Option

Description

FILE

Load
New
Save
Write to ..

Bring an existing file to the screen
Open a new file for editing
Write the current file to disk
Save the current file under the file name you
specify
List files in the specified directory
Change the active drive/directory
Rename a file
Temporarily exit to DOS
Leave Eureka and return to DOS
Create a new file or modify an existing file
Find the value of the specified variable(s)
Evaluate both sides of the equation and compare
the result
Open a window in which arithmetic functions
can be solved
Search another area for the value of the specified
variable
Continue iterating the value of the specified
variable
Print/save a report about an equation file
Specify output device
Put report into formatted form
Log your work to a file on disk
Specify name for capture file
Plot a graph of a user-defined function
Specify output device
Generate a table of values for a user-defined
function
Enter function to allow plotting or graphing
Change value of variable(s) or constants
Open a submenu of user-modifiable settings
Change window colors
Specify where Eureka looks for its files
Let a predefined group of settings be used
Save a group of settings as a file
Open the specified window
Close the active window
Open and make active the next window
Expand the active window to fill the screen
Make all open windows visible and of equal size
Stack open windows at largest possible size
Go to the active window

Directory
Change dir
Rename
OS shell
Quit
EDIT
SOLVE
COMMANDS

Verify
Calculator
Find other
Iterate

REPORT

GRAPH

OPTIONS

WINDOWS

Getting Started

Go
Output
Formatted
Capture
Log file name
Plot
Output
List
Function
Variables
Settings
Colors
Directories
Load setup
Write setup
Open
Close
Next
Zoom
Tile
Stack
Goto

Table 2-2

Summary of Special Function Keys

Function Keys

Description

FI
F2
F3
F5
Alt-F5
F6
F7
FB
Alt-X

Display help text
Save current file
In editor, open new file
Zoom active window
Text zoom for plot (graph) window
Change active window
Go to active window (in editor, Block begin)
In editor, Block end
Exit Eureka
Go to Edit window
Solve the equation file
Activate the calculator
Insert pi character
Return to previous activity
Interrupt a solve
Move/scroll through screen

Alt-E
Alt-S
Alt-C

Alt-P
Esc
Ctrl-Break
Arrow keys and PgUp,

PgDn
Scroll-Lock, Num-Lock,
and arrow keys
Scroll-Lock
and arrow keys

Resize window

Move window to new position

Making a Selection From the Main Menu
There are two ways to select a menu item: the cursor method and the initial letter
method.
• The cursor metlwd: Move to the right or left with the Right arrow and Left arrow
keys. When the item you are interested in is highlighted, press Enter to open a
pull-down menu containing details about that item. Use the Up arrow and
Down arrow keys to move the highlight bar up and down in the menu; press
Enter to select an item from the menu.
• The initial letter metlwd: Press the key for the first letter of the item. For
instance, to select File, press F (lowercase or uppercase); the File pull-down
menu will appear (Figure 2-2). To select Save from the File menu, press S.

iEureka: The Solver Owner's Handbook

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File Menu

Creating a File
Although Eureka allows you to use any ASCII text editor you like, we will use the
one supplied with the program to write a sample file.
To create a file, you need to start at the main menu. Select New by pressing F N.
Figure 2-3 shows the Edit window as it should appear after you have entered some
text into the file.

Getting Started

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Figure 2-3

Edit Window

You can think of a window as a place where things happen - where you interact
directly with the program.
• Some windows display a prompt, requesting more information;
• Others display the results of a particular operation;
• Still others, such as the Edit window, let you enter data or text.

Using the Eureka Editor
Notice the information at the top of the window on your screen:
• "C:NONAME" displays the active drive and the file's name.
• "Line I» and "ColI» refer to the position of the cursor; it is on the first line, in
the first space or column.
• "Insert" tells you the editor is in Insert mode, not Overwrite typing mode.
• "Indent" shows that automatic line indention is available.
To write something into your new file, you type text just as you would if you
were using a typewriter, remembering to press Enter at the end of every line.
(Eureka's editor has no word-wrap feature.) Then, to move the cursor around in the
text, you can use the cursor keys on the numeric keypad or the PgUp and PgDn keys.
Table 2-3 summarizes selected editor commands you may find useful while
working through the tutorials. The editor is discussed in greater detail at the end of
this chapter and in Appendix B.

Eureka: The Solver Owner's Handbook

Table 2-3

Summary of Editor Commands

To Move the Cursor
One space right
To first letter of word on right
To right end of line
One space left
To first letter of word on left
To left end of line
One line up, same column
One screen up, same column
To top of present screen
Beginning of file
One line down, same column
One screen down, same column
To bottom of present screen
To end of file

Ctrl-D or Right arrow
Ctrl-F or Ctrl-Right arrow
Ctrl-Q D or End
Ctrl-S or Left arrow
Ctrl- A or Ctrl-Left arrow
Ctrl-Q 5 or Home
Ctrl-E or Up arrow
Ctrl-R or PgUp
Ctrl-Home
Ctrl-Q R
Ctrl-X or Down arrow
Ctrl-C or PgDn
Ctrl-End
Ctrl-Q C

To Delete Text
One character, wh~re cursor rests
One character, immediately left of cursor
Whole word, immediately right of cursor
Whole line, where cursor rest~

Ctrl-G or Del
Backspace
Ctrl-T
Ctrl-Y

To Insert Text
In between existing text
Write over (replace) existing text

Ctrl-Vor Ins (toggle Insert ON)
Ctrl-Vor Ins (toggle Insert OFF,
Overwrite ON)

Now that you understand how to use the Eureka Editor, write something in
your file. For example, you could type
I'm learning how to use Eureka, and
soon my problem-solving problems will
be over.

Enter
Enter
Enter

Saving A File
With some text in your file (although it doesn't contain any equations yet), this is a
good time to save it and then continue working: Select Save from the File menu.
When prompted, enter TRIAL as the name for your file.
A Eureka file name follows standard DOS conventions: one to eight characters
with an optional one-to-three-Ietter extension. In a file name, you can use any
letters or numbers, and the symbol _.

Getting Started

You can use either uppercase or lowercase letters in the file name; when it
comes to file names, Eureka is not case sensitive. (That means that Eureka doesn't
know the difference between uppercase and lowercase letters; for example, the file
names TEST, test, and Test all look the same to Eureka.)
Although the file doesn't disappear from the screen, a copy of it has been saved
to disk, under the name TRIAL. To bring this saved file to the screen, you would
use the Load command on the File menu.

Tutorial Part 2: Creating and Solving an Equation File
This part of the tutorial demonstrates how to use Eureka to solve a real problem
and print a report. It tells you how to:
• Create a problem file
• Solve the problem
• Modify the equation file to contain a user-defined function
• Plot the graph of the function
• Print a report about the problem
The first step is to write an equation file. When you create an equation file, you
assemble the various parts of a problem in a way that Eureka understands. An
equation file contains one or more equations and any initializations, directives,
functions, and comments that are required.
You will write an equation file that contains an equation and a comment. You will
then solve it, modify it to contain a user-defined function, and plot a graph of that
function.

Creating the Equation File
To write the equation file, ask Eureka to load in the file named TRIAL; type
F L

Eureka will ask you the
Load file name

Enter the name of the file you created earlier:
TRIAL

Enter

Eureka: The Solver Owner's Handbook

A window appears containing the file you previously saved. However, you can
only look at the file from this window; you can't make any changes to what you see
because you are not yet in the Edit window. To edit the file, select the Edit option
from the main menu: press E.
In the next step, you will create a typical equation file containing an equation
and a comment.

Writing an Equation
You write equations, one per line, in standard mathematical syntax and notation,
with a few special symbols. For now, all you need to know about equation syntax is
that:
• The exponentiation operator is the caret symbol,
power is written as x 3.

For instance, x to the third

• Multiplication is not performed implicitly.
• Multiplication is denoted with an asterisk,

* (3 times z is written as 3 * z).

(For more information about equation file syntax, refer to Chapter 5.)
The equation you will solve resembles a polynomial; it mayor may not have
three roots. After solving for the first root, you will use some special techniques to
determine if there are other roots.
Enter the following equation to be solved:
x-3 + 3 * x-2 - 2 * x - cos (x) = 0

Double check that you have typed the equation correctly. If you make a mistake,
use the editor commands given in Table 2-3 to correct it.

Writing a Comment
Comments must either be set off by a semicolon (;) or appear between braces ({ }).
To enter a comment, press Enter (to skip a line, improving legibility), then type
{ Test of equation solving}

You can set off the comment with a semicolon instead:
; Test of equation solving

The text you entered into the file during the first part of the tutorial can also be
converted to a comment. Eureka can't solve the equation file if you just leave the
text as is, so if you don't want to set it off as a comment, delete it. To erase or delete
lines of text, see the editor commands in Table 2-3.

Getting Started

Solving the Equation File
When the file contains only comments (appropriately set off) and the equation,
return to the main menu and select Solve. Eureka will first look directly, then
iteratively, for an answer to the problem. While Eureka searches for an answer, a
Progress window appears on the screen. It displays the amount of time the program is using to find the solution, the current amount of error, and the current
solution.
As Eureka iterates, it modifies the current variable values, then checks to see if
the modified values yield the desired function value. The current amount of error is
a measure of the difference between the desired function value and the function
value calculated by using the current variable values.
When the program finds a solution, the Progress window disappears and the
Solution window (Figure 2-4) takes its place.
The Solution window displays the values Eureka has found for the indicated
variables. For the equation file TRIAL, Eureka determines x to be equal to
0.77442927.
The Progress window is useful if you want to interrupt a lengthy solving process.
To interrupt a Solve and examine the current variable values:
• Press Ctrl-Break
• Examine the current solution
• Resume solving by selecting Iterate from the Commands menu
Notice that the keys listed on the status line change as soon as the Solution
window appears. The new keys affect only the currently active window.
To move the cursor in the Solution window ("scroll the window"), use the arrow
and PgUp and PgOn keys (just as in the Edit window).
After the Solution window appears, you can still tell Eureka to continue looking
for solutions by selecting Iterate from the Commands menu.
Often the solution process is so quick that you do not see the Progress window.
For minimization and maximization problems, after you select Solve, you will see
interim iterative solutions in the Progress window.

Eureka: The Solver Owner's Handbook

----------------------._"
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Figure 2-4

Solution Window

Modifying the Equation File
You have determined one root of the equation to be 0.77442927. But what about the
other roots this equation may have?
You need to examine a graph of the equation to see if the other roots exist. The
Eureka plotting feature allows you to graph the equation and determine if there are
other values of x for which the functionf(x) = O.

If you try to use either the plotting or table (list) feature now, you will find they
are inactive; verify this by pressing Esc then G to select the Graph menu.
Look closely at the Graph menu. Plot, Output, and List are dim, meaning that
you cannot use them yet. You can only access the Function option; Plot, Output,
and List are unavailable because they have no user-defined function to work with.
In order for you to use Plot (or List), an equation file must
• include at least one user-defined function and
• have been solved
or else you must enter a user-defined function directly into the program using the
Function option. Let's modify the file to incorporate a user-defined function and
then solve it.

Getting Started

First get back into the Edit window by pressing F6 until the Edit window is
active (indicated by double bars around it). Then press F7 to go to the window. Use
the editing commands listed in Table 2-3 to change your equation file so it looks
like this:
f(x) := xA3 + 3 * xA2 - 2 * x - cos (x)
f(x) :; 0
{ Test of equation solving}

(The symbol: = indicates thatf(x) is a user-defined function.)
Return to the main menu screen, then solve the problem again by pressing Esc,
then S.
Once you've verified that the solution is the same (x = 0.77442927), continue on
to the next step and plot the function.

Plotting a Graph
Re-select the Graph option from the main menu by pressing G. All the options are
now available because the equation file contains a user-defined function. (None of
the options are dim, and you can move the highlight bar up and down throughout
the whole menu.)
Select the Plot option by pressing P. Now set the left and right endpoints of the
plot's horizontal (x) axis; when the Left endpoint prompt appears, type - 4 Enter;
when the Right endpoint prompt appears, type 4 Enter.

(If you hadn't responded to these prompts, Eureka would have plugged in the
default values for Plot ( - 2 and 10), and would have plotted and displayed a different section of the function.)
Since Eureka automatically scales the vertical (y) axis, a graph of the function
appears immediately (Figure 2-5). Notice that several different sized point-like
characters have been used to draw the plot. This has the effect of smoothing out the
curve: If one type of character were used exclusively, the plot would appear to be
much more jagged.

If you have a graphics card in your computer, you can press the Zoom key (FS) to
see the plot in graphics mode.
To see a larger-scale version of the plot, with scaled axes, press Alt-F5 (To then
return to the smaller version, press Alt-F5 again.)

Eureka: The Solver Owner's Handbook

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First Plot Screen

This plot shows that there are at least two roots to the equation. One root is
close to - 4. A region from about - 1.5 to 1.5 may contain two roots, the
0.77442927 value already solved for, as well as a third root.
To examine this region more closely, select the Plot option again, but this time
use - 4 and 1 as the left and right endpoints. This generates the plot shown in
Figure 2-6.
This second graph shows what the two roots must be. In the next part of this
tutorial, after printing a report, you will select Variables from the Options pulldown menu to solve for the other two roots more exactly.

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Getting Started

Solution
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Second Plot Screen

Printing a Report
Now you will print a report that includes the original equation file with the latest
solution and plot.
Select Report from the main menu. The Report menu offers several ways to store
or print an equation file:
Output

lets you send the report to one of three places: to a file for saving, or
to the printer or the screen for printing.

Formatted

lets you specify a left margin and page breaks.

Use the arrow keys to select Output and then press Enter to display a menu of
the output choices.
To select the printer, press P. To select a formatted report, move the cursor to
Formatted and toggle to yes with Enter. To print the report, use the Up arrow key to
highlight Go and then press Enter. This sends the formatted report to your printer.

Tutorial Part 3: Using Special Features
This part of the tutorial gives instructions for using some of Eureka's special features on the equation file. This tutorial tells you how to:
• Use the editor to add to the equation file, write a directive, and tailor settings to
fit the current equation file
• Change default settings
• Select the Variables option and find other roots
• Evaluate the equation
• Save the equation file
• Use the Window menu to manipulate windows
• Leave the program

Eureka: The Solver Owner's Handbook

Writing a Directive
Now you will add a directive to the equation file TRIAL. A directive is a command
that overrides the defaults for various conditions.
For instance, although the default for solved variables is eight digits, you can use
the digits directive to cause Eureka to display up to 13 digits (the internal limit).
All directives follow the general format of the example directive shown in this
tutorial. To find out more about directives, refer to Chapter 5.
To change the number of displayed digits from eight (the default value) to ten,
for example, add the following line to your equation file:
$ digits = 10

Changing Default Settings
This is a good time to change additional settings. Most of the settings that you can
change with a directive can also be changed directly on the screen, using the main
menu Options command.
In general, you change the settings by entering a number or by pressing Enter to
toggle between yes and 00. Specific conditions that apply to the individual settings
are described on page 72 and in Chapter 5 under Directives.
The accuracy setting is an example of one you can change with a directive or
with the Settings command. Accuracy affects Eureka's iterative solution searches
by defining how much a solution is subject to its attendant constraints.
A constraint is a condition that a solution must meet in order to be valid. For
instance, in our equation file TRIAL,
f(x} = 0

is a constraint.
Eureka will solve for x iteratively, and the smaller the accuracy setting is, the
closer Eureka will try to make the left-hand side to the right-hand side of the
equation.
(The penalty setting also affects the relative importance of the constraints in an
equation file; see the description of penalty on page 93.)

Getting Started

To change the accuracy setting in TRIAL, first select Options from the main
menu, then select Settings from the Options pull-down menu. Next, select Accuracy setting: Highlight Accuracy and press Enter. Then type
1.0e-10

Enter

This new setting replaces the default value of l.Oe - 7. This smaller number
increases the accuracy of the solution when Eureka resolves any constraints that
may exist in the equation file.
Now select Solve from the main menu. Eureka will solve the modified equation
file TRIAL, taking into account the directive to display ten decimal places and the
new, tighter accuracy requirement. The Solution window should read
x = .7744292653

Selecting the Variables Option and Finding Other Roots
With Variables from the Options menu, you can restart Eureka's search for a solution to TRIAL, starting at a value you specify. This is helpful if you have found a
local minimum or maximum and want Eureka to search other regions.
Select the Variables option: press Esc 0 V.
The variables window lists the current solution for x as 0.7744292653. To start
Eureka searching for a root in a different region of the x axis, you must enter a new
value for the variable x. The new value must be close to the value you believe the
alternate root to be.
When you plotted the equation file, you saw that there is another root to the
equation very close to - 4. To find the root that lies just above - 4, type Enter -4
Enter, then press Esc Esc C to activate the Commands menu and select Find other.
This second solution should be
x = -3.620186934

You can solve for the third root by repeating the substitution procedure just
described, using a value for x close to - 1. The third solution should be
x = -0.3288122319

Eureka: The Solver Owner's Handbook

Evaluating the Solution
You can use Verify to check the validity of anyone of the solutions. This command
evaluates each side of the solved equation individually and then compares the two
sides; the difference between the two results should ideally be O.
Eureka may find an unwanted or inappropriate solution (such as a negative root,
when the value you need must be greater than zero). With the Verify command,
you can identify such solutions. Knowing which solutions do not apply, you can
modify the file so that Eureka finds the appropriate solutions.
To use Verify, press Esc, then type C V to select Verify from the Commands
menu. The Verify window appears (Figure 2-7). The Verify command does not find
a new solution; rather, it evaluates each side of the already-solved equation, then
compares and displays the results.
Notice that the left-hand side of the equation has been evaluated to be a very
small number, which results in an error of the same amount. (You may need to
move the cursor down to display the amount of difference.) In this case, the amount
of error is minuscule, so you can disregard it.
. - - - - - - - - - - lureb: TIle SolveI' - - - - - - - - - - ,
File
JUt
Solve
Co••anis
Repol't
Gra,.
Options
DII!!I!l

r------C:TRIAL.

Uit - - - - - - - ,
Line 5
Col 1

Solution - - - - ,
Line 1

III Sol.tion:
C: SOLUTIO".

I(X) := x"3 t 3 • x"Z - Z • x - cos(x)
I(X) = II
$ .i,its = 111
Vel'ily

Val'iahles

C:VERIF'I'.
Line 8
{Test 01 equation solv valuation 01 lOl'llUlas:

.1111l1li11IIIIIIII1

f(x)

411 f el'ence =

I]-Hel,

ID-Zoo.

-.3288122319

Values

FOI'.ulas

'--------111

Values

.IIIIIIIIIIIIIIIIIIII 1 - - - - - - - - '
.11111111111111l1li1

iD-llext lif-Goto Bmm!-Size/.ove

riJlE-Exit

',,----------------------_._-,.

Figure 2-7 Verify Window

Using the Window Menu
The Window menu allows you to manipulate Eureka's windows. You can open,
close, rearrange, and move back and forth between windows by selecting items on
this menu.

Getting Started

Open and Close do exactly what they say: Open opens and makes active the
window you specify, and Close closes the active window and removes it from the
screen. Next is also easy to use; it moves from window to window, making each
. window active in turn. (The function key F6 does exactly the same thing.)
Goto simply moves to the active window.
The other items on the Window menu - Zoom, Tile, and Stack - allow you to
enlarge, shrink, and rearrange your windows.
Let's play around a little with the four windows that are currently on screen:
Edit, Solution, Plot, and Verify. Suppose you'd like the Plot window to be larger.
Press F6 until the Plot window is active. Press the Scroll-Lock and Num-Lock keys.
Now press the Up arrow key until the window is as big as you want it, then press
Esc. If you just want a window to be as big as the entire screen, select Zoom and
the window will expand to fill the screen. (The function key F5, the Zoom key, does
the same thing.) To "dezoom" the window, press F5 again.
Now let's move the Verify window to a different spot on the screen. Press F6
until the Verify window is active, then press the Scroll-Lock key. Use the four arrow
keys to push the window around the screen.
Finally, you can arrange all the windows at once. Select Tile, and all the windows will line up neatly side by side in a tile-like pattern, each the same size. Now
select Stack. All the windows will pile atop each other in a neat stack in the center
of the screen. When windows are either stacked or tiled, you can move from one
window to the other by pressing F6 as usual. When windows are stacked, press F6
to "shuffle" them until the window you are interested in is on top.

Saving the Equation File
This marks the end of the tutorial. If you want to experiment with the equation file
TRIAL (perhaps change other settings, or add directives, or print a report containing all three solutions), you should save it now.
To save the file, press Esc to return to the main menu, then type F S to select Save
from the File menu.

Leaving the Program
Finally, to leave the Eureka program, select Quit from the main menu to return to
DOS, or press Alt-X

Eureka: The Solver Owner's Handbook

Wrap Up
This tutorial has introduced basic features of Eureka:
• You have learned how to create, edit, solve, and save an equation file; how to
move between and manipulate windows; and how to return to DOS.
• You have learned that the usual sequence of operations is to write a problem,
set options, solve the problem, and then evaluate the solution.
• You have learned how to produce both a graph and a written report.
But these are just basics; you will find that Chapter 3 describes useful special
features, and Chapter 6 offers several worked examples. The example files can also
be found on the Eureka disk. In addition, there are twelve other example files on
disk under the file names PROBl through PROB12.
(Note: PROBlO is an example that demonstrates the use of special IBM characters. If you have the APL character ROM installed, you will not be able to use this
example.)
You might want to experiment with these files since they illustrate some of
Eureka's important mathematical capabilities, such as maximizations and minimizations, not shown in the tutorial. In addition, a close reading of Chapters 4, 5, and
6 will show you how to fine-tune your equation file.
When you feel fairly proficient with the program, you may want to try running
some problems in batch mode. See page 42 for information.

Eureka Editor Summary
This section explains some of the editor's basic features, and gives some tips about
using the editor. For descriptions of all the editor commands, refer to Appendix B.
To change any of Eureka's default editor keystrokes, use the EINST program
(described in Appendix F).
Eureka's editor is an ASCII text editor. It does not introduce non-ASCII characters into the file being worked on. For this reason, files created with Eureka are
compatible with other programs.

Getting Started

General Editing Techniques
If you have used a word processing program before, using the editor to create and
modify a Eureka file will be very simple.
If you have never used a word processing program, Eureka's text editor will be
easy for you to learn. One purpose of the tutorials in this chapter is to give you
some practice with Eureka's editor before you dig in and start solving problems.
Basically, you use Eureka's editor like you do a typewriter, except that what you
type appears in the Edit window rather than on a piece of paper.
When you want to move the cursor to a new location in your file, you use Control
commands.

• Ctrl-C means to hold down Ctrl on your keyboard while you press C.
• Ctrl-Home means to hold down Ctrlon your keyboard while you press Home.
The Eureka editor works very much like the Turbo Pascal editor or like WordStarin non-document mode, using many of the same Control commands.

Basic Features of the Editor
Two typing modes
The editor has two typing modes: Insert and Overwrite. In Insert mode, anything
you type is inserted at the cursor position. In Overwrite mode, pressing a key
replaces the current character at the cursor position with the new typed character.
By default, the editor operates in Insert mode (you can change the default with the
EINST program; see Appendix F). To switch modes, press Ins or Ctrl-V.
No word-wrap
Unlike a more standard word processing program, the editor does not have wordwrap. You must press Enter at the end of every line.
Autoindent
The editor has an autoindent feature that allows subsequent lines to automatically
indent. If you leave several spaces at the start of a line, use Ctrl-O I to indent
subsequent lines the same amount. This command sequence is a toggle, so you
only need to press Ctrl-O I again to stop indenting.
The F5 ZOOM key
Sometimes you may want to see more than the area covered by the current Eureka
window. You can "zoom" the screen, temporarily blanking out all windows and
menus by pressing F5. Zooming the Edit window converts the entire screen into an
editor environment. To return to the previous screen, with all windows and menus
intact, press F5 again.

Eureka: The Solver Owner's Handbook

Editing ASCII files
In addition to the equation file, you can edit files created by another ASCII editor,
the report files Eureka generates, and any setup files you create.
Scroll-Lock and the F6 key
To move the Edit window, you must first activate the window by pressing F6. To
move the window, press Scroll-Lock then use the cursor arrow keys. When the Edit
window is in its new position, reactivate the window by again pressing Scroll-Lock.

More About the Editor Cammands
Cursor movement commands control the position of the cursor in the file. You can
edit your file by using only the cursor keys, PgUp and PgDn; the rest of the commands listed here are handy but not critical for small files.
Word left
Ctrl-Left arrow or Ctrl-A
Moves the cursor to the beginning of the word to the left.

Ctrl-Right arrow or Ctrl-F
Word right
Moves the cursor to the beginning of the word to the right.

Insert and delete commands let you insert and delete text and control the typing
mode.
Backspace
Delete character to left
This is the "backspace" key immediately above the Enter key. It moves one character to the left and deletes the character there.

Delete character under cursor
Del or Ctrl-G
Deletes the character under the cursor and moves any characters that are to the
right of the cursor over one position to the left. This command does not work across
line breaks.
Delete word right
Ctrl- T
Deletes the word to the right of the cursor. This command works across line breaks.
Delete line
Ctrl- Y
Deletes the line containing the cursor and moves any lines below one line up.
Insert mode on/off
Ins or Ctrl-V
This command lets you toggle between Insert mode (the default) and Overwrite
mode while entering text. In Insert mode, new text is tucked in at the cursor
position and existing text is moved over to the right. In Overwrite mode, new
characters replace existing characters as you type over them.

Getting Started

Eureka: The Solver Owner's Handbook

Modeling Tips and Techniques

This chapter covers some of the techniques you can use with Eureka to solve
problems. It also discusses Eureka's problem-solving method. The last section
gives some tips for successful modeling.
The techniques covered include:
• maximizing and minimizing
• continuing a search
• modifying constants and variables
• finding alternate solutions
• running problems from batch files
• interface with data files (sample BASIC input file)

How Eureka Works
This section discusses the inner workings of Eureka, covering both the program's
strengths in solving many types of difficult problems and its limitations with regard
to the phrasing of equations.
For examples of the solutions Eureka finds for real problems, refer to Chapter 6.
For more information about error messages, turn to Appendix E.

Eureka's Problem-Solving Metlwds
Mter reading in an equation file, Eureka sorts out the relevant variables and formulas. It then performs symbolic manipulations and substitutions to eliminate any
extraneous variables. If the problem cannot be solved directly, Eureka recasts it as
a minimization problem.
Eureka minimizes functions by the steepest-descent method. This method
involves starting at some initial point, and proceeding along a path which is always
in the direction of the most rapid decrease for the function.

If there are constraints, the solution path goes from the initial point to the constraint manifold, and then lies on the constraint manifold until Eureka finds a
solution.
Mathematically speaking, the solution process is actually more complicated than
this description suggests. The manifold of points has a Riemannian metric which is
derived from the Hessian of the function being minimized. The constraints are
enforced by adding a penalty function.
Many problems have multiple solutions. Eureka's search method finds one at a
time. The program can find other solutions if you insert different starting values or
if you add constraints that exclude undesirable solutions. You can also use the Find
other command, which directs Eureka to sample distant points in hopes of finding
another promising search region.
There is no known efficient algorithm that always yields the solutions to all
problems. Eureka does its best, but its solution path will occasionally get trapped
in a region far from the actual solution.
Because some variables are eliminated before the search begins, it is not necessary to initialize them. Only the active variables influence the search process.

Solving Techniques
This section covers the various solving techniques you can use with Eureka.

Maximizing and Minimizing
In addition to solving equations, Eureka can solve maximization and minimization
problems. This means that, instead of just finding any values for the variables
which satisfy the given constraints, Eureka can find values for the variables which

Eureka: The Solver Owner's Handbook

satisfy the constraints, and maximize or minimize some variable among all possible
values of the variables which satisfy the constraints.
Maximization or minimization is thus an additional requirement imposed upon a
set of equations or inequalities. You impose this requirement by inserting either
the max or min directive in the equation file. Max and min work on variables, not
on functions; if you need to maximize or minimize a function, you first need to
define a variable equal to the function, then direct Eureka to max (or min)imize the
variable.
For instance, suppose you want to find the maximum'value of a profit function,
where the relationships between the variables are expressed by
Profit(x,y) := 4 * x + 2 * x * y - 3 * y + 10

Before Eureka can maximize this relationship, you need to add the proviso that
P = Profit(x,y)

Then the program can follow the directive
$ max(P)

Continuing a Search
If Eureka is unable to solve a set of equations directly, it uses an iterative method.
With the iterative method, the program obtains better and better approximations
until a solution satisfies certain preprogrammed convergence criteria. If you think
Eureka gave up too soon on a given problem, you can select Iterate from the
Commands menu, and the program will continue to look for solutions.
You can interrupt a search by pressing Ctrl-Break. Eureka will then give the
current values of the variables as the solution.

Modifying Constants and Variables
Sometimes Eureka finds a solution other than the intended one, or gets off the
track looking for a solution. Usually this is due to an inadequate initialization. It is
possible to restart the iterative procedure with new initial values by using the
Variables command on the Options menu.
When you select the Variables command, a window opens showing the equation
file variables and constants that can be modified. You select one with the cursor
keys, then type in a new value to replace the current value.

Modeling Tips and Techniques

Finding Alternate Solutions
Many systems of equations have several distinct solutions. Normally, Eureka is
satisfied when it finds one solution. To find other solutions, select Find other from
the Commands menu. This causes. Eureka to try to find a solution different from
the one already obtained. You can use the Find other command repeatedly to find
multiple solutions.
You can also use the Find other command to find other local minima to a minimization problem or to find other local maxima to a maximization problem.

Running Problems from Batch Files
In solving a large number of problems, it may be more convenient to run them
directly from DOS or from a batch file rather than from the Eureka pull-down
menus. The syntax for doing this is
>Eureka ProblemFile OutputFile

where
is the DOS prompt, and
ProblemFile

is the problem file name.
OutputFile

is the output file to be created.
The output is the same as an unformatted Eureka report.
Some interactive capabilities, such as plotting, are not available when you run
Eureka from DOS.

Eureka: The Solver Owner's Handbook

Interface with Data Fil£s: Sampl£ BASIC Input Fil£
Because Eureka uses standard ASCII files, it can take input from a wide variety of
ASCII programs.
For example, suppose you are working in Borland's Turbo Basic, and have
twenty points in an x-y plane. These are stored in arrays X(I) and Y(I), where I runs
from 1 to 20, the X(I) are the points' x-coordinates, and the Y(I) are the respective
y-coordinates.
The following Turbo Basic program will create a file called PROBLEM and
write the data on that file in a format suitable for Eureka to solve:
cls
defdbl a-z
dim X(20), Y(20)
-A = 5
B = 1.2

for i = 1 to 20
X(i) = i/10 - 1
Y(i) = A * exp(-B * X(i)~2) + 0.001 * rnd
next i
open "0", 1, "PROBLEM"
print #1, "$ substlevel = 0"
print #1, "F(X) := A * exp(-B * X~2)"
print #1, "A := 4"
print #1, "B := 1"
for i = 1 to 20
print #1, "F("; X(i); ") = "; Y(i)
next i
close #1

This Turbo Basic program will create the following Eureka equation file:
$ substlevel = 0
F(X) := A * exp(-B * X~2)
A := 4
B := 1
F(-.9 ) = 1.892596768501327
F(-.8 ) = 2.319901816869821
F(-.7 ) = 2.777704759459417

Refer to Chapter 6, Problem 6, for an example problem in which this technique
can be quite useful.

Modeling Tips and Techniques

Modeling Tips for Successful Problem-Solving
This section provides some tips on how to set up problems in ways that Eureka can
handle well.

Helping Eureka Find Correct Solutions
Sometimes Eureka does not find the correct solution or gives an inaccurate solution. When this happens, you can
1. Try a different initialization.
2. Use the Find other command. This is especially useful for finding other local
minima to a minimization problem, or other local maxima to a maximization
problem.

3. Adjust the accuracy setting (select Options/Settings). Often more accurate
solutions can be obtained by merely changing the accuracy to be closer to
zero.
4. Reformulate the problem.
More general considerations to keep in mind when formulating your problem
include:
1. When setting up an equation file, it is important to remember to include all
the constraints.
Variables are not always positive. For instance, if you want to minimize
(x + l/x), you probably want to require that x > O.

2. Put in starting values when possible. Otherwise, Eureka may start its search
at some random point that is very far from the solution and thus may not find
the solution.
3. Use soft constraints where appropriate.
A hard constraint is a condition that absolutely must be met for the solution
to be useful; a soft constraint is more flexible. The follOwing minimization
has only a hard constraint:
$ min(z)
5 ; x +w+y + z
x <; 10
w >; 0
y >; 0

The constraint on x, to be equal to or less than 10, corresponds to a curve
with infinite slope. If this were a cost curve, it would be untenable.

Eureka: The Solver Owner's Handbook

To alleviate this difficulty, you might recast the problem as:
$ min(z)
5 = x +w+y + z
x < = 10 + z
w>=0
y >= 0

The slope now has a definable limit.
4. Assign names to constants.
Instead of this user-defined function containing numeric constants,
f(x) := 4.7

exp(-2.1

+ 1)

use this modification, inserting letters for the constants:
f(x) := A
A = 4.7
B = 2.1

exp(-B

+ 1)

This does not create extra work for Eureka, and it makes it easier for you to
rerun the equation file with different parameters.
5. Try setting the substlevel setting to different values.
Eureka tries to be clever about making symbolic substitutions before it does
a numerical search. Usually, this speeds up the numerical search, but sometimes the substitutions convert the problem to a form in which the search is
more difficult. In these cases, sometimes it helps to lower substlevel- the
number of substitutions Eureka is allowed to make during a search.
Substlevel can have any value from 0 to 6, with 6 as the default. If substlevel
= 0, Eureka performs no substitutions. This can be useful (see Chapter 6,
Problem 6 for an example where substlevel = 0 produces the solution to a
complicated problem).
6. Avoid using badly scaled variables and functions.
For example,
x

and
0.0000001 * x = 0.0000007

have the same solution mathematically, but computers sometimes cannot
handle equations such as the second one very well because the difference in
scale between the constant (0.0000001) and the variable (x = 7) is several
orders of magnitude.
Eureka might conclude that x = 1 is a good approximation, since the difference between the right- and left-hand sides is only 0.0000006.
Refer to Chapter 6, Problem 15, for an example in which poorly scaled variables and functions cause Eureka great difficulty in solving a problem.

Modeling Tips and Techniques

About Error Messages
If Eureka encounters a syntax: error in the equation file, it displays an appropriate
error message. If this happens when you tell Eureka to Solve an equation file, refer
to Appendix E for an explanation of the error message, then modify the equation
file as necessary to correct the error.
Sometimes floating-point errors occur in the process of finding a solution. Floating-point errors do not necessarily invalidate a solution; they may mean only that
Eureka used inappropriate numbers in the search. For example, in solving

l/x = 10
Eureka may try x = 0 before the correct solution of x = 0.1. Because division by
zero results in floating-point overflow, the solution appears with a message:
Warning: floating point overflow

Similarly, Eureka may try to take a square root of a negative number while
searching for a solution and issue a warning to that effect.

If a floating-point error does occur, it's a good idea to use the Verify or Iterate
commands to validate the solution.

Eureka: The Solver Owner's Handbook

Menu Commands and Settings

This chapter describes Eureka's commands and settings that you select from the
main menu or a submenu.
The descriptions are arranged first by main menu item (Commands, Graph,
Report, and so on). Then, under each menu item heading, the descriptions are
arranged in the order they appear on the menu.
Each description includes the name of the command or setting, followed by a
discussion of the command's or setting's purpose, effect, and notable features.
The menu commands provide a way for you to direct the program and to make
things happen. A command specifies a particular kind of operation that the computer is to perform. For instance, the File/Save command instructs Eureka to
incorporate all newly entered text into the current file in memory and then save it
(write it to disk).
Settings are those program parameters that you can change in the equation file,
such as the number of digits displayed for values. Settings remain in effect
throughout the entire work session. When you exit Eureka, the settings revert to
their default values. You can save default settings with the Write setup command.

Main Menu
The main menu displays the fundamental program commands and setting groups:
File

a group of commands that perform operations on files; also allows
you to access DOS.

Edit

allows you to create a new file, or change an existing one.

Solve

finds the value of the variable(s) in the equation file.

Commands

a group of commands that perform operations that supplement
Solve.

Report

a group of commands that produce a report based on the current
equation file.

Graph

a group of commands that generate plots or tables of values for the
current equation file.

Options

allows you to change default settings or selected variables.

Windows

a group of commands that allow you to move and manipulate windows.

To select a command, type either the first letter of its name or cursor to the
command you want and press Enter.
In most cases you select a setting in the same way. However, in some instances
you select a setting by toggling a word (such as YES or NO) with the Enter key. The
screen menu will prompt you for the correct response.
Table 4-1 summarizes the functions of the menu commands and settings, also in
order of screen appearance.
Following Table 4-1, complete descriptions of each command and setting round
out this section.

Eureka: The Solver Owner's Handbook

Tabk 4-1

Main Menu And Submenu Summary

Entry

Option

Description

FILE

Load
New
Save
Write to ..
Directory
Change dir
Rename
OS shell
Quit

Bring an existing file to the screen
Open a new file for editing
Write the current file to disk
Save the current file under the file name you specify
List files in the specified directory
Change the active drive/directory
Rename a file
Temporarily exit to DOS
Leave Eureka and return to DOS
Create a new file or modify an existing file
Find the value of the specified variable(s)
Evaluate both sides of the equation and compare the
result
Open a window in which arithmetic functions can
be solved
Search another area for the value of the specified
variable
Continue iterating the value of the specified variable
Print/save a report about an equation file
Specify output device
Put report into formatted form
Log your work to a file on disk
Specify name for capture file
Plot a graph of a user-defined function
Specify output device
Generate a table of values for a user-defined
function
Enter function to allow plotting or graphing
Change value of variables(s) or constants
Open a submenu of user-modifiable settings
Set stopping point for iterative searches
Let upper and lowercase letters define different
variables
Let variables be complex numbers
Set number of displayed digits
Set odd period as beginning or end
Make financial functions smooth
Initialize variable
Set first table value
Set increment for table
Number of values in table
Override the default list settings

EDIT
SOLVE
COMMANDS

Verify
Calculator
Find other

REPORT

GRAPH

OPTIONS

Iterate
Go
Output
Formatted
Capture
Log file name
Plot
Output
List
Function
Variables
Settings
accuracy
casefold
complex
digits
finanmode
finansmooth
initval
lisLfirst
lisUnc
lisLnum
listdefault

Menu Commands and Settings

Table 4-1 Main Menu And Submenu Summary, continued

WINDOWS

maxtime
penalty
ploLleft
ploLright
plotdefault
radius
rootsign
substlevel
syntax
Colors
Directories
Load setup
Write setup
Open
Close
Next
Zoom
Tile
Stack
Goto

Set time limit for solving
Determine relative weight of constraints
Set left endpoint of plot
Set right endpoint of plot
Override the default plot endpoints
Set range of solve search
Set roots all negative or positive
Set level of internal substitution in solving
Set syntax for arithmetic expressions
Change window colors
Specify where Eureka looks for its files
Let a predefined group of settings be used
Save a group of settings as a file
Open the specified window
Close the active window
Open and make active the next window
Expand the active window to fill the screen
Make all open windows visible and of equal size
Stack open windows at largest possible size
Go to the active window

Eureka: The Solver Owner's Handbook

The File Commands

, - - - - - - - - - - - lureh: file Solver - - - - - - - - - ,
Edit
Solve
Co••~"'s
leport
Gr~.11
Options
Uhdoll

Uit - - - - - - , . . - - - - S o l u t i o n - - - - - - - ,

!lew
Save
Urite to ••
lirectory
Cllnge llir
Ie.~.e

sllell
Quit

Uerify

leport

II!-Help
'\

jE-zoo.

!D-JIext

IZ-Goto

I!Iiim!!-s i xe/.ove

riBE-lxit

Figure 4-1

The File Menu

File
Press

Function

Enables you to select one of the File commands from the File
menu.

Description

The File commands, in order of appearance in the File menu, are:
Load, New, Save, Write to ... , Directory, Change dir, Rename, as
shell, and Quit.
The File commands are described in the order they appear on
the menu.

Looo
Press

Function

Brings a copy of an existing file from disk to the screen, or names
the edit buffer with a new file name.

Description

When you select Load, a prompt appears
Load Fil e name

Menu Commands and Settings

Type the name of the desired file, followed by Enter. You can
specify a file on another drive or in another directory by specifying
the full path name (for example, B:\PROBLEMS\FILENAME).
You can also use wildcards (* or ?) to see a list of files; you can then
choose the file you want to load. For example, you could type
*.EKA to see all files with the .EKA extension.
Once you have entered the file name, Eureka reads the file from
disk (or opens a new file if there is no existing file by that name)
and displays it in the Edit window. Select Edit to activate the
Eureka editor.

New
Press

Function

Opens a new unnamed file for editing and activates the Eureka
editor.

Description

When you select New, Eureka opens the Edit window. You can
then create a new equation file. When you want to save the file,
select Save or Write to ...

Save
Press

Function

Incorporates your latest changes to the named file in the Edit window and writes a copy of the changed file to disk.

Description

When you select Save, Eureka writes a copy of the file in the Edit
window to the current directory or to another directory/drive if you
specify a path name. The file remains on the screen so you can
continue editing or solve the equations.
Save does not save plots or tables that you generate with the
Graph command. To save a copy of a plot or table, use the Report
command.
Save creates a backup file with the extension .BAK each time you
save a file. If for some reason your original file is corrupted, you can
always load the .BAK file.

Eureka: The Solver Owner's Handbook

Write to...
Press

Function

Allows you to give a name to the file you are editing and save it to
disk.

Description

Write to... offers an easy way to make files that are variations of
each other. When you select Write to ... a prompt appears
New name

To save a file incorporating your current changes, type a name
and then press Enter. You can make changes to your file as many
times as you like, each time saving the variations to disk under a
different file name.

Directory
Press

Function

Displays all or some of the files and subdirectories on the current
directory or disk.

Description

When you select Directory, a prompt appears
Enter mask

The term mask, as used in Eureka, refers to a group of characters
that you type in order to search selectively for a file (or group of
files) or some other object.
To view the contents of the current directory, press Enter. A window appears, listing all the files in the directory. To view only certain files in the directory, type an appropriate mask followed by
Enter. Eureka displays a selected listing to be displayed. Pressing
Esc returns you to the main menu.
With a mask, you can screen the contents of a directory by using
the two DOS wildcards: the asterisk (*) and the question mark (?).
The * wildcard tells the computer to find all matching character
strings, starting from the position of the *. For example, you could
use * to display all files starting with the letters AN by typing
AN*.* as the mask. Or, to search for a file that starts with Math and
ends with the extension CAL, you would type Math*.CAL as the
mask.

Menu Commands and Settings

The ? wildcard works similarly to the * wildcard, except ? finds
only one character at a time. For example, to search for the following files
AMT120PA
AMT430PB
AMT890PA
you would enter the mask AMT??OP?; Eureka will list these files
(and any others with the same characters in the first, second, third,
sixth, and seventh positions).
You can also view a list of files in any other directory by specifying the path name of the directory.
To view all the files in a subdirectory, move the cursor to the
subdirectory name in the file list and press Enter.

Change dir
Press

Function

Changes the current drive/directory.

Description

When you select Change dir, a prompt appears
Enter dir name

To change the current drive/directory, type the full path name of
the desired directory (and the drive name if you are changing
drives), followed by Enter. The usual drives are A, B, or C, unless
you have a RAM disk or are using a network (in that case, the
directory can be D: or higher).
Eureka will now look for files on the new directory. If the
requested directory does not exist, Eureka will display an error
message.

Rename
Press

Function

Renames the file in the Edit window.

Description

When you select Rename, a prompt appears:
Enter name

Eureka: The Solver Owner's Handbook

Enter a new name for the file in the Edit window, then press

Enter. Eureka renames the file in memory only. Use Save to save the
file with the new name to disk.

as shell
Press

Function

Allows you to access DOS without removing Eureka from memory.

Description

When you select OS shell, Eureka disappears from the screen (but
not from memory), and the DOS prompt (» comes up.
To execute any DOS command, type the command followed by

Enter.
To return to Eureka, type EXIT.

Quit
Press

Function

Leaves Eureka and returns to DOS.

Description

When you select Quit, you will be returned to DOS. If you have
not saved the file in the Edit window, Eureka will ask you if you
want to save it. If you don't save, any changes you've made to it will
be lost.

Menu Commands and Settings

The Edit Command
/-----------------------..,.
\,

r - - - - - - - - - Eureka: TIle Soiller - - - - - - - - - ,
File

EoIit

Sollie

Co ••aMB

.eport

Gnp.

Options

mmm

r = = = = = = Uit = = = = = = = j I

C:TRIAL.
I(x) := x"3 t 3
I(x) = 8
$ ii,its = III

Line 5
It

x"z - Z

It X

Coil
-

cos(x)

{fest of equation solllhlg}

IIJ-llel,

!D-Zoo.

rD-Next

[ii-Goto !!mlm!!-Size/.o\le

r.JIE-Exit

\'-.._----------------------)

Figure 4-2

Typical Edit Screen

Edit
Press

Function

Activates the Eureka editor. Using the editor, you can either modify
an existing file or create a new one.

Description

When you select Edit, the Edit window becomes active. You can
create a new equation file or edit an existing one.
To create a new file, just type in your file. When you want to save
it, select Write To ... from the File menu.
To edit an existing file, select Load from the File menu, then
type the file's name (using its full path name if necessary) followed
by Enter.
The editor uses a combination of standard control character
sequences and predefined function keys for editing. It is virtually
identical to Borland's Turbo Pascal and SideKick editors. To switch
between Insert and Overwrite modes, press Ins.
More detailed information about the editor can be found in
Chapter 2 and Appendix B.

Eureka: The Solver Owner's Handbook

The Solve Command
,r-.-----------------------...
- - - - - . - - - - . - - - - - Eurek.: TIle Soluer - -..- -..- ..---.-.. --.. - . - - - rile
.. it
Solue
Co ••• n.ls
leport
Gnp.
Options
311m!!
r = = = = = Uit = = = = = 9 1 5 = = = = Solution =====;0

C:TRIAL.

Line 5

CoIl

I(x) := x"3 t 3 • x"2 - 2 • x - cos(x)
I(x) = 8
$ 'i,its = 111

C:SOLUTIOH.
Sol.Uo.:
Vari.hles

Line 1

Values

- _3288122319

{Test 01 equation soluhg}

_ _ Exit

'~-.-------------------------/

Figure 4-3

Typical Solve Screen

Solve
Press

Function

Finds a value for each variable in an equation file that satisfies the
constraints of the equation file.

Description

For each variable, Solve finds the first value that satisfies the constraints of an equation. If a problem cannot be solved directly (by
plugging in a formula), it is recast as a minimization. Eureka then
checks for a user-supplied initialization value (using the value 1 if
none is supplied) and begins to iterate toward a minimum. Once
Eureka finds a solution, the Solution window displays all variables
and their solved values.
/
You can interrupt a search for a solution by pressing Ctrl-Break.
Eureka then gives the current values of the variables. To continue
the search, press Esc and select Iterate from the Commands menu.
Many problems have multiple solutions. Eureka's search method
finds only one at a time. You can find other solutions by using different starting values, by adding constraints that exclude undesirable solutions, or by using the Find other command on the
Commands menu.

Menu Commands and Settings

Eureka can find the roots of polynomials. Consider an equation
file containing a function defined as a polynomial in one variable:

p(x) : = poly(x,l,O,O,O,l)
Applying Solve to the file causes Eureka to find all of the roots of
p(x) as well as to solve for whatever other variables appear in the
file.

Eureka: The Solver Owner's Handbook

The Commands Commands

r---------------------------------------------~~

rile

Ult

Sollie

-,-

lareb: TIle SollieI'
"port

I!lIIIII

Ult
Line 5 CaICtlI.tor
ri.. oUel'
F(x) := x"3 • 3 • x"Z - Z •
Iterate
f(x) = I
$ ii,its = .1
C:TRIAL.

Gr.,.

O,tio..

111.._

Sol.tion
Line 1
c: sOLlm
Solation:

0".

Unldl..
x

U.laes

-.3288lZZ319

{Test 01 e.-at ion sollli.,}

rDE-hit

\.,---------~--------------------./

Figure 4-4 The Commands Menu

Commands
Press

Function

The Commands menu contains commands that perform mathematical operations on elements in the equation file.
The commands, in order of appearance in the Commands menu,
are: Verify, Calculator, Find other, and Iterate.

Description

Verify
Press

Function

Evaluates all expressions in an equation file and compares the
results on an equation-by-equation basis.

Description

You use the Verify command primarily to verify the accuracy of a
previously computed result. A typical command cycle for solving
problems is:
• Edit (to create a problem)
• Solve (to find a solution)

Menu Commands and Settings

• Verify (to verify the accuracy)
• Report (to print the results)
When you select Verify, Eureka goes through all equations and
inequalities in the equation file and evaluates every expression,
substituting in the previously solved values for variables. During
an evaluation, Eureka ignores all directives and initializations,
solves none of the equations, and changes none of the variable
values.
When the evaluation is complete, Eureka displays the results
with any discrepancies in the Verify window.

If the difference indicates that an equation or inequality is
invalid (the difference is greater than that allowed by the accuracy
setting), the difference is flagged as an error.
For example, if a file contains the following obviously incorrect
equation
1/3 = 10 * exp(l)

the Verify window shows this result:
1/3 = 0.33333
10 * exp(l) = 27.18282
difference (error) = -26.84948

Calculator
Press

Function

Lets you perform arithmetic calculations in display-calculator fashion.

Description

The Calculator command acts as a shortcut: use it to evaluate an
arithmetic expression when you don't want to make a new equation
file.
Eureka's Calculator can handle any built-in function as long as it
does not require an undefined variable.
When you select Calculator from the Commands menu, an
empty Calculator window appears on the screen. Type in an
expression such as
10 * exp(1) - 7

followed by Enter, and Eureka displays the value of the expression.

Eureka: The Solver Owner's Handbook

An arithmetic expression is limited to one line of 29 characters,
unless you expand the window with the arrow keys after pressing
Scroll-Lock and Num-Lock.

If there is a syntax error in the arithmetic expression, an error
message appears.
If you have solved an equation file, you can insert variable names
from the file into expressions in the Calculator, such as the variable
Depth shown in the example below:
sin(Depth~2)

The pound symbol (#) has a special meaning in the Calculator: it
contains the value of the previously evaluated expression.

Find other
Press

Function

Tells Eureka to find another solution to the equation file, different
from the one that Solve just generated.

Description

While Eureka is normally satisfied when it finds one solution to an
equation file, many systems of equations have several distinct solutions. The solution you need may not be the first one that Eureka
finds. When you select Find other, Eureka tries to find a solution
that is different from the solution(s) already obtained.
You can use Find other repeatedly to find multiple solutions. For
example, Find other can find additional local minima to a minimization problem. (Note: In such a minimization problem, you can
also use the Variable setting to help Eureka start a new search in a
different range of numbers.)

Iterate
Press

Function

Continues solving a problem that has been Solved at least once.

Description

Sometimes, Eureka is unable to solve a set of equations directly. If
this occurs, the program uses an iterative method to obtain better
and better approximations until the solutions satisfy certain convergence criteria. However, you may think that the program has given
up too soon, settling for a solution that meets the accuracy require-

Menu Commands and Settings

ments, but stops short of the best solution. If you think this has
happened, select the Iterate command and Eureka will continue
solving where it left off.
You can also use the Iterate command to resume a search that
has been interrupted with Ctrl-Break.

Eureka: The Solver Owner's Handbook

The Report Commands

~=====Uit

C:TRIAL.

Line 5

Coil

fIx) := x"3 • 3 • x"2 - 2 • x - cos(x)
fIx) = II
$ ii,its = 111

Screen
VIS

orr

EOIIIA.LOG
- .3288122319

{Test of equation sO)IIill!J}

IlE-lxit

Figure 4-5

The Report Menu

Report
Press

Function

Enables you to select one of the Report commands from the Report
menu. The Report commands prepare a report of the solved equation file and then send it to a specified output device. Reports are
formatted for a standard 66-line, 80-column printer.
A report consists of:

• a header with date, time, and file name information
• the equation file
• the solution (as displayed in the Solution window)
• the evaluation (as displayed in the Verify window)
• the most recently generated graph or table
Description

The Report commands, in order of appearance in the Reports
menu, are: Go, Output, Formatted, Capture, and Log file name.

Menu Commands and Settings

Press
Function
Description

RG
Sends a report in the specified format to the previously specified
output device.
You use Go after you have set the format and output device for the
report (use the Formatted and Output commands to set these
report features). See Chapter 6 for examples of finished reports.

Output
Press
Function

R 0 < s, f, or p >

Determines where Eureka sends a report: to the screen (s), to a file

(f), or to a printer (p).
Description

Select one of three possible values for output by highlighting the
option you want, then pressing Enter. The report destinations are:
• Screen - (the default) sends the report to the screen
• File - sends the report to a disk file
• Printer - sends the report to a printer
When you select File, then select Go, a prompt appears:
Output file name

Type a file name just as you would for an equation file. The
report is saved to disk.
Once saved as a file, you can edit a report like an equation file
with the Eureka editor or any ASCII text editor. However, no
mathematical operations (such as Solve) can be performed on a
report file.

Eureka: The Solver Owner's Handbook

Formatted
Press

R F

Function

Determines whether or not a report is formatted. Default is yes.

Description

A formatted report is designed to go to a printer from your computer and has the following features:
• l/2-inch left margin (rather than printing Hush with the edge
of the paper)
• incorporates page breaks

If you intend to edit the report with a different text editor before
printing it, you should select Formatted/no.
To toggle between Formatted/yes (the default) and Formatted/

no, press Enter.
When Eureka sends a formatted report to a printer, printing
starts at the top of the page. If you want blank lines at the top of the
page, position the printer appropriately.

Capture
Press

Function

Automatically writes work session into a file periodically.

Description

Capture is useful for automatically saving your work to guard
against power failures. It is also useful when you want to solve a
series of problems and save all your work to a file. When you select
Capture, you toggle Capture on and off. With Capture on, your
work will be continuously saved to the file name specified by Log
file name.

wgfile name
Press

Function

Specifies a file name to be used by the Capture command; the
default name is EUREKA.LOG.

Menu Commands and Settings

The Graph Commands

. - - - - - - - - - - - EUl'en: TIle SollieI' - - - - - - - - - - ,
File
Edit
Sollie
Co.......
'epol't
iII!Ill Options lIiNOIl

r======
C:TRIAL.

Llit
Line 5

I,m
Coli
Sol

o.t,.t:
List
FallCtioa

Screen

r
u.'-------------'I

rex) := x"3 t 3 • x"Z - Z • x - cos(x)
rex) = 8
$ ii!Jits = 111

-.32118122319
{Test of e'luation solllhg}

!Il-Hel, ID-Zoo.

iD-lIext li-Goto BDl-Size/1tOVe

Figure 4-6

IJE-Exit

The Graph Menu

Graph
Press
Function

Description

G
Enables you to select one of the commands from the Graph menu.
The Graph commands generate and display on-screen plots of functions and tables of values.
The Graph commands, in order of appearance in the Graph menu,
are: Plot, Output, List, and Function.

Plot
Press

Function

Plots a function on the screen.

Description

Eureka can plot functions on the screen, even on a computer without a graphics card. Eureka can only plot user-defined functions of
exactly one variable. For example, the function sin(x) + 0.1 X x
can be plotted if the following definition of the function appears in
the equation file:
f(x) := sin(x)

+ 0.1 *

Eureka: The Solver Owner's Handbook

When you select the Graph menu, the Plot command is available
only if at least one user-defined function of a single variable
appears in the just-solved equation file. If the Plot command is not
available, it is displayed in half intensity on the Graph menu.

If no appropriate function exists in the equation file or if there is
no just-solved equation file, you can use the Function command to
write a function directly to Eureka.
If more than one user-defined, single-variable function appears
in the equation file, a list of choices appears, prompting you to
select the name of the function you intend to plot.
When you have selected the function to be plotted, a prompt
appears immediately, requesting the left and right endpoints (lower
and upper limits of the variable).
Enter the left value, then press Enter and repeat for the right
value. If you hit Enter without selecting lower and upper limits to
the variable, Eureka either resorts to its default values, - 2 and 10,
or whatever values you have specified in the Settings menu for
ploLleft and ploLright. Eureka scales the vertical axis automatically.
Using your selection ofleft and right endpoints for the horizontal
axis, Eureka displays the plot in a window on the screen; you can
save the plot in a report with the Report commands.
You can change the default values for several settings that govern the plot (see ploLleft, ploLright, and plotdefault under the
Options/Settings menu). If you have a graphics card in your computer, you can see a graphic display of your plot by pressing the
Zoom key (FS).

Output
Press

Function

Determines where Eureka sends a plot: to the screen (s) or the
printer (p).

Description

By default, Eureka will send your plot to the screen. If you press P,
Eureka sends the plot to a printer (must be Epson-compatible).

Menu Commands and Settings

List
Press

Function

Computes a table of values for a function, then displays the table on
screen.

Description

The List command creates a two-column table (a set of ordered
pairs of numbers) for a function. The table represents the same
information that would appear in a graph of the function but with
greater precision.
The List command, like Plot, acts on user-defined functions of
one variable. Eureka prompts you for:

1. the first value of the function argument
2. the increment
3. the number of points (200 is the maximum allowed)
2

For example, given the function fix) : = x , with:
First point: 3
Increment: 1
Number of values: 5
Eureka returns the following table:

fix)
x
3
9.000
4
16.000
5
25.000
6
36.000
7
49.000
You can use the Report menu to save your tables in report files or
print them as hard copy.
You can change the default values for several settings that govern the list (see lisLfirst, lisLinc, and lisLnum on page 73 ).

Eureka: The Solver Owner's Handbook

Function
Press
Function

GF
Allows you to enter a function of one variable that can then be
plotted or listed.

Description

If an equation file does not contain a user-defined function, this
command provides an alternate way to list or plot an equation. You
can use this feature by itself or in conjunction with an equation file.
When you select Function, a prompt appears:
Enter function

Type the appropriate function name; for example, j(x), followed
by Enter. Another prompt appears:
Enter function definition

Type the definition (for example, x-3), followed by Enter.
The function name and the definition are each limited to 18
characters.
When you have entered the function name and definition, use
the cursor keys to move to the Plot or List options.

If your equation file contains a user-defined function, you can
bypass the Function command and go directly to Plot or List.

Menu Commands and Settings

The Options Commands

r - - - - - - - - - Eurek.. : TIM! Solller - - - - - - - - - ,
File

Edit

Sollie

Co....nis

leport

Gr.,.

mID

lIiMoN

ltJMIiI'·

C:TRIAL.
f(x) := x""3 t 3 • x"Z - Z • x - cos(x)
rex) = 8
$ digits = 18

Settings
Colors
Directories
Load set.p
IIrite set.p

{Test of equation SOlllhlg}

Version 0.98h

~Ip

ID-Zoo. iD-lIext

II-Goto IiIilIiD!-Size/.o\le

DJE-Exit

,,~------------------------j

Figure 4-7 The Options Menu

Options
Press

Function

Enables you to select one of the commands from the Options menu.
The Options commands allow you to temporarily change some of
Eureka's internal information and screen appearance, as well as
write or read a setup file. Settings remain in effect throughout the
work session. When you exit Eureka, the settings revert to their
default values (unless you select Write setup).

Description

The Options commands, in order of appearance in the Options
menu, are: Variables, Settings, Colors, Directories, Load setup, and
Write setup.

Variables
Press

Function

Lets you restart Eureka's solving procedure using a new initial
value; it also lets you change the value of a constant.

Eureka: The Solver Owner's Handbook

Description

After Eureka solves an equation file, select Variables to display a
window of any active variables and constants.
To modify any of the displayed items:
1. Use the cursor keys to select the desired item. Press Enter.

2. Type in a new value.
3. Press Enter.
To reactivate the main menu and re-solve the problem, press Esc
Esc, then press S. If the problem has already been solved, press Esc
Esc C I to reactivate the main menu.
The Variables command is useful when the first solution Eureka
finds is not the one you want. (Problems often have more than one
solution, though only one may be appropriate.)
For example, in the case of a polynomial, Eureka starts from
some initial solution and iterates toward a final solution. While
most equations are readily solved this way, others are not. This is
because an initialization value might be selected that leads Eureka
off into a local minima that is not the true solution. Whenever you
suspect this is the case, use Variables to re-examine the problem.
This is an example of a situation where you could use the Variables command:
Suppose that Eureka is to minimize (exp(x) - a X 2) for various
values of a. You could type this into the equation file:
y = (exp(x) - a * 2)
a =3
$ min(y)

After the solution appears, choosing Variables reveals that x and
yare active variables and that a is a constant.

Note: If a had been initialized to 3 (a : = 3) rather than set equal
to 3, it would appear as an additional active variable and not as a
constant.

If you change a to some other value and select Iterate, Eureka
can find a minimum value for y using the new value of a.
Using the Variables command, a whole sequence of problems can
be solved without changing the equation file.

Menu Commands and Settings

Settings
Press
Function

OS
Lets you alter selected settings for the current equation file.
When you select Settings, a submenu appears, listing each modifiable setting, the setting's current default value, and the value's
type (real number, integer, and so on). One of the settings is highlighted.
To make a selection from this menu:
1. Use the arrow keys to highlight the setting you plan to
change.

2. Type the new value for the setting (or press Enter for yes/no
toggles).
3. Press Esc.
Enter the new value in the same format as the default value, which
is the middle item in the highlighted line.
For example, to set complex, press Enter to toggle it to yes.
Description

A number of settings can be changed from the Options/Settings
menu item. You can also change most of these ,settings by inserting
a directive in the equation file. There are six settings that cannot be
changed via this menu: include, maximize, minimize, settings, solve,
and unit conversions. See Chapter 5, "Directives," for information
about these directives.
The settings that can be controlled from the Settings menu are:
accuracy
casefold
complex
digits
finanmode
finansmooth
initval
lisLflrst
lisLinc
lisLnum

listdefault
maxtime
penalty
ploLleft
ploLright
plotdefault
radius
rootsign
substlevel
syntax

Eureka: The Solver Owner's Handbook

All of these settings except lisLfirst, lisLinc, lisLnum, ploLleft,
and ploLright can be used as directives (see "Directives" in Chapter 5 for more information about these settings).
The seven non-directive settings are described as follows:
listdefault Options: < yes or no >
Default: no
Determines whether values specified for lisLfirst, lisLinc, and
lisLnum are used. When set to yes, these values are used; when
00, you are prompted to enter values.
lisLfirst

Options: < any negative or positive number>
Default: 0.000000

Sets a new default value for the first variable to be solved for in a
table of values for the listed function.
lisLinc

Options: < any negative or positive number>
Default: 0.500000

Sets a new default value for the increment between values to be
solved for in a table of values for the function.
lisLnum

Options: < the integers 1 through 200 >
Default: 10

Sets a new default value for the number of values to be solved for
in a table of values for the function.
plotdefault Options: < yes or no>
Default: no
Determines whether values specified for ploLleft and ploLright
are used. When set to yes, these values are used; when 00, you are
prompted to enter values.
ploLleft

Options: < any negative or positive number>
Default: -2

Sets a new default value for the left-most point at which the plot
of the function begins.
ploLright Options: < any negative or positive number>
Default: 10
Sets a new default value for the right-hand side of the plot of the
function. The value must be greater than the value set for ploLleft.

Menu Commands and Settings

Colors
Press

0 C

Function

Lets you change the window colors.

Description

When you select Colors, Eureka calls up a windows menu to the
center of the screen. Move the highlighted bar up and down in the
window with the arrow keys. When the block highlights the name
of the window to be changed, press Enter. A blank sample version of
the selected window appears on the screen.
To change the window's colors, use the up and down arrow keys
to select background, text, or margin (items in the status line), then
use the left and right arrow keys to toggle the color of the selected
item in the sample window. When the item displays in the color
you want, press Enter.
To save these changes in a setup file, select Write setup.

Directories
Press

0 D

Function

Determines where Eureka stores its files.

Description

When you select Directories, Eureka presents you with a list of two
kinds of files: Eureka dir and Problem files. Use the arrow keys to
highlight a selection and press Enter. Eureka then prompts you for
the path name of the directory where you want to store your files.
For instance, suppose you want to store your equation files in a
subdirectory called C:\EUREKA\PROBLEMS and the Eureka
program itself in a directory C:\EUREKA. Enter the full path
name for each type of file. From now on, your equation flIes will be
automatically saved in the C:\EUREKA \PROBLEMS directory
when you select Write to .. or Save on the Files menu, and Eureka
will look for its program, help, and setup files in the Eureka directory.

Eureka: The Solver Owner's Handbook

Inad setup
Press
Function

0 L
Brings a file (created with the Write setup command) containing
various setup parameters to the screen. These setup parameters
will be used until you exit Eureka.

Description

When you select Load setup, a prompt appears
Enter file name

Type the full path name of the desired file, followed by Enter. If
you specify no extension, the extension .EKA is automatically
appended.

Write setup
Press

OVV

Function

Saves a file containing various screen and printer parameters.

Description

When you select Write setup, Eureka saves the current setup file
to disk. By default, Eureka looks for a file named INITIAL.EKA
when it is first loaded. If you want to use a standard set of setup
parameters when you load Eureka, name your file INITIAL.EKA.
If you specify no extension, the file name extension .EKA is automatically appended.
The modifiable Setup parameters include those for window size,
placement, and screen colors.

Menu Commands and Settings

The Window Commands

, - - - - - - - - - Eoreb: Tile Solver - - - - - - - - - - - .
rile
I.lit
Solve
CO ••AM.
leport
GrAph
Options
JIIIII!!

llext
ZTile
Stack
Goto

riJlE-lxit

~~-----------------------------------./

Figure 4-8

The Window Menu

Window
Press

Function

Selects one of the commands from the Window menu. These commands allow you to move and manipulate on-screen windows.

Description

The Window commands, in order of appearance on the Window
menu, are: Open, Close, Next, Zoom, Tile, Stack, and Goto.

Open
Press

VV 0

Function

Opens the specified window.

Description

Opens one of Eureka's windows. When you select Open, Eureka
presents you with a menu of all of Eureka's windows. Move the
cursor down to the window you want to open, then press Enter.

If several windows are currently displayed on the screen, you
can move between them by pressing F6.

Eureka: The Solver Owner's Handbook

Close
Press
Function
Description

VVC
Closes the currently active window.
When you select Close, Eureka closes the currently active window
and removes it from the screen. If you have several windows open,
you can select Close repeatedly until all on-screen windows are
closed.

Next
Press
Function
Description

VVN
Opens and makes active the next on -screen window.
Use Next to move back and forth between on-screen windows. If
you select Next repeatedly, Eureka will activate each window in
turn, moving in a clockwise direction.
The function key F6 does exactly the same thing as the Next
command.

Zoom
Press
Function
Description

VV Z
Expands the active window to fill the screen.
Use Zoom when you want the window you are currently working
with to fill the screen. This might come in handy when you are
working with an equation file that is too long to see with the default
size Edit window. To "dezoom" the window, select Zoom again. The
F5 key does exactly the same thing as the Zoom command.

Tile
Press
Function
Description

VV T
Makes all open windows visible and of equal size.
The Tile command rearranges all on-screen windows so that that
are aligned side by side in a tile-like pattern, each the same size.

Menu Commands and Settings

7,7

Stack
Press
Function

WS
Stacks all on-screen windows on top of each other.

Description

The Stack command stacks all on-screen windows atop each other
in the center of the screen. Windows are expanded to fill the entire
screen.
You can move back and forth between windows by pressing F6 or
by selecting the Next command. This "reshufHes" the windows so
that the selected window is on top.

Goto
Press
Function

WG
Puts the cursor in the last active window.

Description

The F7 key does the same thing as Coto.

Eureka: The Solver Owner's Handbook

The Equation File:
Syntax, Directives, and Functions

This chapter describes the equation file:
• what goes into an equation file
• what the syntax of an equation file must be
• what the equation file directives are and what they do
• what built-in and user-defined functions are; how they are alike, how they differ
• what causes Eureka to return an error message
Chapter 2 explains the Eureka editor and how to create the equation file; it
includes a discussion of predefined keys and illustrations of the Edit window, the
Solution window, and the Verify window. Chapter 3 provides tips for writing good
equation files. Appendix C describes all the built-in functions.

Contents of tlw Equation Fik
An equation file must contain one or more equations, as well as initialization values
and user-defined functions when needed. It may also contain directives and comments.

Equations
Equations are mathematical statements that consist of two expressions (sometimes
more) connected by an equal sign (=) or a relational operator « or ». The
following are examples of equations that can be used in a Eureka equation file:

+2 * x +x - 4
sin«x + 2 * pi)/3) > cos«x - pi)/4)
y-2 - x < x-2 + 2 * x * y
y = x-3

An expression is composed of one or more terms connected by an algebraic
operator.

A term is either a numeric value, a variable name, or a function.
The argument of a function may in turn be a variable, a constant, or a term.
Algebraic operators are:

.multiply

divide

add
subtract
raise to the power

Typically, you write one equation per line. However, you can use a colon (:) to
separate equations appearing on the same line. The colon is useful when you want
to define the constraints that limit a variable. For example, you could enter the
constraint

o,< >,< =, > =

2
3
4
5 (last)

The Equation File: Syntax, Directives, and Functions

For example, Eureka evaluates the following expression:
- U

* V-

y/ z

+ AABAC

See the description of the syntax directive in this chapter for more information
about parentheses in order of evaluation.

Syntax Rules
Eureka is fairly lenient about what it considers to be a syntactically correct equation file. You need not enter the different parts of the equation file in any specific
order as long as each equation, comment, directive, initialization, and user-defined
function is properly set up.
Eureka follows these syntax rules:

1. All spaces and tabs are ignored. Blank lines are permitted.
2. Multiple formula relations are allowed, as with
t = rA2 = xA2 + yA2
or

o< x

3. A colon (:) can be used to separate formulas on the same line. For example,
the above multiple relations are equivalent to
T = r A2 : r A2 = xA2 + yA2

and

o in
$end

: x / 2.54

Eureka: The Solver Owner's Handbook

5. Comments may either be inserted to the right of a semicolon (;) on any line
or enclosed in braces ({ }), as in
r A2

= xA2

+ yA2 ; Thi sis a conment

or
{ This file solves the ideal gas law, }
{ using mks units}

Mixing the two notations for comments can have unpredictable results and is
not recommended.
6. An underscore character (_) means that the formula continues to the next
line, as in
z = (x - a)A2 _
+ (y - br2

The underscore character is unnecessary if the upper line is an incomplete
formula, as in
z = (x - a)A2
(y _ b)A2

7. A tilde (-) gives the complex conjugate of a function, as in
z = re(z) + im(z) * sqrt(-l)
-z = re(z) - im(z) * sqrt(-l)
abs (z) = sqrt (z * - z)

8. Variables may be initialized with any valid arithmetic expression, such as
x := exp(l) * 3.14159/2

9. A directive is a line that has a dollar sign ($) as its first nons pace character.

Special Symbols
Most equation files consist of ordinary ASCII characters, but there are a few special IBM graphics characters that Eureka recognizes. These special symbols and
their decimal values are:
• square root sign (251)
• greek letters (224 through 238, except 236)
• divide sign (246)
• greater-than-or-equal, less-than-or-equal (242, 243)
• two (253)
• one half, one quarter (171, 172)
• pi (227) (You can also get the pi symbol by pressing Alt-P.)

The Equation File: Syntax, Directives, and Functions

To use a special symbol in an equation file, hold down Aft and Shift while typing
the symbol's value on the numeric keypad to the right of the keyboard (the one set
up like a lO-key adding machine). For SuperKey users, the file EUREKA.MAC on
the distribution disk contains macros for these special symbols.

Directives
A directive is a command to the Eureka program; it is embedded within an equation file and affects only that file. Most directives in Eureka affect the settings that
are user controlled. There are a few others that perform different kinds of functions.
You can also change some settings from the Options menu. The settings stay in
effect until you exit Eureka. You can also use the Load setup and Write setup
commands to save and load settings. You can only make permanent setting changes
within a given equation file by using directives.
The only directives (settings) that canrwt be changed from the Options menu
are: include, maximize, minimize, settings, solve, and unit conversions.
A directive consists of the symbol $ followed by the directive name, an equal
sign (usually), and some value. For example, you can set the setting syntax to 2 in
an equation file with the directive
$ syntax = 2

You can use the settings directive to enter a subgroup of the directives (those
which affect Eureka's internal default settings) in an alternate format, illustrated
here:
$ settings
syntax = 2
digits = 10
accuracy = le-13
penalty = 50
$ end

This format is convenient for changing multiple settings, since only two $ symbols are required. You can include as many directives as you need between settings
and end.

If you have a group of non-default settings common to a number of problems,
you can store them in a separate file. You can then insert this separate settings file
into an equation file with the include directive.

Eureka: The Solver Owner's Handbook

For instance, the following directive would include a settings file called
DEFAULTS:
$ include DEFAULTS
II

The directives are individually described in alphabetical order on the following
pages. Each description includes the directive name, syntax (exactly how to type
it), default value, function, and any pertinent remarks.

The Equation File: Syntax, Directives, and Functions

accuracy
SY11tax

$ accuracy =

Default

.0000001 (tOe - 7)

Function

Specifies a value that must be met before an iterative search stops.

Description

When Eureka reports a solution, it is not necessarily exact. In most
cases, Eureka's iterative procedure searches for a solution until the
accuracy matches the accuracy setting value.
The default value (.0000001) results in an answer that is accurate
to six or seven decimal places. Eureka can achieve greater accuracy
if you set the accuracy directive to a smaller value, such as
$ accuracy = 1.De - 12

However, there are tradeoffs to using very small values for the
accuracy. Smaller values increase computer processing time and
could undermine the overall reliability of the solution procedure.

casefold
Syntax

$ casefold

Default

Function

Changes Eureka's sensitivity to the case of variable names.

Description

The default allows uppercase and lowercase letters to be used as
different, unique characters; by default, Eureka considers pressure
:md Pressure to be two different variables.

If you change casefold to yes, Eureka will consider the uppercase and lowercase versions of any letter as only one variable.
Key words (directives, and so on) and built-in functions are
never affected by casefold and are not case sensitive. For example,
the exponential function can be written exp(x), Exp(x), or EXP(x).

complex (numbers)
Syntax

$ complex =

Default

Function

Makes all variables complex.

Eureka: The Solver Owner's Handbook

Description

When complex = no, Eureka cannot solve equations whose solutions have a non-zero imaginary component, except for polynomials
solved with the poly function.
For example, it is impossible to solve the following equation
without using complex numbers, since its solutions are x = (3 + i)
and x = (3 - i):
However, Eureka can find a complex solution if you change the
mode to complex, either from the Settings menu or by putting the
complex directive in the equation file.
While Eureka is operating in complex mode, the functions exp,
ln, sin, cos, tan, and sqrt refer to their analytic continuations. (A
branch cut is used for in, sqrt, and exponentiation.)

Re and im are the real and imaginary parts, respectively. The
tilde ( - ) gives the complex conjugate. By this definition,
z = re(z) + im(z) * sqrt(-1)
- z = re ( z ) - i m(z ) * s q rt ( -1)
abs (z) = sqrt (z * - z)

When maximizing or minimizing a complex function, Eureka
uses only the real part.
Similarly, <, >, and polar refer only to the real parts of the
expressions they operate on. Usually, however, these expressions
are arranged so as to be real, as in
abs(z) < 100

The best way to obtain complex constants is to assign an imaginary value to some variable (typically i or I). The following example
illustrates this technique:

= sqrt( -1)
z=2+3*i

digits
Syntax

$ digits

Default

Function

Sets the number of digits displayed for solved variables.

Description

The digits directive tells Eureka to display only the specified number of digits for values in the Solution window.

The Equation File: Syntax, Directives, and Functions

This directive only affects the way in which Eureka displays real
numbers and is unrelated to the underlying accuracy of the solutions.

end
Syntax

$ end

Default

None

Function

Completes a list of directives begun with either a settings or units
directive.

Description

See the descriptions for settings and units.

finanrrwde
Syntax

$ finanmode =

Default

End

Function

Sets the odd financial period (if there is one) to be at either the end
or the beginning of the series of periods used for calculations made
with the financial functions.

Description

Changingfinanrrwde to begin puts the odd period at the beginning
and results in slightly different values for the functions pval, paymt,
andfval.

finansrrwoth
Syntax

$ finansmooth =

Default

Function

Determines whether the pval, paymt, and fval functions are
smooth.

Description

The financial functions, pval, paymt, and fval, are discontinuous as
functions of time. This is because the value of a loan contract
changes at the end (or beginning) of each period, every time a
payment is made. These discontinuities make solving for the time
variable difficult or impossible.

Eureka: The Solver Owner's Handbook

You can use the finansrrwoth directive to circumvent this problem. When you set finansrrwoth to yes, pval, paymt, and fval
become smooth functions. These smoothed functions are exactly
correct if the time is an integer; if time is not an integer, they return
interpolated values which may differ somewhat from the true
values.

include
Syntax

$ include ""

Default

None

Function

Inserts a second file into the body of a first.

Description

The way files are included in Eureka is similar to the way they are
commonly included in programming languages. This directive is
provided as an editing convenience. When Eureka processes an
equation file with an include directive, it replaces the directive with
the named file. This keeps your equation files from becoming
unnecessarily large and unwieldy.
Include files are primarily useful for storing unit conversions and
set-up options separately, so they can easily be used in a number of
problems. They are also useful for storing frequently used data,
such as chemical molecular weights for a series of stoichiometry
problems.
Include files can be nested.

initval
Syntax

$ initval

Default

Function

Provides an alternative way to initialize the starting value for a
variable.

Description

To start the solution procedure, Eureka requires a starting value for
every variable. If a variable is not given an explicit initialization in
the problem file, it is initialized with the default value, 1.

If some variables in the equation file are not explicitly initialized
in the file, and if you include initval in the file, Eureka will use the
initval value to initialize those variables.

The Equation File: Syntax, Directives, and Functions

listdefault
SY1ltax

$ listdefault =

Default

Function

Lets you override the default values for the List function.

Description

If you set listdefault to no, a series of prompts appears in response
to the List command. These prompts allow you to override the
default values set for List.

If you set listdefault to yes, Eureka automatically plugs in the
default values for lisLfirst, lisLinc, and lisLnum, and generates
the corresponding table.
Note: lisLfirst, lisLinc, and lisLnum are not directives. See
Chapter 4 under the Settings menu.

max (imize)
SY1ltax

$ max (variable)

Default

None

Function

Maximizes the specified variable.

Description

When the max directive is in an equation file and you select Solve,
Eureka finds a maximum for the specified variable. For example,
with the following in an equation file
y =

-x~2

- 4

x +3

$ max (y)

Eureka will find the value of x that yields the maximum value of
y. Without a max directive, Eureka solves for the roots of y: the
value(s) of x that yield a zero value for y [ - 2 ± sqrt(7)].
You can use the max directive only once in an equation file, and
you cannot use both max and min in the same file.

Eureka: The Solver Owner's Handbook

maxtime
Syntax

$ maxtime = (positive integer)

Default

200

Function

Sets a limit, in seconds, to the amount of time Eureka spends solving an equation.

Description

When you place the maxtime directive in an equation file, you set a
limit to the amount of time Eureka spends seeking a solution.

min (imize)
Syntax

$ min (variable)

Default

None

Function

Minimizes the specified variable.

Description

When you place the min directive in an equation file, it supersedes
the Solve command (which searches for roots-zeroes-of equations). Eureka will find a minimum value for the specified variable
or function rather than a solution for all variables that make equations in the file equal zero.
For example, with the following in an equation file
y =

x~2

+4 *

$ min (y)

Eureka will find the value for x that yields the minimum value of
y. Without the min directive, Eureka would solve for those values of
x which make y = 0 (-3 and -1).
You can use the min directive only once in an equation file, and
you cannot use both max and min in the same file.

penalty
Syntax

$ penalty =

Default

Function

Sets a weight (relative importance) for the constraints affecting an
equation.

Description

Eureka uses the value of penalty to calculate the effect of the constraints in an equation file. Changing the default affects the relative
strength of the constraint criteria.

The Equation File: Syntax, Directives, and Functions

plotdefault
SYJltax

$ plotdefault =

Default

Function

Lets you override the default values for the Plot command.

Description

If you set plotdefault to no, a series of prompts appears in response
to the Plot command, allowing you to override the default values
set for Plot.
If you set plotdefault to yes, Eureka automatically plugs in the
default values for ploLleft and ploLright, and generates the corresponding plot. The default values for ploLleft and ploLright are
- 2 and 10, respectively.
Note: ploLleft and ploLright are not directives. See Chapter 4,
page 72 for more information.

radius
Syntax

$ radius =

Default

~one

Function

Sets a limit to the range over which Eureka seeks a solution.

Description

If you set radius to no, Eureka will search an indefinite range of
numbers, limited only by its internal constraints, accuracy, and the
maxtime directive.

If you set radius to a real number, Eureka will only search a
range of numbers that are equal to or less than the number speci ..
fied.

rootsign
SYJltax

$ rootsign =

Default

Pos

Function

Toggles between displaying the roots of an even-powered number
as negative or positive numbers.

Description

If you set rootsign to pos, Eureka displays roots of even powers as
all positive values; if you set it to neg, Eureka displays the roots as
all negative values.

Eureka: The Solver Owner's Handbook

settings
Syntax

$ settings

$ end

Default

None

Function

Allows you to enter several directives in a series without having to
set each one off with a $ symbol.

Description

You can think of the settings directive as the first, or opening, step
in a three-step process for changing multiple settings.
The second step is to type in the directives for the settings that
are changing (without typing in the initial $ symbol each time).
The final step is to close the change session by typing in the end
directive.

If you have a group of frequently used settings, you can store
them in a separate file. You can then introduce this separate Settings file into an equation file with the include directive.
Refer to the description of the include directive in this chapter
for more information.

solve
Syntax

$ solve (variable)

Default

All

Function

Specifies which variable in an equation file Eureka will solve for.

Description

By default, Eureka solves for all variables in an equation file. However, you can use the solve directive to tell Eureka to solve for only
the designated variable and ignore the rest.

The Equation File: Syntax, Directives, and Functions

substlevel
Syntax

$ substlevel

Default

Function

Sets the maximum amount of internal variable substitutions
Eureka can perform when solving an equation file.

Description

The default, 6, allows Eureka to make the maximum number of
substitutions during a Solve process. Substitution level 5 allows
Eureka to make fewer substitutions; level 4 allows less than level 5,
and so on down to substitution level 0, which allows no substitutions.

<0, 1, 2, 3, 4, 5, or 6>

syntax
Syntax

$ syntax

Default

Function

Specifies one of three possible options for the syntax of arithmetic
expressions used in the equation file.

Description

There are three options for the syntax of arithmetic expressions:

<0, 1, or 2>

0: The default syntax, which requires the * symbol to represent
multiplication. The default syntax allows scientific notation.
1: Does not allow scientific notation, because a number placed
before a variable is treated as a multiplier.
2: Does not allow scientific notation, for the same reason as syntax level 1. Also restricts variables to one letter each, rather than
the default choice of up to 60 alphanumeric characters.

Examples of each syntax level
Syntax = 0: The default syntax is the standard used throughout
this manual.
Syntax = 1: Scientific notation (such as 2.3e - 4 to represent
0.00023) is not allowed because Eureka interprets it as 2.3 X (a
variable called e) - 4. A number before a variable name is interpreted as a multiplier.

Eureka: The Solver Owner's Handbook

For example, when syntax

1, Eureka treats the expression

2.7 water

as an abbreviation for
2.7 * water

As when syntax = 0, Eureka ignores spaces. However, the program still treats a character string (such as dog34) as a variable
name.
Syntax = 2: Variables are restricted to a single letter. All juxtapositions of numbers and variables are treated as products.

For example, a problem to find the intersection of two hyperbolas in the x - y plane could be formulated as:
$ syntax = 2
x-2 - 4xy + 3y-2 = 7
2xy - 5y + 3 = 0
{ exact solution is x = 2, y = 3 }

When syntax = 2, variable names must be distinct from function
names; otherwise, an expression such as f(x + 1) is ambiguous
(Le., is f a variable multiplying x + 1, or is it a function of x + I?).

If the equation file contains a user-defined function, the function
definition must precede the first use of the function.
Refer to the section "Notes On Equation File Syntax" (page 82)
for more information about writing equations using the default syntax level.

units (conversion)
Syntax

$ units
->

$ end

Default

No automatic conversion

Function

Defines unit conversions.

Description

Eureka can automatically associate units to variables, and then
convert the values of the variables to other units in the Solution
window.

The Equation File: Syntax, Directives, and Functions

For example, the variable dist can appear in the equation file
expressed in feet, but can be converted to meters and yards when
Eureka solves for its value.
The units conversion directive lets you define as many unit conversions as needed. For example,
$ units
em -> in : x/2.54
F. -> C. : (x - 32) * 5/9
feet -> yds : x/3
km/hr -> mi/hr : 0.6214 * x
$ end

Each line after $ units gives a formula for converting from one
unit of measurement to another.
The nrst formula converts from centimeters (em) to inches (in); x
is simply a dummy variable. The formula states that if x is in centimeters, dividing x by 2.54 will convert it from centimeters to
inches.
The names for the units can be any sequence of characters,
including punctuation and math symbols, as long as the symbol
does not include a hyphen (-) or a colon (:). Eureka ignores blanks.
The syntax for expressions is the standard Eureka syntax.
You associate units with the variables in the initialization statement. For example, the initialization
dist := 122.7 [em]

initializes the variable dist to 122.7 centimeters. When solving for
dist, Eureka nrst gives the solution in centimeters, then converts
the value in centimeters to any other units for which a units conversion directive exists.
Here is a typical solution for this example:
dist = 110.0235 em
= 43.3163 in

Eureka can invert any simple unit conversion. If the equation
:6le contains the formula for converting from centimeters to inches,
it is not necessary to also give the formula for converting from
inches to centimeters.

Eureka: The Solver Owner's Handbook

Eureka Functions
Besides providing a wide variety of standard mathematical and special built-in
functions, Eureka also allows you to define and use your own functions. Most of the
built-in functions need little explanation; they are described in Appendix C. However, the financial functions require some in-depth explanation; they are discussed
in detail in this chapter. After the section about financial functions in this chapter,
we give an explanation of user-defined functions.

Financial Functions
Eureka has three functions for performing interest-rate calculations: ivaI, paymt,
and pval.
These functions are designed for situations in which a person enters into a contract by which a fixed sum is paid at regular intervals. We assume that money earns
interest at some fixed rate, with the interest compounded in each interval.
In general, there are five variables involved in these financial calculations:

n
pval
paymt
ivaI

number of payment intervals
interest rate per payment interval
present value of the contract
payment at the end of each interval
future value of the contract

Given any four of these five variables, Eureka can solve for the fifth. The variables are related by the financial functions in the following ways:
pval = pval(i,n,paymt!val)
paymt = paymt(i,n,pval!val)
ivaI
= ival(i,n,pval,paymt)

nand i can be derived directly from the values of the other variables using Solve.

The Equation File: Syntax, Directives, and Functions

Variables
The number of payment intervals, n, is normally an integer, but it can be any
positive number.
If n is not an integer, there is a fractional period that is assumed to occur before
all other periods. Interest is compounded at the end of, but not during, the fractional period. The finanrrwde directive can place the fractional period at the beginning or end of the series.
i, the interest rate, should not be converted to a percentage rate.

For example, i = 0.12 if the interest rate is 12%. The interest is compounded at
the end of each payment interval.

Functions
pval, paymt, andfval all represent money that changes hands. By convention, the
number is positive for money received and negative for money paid out.
pval, the present value, is defined as the principal: the money received (loaned)
at the beginning of the contract. In the case of a home mortgage, pval is the price of
the home minus the down payment.
paymt, the payment, is the money received at the end of each payment interval.

If the person pays the money out, as with a home mortgage, paymt is a negative
amount.
fval, the future value, is the money received at the end of the contract. In the
case of a home mortgage, fval is either zero or a negative number that represents
the balloon payment.

Example
A typical use of the financial functions would be to compute the payments on a
mortgage.
For example, suppose a house costs $100,000; the interest rate is 12%; and the
mortgage is for a period of 30 years. Assume that the down payment is 10%, there is
no balloon payment, and payments and interest compounding are monthly.
Using paymt, the monthly payment can be calculated as follows:
paymt(0.12/12,30 X 12,90000,0)
where 0.12/12 is the monthly interest rate, 30 X 12 is the number of payments,
90000 is the amount of principal, and 0 is no balloon.
Note: The last argument would be non-zero only if there were a balloon payment
at the end of the 30 years.
100

Eureka: The Solver Owner's Handbook

By default, Eureka assumes that payments are made at the end of each interval
rather than at the beginning. This is because the settings parameter finanmode has
the following default value:
finanmode = end

You can change this setting to begin either from the Options/Settings pull-down
menu or by directive in the equation file.
Because banks often have their own unique methods for computing interest
rates, Eureka's built-in financial functions may vary slightly from any particular
bank's methods.

User-defined Functions
Eureka deals primarily with variables and the relationships among them. It is often
convenient to think of variables as functions of one another, and Eureka does so
with its set of built-in functions.
You can add to Eureka's set of functions by defining your own for a given equation file; these are known as user-defined functions. The following discussion
describes the relationship between variables and functions and then explains various aspects of user-defined functions.

Relationship of Variables and Functions in Eureka
Consider an equation with one or more variables in which it is possible to solve for
one variable in terms of the others. This solved variable could, in effect, be
expressed as a function of the other variables. However, to Eureka, it is still just a
variable.
For example, the function sin(x)
where

+ cos(x) can be maximized by maximizing f

f = sin(x) + cos(x)
Here, f can be conceived as a variable that happens to be related to x.
In another example, the equation
x2 + y2 = 25
might be transformed by Eureka internally to either

x = sqrt(25 _ y2)
or

The Equation File: Syntax, Directives, and Functions

101

In other words, x can be considered as a function of y, or y as a function of x. To a
user, however, x and yare simply related variables.
Eureka recognizes two types of functions:
• built-in functions (such as cos and sin)
• user-defined functions
Built-in functions have no values unless arguments are supplied, and it makes no
sense for Eureka to solve for them.
User-defined functions describe configurations of variables that you wish to capture in shorthand fashion.
Note: User-defined functions affect only the equation files in which they are
defined.

Writing User-defined Functions
The symbol : = Hags a user-defined function as in
trig(x) := sin(x) + cos(x)

This example defines a new function called trig that acts in the same way as a
built-in function. The variable x is a dummy variable that merely serves to define
the function and is unrelated to any other x that may occur elsewhere in the equation file.
When an equation file contains this user-defined function, trig(x), you can insert
tri 9(7)

or
trig(a * b + 3)

into an equation rather than inserting the lengthier expressions
(sin(7) + cos(7»

or
(sin(a * b + 3) + cos(a * b + 3»

The lexical rules regarding function names are the same as those for variable
names (begin with a letter or certain special characters, limited to 60 alphanumeric
characters). Although you can give the same name to a variable and a function, this
practice is not recommended. You can define a function of several variables (or
even no variables).

102

Eureka: The Solver Owner.'s Handbook

For instance, defining a function by
gas(x,y) := x

+y *

+ 10

causes the expression 2
2

* (3 +

+ 1) *

* gas(3,a +

1) to be treated as

+ 10)

For most equation solving purposes, user-defined functions are essentially editing conveniences. However, several Eureka features require user-defined functions:
• making plots or tables
• finding roots of polynomials
• solving differential calculus problems

Error Messages
When writing user-defined functions, keep in mind that Eureka returns an error if:
1. You use a function without defining it somewhere in the equation file.

2. You redefine a function in the equation file.
3. A function calls for a certain number of arguments, but you use it with a
different number of arguments than those defined, as in
f(x,y,z) := x~2 - y~2 + z~2
f(a,b) = 100

4. You define a function in a circular way, as in
f(x) := x~2
g(x) := f(x

+ 3 * g(x)
+ 1) - 7 *

The Equation File: Syntax, Directives, and Functions

103

104

Eureka: The Solver Owner's Handbook

Worked Examples

This chapter provides several worked problems. These examples are designed to
demonstrate Eureka's power in solving a variety of mathematical problems. The
problems are contained on your distribution disk under the following file names:
Problem 1 Quick Demo

Problem 2 Loan Payoff
Problem 3 Ladders in an Alley
Problem 4 Distribution of Line Charges
Problem 5 Radioactive Decay
Problem 6 Least-Squares Fit
Problem 7 Motion of a Projectile
Problem 8 Charged Masses
Problem 9 Using the Poly Function
Problem 10 Maximization (House Design)
Problem 11 Maximization (Linear Program)
Problem 12 Equity in a House
Problem 13 Probability in a Card Game

CLASSICS.EKA,
DERINT.EKA,
MAXMIN .EKA,
COMPLEX.EKA
CARLOAN .EKA
LADDERS.EKA
LINECHG.EKA
DECAY.EKA
LSQRFIT.EKA
PROJCTL.EKA
MASSCHG.EKA
POLYNOM.EKA
HOUSE.EKA
LINPROG.EKA
EQUITY.EKA
CARDS.EKA

105

Problem 14 Income Distribution (Integ Function)
Problem 15 Chemical Calculations

INTEC.EKA
CHEMBAL.EKA,
CHEMEQl.EKA,
CHEMEQ2.EKA,
CHEMEQ3.EKA

The first series of problems demonstrate Eureka's basic features:
• solving systems of nonlinear (and linear) equations
• solving and plotting user-defined functions
• calculating a least-squares fit
• solving transcendental functions
Later problems incorporate some of Eureka's special capabilities and built-in
functions:
• maximizations and minimizations
• definite integrals
• factorials
• polynomial solutions
• financial calculations

If you want to get an idea of how to set up and solve problems with Eureka, or if
you want to see how we have incorporated some of Eureka's capabilities and features into an assortment of problems, we suggest you look over and study some or
all of the examples in this chapter. They include a sampling from a variety of fields
and topics: chemistry, finance, probability, physics, electrostatics, and economics, to
name a few.
Each worked problem consists of:
• a statement of the problem
• an explanation of the model equations
• a copy of the report generated after Eureka solves the problem
All of the reports contain the equation file and the contents of Eureka's Solution
window for the problem. Some reports also include the contents of Eureka's Verify
window, comparing the solution values with the original equations and constraints.
Other reports include a two-column table (generated with Eureka's Graph/List
command) that compares values of a specified function from the equation file with
the function variable.
Remember - Eureka: The Solver is designed to help you solve a whole world of
mathematical problems: This chapter presents but a small sample of what Eureka
can do!

106

Eureka: The Solver Owner's Handbook

Problem 1
A Quick Demonstration of Eureka's
Mathematical Capabilities
This problem (CLASSICS.EKA, DERINT.EKA, MAXMIN.EKA, and COMPLEX.EKA on the distribution disk) highlights some of Eureka's capabilities. It
demonstrates how Eureka:
• quickly solves basic mathematical identities
• calculates and plots definite integrals and derivatives
• solves integral problems
• maximizes or minimizes functions over a given range
• solves for complex variables
This example is broken into four sets of problems, covering four topics:
• classic identities
• integrals and derivatives
• minimizations and maximizations
• complex variables
In the first set of problems, Classic Identities, Eureka solves a few trigonometric
and transcendental equations. You already know the solutions to the equations; our
purpose in demonstrating how Eureka handles them is to familiarize you with how
fast (and how well) the equation solver works.

The second set of problems, Derivatives and Integrals, illustrates how you use
Eureka's built-in integ and deriv functions, and shows how to set up user-defined
functions (delimited with the: = symbol) so Eureka can plot them.
Maximizations and Minimizations, the third set of problems, demonstrates how
to incorporate Eureka's max and min directives in your equation files. Maximizations (and minimizations) are not an easy task for a computer program, especially
when the variables are subject to certain constraints. However, as you will see in
this set of problems, Eureka solves them with ease.
The final set of problems, Complex Variables, shows how Eureka solves for both
the real (re) and imaginary (im) part of problems such as
y2 + 1 = 0 and e iXpi = theta.

Worked Examples

107

Statement of the Problem
Suppose you were given the following (admittedly simple) problems to solve with a
hand-held calculator. How long would it take you to solve for the given variables,
verifY that your answers were appropriate, and plot the functions requested?

Classic Identities
1.

Solve for x where In(x) = 1; plot the function for
0.1 < x < 5

=1
=1

Solve for y where sin(y/2)

Solve for z where tan(z/4)

Plot F(a) = sin(a)/a for -10 < a < 10
Solve for j where F(j) = 1/2

Derivatives and Integrals
5.

Given the following functions A, B, C, and D:
A(x) = (COS(X))2
B(x) = sin(x)
C(x) = cos(x/2)
D(x) = A(x) + B(x) + C(x)
a. Plot D(x) for -10 < x < 10
b. Calculate the integral of A(x) for 0 < x < 5
c. Calculate the integral of D(x) for 0 < x < 5
d. Find a zero of dA(x)/dx
e. Find a zero of dB (x)/dx

£ Find a zero of dC(x )/dx
g. Find two zeroes of dD(x)/dx: one negative, one non-negative

Maximizations and Minimizations
6.

For the function F(a) in problem 4:
a. Find the minimum value of F(a) over the range 0.1 < a < 10
b. Find the maximum value of F(a) over the range -10 < a < - 4

108

Eureka: The Solver Owner's Handbook

At what point x > 0 does the integral of [sin(a) da] reach the value I?

Maximization: at what point x > 0 does the integral of [sin(a) da] reach a
maximum value?

Maximization: at what point c on the range - 4 to 4 does sin(c) reach a maximum?

10.

Minimization: at what point a on the range - 4 to 4 does sin(a) reach a minimum?

11.

Plot sin(a) over the range - 4 < a < 4

Complex Variables
12.

Solve for the variables x, y, and theta in the following equations:

x2 = -1
y2 + 1 = 0
exp(x X pi) = theta (pi

= 3.1415926)

Would you look forward to solving, evaluating and plotting these functions with
only the aid of a calculator and a pencil? Probably not.
Not only can Eureka solve for the given variables; the equation solver can also
plot the requested functions and evaluate the solutions (flagging any that don't
satisfy the given constraints). Not only that, Eureka will generate complete, formatted written reports giving the original problems, the solutions, the evaluations
and the plots. And you never have to lift a pencil.

Equations
For the given problems, Eureka will solve these corresponding equations:

L(x) : = In(x)
L(A) = 1.00000

S(x) : = sin(x/2.0)
S(B) = 1.00000
T(x) : = tan(x/4.0)
T(C) = 1.00000
F(x) : = sin(x)/x
F(j1) = 0.50000 : j1 < 0
F(j2) = 0.50000 : j2 > 0

3.
4.

Worked Examples

109

A(x) : = (COS(X))2
B(x) : = sin(x)
C(x) : = cos(x/2)
D(x) : = A(x) + B(x) + C(x)
D(ddl) =
< ddl < 10
D(dd2) =
dd2 : = - 1

°°::°

Ap(x) : =
Bp(x) : =
Cp(x) : =
Dp(x) : =

deriv((cos(x))2, x)
deriv(sin(x), x)
deriv(cos(x/2), x)
deriv(D(x), x)

°°
°°

Ap(ap) =
Bp(bp) =
Cp(cp) =
Dp(dp) =
Dintl = integ(D(x),x,0,5)
Aintl = integ(A(x),x,0,5)
Eureka will solve for ap, bp, cp, dp, ddl, dd2, Aintl, and Dintl. Those equations containing the: = symbol are "user-defined" functions; refer to Chapter 5 for more information about these and other special Eureka functions.

F(z) : = sin(z)/z
$ min (K)
= F(z)
OJ < .z < 10

$ max (L)
L = F(z)
-10 < z < -4
7.

1.000 = integ(sin(a), a, 0, w/2)

w/2

$ max (1)
1 = Maxlnt(x)
Maxlnt(a) : = integ(sin(a), a, 0, x)

$ max (z)
z = S(c)
S(c) : = sin(c)
-4 < c < 4

d = 2c
10.

$ min (y)
y = S(a)

S(a) : = sin(a)

110

Eureka: The Solver Owner's Handbook

-4 < a < 4
b = 2a

11.
12.

S(a) : = sin(a)
x2 = -1
y2 + 1 = 0
exp(x X pi) = theta: pi = 3.1415926

To solve these problems, select one of the equation files (use File/Load from the
main menu), then select Solve.

Equation Files and Solutions
The print-outs that follow show the equation files and Eureka's solutions for the
examples in this problem.

Graphing the Functions
Eureka also offers the ability to plot functions of one variable. To graph anyone of
the functions in these problems:
• Solve the equation file.
• Select the Graph option from the main menu.
• From the Graph menu, choose Plot.
• Select one function from the mini-menu listing of all user-defined functions.
• Enter values for the left and right x axis endpoints.
Eureka will automatically plot the function.

Generating Reports
You can also direct Eureka to generate your own written reports; after solving an
equation file (also evaluating the solutions, if you choose) and plotting a function,
select Report from the main menu. Eureka will place the report on screen, write it
to a disk file, or send it directly to your printer, as you choose.
Note: We used Borland's SideKick to capture the "zoomed" plots and insert
them in the reports that follow.

Worked Examples

III

*******************************************************************
Eureka: The Solver, Version 1.0
Thursday February 12, 1987, 10:13 am.
Name of input file: CLASSICS.EKA
*******************************************************************

These examples show how Eureka quickly
solves some classic problems
L(x) := In(x)
L(A) = 1.00000
plot L(x) for 0.1 < x < 5
Sex) := sin(x/2.0)
S(B) = 1.00000
T(x)
T(C)

:= tan(x/4.0)
= 1. 00000

After solving this equation file, plot
the function F(x) for -10 < x < 10
F(x) := sin(x)/x
F(j1) 0.50000 j1 < 0
F(j2) = 0.50000 : j2 > 0
*******************************************************************

Solution:
Variables

Values

2.7182818

3.1524423

3.1416042

-1.8954943

1.8954943

Maximum error is .000014714291
*************************************************************************

112

Eureka: The Solver Owner's Handbook

1.87

-c~·------------~I
.111.5.11

-2.51
*******************************************************************

18.2

..
.
.
.
.
t--.....••--. - ...... ---. - ......•• __._-.....•• __ .--.. ...•• --+
-38.8

•

••

•

38.8

-10.2

*******************************************************************

Worked Examples

113

r
e

. .

.e .

l·--·-----+------·--·l
-11.1

11.1-

-.289
*******************************************************************

1.16~

f-.----

-18.8

-----'.-+
111.8·

*******************************************************************

114

Eureka: The Solver Owner's Handbook

*******************************************************************

Eureka: The Solver, Version 1.0
Thursday February 12, 1987, 11:23 am.
Name of input file: DERINT.EKA
*******************************************************************

These problems demonstrate how to use
Eureka's INTEG and OERIV functions, and
include examples of plotting the
functions
A(x) : = (cos (x»~2
Sex) := sin(x)
C(x) . - cos (x/2)
O(x) := A(x) + Sex) + C(x)
0(dd1) 0
< dd1 < 10
D(dd2) =
dd2 := -1

°: °

Ap(x) := deriv«cos(x»~2, x)
Sp(x) .- deriv(sin(x), x)
Cp(x) .- deriv(cos(x/2), x)
Op(x) .- deri v(0 (x), x)
Ap(ap)
Sp(bp)
Cp(cp)
Op(dp)
Ointl
Aintl

°°0
integ(D(x),x,0,5)
integ(A(x),x,O,5)

*******************************************************************

Solution:
Variables

Values

Aintl

2.3639947

1.5707964

-1.8097622e-10

dd1

3.6612084

dd2

-2.2077911

Ointl
dp
Maximum error is

Worked Examples

4.2772768
-349.75174
.067092381

115

*******************************************************************

Evaluation of formulas:
Formulas

Values

D(dd1)

-6.1014156e-08
.00000000
difference (error)
-6.1014156e-08

o
o

dd1
difference

.00000000
3.6612084
-3.6612084

10
dd1
difference =

10.000000
3.6612084
-6.3387916

D(dd2)

= 4.0720587e-08
.00000000
difference (error) = 4.0720587e-08

dd2

-2.2077911

Ap(ap)

8.7422780e-08
.00000000
difference (error) = 8.7422780e-08

Bp(bp)

= -4.3711390e-08
.00000000
difference (error) = -4.3711390e-08

Cp(cp)

= 4.5244055e-11

.00000000
difference (error) = 4.5244055e-11

Dp(dp)

-.067092381
.00000000
difference (error)
-.067092381

Dintl
integ(D(x) ,x ...
difference =

4.2772768
4.2772768
.00000000

Aintl
integ(A(x) ,x ...
difference =

2.3639947
2.3639947
.00000000

Maximum error is

.067092381

116

Eureka: The Solver Owner's Handbook

*******************************************************************

•

1-. ------. -- 4-------.--+
-11.1

-1. 92

18.8

******************************************************************~

~.-----.-----

-18.8.

.--~r-------------------

------------+1
18.8

-2.81

*******************************************************************

Worked Examples

117

*******************************************************************

Eureka: The Solver, Version 1.0
Thursday February 12, 1987, 11:48 am.
Name of input file: MAXMIN.EKA
*******************************************************************

These examples illustrate how to use
Eureka's $max and $min directives to
maximize and minimize any variable that
is defined to be the result of a given
function. One of these examples also
demonstrates how to solve an integral
equation with Eureka.
NOTE: To solve anyone of these examples,
remove the ";" symbol at the beginning of
each equation line for that example.
Eureka will only solve for ONE $min or
$max in an equation file. To solve a
different example, first reinsert the ";"
symbols from the previously-solved
example.
Part 1: A well-known example:
F(z) ::: sin(z)/z
Problem I-a:
$min (K)
K :: F(z)
0.1 < z < 10
Problem I-b:
$ max (L)
L :: F(z)
-10 < z

-4

Part 2: Integral of Sin(x) dx
Problem 2-a:
Solving an integral equation: at what
point x > 0 does the integral of
[sin(a) da] reach the value I?
(Let w/2 equal x.)
1.000 :: integ(sin{a), a, 0, w/2)
w/2 :: x
Problem 2-b:
Maximization: at what point x > 0 does
the integral of [sin(a) da] reach a
maximum value?
$max (I)
I :: Maxlnt(a)
Maxlnt(a) ::: integ(sin(a), a, 0, x)
Part 3: Max and Min of Sin{x)

118

Eureka: The Solver Owner's Handbook

Problem 3-a:
Maximization: at what point c on the
range -4 to 4 does sin(c) reach a
maximum? (Note: The function S(c) is
included for plotting purposes - Eureka
does not need it to solve for c or z.)
$max (z)
z = S(c)
S(c) := sin(c)
-4 < c < 4
d = 2*c
Problem 3-b:
Minimization: at what point a on the
range -4 to 4 does sin(a) reach a
minimum? (Note: The function Sea) is
included for plotting purposes - Eureka
does not need it to solve for a or y.)
$min (y)
y = Sea)
Sea) := sin(a)
-4 < a < 4
b = 2*a
*******************************************************************

Solution: (I-a)
Variables

Values

-.21723363

4.4934066

Confidence level
98.9%
All constraints satisfied.
*******************************************************************

1·16t

.-.

t·-·---+-----·-·t

-11.1

11.1-

-.289
*******************************************************************

Worked Examples

119

1.1rtf-

__----C-.-+

1------'

11.1·

-II.'

-.311

*******************************************************************

Solution: (2-a)
Variables

Values

3.1415882

1.5707941

Maximum error is 2.2415074e-06
*******************************************************************

Evaluation of formulas:
Formul as

Values

1.000
integ(sin(a) •••
difference (error)
w/2
x
difference

1.0000000
.99999776
2.2415074e-06

1.5707941
1.5707941
.00000000

Maximum error is 2.2415074e-06
*******************************************************************

120

Eureka: The Solver Owner's Handbook

Solution: (2-b)
Variables

Values
2.0000000
3.1415927

Confidence level
100.0%
All constraints satisfied.
*******************************************************************

Solution: (3-a)
Variables

Values

1.5707963

3.1415927

1.0000000

Confidence level
98.7%
All constraints satisfied.
*******************************************************************

s
1. 111

4---.---- ---_.-+

-4.11-

4. •

•

-1.11
*******************************************************************

Solution: (3-b)
Variables

Values

-1.5707963

-3.1415926

-1.00000000

Confidence level = 98.7%
All constraints satisfied.
*******************************************************************
Worked Examples

121

*******************************************************************
Eureka: The Solver, Version 1.0
Thursday February 12, 1987, 1:35 pm.
Name of input file: COMPLEX.EKA
*******************************************************************
This example demonstrates how Eureka
solves for complex variables

$ complex = yes

theta
exp(x*pi)
pi = 3.1415926
*******************************************************************
Solution:
Variables
re pi
im pi

Values
3.1415926
.00000000

re theta
im theta

-1.00000000
5.3589793e-08

re x
im x

6.1257423e-17
1.0000000

re y
1.078372ge-27
im y
1.0000000
*******************************************************************

122

Eureka: The Solver Owner's Handbook

Problem·2
Loan Payoff
This problem (CARLOAN.EKA on the distribution disk) demonstrates how to set
up equations for a financing situation and use Eureka's Graph/List capability to
generate a list of function values vs. the variable, time.

Statement of the Problem
The car you purchased cost $10,900. When you bought it, you made a down payment of $2,900 and financed the rest ($8,000) for 48 months at 13.5% annually.
Now it is a year and a half later, and you are thinking about buying a new car.
• How much would it cost to pay off the loan?
• How long should you continue to make payments?
These questions can be answered by a few mathematical calculations. To start
with, define some terms:

Pay = periodic payment
Int = interest per period
Prin = principal amount borrowed
ntot = the total number of periods for the original loan
n
= the number of periods actually paid before payoff
FofP = the future value of the principal at month n, given a present value
FofA = the future value of a payment series at month n, given a periodic
payment
Using these terms, you first need to calculate the original payment. You could
use the built-in function paymt, but this time you are going to use the actual
mathematical formula to see how it is done.
Once you know the monthly payment, you need to find the remainder of the
principal at a given month, then subtract the value of the sum of remaining payments.
To do this, use the time at which you borrowed the money as a reference point.
From this point, find the future value of the principal (FofP) at the time of the
payoff and the future value of the payment series (FofA) at the time of the payoff.
After subtracting FofA from FofP, you have the payoff amount.
By making the formula for PayOff a function of n, you can generate a list of 20
possible payoff points with the Eureka Graph/List function. This can help you
make your decision.

Worked Examples

123

Equations
The first step when setting up the equations for this problem is to define the
variables and assign values to them. The variables are:
Monthly interest (IntM) is 13.5 annual percentage rate divided by 12 months per
year:

IntM = (13.5/100)/12 = 0.01125
Principal, the amount borrowed:

Prin

8,000.00

Total number of payments for the loan:

ntot = 48
Monthly payment (to be calculated from given variables):

PayM
The equations required for solving the two questions posed by this problem are:
Calculate the monthly payment:

PayM = Prin X IntM X ((1

+ IntMttot - 1)/((1 + IntM) -1)

Calculate the future value of the original loan amount at the specified time n,
where n indicates one of the 48 months:

FofP(n) : = (Prin

+ IntM)n)

Caiculate the future value of the payments at the same specified month, n:

FofA(n} = (PayM

((1

+ IntMt -1)/IntM)

(Note that these two calculations are made from user-defined functions.)
Finally, for a given month, n, calculate the Payoff: What is the principal that
remains outstanding after the payment n?

Pay()jJ{n} : = FofP(n} - FofA(n)

Equation File and Solution
The equation file and solution for this problem appear in the print-out that follows,
along with the table generated using Eureka>s Graph/List command. The table lists
the function, Payoff, vs. months 15 through 34.

124

Eureka: The Solver Owner's Handbook

*******************************************************************

Eureka: The Solver, Version 1.0
Thursday November 6, 1986, 5:47 am.
Name of input file: CARLOAN.EKA
*******************************************************************

$ finansmooth = yes

IntM .01125
Prin 8000.00
ntot 48
n = (l+IntM)-ntot
PayM = Prin * IntM * «l+IntM)-ntot) / ({l+IntM)-ntot - 1)
FofP{n) := (Prin * (ltIntMrn)
FofA(n) : = {PayM * ({l+IntMrn - 1) / IntM)
PayOff{n) := FofP{n) - FofA{n)
*******************************************************************

Solution:
Variables

Values

IntM

.011250000

ntot

48.000000

PayM

216.61058

Prin

8000.0000

1.7108410

*******************************************************************

Worked Examples

125

List of function values.
n

PayOff(n)

15.000000
16.000000
17.000000
18.000000
19.000000
20.000000
21.000000
22.000000
23.000000
24.000000
25.000000
26.000000
27.000000
28.000000
29.000000
30.000000
31.000000
32.000000
33.000000
34.000000

5943.7392
5793.9957
5642.5676
5489.4359
5334.5814
5177.9849
5019.6267
4859.4869
4697.5455
4533.7823
4368.1768
4200.7082
4031.3556
3860.0977
3686.9133
3511.7804
3334.6774
3155.5819
2974.4716
2791.3239

*******************************************************************

126

Eureka: The Solver Owner's Handbook

Problem 3
Ladders in an Alley: Solving Simultaneous
Nonlinear Equations
This problem (LADDERS.EKA on the distribution disk) demonstrates Eureka's
ability to quickly and accurately solve simultaneous nonlinear equations.

Statement of the Problem
You have two ladders, one 35 feet long, the other 45 feet long. You need to arrange
them in an alley between two skyscrapers so that they cross 10 feet above the
ground. How wide should the alley be?

Equations
First, assign variables to all the known and desired quantities:

x = width of the alley
a = height of the 45-foot ladder at the wall

b = height of the 35-foot ladder at the wall
h = the point on the ground immediately below wher~ the ladders meet.

Equation File and Solution
The print-out that follows shows the equation file and solution for this problem.
Notice the initialization values for the four variables. If you have a general idea
about the region where the solution(s) lie, it's a good idea to start Eureka off within
or close to that region. Eureka solves problems such as this one by iteration, getting
closer and closer to an error-free solution. However, the default value for all initializations is 1.
When you are confident that the solution(s) are significantly larger (or smaller)
than 1, you can initialize the variables with some plausible values, then tell Eureka
to Solve. This way, Eureka will not spend a lot of time covering unproductive
ground.

Worked Examples

127

When you are confident that the solution(s) are significantly larger (or smaller)
than 1, you can initialize the variables with some plausible values, then tell Eureka
to Solve. This way, Eureka will not spend a lot of time covering unproductive
ground.

If Eureka's first round of iterations produces an unsatisfactory solution, you can
use other techniques to get the program re-solving in a different region. These
other techniques are covered in Chapters 3 and 5 of this manual.

128

Eureka: The Solver Owner's Handbook

*******************************************************************
Eureka: The Solver, Version 1.0
Thursday November 6, 1986, 6:47 am.
Name of input file: LADDERS.EKA
*******************************************************************

This example demonstrates how Eureka can solve
a system of nonlinear equations iteratively, after
you initialize the iteration.
System of nonlinear equations:
x~2
x~2

+ a~2
+ b~2

45~2

35~2

y/10 = x/b
(x-y}/10 = x/a
Initializations:
x := 10
y .- 10
a := 20
b := 20
constraints:
a>O
b>O
x>O
x>y
y>O
Exact solutions are: x = 31.817459 , Y
and a = 31.822151 , b = 14.582500

21.818934

*******************************************************************

Solution:
Variables

Values

31.822151

14.582500

31.817459

21.818934

Maximum error is 6.8212103e-13
*******************************************************************

Worked Examples

129

Problem 4
Distribution of Line Charges
This problem in electrostatics (LINECHG.EKA on the distribution disk) illustrates Eureka's ability to solve several simultaneous nonlinear equations.

Statement of the Problem
A system of several parallel positive line charges is aligned along an axis within an
electric field so that the total force exerted on each line charge is zero~
As shown in Figure 6-1, a line charge is fixed at the origin, lying on the z axis.
Three free-line charges lie on the x - z plane (positive x) parallel to the first. The
electric field is constant in magnitude and aligned in the negative x direction. From
physics, you know that the force on a charge is proportional to the electric field and
like charges repel.
The problem is to distribute the line charges along the x axis so that the sum of
forces on each of the three free charges equals zero.
¢.

I Y axis I

Figure 6-1

Line Charges in Equilibrium

This problem requires you to solve a set of nonlinear equations. (An interesting
note: The same equations which govern this problem also apply to the distribution
of dislocations under uniform shear stress in the study of the fracture of solids.)

130

Eureka: The Solver Owner's Handbook

Equations
From basic electrostatic theory, you know that the electric field produced by a line
charge varies
• inversely with the first power of the distance from the line, and
• linearly with the charge density of the line
The field produced by a line charge at some point p (or infinitesimal length p)
has the form
EOinecharge) = 2l!r
where

1 = charge density along the line (assumed to be invariant)
and
_ r = distance from point p to the line
The force F from a line charge at point p is the field at p multiplied by the probe's
line charge density:

F = 2l!r

In this example, you have an array of line charges. Each line charge produces a
field that affects all its neighbors; the line charge, in tum, is affected by all its
neighbors and the constant field, E(constant).
The force acting on any of the movable lines, Fi, is the sum of each other line's
individual force acting on that line (where Xi - Xj = r(i - j)):

Fa = 1 X Ea(l) = 1 X (2l!r(a - 0) + 2l!r(a - b) + 2l!r(a - e))
Fb = 1 X Eb(l) = 1 X (2l!r(b - 0) + 2l!r(b - a) + 2l!r(b - e))
Fe = 1 X Ee(l) = 1 X (2l!r(e - 0) + 2l!r(e - a) + 2l/r(e - b))
Fi = (sum Une i, 0

j S 3] 2l /(X

X)) - E(constant)

for each infinitesimal length of the line (assuming each line has the same charge
density, l).
To find the solution to this problem, derive a set of coupled equations that must
be solved if the force on each line charge is zero. Using the notation in Figure 6-1,
derive the following equations:

lI(a - b) + lI(a - e) + l/a = E(constant)/l2
lI(b - a) + lI(b - e) + lib = E(constant)/l2
lI(e - a)

+ 1/(e - b) + lie = E(constant)/l2

For simplicity, select values for E(constant) and 1 such that E(constant)/l2 = 1,
then solve the resulting set of equations.
Worked Examples

131

Equation File and Solution
The print-out that follows shows the equation file and solution for this problem.
Notice that although Eureka generates solutions for a, b, and c, the order of the
variables is not the same in the solution as it is in the equation file. This is because
the equations are symmetrical with respect to interchange of the variables, and
Eureka's solution process happens to produce the given order.

132

Eureka: The Solver Owner's Handbook

*******************************************************************
Eureka: The Solver, Version 1.0
Saturday November 8, 1986, 1:28 pm.
Name of input file: LINECHG.EKA
*******************************************************************

This example demonstrates Eureka's ability to solve
a system of nonlinear equations. These equations
represent the net force on each line charge in a
system of 4 parallel, coplanar line charges aligned in
an electric field.
l/(a-b)

l/(a-c)

l/a

l/(b-a)

l/(b-c)

l/b

l/c
l/(c-b)
l/(c-a)
*******************************************************************
Solution:
Variables

Values

a
b

3.8793852
.46791111
1.6527036

Maximum error is -1.1102230e-15
*******************************************************************

Worked Examples

133

Problem 5
Radioactive Decay
In this problem (DECAY.EKA on the distribution disk), you use Eureka to solve a
transcendental function of time.

Statement of the Problem
Radioactive elements, such as uranium, decay according to an exponential law.
This means that the rate of decay is constant and proportional to the amount of the
material.
In the case of uranium-238, a given amount of uranium will decay into lighter
elements at a rate such that half of it will have decayed every 10 million years. If
100 kilograms of uranium-238 sits for 10 million years, at the end of that period
only 50 kilograms of radioactive material will remain. At the end of another 10
million years, 25 kilograms of radioactive uranium will be left.
This time period for the uranium to be reduced to half its original amount is
called the half-life. Given 100 kilograms of uranium-238, how much radioactive
material will remain after 1,000,000 years?

Equations
The rate of mass decrease as a function of time is proportional to the mass of
radioactive substance at that time. Symbolically, in terms of differential calculus,

dm/dt = -km
where
m = mass
t = time
k = proportionality constant
According to the laws of calculus, the only functions to satisfy this formula are of
the form m = a X exp( - kt) where a is some constant. HalfLife and k are related
by the formula

k = In(2)IHalfLife

134

Eureka: The Solver Owner's Handbook

Mathematically, the following formula gives the amount of a radioactive substance at any particular time:

Mass(time) = InitMass

2(-timeIHa lfLife)

where

InitMass = initial mass (or weight) of the substance
Mass (years ) = mass of the substance after a given time; in this case, years
HalfLife = half-life of the substance

Equation File and Solution
The print-out that follows shows the equation file and solution for this problem.
Because a user-defined function is entered for mass as a function of years, you
can use Eureka's Graph/List capability to generate a list of the remaining radioactive mass vs. time. This list, covering the period from 500,000 to 5,250,000 years
elapsed in increments of 250,000 years, completes the print-out.

Worked Examples

135

*******************************************************************
Eureka: The Solver, Version 1.0
Wednesday January 28, 1987, 11:30 am.
Name of input file: DECAY.EKA
*******************************************************************
This problem demonstrates how to solve a user-defined
transcendental function of time.
The problem is to calculate MassUnknwn, the mass of
100 kg Uranium 238 after one million years.
Mass(Years) := InitMass * 2
InitMass = 100
Half Life = 10 * 1e6
MassUnknwn = Mass(le6)

(

Years / Half Life)

*******************************************************************
Solution:
Variables

Values

Half Life

10000000.

InitialMass

100.00000

MassUnknwn

93.303299

List of function values.
Mass
x
500000.00
750000,00
1000000.0
1250000.0
1500000.0
1750000.0
2000000.0
2250000.0
2500000.0
2750000.0
3000000.0
3250000.0
3500000.0
3750000.0
4000000.0
4250000.0
4500000.0
4750000.0
5000000.0
5250000.0

Mass(x)
96.593633
94.934212
93.303299
91. 700404
90.125046
88.576752
87.055056
85.559503
84.089642
82.645032
81.225240
79.829839
78.458410
77 .110541
75.785828
74.483873
73.204285
71.946679
70.710678
69.495911

*******************************************************************

136

Eureka: The Solver Owner's Handbook

Problem 6
Least-Squares Fit
This problem (LSQRFIT.EKA on the distribution disk) demonstrates how you can
direct Eureka to find values for the constants in a function so that the function
closely fits predetermined data points. You do this by limiting the program's substitution level (using the substlevel directive) during a Solve.

Statement of the Problem
You have a set of points (x,y) from empirical data, and you know that the data
ideally should fit a curve of the form

= f(x) =

e(Ax

where A, Band N are constants. You need to determine those values for A, B, and N
that produce the best-fit curve for your data.
The data points are:
Xn

1
2
3
4
5

Yn
1.49
2.35
4.26
8.59
19.80

Equations
Normally, in the process of solving a problem, Eureka makes symbolic substitutions in the equation file. However, you can limit the number of times the program
will substitute for terms in the file by adding a substlevel directive to the file.
Setting substlevel to 0 directs Eureka to make no substitutions; this is the technique you will use here.
In this problem, you need to tell Eureka three things:
• the general form of the equation
• the data points (x,y)
• the level of substitution (none)

Worked Examples

137

This user-defined function represents the general form of the equation:

f(x) = exp(AxN + B)
These equations give Eureka the empirical data points with the y values given as
functions of x:

f(l)
f(2)
f(3)
f(4)
f(5)

1.49

= 2.35
= 4.26
= 8.59
= 19.80

This directive sets the level of substitution to zero:
$ substlevel = 0

For Eureka to successfully find a unique solution for the problem, the number of
equations (data points) must equal or exceed the number of unknown constants in
the user-defined equation. Notice that there are five equations (f(1),f(2),f(3),f(4),
andf(5)) in this problem and three unknowns (A, B, and N).
If you do not set the substitution level to zero, Eureka will solve for those values
of the three unknowns (A, Band N) which satisfy the first three equations (f(1) =
1.49,f(2) = 2.35, andf(3) = 4.26) without looking at the equations for f(4) and
f(5).
With the substitution level at zero, Eureka looks at all the given equations
(f(x) = y) and automatically minimizes the least-squares function F where

F(x,y) = ((f(x) - y/ + (f(x 2 ) - Y2)2 + (f(x 3 )
+ (f(x) - y/ + (f(x s) - y/)

Y3)2

This function, F, represents the sum of the vertical (ordinate) distances of the
points (xn'y) from the points (x(n),f(x(n)) on the plot of the sought function. Minimizing this sum is equivalent to finding the best-fitting curve to match the plotted
data.
You could enter up to 20 data points into a problem of this type, although the
computations involved would make the Solve process quite lengthy. Ten data
points (or less, where possible) is a more practical limit.

138

Eureka: The Solver Owner's Handbook

Additional Techniques: Inputting Data From Another Program
This is the type of problem where you can make good use of Eureka's ability to
solve problems created in another software environment. For example, if you have
a Turbo Basic program that generates the points xn'Y n used in this problem and
those point values are stored in two arrays, X(I) and Y(I), you can write a Turbo
Basic program that creates the Eureka equation file for this problem.
The following Turbo Basic program will create a file called PROBLEM and
write the data from the arrays into the file in a format suitable for Eureka to solve.
OPEN "0", 1, "PROBLEM"
PRINT #1, "$ substlevel = 0"
PRINT #1, "f(x) : = exp(A * (x~N) + B)"
FOR 1= 1 TO 5
PRINT #1, using "\\#\ \#11.##"; "f("; X(I);"):="; Y(I)
NEXT I
CLOSE #1
END

Equation File and Solution
The equation file, the solution, and a list of calculated values for this problem
appear in the print-out that follows.

Worked Examples

139

*******************************************************************
Eureka: The Solver, Version 1.0
Sunday November 9, 1986, 3:33 pm.
Name of input file: LSQRFIT.EKA
*******************************************************************

This example demonstrates Eureka's ability to find
values for constants in a function that make the
function best fit empirical data. Because the equation
file includes a $ substlevel = 0 directive, Eureka will
perform a Least Squares Fit to find the function (of
the required form) that best matches the points
(x, f(x» given.
In this example, the function is
f(x) : = EXP(A * x~N + B)
where the ideal solution is A = 0.25, B
N = 1.5
f(x) .- EXP(A *
f(1)
f(2)
f(3)
f(4)
f(5)

x~N

0.15 and

+ B)

1.49
2.35
4.26
8.59
19.01

$ substlevel = 0
*******************************************************************

Solution:
Variables

Values

.25247836

.14432763

1.4951057

Maximum error is -.00011769096
*******************************************************************

List of function values.
x

f(x)

.00000000
1.0000000
2.0000000
3.0000000
4.0000000
5.0000000
6.0000000
7.0000000
8.0000000
9.0000000
10.000000

1.1552625
1.4870674
2.3537876
4.2597419
8.5891228
19.010163
45.722423
118.62542
330.10583
980.76072
3099.2752

*******************************************************************

140

Eureka: The Solver Owner's Handbook

Problem 7
Motion of a Projectile in a Gravitational Field
with Air Resistance
This is a problem in elementary dynamics (PROJCTL.EKA on the distribution
disk). You are to find the motion of a projectile in the atmosphere; air resistance and
gravity eventually bring the projectile to earth. This problem demonstrates
Eureka's ability to solve transcendental functions as well as automatically convert
units of measurement.

Statement of the Problem
Consider a projectile that is fired into the air at an angle A with an initial velocity
Vinit.

If you neglect air resistance, it is easy to show that the path of the projectile is a
parabola. With air resistance, the path is a more complicated function (which you
derive below). Eventually, the projectile will hit the ground after it has travelled a
distance R (the range). In this problem, you find the range (R) and the total flight
time (T).

Equations
You will consider the x and y coordinates separately. The initial conditions are:

x(t) = 0 when t = 0
y(t) = 0 when t = 0
x'(O) = Vinit(O)cos(A) = U
y'(O) = Vinit(O)sin(A) = V
where x' and y' are the first time derivatives (that is, the velocity).
The equations of motion are:

mx" = -kmx'
my" = -kmy' - mg (the y-deceleration)
The first of these equations states that the x-deceleration is proportional to the
velocity; this is due to air resistance.
The second equation states that the y-deceleration (remember, the projectile is
fired upwards) has two terms - an air resistance term and a gravitational term.
The constant k, the drag coefficient, measures the strength of the resistance; g is
the acceleration of gravity; and m is the mass of the projectile.
Worked Examples

141

These differential equations can each be solved by integrating twice; the results
are:

x(t) = (U/k) X (1 - e -kt)
y(t) = gt/k + ((kV + g)/k2)

(1 - e- kt )

where U and V are the horizontal and vertical components, respectively, of the
projectile's initial velocity, Vinit.
The total flight time, T, can be calculated from the equation for y(t) by noting
that y = 0 at the end of the flight; therefore, y(t = T) = O. Solving for T, you get

y(T) = 0 = (gT/k) + (((kV + g)/e)
T = - ((kV + g)/gk) X (1 - e -kT)

(1 - e-

))

The range, R, is the distance travelled in the x direction at t

R = (U/k)

(1 - e-

= T, so

)

For typical ballistics problems involving artillery:
• drag coefficient = 0.1 X (l/sec)
• 1,000 ft/s < Vinit < 3,000 fUs (30,000 cm/sec < Vinit < 90,000 cm/sec)

Let Vinit

= 980 cm/(sec
= 50,000 cm/sec, and let A =

• acceleration of gravity

)

45 degrees, then

U = V = 35,355 cm/sec
1
T = ((0.1 X 35,355) + 980)/(980 X 0.1) X (1 - e-· x T) sec
R = (35,355/0.1) X (1 - e -0.1 x T) cm

You will use Eureka to solve for T and R.

Equation File and Solution
The print-out that follows shows the equation file and solution for this problem. To
simplify the equation for T, we show the calculation of an intermediate constant
c = (k * V + g)/(g * k)

as a separate line. We also use Eureka's unit conversion directive, so the solution
will be displayed in several convenient units.
You can experiment with this problem and see how varying the initial parameters alters the solution. In particular, notice that a significant change to the drag
coefficient changes the range of the projectile significantly, while Hight time
changes slowly.
For example, changing k from 0.1 to 0.001 increases the range by more than 7
times (to 15 miles), but only increases the Hight time to 71 seconds.

142

Eureka: The Solver Owner's Handbook

*******************************************************************
Eureka: The Solver, Version 1.0
Saturday November 8, 1986, 2:22 pm.
Name of input file: PROJCTL.EKA
*******************************************************************
This example demonstrates solving a typical ballistics
problem, and Eureka's unit-conversion capability.
c

((k*V + g)/(g*k))

k 0.1
V 35355
V := 35355 [cm/sec]
g 980

intermediate constant to
simplify equation for T
drag coefficient
define units of initial velocity
force of gravity

solve for total flight time
T c*(l - exp(-k*T))
T : = 40 [sec]
R = (V/k)*(l - exp(-k*T)) ; solve for projectile's range
R : = 1 [cm]

$ units
cm -> feet: x / (2.54*12)
cm/sec -> feet/sec: x / (2.54*12)
feet -> miles: x / 5280
feet/sec -> miles/hour: x / (5280/3600)
$ end
*******************************************************************
Solution:
Variables

Values

46.076531

980.00000

.10000000

349848.99 cm
11477 . 985 feet
2.1738608 miles

45.594195 sec

35355.000 cm/sec
1159.9409 feet/sec
790.86883 miles/hour

Maximum error is 7.1054274e-15
*******************************************************************

Worked Examples

143

Problem 8
Charged Masses in a Gravitational Field
This is a typical problem in electrostatics (MASSCHG.EKA on the distribution
disk), illustrating Eureka's ability to find the minimum of a function.

Statement of the Problem
Four negatively charged mass points (for example, pith balls) are constrained to a
vertical line (for example, by a non-conducting wire) in a gravitational field. The
lowest ball rests on the ground. The problem is to find the equilibrium positions of
the other three balls.
As shown in Figure 6-2, the balls lie along the y axis and gravity acts in the
negative y direction. The charges repel each other, but the balls are also pulled
together by gravity.
The problem is to find the positions where the gravitational pull and electrostatic
repulsion balance.

......

ball #2 t""t"\
x(3) :: Z

\toii~

x(2) :: Y

ball #1

6:[')

'liliiP'
ball #0 fl)

x(1):: X

Figure 6-2

144

Charged Balls

Eureka: The Solver Owner's Handbook

Equations
You will solve this problem by minimizing the total potential energy of the system.
The total potential energy of the system is the sum of the gravitational and electrostatic potential energies of each ball.
The gravitational potential energy of a ball is equal to its mass multiplied by its
distance from the ground multiplied by the acceleration of gravity. This value is
zero at the ground and increases with increasing height.
The electrostatic potential energy of a ball due to the charge of another is just
the charge of the first ball multiplied by the electrostatic potential produced by the
other one.
The total potential energy of a ball is the sum of the potential energies due to
each of the other balls in the system. This value is defined to be zero for the ball on
the ground and positive for all other balls.
Thus, if the four balls are numbered 0 to 3, with number 0 being the ball on the
ground, and if
• the masses are mo to m3
• the charges are eo to e3
• the distances from the origin are Xo to

(with Xo being zero),

then the following equations express the electrostatic potential energies E of each
ball:
j

Eo = 0
EI = eJejx I + e/(x2
E2 = e2 [ejx 2 + eJ(x2
E3 = e3[ejx3 + e/(x3

+ e/(x3
X) + e/(x3
Xl) + e/(x 3

Xl)

x)]
x2)]
x2)]

The individual gravitational potential energies, G(i), have the following form:
G = mjgxj
j

where g is the acceleration of gravity (980 cm/sec ). The total potential energy of
the system, T, is just the sum of all the E and G
j

LE +
j

i =

In order to find the solution, you will use Eureka to find the values for x(i) that
minimize T.

Worked Examples

145

Assume that the balls all have a mass of 1 gm and a charge of 80 esu. To avoid the
use of subscripts, we will use x, y, and z for the distances Xi. Mter combining
common terms, the expression for T is:

980 X (x + y + z) + 6400
+ 2/(z - x) + 2/(z - y)]

[lIx

+ lIy + liz + 2/(y - x)

Since we are using the CGS system, T is measured in ergs, and x, y, and z are in
centimeters.

Equation File and Solution
The print-out that follows shows the equation file used to minimize T. Notice that
you must supply the additional constraints z > y > x > 0 and T > 0, and initial
values for x, y, and z. Without these constraints and initial values, Eureka can get
lost.

146

Eureka: The Solver Owner's Handbook

***************************************************************

Eureka: The Solver, Version 1.0
Sunday December 7, 1986, 6:23 pm.
Name of input file: MASSCHG.EKA
****************************************************************

$ min (T)

T = 980*(x + y + z) + 6400*(1/x + l/y + l/z + 2/(y~x)
2/(z-x) + 2/(z-y))
T>0
T := 29000
x >0
x : = 2 [em]
y >x
y : = 5 [em]
z >y
z := 10 [em]
y-x > 0
z-x > 0
z-y >0
****************************************************************

Solution:
Variables

Values

29900.376

1.5793519 em

4.6531267 em

9.0228156 em

96.6%
Confidence level
All constraints satisfied.
****************************************************************

Worked Examples

147

Problem 9
Using tire Built-in Polynomial Function
to Find Real and Complex Roots
This problem (POLYNOM.EKA on the distribution disk) demonstrates how to use
Eureka's built-in poly function to find the roots of nth degree polynomials in one
variable and illustrates methods for plotting polynomial functions. The poly function can find real and complex roots, and, with the Graph/Plot option, can plot the
function over a specified, definite range.

Statem£nt of the Problem
Consider the following functions of x:

R{x) = X4 - 2x3
S{x) = x2

13x2 + 14x + 24

2x + 5

T{x) = 2l + 3x + 9
U{x) = 2X4 - x3 + 13x2
V{x)

= 2x

W{x) = 3x + 5x
5

P{x) = x

3x + 45

3x - 36

7x + 24

The problem is to find all roots of these polynomials. One of these functions has
all real roots, two have all complex roots, and four have a combination of real and
complex roots. When you use poly, it is not necessary to change the Complex
setting to yes in order to find the complex roots; Eureka can solve all these problems with a single Solve command.

Equations
To use the poly function in Eureka, you only need to enter one equation for each
polynomial function to be solved; these are user-defined functions designated by
the symbol : = .
Each user-defined function states, in terms that Eureka understands, that" some
function of x is defined as the polynomial in x that has the following listed coefficients."

148

Eureka: The Solver Owner's Handbook

When setting up a poly function, remember to:
• first list the variable (typically x, but can be any valid variable)
• then list all coefficients of x; start with the highest degree of x, then the next
highest, and so on to the constant.

If a polynomial has no coefficient for a given degree of x, then you must enter
into the poly function for that degree's position, so that Eureka can keep track of
each degree of the variable.
For example, using the functions of x given above, the user-defined functions
would be written as:
R(x) : = poly(x,l, - 2, -13, 14, 24)
S(x) : = poly(x,l, - 2, 5)

T(x) : = poly(x, 2, 3, 9)

U(x) : = poly(x,2, -1, 13, - 3,45)
V(x) : = poly(x, 2, - 5, - 3, - 36)
W(x) : = poly(x, 3, 0, 5, 0, - 7, 24)
P(x) : = poly(x, 1, 0, 0, 0, 1)

Graphing the Functions
Eureka also offers the ability to plot functions of one variable. To graph one of the
functions in this problem:
• Solve the equation file.
• Select the Graph option from the main menu.
• From the Graph pull-down menu, choose Plot.
• Select one function from the mini-menu listing of all user-defined functions.
• Enter values for the left and right x axis endpoints.
Eureka will automatically plot the function.

Equation File and Solutions
The print-out that follows shows the equation file and Eureka's solutions for the
seven polynomials in this problem.

Worked Examples

149

********************************************************************
Eureka: The Solver, Version 1.0
Saturday November 8, 1986, 8:44 pm.
Name of input file: POLYNOM.EKA
********************************************************************
This problem demonstrates Eureka's built-in Poly
function, which finds the roots of n-th order
polynomials of one variable.
For polynomials in one variable with complex roots,
you do not need to change the Complex setting to
yes - Eureka can automatically find complex roots
when using the Poly function.
all real roots
; R(x) = (1*x A4 - 2*xA3 - 13*xA2 + 14*x + 24)
R(x) := poly(x,l,-2,-13,14,24)
; all complex roots - Eureka solves for the real part
; of the root being < or = 1
Sex) := poly(x,l,-2,5)
T(x) .- poly(x,2,3,9)
; complex and real roots
U(x) := poly(x,2,-l,13,-3,45)
Vex) := poly(x,2,-5,-3,-36)
; some coefficients are zero - they still must be entered
; into the list for Poly
W(x) := poly(x,3,O,5,O,-7,24)
P(x) := poly(x,l,O,O,O,O,l)
*******************************************************************

150

Eureka: The Solver Owner's Handbook

*******************************************************************

Roots to the polynomial R
Real part
Imaginary part

1
2
3
4

-1.0000000
-3.0000000
2.0000000
4.0000000

.00000000
.00000000
.00000000
.00000000

Roots to the polynomial S
Real part
Imaginary part

1
2

1.0000000
1.0000000

2.0000000
-2.0000000

Roots to the polynomial T
Real part
Imaginary part

#
1
2

-.75000000
-.75000000

1.9843135
-1.9843135

Roots to the polynomial U
Real part
Imaginary part

1
2
3
4

1.0000000
1.0000000
-.74999999
-.74999999

2.0000000
-2.0000000
1.9843135
-1.9843135

Warning: Some roots inaccurate.
Roots to the polynomial V
Real part
Imaginary part

1
2
3

4.0000000
-.75000000
-.75000000

.00000000
1.9843135
-1.9843135

Roots to the polynomial W
Real part
Imaginary part

3
2
1
4
5

-1.4464361
-.28746570
-.28746570
1.0106838
1.0106838

.00000000
1.7610191
-1.7610191
.84598193
-.84598193

Roots to the polynomial P
Real part
Imaginary part

1
2
3
4
5

-1.0000000
-.30901699
-.30901699
.80901699
.80901699

.00000000
.95105652
-.95105652
.58778525
-.58778525

********************************************************************

Worked Examples

151

Problem 10
Maximization: Designinc ~ Maximum-Square-Footage
House Plan While Satisjying Several Constraints
This problem (HOUSE.EKA on the distribution disk) shows how to use Eureka to
maximize a variable that is a nonlinear function of several other variables, while
satisfying constraints placed on the function and on the variables.

Statement of the Problem
In this problem, you are to design an L-shaped house with as much square footage
(Hoor area) as possible, within the limitations imposed by city codes and certain
aesthetic considerations. A rectangular second story will be built above the largest
part of the first Hoor, extending from the back wall but only overlapping half of the
front wing.
Limitations and Constraints
1. The foundation must encompass no more than 3000 square feet.
2. The width of the front wing (dimensions xy, where x is width) must be within
one-third to one-half of the total house width. The total house width is b, and
the dimensions of the back wing are abo
3. The front and back wings will not be disproportionately sized.
4. The house will sit on a 90 x I50-foot lot with 10 foot minimum setbacks on
either side and 25 foot minimum setbacks front and back.
5. Leave at least 1,500 square feet in the inside corner of the L for a pool and
patio.

Equations
These equations state (1) the purpose of the problem (to maximize the total.Hoor
area of the house you design) and (2) the limitations and constraints.
Areas
Area of first Hoor (covered by foundation):
B = (xy

152

+ ab) sq ft

Eureka: The Solver Owner's Handbook

Maximum foundation area allowed by city code:
B < 3,000 sq ft

Area of second floor (located above front wing of first floor, leaving room for a
balcony at front of house):
A2 = xz sq ft

Total area of house (sum of areas of first and second floors):
A = (xy

+ ab + xz)

= B

+ xz sq ft

Aesthetic constraints
Front wing width is to be between one-third and one-half of total house width b:

x > b/3
x < b/2
The patio/pool area will occupy the area bordered by the inner walls of the two
wings. The dimensions of this area are (b - x) feet by (y) feet, and the total patio/
pool area must be at least 1,500 sq ft:

(b - x) > 1,500

The house width (b) must be less than the lot width (90 ft) minus the two 10-foot
setbacks, but it must be at least 40 feet wide:

b < 68
b > 40
The total house length (y + a) must be less than the lot length (150 ft) minus the
two 25-foot setbacks, but it must be at least 56 ft long:

+ a < 100
y + a > 56

The second story, length (z), must leave half of the extending part of the front
wing uncovered:

=a+

y/2

The smaller, back-wing extension must extend out at least half as far as the frontwing extension (so the wings will not be disproportionate):

a > y/2

Worked Examples

153

Equation File and Solution
The print-out that follows shows the equation file and solution for this maximization problem, along with an evaluation of all the solutions found.
Note the directive $ max (A); that is, the command for Eureka to find the maximum combined first- and second-Hoor area possible within the listed constraints.
Also note that the equation file includes initial values for band y. Since this
problem contains many conditions that the few variables must satisfy, it is a good
idea to get Eureka started in the right direction with some good first guesses;

If you run this problem on your computer, you may notice that it seems to take a
relatively long time to reach the solutions. This is normal for a problem like this,
which contains many constraints for just a few variables. Watch the Progress window for information about how Eureka is handling the problem.

154

Eureka: The Solver Owner's Handbook

*******************************************************************
Eureka: The Solver, Version 1.0
Saturday November 8, 1986, 9:02 pm.
Name of input file: HOUSE.EKA
*******************************************************************
This problem illustrates maXlmlzlng a nonlinear
function while satisfying several constraints on the
function and on the variables. Notice that we maximize
a variable, A, which is related to the other variables
x, y, z, a and b according to the first equation.
$ max (A)
; maximize area within limits of city code
A (x*y + a*b + x*z) = B + x*z
B (x*y + a*b)
B < 3000
aesthetic constraints
x > b/3
x < b/2

; leave room for patio/pool
y*{b-x) > 1500
; set-backs and more aesthetics
b < 68
b > 40
y+a < 100
y+a > 56
z a + y/2
a > y/2
initial values
b:=68
y:=56
*******************************************************************

Worked Examples

155

*******************************************************************
Solution:
Variables

Values

4500.0000

23.677311

3000.0000

63.351788

31.675894

47.354622

Confidence level = 92.3%
All constraints satisfied.
Evaluation of formulas:
Formulas
A

(x*y+a*b+x*z)
difference
B+x*z

Values
4500.0000
4500.0000
.00000000

difference

4500.0000
.00000000

(x*y+a*b)
difference

3000.0000
3000.0000
.00000000

3000.0000
3000.0000
difference
-4. 9408300e-08
*******************************************************************
B

3000

156

Eureka: The Solver Owner's Handbook

*******************************************************************
x

b/3
difference =
x
b/2

31.675894
21.117263
10.558631

31.675894
31.675894
difference = -2.9878677e-10

y*{b-x)
1500.0000
1500
1500.0000
difference (error) = -4.0927262e-12

difference =

63.351788
68.000000
-4.6482119
63.351788
40.000000
23.351788

difference

70.031933
100.00000
-28.968067

b
68
difference
b
40
y+a
100
y+a
56
difference =

70.031933
56.000000
15.031933

difference =

47.354622
47.354622
.00000000

z
a+y/2
a
y/2

23.677311
23.677311
difference (error) = -1.0976464e-10

b
y

63.351788
47.354622

Maximum error is -1.0976464e-10
*******************************************************************

Worked Examples

157

Problem 11
Maximization: Linear Programming
in Manufacturing
This example, a problem in linear programming (LINPROG.EKA on the distribution disk) demonstrates Eureka's ability to solve simultaneous linear equations in
multiple variables and to maximize the objective function in order to reach an
optimum solution.

Statement of the Problem
Suppose a company makes two types of widgets, one wooden and one plastic. To
make a wooden widget, the producer uses machine A for 2 hours, machine B for 1
hour and machine C for 1 hour. To make a plastic widget requires 1 hour of machining on A, 2 hours on B, and 1 hour on C. The maximum amount of time (per month)
available on machine A is 180 hours; on B, 160 hours; and on C, 100 hours. The
profit on each wooden widget is $4.00 and on each plastic one $6.00.
Assuming that all widgets made will sell, how many of each type should the
company make in order to maximize the monthly profit?

Equations

Machine

Hours
To make wooden widgets
To make plastic widgets
Available per month

Machine
B

Machine
C

Profit
(per unit)

2
1
180

1
2
160

1
1
100

$4.00
$6;00

Profit Function (the Objective Function)
First, assign variables to each model of widget, with x being the monthly manufactured amount of the wooden variety and y being the same for the plastic widgets. Monthly profit is then the sum of the individual profits:
Profit

158

= 4x +

Eureka: The Solver Owner's Handbook

This is the objective function that you are going to maximize. To maximize the
objective function, use Eureka's built-in maximization function. Add the max
directive to the file:
$ max (Profit)

Remember that the max directive can be used only once per equation file.
To avoid a solution which says that only the higher-profit plastic model will be
manufactured, set
x > 0 ; wooden widgets
y > 0 ; plastic widgets

Since the total time on machine A can't exceed 180 hours per month and the
manufacturing processes require 2 hours of machine A's time for a wooden widget
or 1 hour for a plastic widget, write the following constraint to represent the time
limitation:
2 * x + Y < 180

Similarly, write constraints for the manufacturing time on machines Band C:
x + 2 * Y < 160 ; total time on machine B can't exceed 160 hours per month
x + y < 100
; total time on machine C can't exceed 100 hours per month

Equation File and Solution
The print-out that follows shows the equation file and solution for this problem.
Notice that we have started Eureka off with some initial values for x and y. We
chose initial values that represented the maximum possible value for each variable
that would satisfy the manufacturing-time constraints. Although this is not necessary, it may help to speed up Eureka's search for a solution; the default value for
both x and y at the beginning of the search is 1.0.
Notice also that we did rwt set the Profit function up as a user-defined function;
we used the equal sign ( =) for an assignment operator, not the colon-equals operator (: =). This is an important distinction; if you were to write the objective function as Profit := (4 * x +6 * y), Eureka would interpret that as meaning that Profit is
a variable to be maximized (due to the max directive) and it merely has a starting
point value of 4 * x +6 * y. The result would be that Eureka maximizes Profit to the
rather high value of 1.1529215 * 10 20 (dollars).
A

Worked Examples

159

*******************************************************************
Eureka: The Solver~ Version 1.0
Saturday November 8, 1986, 9:06 pm.
Name of input file: LINPROG.EKA
*******************************************************************
This example, a problem in linear programming, demonstrates Eureka's ability to solve simultaneous linear
equations in multiple variables, and to maXlmlze the
objective function in order to reach an optimum
solution.
Equations
Profit function (the objective function)
Profit = (4*x) + (6*y)
$ max (Profit)
x >0
y > 0

wooden widgets
plastic widgets

x := 90
y := 80

2*x + Y < 180 ; total time on machine A can't exceed 180
x + 2*y < 160 ; total time on machine B can't exceed 160
x + y < 100
total time on machine C can't exceed 100
*******************************************************************
Solution:
Variables

Values

Profit

520.00000

40.000000

60.000000

Confidence level = 93.1%
All constraints satisfied.
*******************************************************************

160

Eureka: The Solver Owner's Handbook

Problem 12
Equity in a House: the paymt, pval and foal functions
This example (EQUITY.EKA on the distribution disk) is a simple problem in
finance. Using some of Eurekas built-in functions, you will calculate the growth in
the equity in a house purchased with a mortgage.

Statement of the Problem
Suppose you purchase a house on January 1 for $200,000, using a $20,000 down
payment and a 13% 30-year fixed-rate mortgage for $180,000. Assume that the
mortgage interest compounds monthly and that your payments are made monthly.
Use Eureka to calculate:

1. Your monthly payments.
2. How long it will take for your equity in the house to increase to $100,000.

Equations
Eureka has several built-in functions that are useful in financial calculations. These
functions compute the key values in interest-rate calculations. In this example, you
will use the paymt and fval functions.

paymt calculates the payment in each interval on the loan, using as inputs the
interest rate, the number of intervals, and the present and final value of the loan.
fval calculates the future value of a loan, using as inputs the present value, the
payment per interval, the number of intervals until the future date, and the interest rate.
These are the input parameters:
number of payment intervals
interest rate per interval
price of house
down payment
value of mortgage (principal)
amount of balloon payment due

Worked Examples

N
= 30 X 12
I
= .13/12
P
= 200,000
D
= 20,000
pval = 180,000
B
=0

= 360 months

161

You need to calculate the size of the monthly payments (mp) and the number of
months (time) until the equity is $100,000. The monthly payment can be found by
simply plugging the input parameters· into paymt in the following order:
mp
mp

= paymt(J, N, P-D, B) which leads to
= paymt(0.13/12, 30 * 12, 180000, 0)

where the final is the final value of the mortgage after 30 years (in this example,
there is no balloon payment).
Equity in the house is simply the difference between what the house is worth
and what you still owe the bank. The equity will be $100,000 at some future time
that can be calculated using the fval function:
Future Value of Mortgage

= fval(J, time, L, mp)

which gives
100,000

= 200,000 +

fval(.13/12, time, 180000, mp)

You can now solve these two equations for mp and time to determine the regular
monthly payment and the elapsed time (in months) until the equity reaches
$100,000.00.

Equation File and Solution
The Eureka equation file corresponding to these equations is shown in the printout that follows.
Notice that the first entry in the equation file is a directive (set off with the $
symbol) to set finansmooth to on. This is necessary because financial functions are
typically discontinuous in time. With finansmooth on, Eureka automatically
smooths out the discontinuities and converts the financial function to a smooth
curve instead of a series of step functions.

162

Eureka: The Solver Owner's Handbook

*******************************************************************
Eureka: The Solver, Version 1.0
Tuesday February 10, 1987, 5:33 pm.
Name of input file: EQUITY.EKA
*******************************************************************

This problem shows how to use Eureka's
built-in financial function PAYMT to
calculate monthly payments for a 30-year,
$180,000.00 mortgage, with a $20,000.00
down payment and a yearly interest rate
of 13%.
This problem also demonstrates using
the built-in function FVAL to calculate
when the equity reaches $100,000.00, and
calculates the balance of the loan at
any given time.
Note: this problem incorporates zero
market appreciation.
$ finansmooth = yes
variables assigned to the given data:
DownPay = 20000
Loan = 180000
Duration = 30 * 12
determining monthly interest rate from
the annual rate
Yrate = 0.13
Mrate = Yrate / 12
finding the monthly payment
Payment PAYMT(Mrate, Duration,
Loan,O)
formula for determining Equity at a
given time:
FutureValue(GivenMonth):=FVAL(Mrate,_
GivenMonth,Loan,Payment)
Equity(GivenMonth) := DownPay + Loan + FVAL(Mrate,_
GivenMonth, Loan, Payment)
Equity(HowSoon) = 100000
formula for determining total amount
paid at the end of the life of the
loan
TotalPaid = Payment * Duration
Amount still owed on loan at any given
month
AmtOwed(GivenMonth) := Loan + DownPay - Equity(GivenMonth)

Worked Examples

163

Solution:
Variables

Values

Down Pay

20000.000

Duration

360.00000

HowSoon

287.10773

Loan

180000.00

Mrate

.010833333

Payment

-1991.1591

Total Paid

-716817.29

Yrate

.13000000

Maximum error is .011900878
List of function values.
Equity
x

12.000000
24.000000
36.000000
48.000000
60.000000
72.000000
84.000000
96.000000
108.00000
120.00000
132.00000
144.00000
156.00000
168.00000
180.00000
192.00000
204.00000
216.00000
228.00000
240.00000
252.00000
264.00000
276.00000
288.00000
300.00000
312.00000
324.00000
336.00000
348.00000
360.00000

Equity(x)
20524.427
21121.243
21800.438
22573.385
23453.023
24454.080
25593.315
26889.801
28365.245
30044.348
31955.222
34129.859
36604.665
39421.076
42626.243
46273.827
50424.896
55148.947
60525.071
66643.275
73605.989
81529.784
90547.320
100809.57
112488.34
125779.17
140904.55
158117.74
177706.90
200000.00

*******************************************************************

164

Eureka: The Solver Owner's Handbook

Problem 13
Probability in a Card Game
This problem (CARDS.EKA on the distribution disk) illustrates Eureka's built-in
factorial function, fact, and demonstrates how to set up a user-defined function.

Statement of the Problem
In a game of Bridge, find the probability that a trump suit with six trumps outstanding will split three-three.
For those who do not play Bridge, here is a synopsis of how to play the game:
There are 13 cards dealt to each of 4 players. Of these, 13 of the cards are
trumps. When a player says s/he is hoping for a three-three trump split, it is
because s/he controls 2 of the hands and 7 of the trumps, with the remaining 6
trumps divided between the remaining 2 hands in an unknown manner. The
trumps are said to split three-three if there are exactly 3 trumps in each of the
unknown hands.

Equations
Stating the problem in different terms, we can ask how many combinations of N
clwose M exist for this problem, where N choose M represents the number of
unique ways to choose a subset of size M from a set of N items.
In standard mathematical notation, this is expressed as a "combinatorial coefficient'" (CofNandM) where
_
CofNandM - (MI)

(NI)
((N - M)I)

If you (perhaps naively) assume that the probability of distributing six things
randomly yields a three-three split, the equation for the (approximate) probability
is:
ProbApprox

= C(6,3) /2

Worked Examples

165

If you assume that the probability of random distribution does not yield a threethree split, the exact probability for an even split is:
ProbExact

C(26,13)
C(20,lO)

C(63)
,

Equation File and Solution
The print-out that follows shows the equation file and solution for this problem.

166

Eureka: The Solver Owner's Handbook

*******************************************************************

Eureka: The Solver, Version 1.0
Saturday November 8, 1986, 10:46 pm.
Name of input file: CARDS.EKA
*******************************************************************

This example demonstrates how to use Eureka's built-in
FACT function (FACT(x) = x factorial). The problem
is:
In a Bridge game, one player controls 7 trump cards.
We must determine the probability that the remaining
6 trump cards are evenly distributed (3 and 3) among
the other two players. We calculate both the
approximate and the exact probability in this example.
A(x) : = FACT (x)
C6 = A(6)
C3 = A(3)
C10 A(10)
C20 A(20)
C26 A(26)
C13

A(13)

The combinatorial coefficients for total available
cards and cards per remaining hand:
Cof6and3 = A(6) / (A(3) * A(6-3»
Cof20and10 = A(20) / (A(10) * A(20-10»
Cof26and13 = A(26) / (A(13) * A(26-13»
; approximate probability of 3-3 split:
ProbApprox = Cof6and3 / 2~6
exact probability of 3-3 split:
ProbExact = Cof6and3 * Cof20and10 / Cof26and13
The correct answers are
ProbApprox = .3125
ProbExact = .355279503

Worked Examples

167

Solution:
Variables

Values

Cl0

3628800.0

C13

6.2270208e+09

C20

2.4329020e+18

C26

4.0329146e+26

6.0000000

720.00000

Cof20andl0 =

184756.00

Cof26and13

10400600.

Cof6and3

20.000000

ProbApprox

.31250000

ProbExact

.35527950

*******************************************************************

168

Eureka: The Solver Owner's Handbook

Problem 14
Income Distribution: The integ Function
This problem (INTEG.EKA on the distribution disk) demonstrates how to use
Eureka's built-in integ function to calculate the definite integral of functions.

Statement of the Problem
In the study of income distributions, one useful quantitative measure is the
Lorentz Curve, a function relating the cumulative percentage of income recipients
(x) to the cumulative percentage of income (y). The equality of income distribution
given by the line x = y represents the ideal state, where income is evenly distributed among the whole population: 10% of the people receive 10% of the income,
30% receive 30%, and so on.
A more realistic, non-ideal income distribution is the function
y = (20/21)x

+ (l/21)x

The coefficient of inequality measures the degree of deviation from equality; this
coefficient is defined as the area between the income-distribution curve and the
diagonal (x = y), divided by the area under the diagonal. When all incomes are
evenly and equally distributed, the coefficient of inequality becomes zero.
The problem: For the income-distribution function given above, what is the
coefficient of inequality?

Equations
To solve this problem, you must integrate two functions from 0 to 1 (0 to 100%),
then calculate a quotient using the two integrals. With Eureka's built-in integ
function, this is an easy exercise. From the definition of the coefficient of inequality
given above, define the numerator (Num) and denominator (Venom) as the integrals of the two functions given:

Num is the difference between the definite integral of the diagonal (x
the definite integral of the function (y(x)):

= y) and

Num = [integral(x) - integral((20/21)x2 + (l/21)x)] from x = 0 to x = 1
Venom is the definite integral of the function:
y = x from x = 0 to x = 1

Worked Examples

169

The coefficient of inequality is the quotient
CoefflnEq = Num/Denom

Using the integ function, this is how you set up the two integrals:
Num = integ(x, x, 0, 1) - integ«20/21)*x 2 + (1/21)*x, x, 0, 1)
Denom = integ(x, x, 0, 1)
ft

Equation File and Solution
The print-out that follows shows the equation file and the solution for the problem.
The table at the end of the print-out shows the result of selecting Graph/List.

170

Eureka: The Solver Owner's Handbook

*******************************************************************
Eureka: The Solver, Version 1.0
Saturday November 8, 1986, 11:40 am.
Name of input file: INTEG.EKA
*******************************************************************
This problem demonstrates how to use Eureka's built-in
INTEG function to perform definite integrals. The
example comes from the study of income distributions.
income distribution function (Lorentz curve - non-ideal)
y(x) := (20/21)*x-2 + (l/21)*x
Num = INTEG(x, x, 0, 1) - INTEG«20/21)*x-2 + (1/21)*x, x, 0, 1)
Denom = INTEG(x, x, 0, 1)
coefficient of inequality:
CoeffInEq = Num/Denom
*******************************************************************
Solution:
Variables

Values

CoeffInEq

0.31746032

Denom

0.5000000

Num

0.15873016

List of function values.
x
.00000000
.10000000
.20000000
.30000000
.40000000
.50000000
.60000000
.70000000
.80000000
.90000000
1.0000000

y(x)
.00000000
.014285714
.047619048
.10000000
.17142857
.26190476
.37142857
.50000000
.64761905
.81428571
1.0000000

*******************************************************************

Worked Examples

171

Problem 15
Chemical Calculations
This problem (CHEMBAL.EKA, CHEMEQ1.EKA, CHEMEQ2.EKA, and
CHEMEQ3.EKA on the distribution disk) illustrates how you can use Eureka to
solve problems in chemistry, from the simple-but-tedious (balancing chemical
equations) to the more sophisticated (calculating equilibrium concentrations).
The first part of this problem:
• demonstrates Eureka's ability to solve some simple systems of linear equations
• identifies a situation where the include directive would be useful
• gives a quick method for balancing chemical equations and determining
unknown stoichiometric quantities
The second part of the problem gives three examples of calculating equilibrium
concentrations, including a classic difficult problem where poorly-scaled variables
hamper Eureka's solution process.
Part I: The Reaction of Iron Sulfide with Oxygen to Produce Iron Oxide and
Sulfur Dioxide
This problem in stoichiometry (CHEMBAL.EKA on the distribution disk) asks
you to find out how much of a reactant is needed for a chemical reaction to produce
a certain amount of product.

Statement of the Problem
Consider the chemical reaction between FeS and 02' written here as an unbalanced chemical equation:

+ 02

... Fe20 3 + S02
How much iron sulfide (in grams) is required to produce 100 grams of iron oxide
in this reaction?
FeS

To solve this problem, you will first need to balance the chemical equation. In
addition, you will need to convert the given mass information (in this case, 100
grams of Fe2 0) into its molar equivalent.
A mole is defined as the formula weight (in grams) of an element or compound. It
is also, by definition, the mass of a specified number of molecules of a substance.
23
The specified number is Avogadro's number (6.0221 X 10 molecules.)

172

Eureka: The Solver Owner's Handbook

Once you know the ratios between the different substances in the reaction and
the given number of moles (of product, in this case), you can determine the number of moles of other substances that satisfy the given requirement.

Equations
You can assign variables for all the unknown amounts in the unbalanced chemical
equation and set up the following algebraic expression to represent the balanced
state:
a * (Fe + S) + b * (O * 2) - c * (Fe * 2 + 0 * 3) - d * (S + 0 * 2)

Since elements cannot just appear or disappear, the net amounts of each element
must be the same on each side of the balanced equation. This means that the
variables for each of the elements have the following relationships:
for Fe, a = 2c
for S, a = d
for 0, 2b = 3c

The solutions to these variables will yield a balanced chemical equation, which
specifies the number of moles of reactants used in the chemical reaction and the
number of moles of products that result.
The next step is to write an equation for converting the given information (100
grams of Fe20 3 ) into moles.
Remember that chemical symbols (Fe, K, Ca, Na, 0, S, and so on) are just thatsymbols - and can be manipulated just like variables in equations. You can combine these symbols with each other and with constants representing mass, valence,
atomic number, or any other property that you need to factor in when solving
chemical equations.
For example, consider Fe 20 3 and the following definitions:
mFe
zFe
vFe

mO
zO

= mass of one mole of Fe (55.847 grams)

= atomic number of Fe (26)
= valence of Fe (2 or 3; depends on particular compound)
= mass of one mole of (15.9994 grams)

°
°

= atomic number of (8)
= valence of (typically, - 2)

vO
mFe20 3 = mass of one mole of Fe20 3 = (mFe

+ (rnO

similarly,
mFeS = mFe + mS
mS0 2 = mS + (mO

Worked Examples

173

To convert measured amounts of substances into their molar equivalents, you
divide the measured amount by the mole weight:
100.00 gm of Fe20 3
100.00
159.69 gm/mole = 159.69 = 0.6262 moles of Fe2 0 3
The amount of each substance used or produced in the chemical reaction can be
represented by a "how much P" variable, written by adding h to the beginning of
each chemical formula. In this example, "how much Fe2 0}" is represented by
hFe2 0 3 and is the given quantity-100 grams. Similarly, hFeS represents "how
much FeSP", the amount (in grams) ofFeS used, and h02 represents the amount of
2 consumed.

In terms of the variables in this problem, the number of moles of Fe20 3 is calculated by
howmuch Fe2 0 3
mass of one mole of Fe2 0 3
where mFe2 0 3

hFe2 0 3
mFe20 3

= 159.69 and hFe20 3 is the given, 100 grams.

You can see that the number of moles of Fe20 3 produced is equal to the number
of moles of FeS supplied. This can be rephrased as:
a moles of FeS produce c moles of Fe20 3

so the ratio of (moles of FeS):(moles of Fe2 0) is the ratio a:c. The equation for this
is
moles of FeS
a
----------= -which leads to the equation

c X (moles of FeS) = a X (moles of Fe20 3)
By substituting the number of moles equation from above, you get
c

hFeS

x--- =
mFeS

hFe2 0 3

x-~--=-

mFe2 0 3

Eureka will solve for the variables a and c in the first part of this problem; mFeS
and mFe20 3 are constants and hFe20 3 is the given, so you can easily solve for the
unknown quantity, hFeS.

174

Eureka: The Solver Owner's Handbook

Equation File and Solution
The print-out that follows shows the Eureka equation file and solution for this
problem.
Eureka will solve this system of linear equations and give values for a, b, c, and d
that balance the chemical equation.
To get Eureka started, you assign a value to one of the variables and see what
solutions the program finds. This is not necessary, but it helps to keep the solutions
for a, b, c, and d as whole numbers.
You can see from the equation for the variable b that c should be an even number
so b will be an integer; that is why we set c equal to 2.
Besides giving a value to one of the variables, you must include the values for the
given amount of product (100 grams of Fe20 3) and the constant values. In this case,
the constant values are the mole weights of the elements Fe, 0, and s.

Using the $ include Directive
You can experiment with this problem, or similar ones, to explore Eureka's usefulness in stoichiometry. If you plan to be solving many problems such·as this one, you
will probably find it helpful to create a separate file containing the mole weights for
all the elements that you will be dealing with. Keeping such a file as a permanent
part of your Eureka problem disk, you could use the include directive in your
equation file to call up the mole weight information when it is needed instead of
entering it over and over.

Worked Examples

175

*******************************************************************
Eureka: The Solver, Version 1.0
Thursday November 6, 1986, 6:11 am.
Name of input file: CHEMBAL.EKA
*******************************************************************
This is an example of balancing a chemical equation
and then finding the amount of reactant, FeS, needed to
yield 100.00 grams of Fe203 product
the unbalanced chemical equation:
FeS t 02 --> Fe203 t S02
algebraic expression for the balanced equation:
a*FeS t b*02 --> c*Fe203 t d*S02
a = 2*c
d =a

b*2 = 3*c t 2*d
c = 2.000
; mole weights (grams/mole)
mFe
mS
mO
mFeS
mFe203
m02
mS02

55.847
28.086
15.9994
mFe t mS
mFe*2 t mO*3
mO*2
mS t mO*2

; amount of product required is 100 grams
hFe203 = 100.00
; ratio of moles of FeS to moles of Fe203 is a:c
c*(hFeS/mFeS) = a*(hFe203/mFe203}
; correct answer is a = d = 4, b = 7, c

176

Eureka: The Solver Owner's Handbook

*******************************************************************

Solution:
Variables

Values

4.0000000

7.0000000

2.0000000

4.0000000

hFe203

100.00000

hFeS

105.11847

rnFe

55.847000

rnFe203

159.69220

rnFeS

83.933000

rnO

15.999400

rn02

31.998800

rnS

28.086000

rnS02

60.084800

*******************************************************************

Worked Examples

177

Part II: Calculating Chemical Equilibrium Concentrations
The second part of this problem (CHEMEQ1.EKA, CHEMEQ2.EKA, and
CHEMEQ3.EKA on the distribution disk) demonstrates Eureka's ability to solve
high-order polynomials by finding equilibrium concentrations for the reactants and
products in some sample reversible chemical reactions.

Statement of the Problem
Many chemical reactions do not proceed to completion (that is, all the available
reactants are not converted into products) because the reaction is reversible. Some
of the products actually react chemically and convert back into their original constituents.
Some examples of such reversible reactions are:

+ 12(gas)
N2(gas) + 3H2(gas)

H 2(gas)

<=> 2HI(gas)
<=> 2NH3(gas)

When the rate of forward reaction (in these examples, the combining action of
the gases) and the rate of decomposition reach a balance, the chemical reaction is
said to be in equilibrium.
At equilibrium at a given temperature, such a reversible chemical reaction can
be characterized by an equilibrium constant, Keq' The equilibrium constant is the
ratio of two numbers: The first is a product of the concentrations of all the substances on the right-hand side of the chemical equation, and the second is a product of the concentrations of all the substances on the left-hand' side. (If a substance
in the chemical equation has a coefficient a, then the concentration for that substance is raised to the ath power when calculating Keq for the reversible reaction.)
Laboratory scientists have determined Keqs for many reactions at specific temperatures. If you know the Keq for a reaction and have enough additional information about the reactants or the products, you can determine the equilibrium
concentrations of all the substances involved in the reaction.
First Example
Consider the reaction of carbon monoxide (CO) and steam:
H 20 <=> CO2 +' H2
At 800°C, the Keq for this reaction is 1.2. If you put 2.0 moles of CO and 2.0 moles
of H 20 into a 10-liter container at 800°C, what will be the equilibrium concentrations of all the substances?
CO

178

Eureka: The Solver Owner's Handbook

Equations
The first step in setting up this problem is to write out a table that lists the starting
and equilibrium concentrations for the substances involved.
Starting Concentration

Equilibrium Concentration

0.2 moles/L
0.2 moles/L

0.2 - X moles/L
0.2 - X moles/L
X moles/L
X moles/L

o
o
By definition of the equilibrium constant,

cco X CH2
cco X cH20

2
= --=---.;;.

K
eCJ

Substituting in the values for KeCI and the four concentrations from the table
yields

12 =
. (X)(X)
.
(0.2 - X)(0.2 - X)

1.2 = (0.2 _ X)2
You will rewrite these equations as a secOnd-order polynomial and let Eureka
solve for X; then, knowing X, Eureka will calculate the equilibrium concentrations
of all the reactants and products.

Equation File arul Solution
The print-out that follows shows the equation file and the solution for this problem.
Note that we have given an initial value to Xin the equation file (X: = 0.19). We
made an assumption that at least 5% of the original substances were converted into
products. Although it was not theoretically necessary to provide Eureka with that
starting point, it turned out to be a practical action. Without a reasonable initial
value for X, Eureka is capable of finding some solutions that are unique butwrong
(such as concentrations less than 0). The easiest way to avoid such solutions is to
provide Eureka with a reasonable starting point of where you expect the answer to
be, if possible.

Worked Examples

179

*******************************************************************
Eureka: The Solver, Version 1.0
Thursday November 6, 1986, 6:23 am.
Name of input file: CHEMEQl.EKA
*******************************************************************

Chemical Equilibrium Calculations - First Example
This problem demonstrates solving a second-order
polynomial in order to calculate equilibrium
concentrations in a reversible chemical reaction
The reaction is: CO + H20 <===> C02 + H2
At 800 degrees C, the Keq for this reaction is 1.2
cC02 = X
cH2 = X
cCO = 0.2 - X
cH20 = 0.2 - X

equil.
equil.
equil.
equil.

conc.
conc.
conc.
conc.

of
of
of
of

C02
H2
CO
H20

; Keq = (cC02 * cH2) / (cCO * cH20)
1.2 = X~2 / (0.2 - X)~2
: initial value given for X
X:= 0.19
correct answers are: cC02 = cH2
cCO = cH20 = 0.2 - X = .0954451

.104555

*******************************************************************

Solution:
Variables

Values

cCO

.095445115

cC02

.10455488

cH2

.10455488

cH20

.095445115

.10455488

Maximum error is -1.092463ge-l0
*******************************************************************

180

Eureka: The Solver Owner's Handbook

Second Example
Consider the reversible reaction of nitrogen (N2) and hydrogen (H2) to form
ammonia:
N2(gas)

+ 3H2(gas) ~ 2NH3(gas)

Empirical evidence shows that log(base 10) of Kf' the equilibrium constant for
the forward reaction of N2 and H2 to form NH3 , is 2.914 at 25°C.
K = 102 .914
"f

(cNH/
= 102 .914
cN2 X (CH2)3
In this problem, you start with .1 mole of N2 and .3 moles of H2 in a 10-liter flask at
25°C:
Starting Concentration

Final Concentration

0.1 moles/L
0.3 moles/L
0.0 moles/L

(0.1 - X) moles/L
(0.3 - 3X) moles/L
2X moles/L

What will be the concentrations of all the constituents at equilibrium?

Equations
The following polynomial results from substituting the concentration values from
the table into the equation for IS-:
2

(2X)
(0.1 - X)((0.3 - 3X)3)

102.914

which simplifies to

27(0.1 - X)4) - (2X)2

Worked Examples

181

Equation File and Solution
The equation file and solution for this problem appear in the print-out that follows.
Notice the constraints placed on the value X (which represents the equilibrium
concentration of NH3):
• X > 0 (There must be some NH3 produced)
• X < 0.1 (You can't have more products than the reactants)

182

Eureka: The Solver Owner's Handbook

*******************************************************************
Eureka: The Solver, Version 1.0
Thursday November 6, 1986, 6:27 am.
Name of input file: CHEMEQ2.EKA
*******************************************************************
Chemical Equilibrium Calculations - Second example
This is an example of solving a fourth-degree
polynomial, in order to find the equilibrium
concentrations of N2, H2 and NH3 in the reversible
chemical reaction
N2 + 3H2 <===> 2NH3
at 25 degrees C, where log(base 10) of K(f) (the
equilibrium constant for the forward reaction) is
2.914.

equil. cone. of N2
equil. conc of H2
equil. cone. of NH3

cN2 = 0.1 - X
cH2 = 0.3 - 3*X
cNH3 = 2*X

equation for equilibrium concentrations:
K = (cNH3)A2 / (c~!2* (cH2r3)
4*(XA2) = K * (0.1 - X) * (0.3 - 3*X)A3
constraints so Eureka finds appropriate roots

o < X < 0.1
*******************************************************************
Solution:
Variables

Values

cH2

.091649822

cN2

.030549941

cNH3

.13890012

820.35154

.069450059

Maximum error is -4.234442ge-10
*******************************************************************

Worked Examples

183

Third Example
This example is a classic case of poorly scaled variables and equations and
clearly demonstrates where and how things could go wrong in your equation files.
Consider the dissociation of hydrogen peroxide at 800°C to yield water (H2 0)
and oxygen (02):
2

H 20 2 <=> 2

H 20

At 800°C, the Keq for this reaction is 3.6 X 1013 • If you start with 0.1 mole of
H 20 2 in a 10-liter container at 800°C, what will be the equilibrium concentrations
of all the substances?

Equations

Starting Concentration

Equilibrium Concentration

0.01 moles/L

0.01 - 2X moles/L
X moles/L
2X moles/L

o moles/L

3.6

1013 =

(X) (2X)2
(0.01 _2X)2

(0.01 - 2X)2
13

Since 3.6 X 10 is a very large number with respect to the other quantities in
this problem, the equation is poorly scaled. Eureka runs into difficulty when trying
to solve an equation that spans so many orders of magnitude (14 in this case). For
example, Eureka might find a solution that yields an evaluation difference of 106 •
6
13
Because 10 is relatively tiny compared to 10 , Eureka might be satisfied with the
solution, even though that solution would be patently wrong.
Given these problems inherent with spanning large orders of magnitude, it
would seem that you could convert both sides of the equation to the natural logs
and ask Eureka to solve the following:
1n (3.6

184

* 10A13)

= 1n (4)

+ 1n

A
(X 3) - 1n «0.001 - (2

X))A2}

Eureka: The Solver Owner's Handbook

However, rewriting the original equation to this form does not work well,
because Eureka must calculate each of the natural logs used; the problem in this
form becomes unworkable.
Your best approach to this quandary is to formulate an intermediate equation
that spans fewer orders of magnitude. To accomplish this, create the following set
of intermediate variables and equations:

1. substitute Keq = W3 so W= 36
Y = (0.01 - (2 *
W3 = 4X 3/Y 2 so W= X * «2/Y)
A

(1/3)

* 10 4
A

X»

(2/3»

2. substitute B = (Y/2V(2/3) so B * W= X

3. set constraints and initializations

o < X < 0.005,

W> 0, B > 0,

X := 0.0001

By redefining the problem as B* W= X, you create a situation where the difference
between the largest and smallest values is on the order of 107 (from 104 to 10- 3),
which Eureka is able to solve in just a few seconds.
Note: It is important to set up the equations in the order shown on the next
page. If you present the equations in a different order, Eureka gives an "Equations
are inconsistent" error message and cannot solve the problem.

Equation File and Solution
The print-out that follows shows the equation file and Eureka's solution for this
problem.

Worked Examples

185

*******************************************************************
Eureka: The Solver, Version 1.0
Thursday November 6, 1986, 6:32 am.
Name of input file: CHEMEQ3.EKA
*******************************************************************
Chemical Equilibrium Calculation - Third Example
This example demonstrates how to handle a poorly-scaled
problem. The original equation was
Keq = 3.6 * 10A13
Y = (0.01 - (2*X»
4*XA3 / yA2 = Keq
To reduce the total span of 14 orders of magnitude,
substitute W (= KeqA(1/3» and B (= (Y/2)A(2/3»
W 36A(1/3) * 10A4
Keq W3
A

B*W
B
Y

X
(Y /2)A(2/3)
(0.01 - (2*X»

X 0
X 0.005
W> 0
X := 0.0001
B >0

Solution is X = 0.0043698957, Y = 0.0012602085 A
B = 1.323437ge-07, W= 33019.272, Keq = 3.6 * 10 13
*******************************************************************
Solution:
Variables

Values

1.323437ge-07

Keq

3. 6000000e+ 13

33019.272

0.0043698957

0.0012602085

Maximum error is 1.5178830e-18
Warning: floating point overflow
*******************************************************************

186

Eureka: The Solver Owner's Handbook

Eureka Quick Reference Table

This appendix provides brief explanations, in alphabetical order, for many of
Eureka's features.

msqrt(x)
ncum(x)
paymt(i,n,pval,fval)
piO
polar(x 1,x2)
poly(x,series)
pos(x)
pval(i,n,paymt,fval)
re(x)
sgn(x)
sin(x)
sinh(x)
sqrt(x)
sum(exp,var,xl,x2 < n > )
tan (x)

Built-in functions

abs(x)
atan2(xl,x2)
cos (x)
cosh(x)
deriv(x 1,x2)
exp(x)
fact(x)
Hoor(x)
frac(x)
fval(i,n,pval,paymt)
im(x)
integ(exp, var,x 1,x2 < n > )
In(x)
loglO(x)

Calculator operations

Press C C to select Calculator from the Commands
menu. Press Num-Lock, then use the number pad keys
located at the right of your keyboard to make entries,
then press Enter. Eureka works just like a hand-held
scientific calculator to solve arithmetic and built-in
functions.
187

Default settings

Change default settings in one of two ways:

1. Embed directives in the equation file.
2. Change settings from the Options menu:
Press 0 S to select Settings from the Options menu.
Scroll through the menu and change the desired settings by pressing Enter to toggle between yes and no,
or by entering an appropriate number. Then press Esc.
Load and save setup files with Load setup and Write
setup.
Directives

accuracy, casefold, complex, digits, finanmode, finansmooth, include, initval, listdefault, max, maxtime,
min, penalty, plotdefault, radius, rootsign, setting,
solve, substlevel, syntax, unit

Directory

Press F D from the main menu, enter optional directory
mask, and press Enter (* and? are wildcards). By
default, Eureka will list all files in the directory.
Select Edit to input the equation you want to solve.
The editor works like the Turbo Pascal and SideKick
editors or WordStar in non-document mode. A brief
listing of editor commands follows:

Edit an equation file

Word left
Word right
Delete character to left
Delete character under cursor
Delete word to right of cursor
Delete line
Insert mode on/off
Equation file format

Equations. Equations are written one per line in standard mathematical syntax and notation. Multiplication
is not implicit.

188

Ctrl-Left arrow or
Ctrl-A
Ctrl-Right arrow or
Ctrl-F
Backspace
Ctrl-G
Ctrl-T
Ctrl-Y
Ins or Ctrl-V

multiplication operator
exponentiation operator
user-defined functions and initializations
used to separate formulas on the same line
indicates formula continued on following line

Eureka: The Solver Owner's Handbook

Directives. Directives begin with a dollar sign and follow the general format of the directive shown here:
$ digits = 6

Comments. Comments must be set off by a semicolon
(;) or appear between braces ({ }).
File names

Press F L to select Load from the File menu, then enter
a file name. File names follow DOS conventions (up to
eight characters, and may also include an optional 1 to
3 character extension).
Use the Write to ... option on the File menu to make
an alternate version of the current file.

Find other roots, modify Press 0 V from the main menu to select Variables from
variables and constants
the Options menu. Enter a value, and press Enter.
Press Esc twice to return to the main menu, then
choose Iterate from the Command menu to solve for
the new variable. In seeking another root, the value
should be close to but not greater than that of the second root. Press Esc, then press S to return to the main
menu and Solve.
Function keys

FI
F2
F3
F5
Alt-F5
F6

FB
Esc

Ctrl-Break
Up arrow, Down arrow,
Left arrow, Right arrow,

Display help text
Save the current file
New file
Zoom to full screen editor
Text zoom in plot window
Change active window
Go to active window or block
Begin (edit function)
Block end (edit function)
Select menu
Interrupt solve
Move/scroll through the screen

PgUp, PgDn,
Alt-X

List a table of values

Eureka Quick Reference Table

Exit Eureka

Press G L from the main menu to select List from the
Graph menu. Respond to the prompts by entering the
function (if necessary) and the initial value, increment, and number of values; then press Enter. See
comment under "Plot a graph."

189

Load a file

Plot a graph

Press F L from the main menu to select Load from the
File menu. At the prompt, type in the name of the file
to be loaded; press Enter.
Press G P from the main menu to select Plot from the
Graph menu. Respond to the prompts by entering the
left and right endpoints and then press Enter.
(Note: A file must have at least one user-defined
function or you must enter one with the Function command before you can select Plot or List.)

Print a report

Press R 0 to select the Output option from the Report
menu. Indicate the desired output device and whether
you want a formatted header (Enter toggles between
the various choices). Select Go and press Enter.

Quit the program

Press Q from the File menu, or press Alt-X.

Return to DOS

Press F 0 to select OS Shell from the File menu.

Return to main menu

From a window or submenu, press Esc.

Save a file

Press F S from the main menu to select Save from the
File menu. This saves a copy of your file on your active
drive.

Select menu items

Cursor Method Move cursor to desired menu item
and press Enter.
Initial Letter Method Type the capital letter of the
desired option. For example, to select the Calculator
option on the Commands menu, you would press Cc.

Settings

In addition to those listed under "Directives:"
lisLfirst, lisLnum, lisLinc, ploLleft, ploLright.

Start the program

Log on to the drive or directory containing the program and type EUREKA Enter.

Solve a problem

Press S from the main menu to select Solve.
To solve the problem another time, press C I from the
main menu to select the Iterate command from the
Commands menu.
To look in a different region for an answer, press C F
from the main menu to select the Find other command
from the Commands menu.

Verify a solution

190

Press C V from the main menu to select Verify from the
Commands menu.

Eureka: The Solver Owner's Handbook

Eureka Editor Commands

All Eureka editor commands are described in this appendix. Each description consists of:
• a heading defining the command
• the keystrokes that activate the command
• a brief explanation of what the command does

If there are two ways of giving a command, that is, one using the IBM cursor
keys and the other using Ctri, both ways are listed, with the IBM cursor key
method listed first.
If you are familiar with WordStar or the Turbo Pascal and SideKick editors, you
should have no trouble with the Eureka editor, since it is virtually identical to these
editors.
For hands-on practice with using the editor, see the tutorials in Chapter 2. To
change default editor keystrokes, use the EINST program, described in Appendix F.

191

Cursor Movement Commands
These commands control the position of the cursor in the file. You can manage
fairly well with just the cursor keys and PgUp and PgDn; the other commands are
handy but may not be critical for small files.
Character left
Left arrow or Ctrl-S
Moves the cursor one character to the left without affecting the character there.
This command does not work across line breaks; when the cursor reaches the left
edge of the window, it stops.
Character right
Right arrow or Ctrl-D
Moves the cursor one character to the right without affecting the character there.
This command does not work across line breaks; when the cursor reaches the right
edge of the window, the text starts scrolling horizontally until the cursor reaches
the extreme right of the line in column 250, where it stops.
Line down
Down arrow or Ctrl-X
Moves the cursor to the line below. If the cursor is on the second-to-the-Iast line,
the window scrolls up one line.
Line up
Up arrow or Ctrl-E
Moves the cursor up to the line above. If the cursor is on the top line, the window
scrolls down one line.
Page down
PgDn or Ctrl-C
Moves the cursor one page (window) down with an overlap of one line. The cursor
moves one window less one line forward in the text.
Page up
PgUp or Ctrl-R
Moves the cursor one page (window) up with an overlap of one line. The cursor
moves one window less one line backward in the text.
To end of file
Moves the cursor to the last character of the text.

Ctrl-PgDn or Ctrl-Q C

To top of file
Moves the cursor to the first character of the text.

Ctrl-PgUp or Ctrl-Q R

To last cursor position
Ctrl-Q P
Moves the cursor to the last position it occupied. This command is useful, for
example, to move back to the last position after a Find/Replace operation.
To left margin
Home or Ctrl-Q S
Moves the cursor all the way to the left edge of the window (column 1).
To right end of line
End or Ctrl-Q D
Moves the cursor to the end of the line; in other words, to the position following the
last printable character on the line. Trailing blanks are always removed from all
lines to save space.
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To top of window
Moves the cursor to the top of the editor window.
To bottom of window
Moves the cursor to the bottom of the editor window.

Ctrl-Home or Ctrl-Q E
Ctrl-End or Ctrl-Q X

Ctrl-Left arrow or Ctrl-A
Word left
Moves the cursor to the beginning of the word to the left. A word is defined as a
sequence of characters delimited by one of the following characters: space < > , ;
•

()

[

]

* + _ / $.

Word right
Ctrl-Right arrow or Ctrl-F
Moves the cursor to the beginning of the word to the right. A word is defined as a
sequence of characters delimited by one of the following characters: space < > , ;
* + _/ $.
•

()

[

]

Insert and Delete Commands
These commands let you insert and delete text by character, word, and line, and
control the Insert/Overwrite typing mode.
Delete character to left
Backspace
This is the backspace key just above Enter. It moves one character to the left and
deletes the character there. Any characters to the right of the cursor are moved to
the left.
Delete character under cursor
Del or Ctrl-G
Deletes the character under the cursor and moves any characters to the right of the
cursor one position to the left. This command does not work across line breaks.
Delete word right
Ctrl- T
Deletes the word to the right of the cursor. A word is defined in "Word left" under
cursor movement commands.
Delete to end of line
Deletes all text from the cursor position right to the end of the line.

Ctrl-Q Y

Insert mode on/off
Ins or Ctrl-V
This command lets you switch between insert, the default, and overwrite mode
while entering text. The current mode is displayed on the status line of the editor
window. In Insert mode, new text is tucked in at the cursor position, and existing
text is moved to the right. Overwrite mode is convenient for replacing old text with
new. In this mode, existing characters are replaced by the new characters typed
over them.

Eureka Editor Commands

193

Line deletion
Ctrl- Y
Deletes the line containing the cursor and moves any lines below one line up. The
cursor moves to the left edge of the window. Be careful; there is no way to restore a
deleted line.
Line insertion
Inserts a line break at the cursor position. The cursor remains stationary.

Ctrl-N

Block Commands
Block commands allow you to maneuver chunks of text. A block can be any amount
of text from a single character to an entire file. A block is marked by placing a
"begin block" marker before the first character and an "end block" marker after the
last character of the desired portion of text. So marked, a block can be copied,
moved, deleted, or written as a file.

Ctrl-K C
Block copy
Places a copy of a marked block starting at the cursor position. The original block is
left unchanged.
Ctrl-K V
Block move
Moves a marked block from its original position to the cursor position. The block
disappears from its original position.
Block delete
Ctrl-K Y
Deletes a marked block. Be careful? no provision exists for restoring a deleted block
of text.
Hide/display block
Ctrl-K H
This command causes the visual marking of a block to be toggled off and on. The
copy, move, delete, and write to a file block manipulation commands work only
when the block is visibly marked.
Mark block begin
Ctrl-K B
Marks the beginning of a block. The marker itself is not visible on the screen, and
the block only becomes visibly marked (displayed in lower-intensity text) when the
end block marker is set. You can also use the begin block marker as a reference
point in your text and jump directly to it with the Ctrl-Q B command.
Mark block end
Ctrl-K K
Marks the end of a block. As above, the marker itself is not visible on the screen,
and the block only becomes visible after the begin block marker is set. You can also
use the begin block marker as a reference point in your text and jump directly to it
with the Ctrl-Q K command.

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Mark single word
Ctrl-K T
Mark a single word as a block. If the cursor is placed within a word, the word is
marked; if the cursor is not on a word, the word to its left is marked. A word is as
defined in "Word left" under cursor commands.
Read block from disk
Ctrl-K R
This command is used to read a file into the current text at the cursor position,
exactly as if it were a block that was being copied or moved.
Write block to disk
Ctrl-KW
This command is used to write a previously marked block to a file. The block is left
unchanged, and the markers remain in place. When you use this command, you are
prompted for the name of the file to write to. If the file name specified already
exists, a warning is given before the existing file is overwritten.

Miscellaneous Editing Commands
Abort operation
Ctrl-U
Abort the current editing operation when it pauses for input (for example, when
Search and replace asks "Replace YIN?" or during the entry of a file name for a
block read).
. Ctrl-P
Control character prefix
The editor allows you to enter control characters into the file by prefixing the
desired control character with Ctrl-P. Control characters are displayed as halfintensity capital letters.
For instance, to cause an Epson printer to print condensed type, enter Ctrl-P
Ctrl-O. Consult your printer's manual for more information about printing control
commands. You can also use Ctrl-B (for bold) or Ctr/~S (for underline) to emphasize
text. These are toggle characters, so you turn off bold, for example, by entering
Ctrl-P Ctrl-B after the last character that should be printed in bold.
Find
Ctrl-Q F
This command lets you search for a string of characters. A prompt appears above
the status line, requesting a search string. If you make an error while typing the
string, use the Backspace key to back up and correct it. The string may contain any
characters, including control characters. Control characters are entered into the
search string with the Ctrl-P prefix. To find a line break, search for Ctrl-M Ctrl-J.
After the search string has been specified, you are asked for search options.
Enter the required options (if any), then press Enter to start the search. If the text
contains a target matching the search string, the cursor is positioned at the end of

Eureka Editor Commands

195

the target. To repeat the search operation, use the repeat last find command, Ctrl-L.
The following options are available (and need not be entered in uppercase):
B

Search backwards from the current cursor position towards the beginning of
the file

n = an integer. Type the number of occurrences of the string for which the
search should take place, counted from the current cursor position.

Ignore uppercase or lowercase. Regard uppercase and lowercase alphabeticals as equivalent.

Search for whole words only. Skip matching patterns that are embedded in
other words.

Find and replace
Ctrl-Q A
This command lets you search for a string of characters, and then replace it with
another. A prompt appears above the status line, requesting a search string. A
further prompt requests the replacement string. In nearly every other respect, this
commands functions like Find, described previously. There are two additional
options:
G

Global search and replace. Search and replace throughout the entire text, irrespective of the current cursor position.

Replace without asking. Does not stop and query whether or not the replace
should be carried out at every occurrence of the search string.

Page break
Inserts a page break (an ASCII form feed) at the cursor position.

Ctrl-P Ctrl-L

Ctrl-L
Repeat last find
Repeats the last Find or Find and replace operation exactly as if all the information
had been re-entered.
Restore error message
Redisplay last error message.

Ctrl-Q W

Restore line
Ctrl-Q L
Restores a line to its original state regardless of what changes you have made;
however, you cannot have left the line for the command to be effective.

Tab or Ctrl-I
Tab
Moves the cursor eight spaces to the right (hard tabs) or moves cursor to next
column based on spacing in the line above (soft tabs).
Tab mode toggle
Toggles tabs on and off.

Ctrl-O T

Set tab
Sets tab at cursor position.

Ctrl-O I

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Eureka: The Solver Owner's Handbook

Built-in Functions

Eureka's built-in functions are described in alphabetical order in this appendix.
The standard mathematical and other functions are listed below:
Standard Mathematical Functions
abs
atan2
cos
cosh
exp
fact

In
log 10
pi ()
sin
sinh
tan
tanh

Other Functions
deriv

floor
frac
fval
im
integ
ncum
paymt
polar
poly
pos
pval
re
sgn
sqrt
sum

197

The following notations are used in describing the Junctions:
expr
series
var
x

abs

must be an expression
series of values or expressions
variable identifier
value or expression

Absolute Value

abs(x)

This function returns the absolute value of x. If x is positive, x is returned. If x is
negative, - x is. returned.
Examples
abs(5) returns 5
abs(-15.5) returns 15.5

atan2

Arc Tangent

atan2(xl,x2)

This trigonometric function returns the inverse tangent of xl/x2. This is the angle,
in radians, formed by x2 and xl. This function offers an extension to the usual Arc
Tangent in that it applies to all four quadrants. The rules that one used· in determining e = atan2 (xl, x2) are as follows:
Quadrant
I
ifxl
II
if xl
III
if xl
IV
if xl

and x2 > 0, then 0 < e < -rr/2
> 0 and x2 < 0, then -rr/2 < e < -rr
and x2 < 0, then--rr < e < -'IT /2
< 0 and x2 > 0, then - -rr /2 < e < 0

The quadrants progress counterclockwise, with I at the upper right-hand corner.
Example
atan2(1,-1) returns 2.35619449

cos

Cosine

cos (x)

This trigonometric function returns the cosine of x. x is assumed to be the angle in
radians.
Example
cos(1) returns 0.54030231

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Eureka: The Solver Owner's Handbook

cosh

Hyperbolic Cosine

cosh(x)

This function returns the hyperbolic cosine of x.
Example
cosh(2) returns 3.76219569

deriv

Partial Derivative

deriv (xl, x2)

This function returns the partial derivative of the first argument with respect to the
second. The second argument must be a variable. For example, solving

z = deriv(x~3 + 5 * x,x)
x =1

gives
x = 1.00000000
z = 8.00000000

The derivative of x + 5x is 3x + 5, which has the value 8 when x = 1. If other
variables besides x were to appear in the expression, they would be treated as
constants in the derivative to be evaluated. For example, solving
3

z = deriv(x~3 + y~2 + 5 * x,x)
x =1
y =2
gives
x = 1.00000000
= 2.00000000
z = 8.00000000

However, while Eureka makes substitutions internally in order to solve equations, such substitutions do not affect derivatives. For instance, solving

y = x~3 + 5 * x
z = deriv(y,x)
x =1
gives
x = 1.00000000
y = 6.00000000

z = 0.00000000

Built-in Functions

199

This result occurs because deriv treats x and y as independent variables. To
differentiate an expression that occurs frequently, you should define an appropriate
function. Here this function could be
f(x) : = x-3 + 5 * x
z = deriv(f(x),x)

x =1
gives
x = 1.00000000
z = 8.00000000

exp

Exponentiation

exp(x)

This exponential function raises the number e to the power x.
Examples
exp(O} returns 1.00000000
exp(l) returns 2.71828183

fact

Factorial

fact(x)

This function calculates the factorial of the positive integer specified by x. The
integer x must be less than or equal to 170.
Examples
fact(6) returns 720.000000
fact(3) returns 6.00000000

floor

Floor

floor(x)

This function rounds x down to the nearest integer.
Examples
floor{-2.5) returns -3.00000000
floor(2.999 + 2.999} returns 5.00000000

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Eureka: The Solver Owner's Handbook

frac

frac(x)

Fraction

This function returns as a positive number the part of x that is not an integer.
Examples
frac(2.5) returns 0.50000000
frac(1.33 + 1.55) returns 0.88000000

Future Value

fval

fval(i, n,pval,paymt)

This financial function returns the future value of a mortgage or similar contract
(the money received at the end of a contract). n is normally an integer representing
the number of payment intervals; i is the interest rate per payment expressed in
hundredths; pval and paymt are described elsewhere in this section. fval is either
zero or a negative number representing the balloon payment. See page 99 in
Chapter 5 for more details.
Example

If you borrow $250,000 at 10% and make ten annual payments of$35,000, how
much money do you owe the lender at the end of the contract?
fval(0.10,10,250000,-35000) returns -90625.75

Imaginary Number

im(x)

This function returns the imaginary part of a complex number. (Note: The complex
number setting must be set to yes.)
Example
$ complex = yes
im(3 + 2 * sqrt(-l»

integ

Integral

returns 2

integ(exp,var,xl,x2 < n > )

This function returns the integral of the first argument with respect to the second.
The second argument must be a variable. The third argument is the lower limit of
integration, and the fourth argument is the upper limit. Improper integrals are not
allowed.

Built-in Functions

201

This function returns the definite integral for the variable (var) varied in the
specified expres~ion between the limits xl and x2. Integ uses Simpson's Rule to
compute the integral. < n > is the number of subdivisions used for the integral. If
n is not specified, 10 subdivisions are used. (See deriv for more discussion of differential calculus.)
Expressions can also be integrated with respect to a given variable. For example, solving
z
pi

=
=

integ(sin(x),x,O,pi)
3.14159265359

gives the integral of sin(x) as x goes from 0 to pi:
pi
x
z

=
=

3.14159265359
1.23900000
2.00000000

In this case, the value for x is meaningless because it is not subject to any
constraints. As with derivatives, variables other than the variables of integration
are treated as constants, but functions can be used. The last two arguments to
integ, lower and upper limits of integration, can be arbitrary expressions.
Example
integ(x-2 + 3 * x + 4,x,2,7) = y

returns the area under the function x

4, between

x = 2 and x = 7, ory = 199.16666667.

Natural Logarithm

In{x)

This function returns the natural logarithm (base e
greater than o.

2.71828183) of x. x must be

Examples
1n(exp(l» returns 1
1n(15) returns 2.70805020

lnglO

Logarithm base 10

logl 0 (x)

This function returns the logarithm base 10 of x. x must be greater than

Example
10g10(l) = 0
10g10(12) = 1.0791812

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Eureka: The Solver Owner's Handbook

msqrl

Negative Square Root

msqrl(x)

This function returns the negative (minimum or minus) square root of x.
Example
msqrt(9) returns -3

ncum

Normal Cumulative

ncum(x)

This function returns the probability that a normal random variable will be below
its mean plus x standard deviations. It can also be expressed as

ncum(x) = '\f(~'TrT

J~X~(

-t'/2) dt

ncum is the basic tool used in analyzing the variance of normal distributions in
statistics and probability. The importance of the normal distribution is primarily
due to the Central Limit Theorem. This theorem states that any sum of independent random variables approximates a normal random variable, with the approximation becoming more accurate as more variables are summed. In the
experimental sciences, the deviation from theory and experiment can be explained
as an error which is the sum of error contributions from many sources. The resulting error is usually quite well approximated as a random variable.
As an example of the use of ncum, consider a chemist who is measuring pollutants in the air. After taking a number of readings, s/he computes the average pollen
concentration at 150 parts per million (ppm), with a standard deviation of 10 ppm.
The probability of making a reading less than 170 ppm is ncum( + 2), since 170 is 2
standard deviations above 150. Also, the chemist can be very confident that a reading will be less than 150 + lOx where
ncum(x)

0.99

Example
ncum{l) returns 0.84134456

Built-in Functions

203

paymt

PaY111£nt

paymt(i,n,pval,jval)

This financial function returns the payment on a mortgage or similar contract. n is
normally an integer representing the number of payment intervals; i is the interest
rate per payment expressed in hundredths; pvai and fvai are described elsewhere
in this section. See the section on financial functions in Chapter 5 for more details.

pi ()

This function returns the value of pi (3.14159265359)

polar

Polar Coordinate

polar(xl,x2)

This function returns the angle in the rectangular to polar coordinate conversion in
radians. The conversion

x = r
y = r

cos (theta)
sin(theta)

is inverted by
r = sqrt(x2 + y2)
theta = polar(x,y)
except for the restriction

< theta <

'TT'

= 3.14159265359

'TT'

If z is complex, these transformations are equivalent to
x
= re(z)
y
= im(z)
r
= abs(z)
theta = im(ln(z))
Polar(x,y) is the same as the Fortran function atan2(y,x), and is the same as the
arc tangent of y/x with the quadrant chosen carefully (see arctan2 for more details).
Example
polar(-1,O) returns 3.14159265

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Eureka: The Solver Owner's Handbook

poly

poly (x, series)

Polynomial

This function returns the value of the polynomial of x using the specified series of
coefficients. Coefficients are listed in descending order of the variable exponent.
For example, the following polynomial equation
X4

+ 2x + 1

would be expressed as follows:
p(x) := poly(x, 1, 3, -1, 2, 1)

Applying Solve to a file containing poly causes Eureka to find all the roots, as
well as solve for whatever other variables are in the file.
A root to a polynomial is a complex number that yields zero when plugged into
the polynomial. A polynomial of degree n always has exactly n roots, counting
repetitions. Since the polynomial here has real coefficients, the roots are either real
or occur in complex conjugate pairs. The above polynomial has the following roots:
No.

Real Part
- 0.29105149
0.50677653
0.50677653
- 2.72250157

1
2
3
4

pos

Positive

Imaginary
0.00000000

1.00258931
- 1.00258931
0.00000000

pos(x)

This function returns the positive value of x.
Examples
pos(I.2) returns 1.20000000
pos(-12) returns 0.00000000

Built-in Functions

205

pval

pval(i, n,paymt,jval)

Present Value

This financial function returns the present value on a mortgage or similar contract.
(For a mortgage, this is the amount of the loan minus the down payment.) n is
normally an integer representing the number of payment intervals; i is the interest
rate per payment expressed in hundredths; paymt and fval are described elsewhere in this section. See page 99 in Chapter 5 for more details.

Real Number

re(x)

This function returns the real part of a complex number. (Note: Complex numbers
must be set to yes.)
Example
$ complex:: yes
re(3 + 2 * sqrt(-1) returns 3

Sign

sgn

sgn(x)

This function returns 1 if x > 0 or if x = 0; otherwise, it returns -1 if x < O.
Examples
sgn(10) returns 1
sgn(O) returns 1
sgn(-I) returns -1

sin

Sine

sin (x)

This trigonometric function returns the sine of x. x is the angle in radians.
Example
sin(1) returns 0.84147098

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Eureka: The Solver Owner's Handbook

sinh

Hyperbolic Sine

sinh (x)

This function returns the hyperbolic sine of x.
Example
s;nh(l} returns 1.17520119

sqrt

Square Root

sqrt(x)

This function returns the positive square root of the argument x.
Example
sqrt(81} returns 9.00000000

Sum

sum

sum (exp, var, xl, x2)

This function returns the sum of the series defined by replacing the variable in the
second argument in the function of the first argument over the range expressed in
the third and fourth arguments, or
n

Lf(x)
x = 1

where f(x) = exp

= var

= xl

n = x2
The second argument must be a variable. The third argument is the lower limit
of the summation, and the fourth argument is the upper limit.

Example
sum(4 * x,x,I,3} returns 24

Built-in Functions

207

tan

Tangent

tan (x)

This trigonometric function returns the tangent of x, where x is an angle measured
in radians.
Examples
tan(O) returns 0.00000000
tan (1) returns 1.55740772

tanh

Hyperbolic Tangent

tanh (x)

This trigonometric function returns the hyperbolic tangent of x.
Example
tanh(l) returns 0.76159416

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Eureka: The Solver Owner's Handbook

A DOS Primer

If you are new to computers or to DOS, you may have trouble understanding
certain terms used in this manual. This appendix provides you with a brief overview of the following DOS concepts and functions:
• What DOS is and does
• The proper way to load a program
• Directories, subdirectories, and the path command
• Using AUTOEXEC.BAT files
This information is by no means a complete explanation of the DOS operating
system. If you need more details, please refer to the MS-DOSTM or PC_DOSTM
user's manual that came with your computer system.
Eureka runs under the MS-DOS or PC-DOS operating system, version 2.0 or
later.

What Is DOS?
DOS is shorthand for Disk Operating System. MS-DOS is Microsoft's version of
DOS, while PC-DOS is IBM's rendition. DOS is the traffic coordinator, manager,
and operator for the transactions that occur between the parts of the computer
system and the computer system and you. DOS operates in the background, taking

209

care of many of the menial computer tasks you wouldn't want to have to think
about - for instance, the How of characters between your keyboard and the computer, between the computer and your printer, and between your disk(s) and internal memory (RAM).
Other DOS transactions are ones that you initiate by entering commands on the
DOS command line; in other words, immediately after the DOS prompt. Your
DOS prompt looks like one of the following:
A>
B>
C>

The capital letter refers to the active disk drive (the one DOS and you are using
right now). For instance, if the prompt is A>, it means you are working with the
files on drive A, and that commands you give DOS will refer to this drive. When
you want to switch to another disk, making it the active disk, all you do is type the
letter of the disk, followed by a colon and Enter. For instance, to switch to drive B,
just type .B:fnter.
There are a few commands that you will use often with DOS, if you haven't
already, such as
DEL or ERASE
DIR
COPY

EUREKA

To erase a file
To see a list of files on the logged disk
To copy files from one disk to another
To load Eureka: The Solver

DOS doesn't care· whether you type in uppercase or lowercase letters, or a
combination of both, so you can enter your commands however you like.
We'll assume you know how to use the first three commands listed; if you don't,
refer to your DOS manual. Next, we will explain the proper way to load a program
like Eureka: The Solver, and that involves the last command-EUREKA.

How to Load a Program
On your distribution disk, you'll find the main Eureka program, under the file
name EUREKA.EXE. This program file is necessary for all functions, so you
always need it when you first start the program. A file name with the "last name" or
extension .COM or .EXE means a program file that you can load and run (use) by
typing only its "first name" on the DOS command line. So, to invoke Eureka, you
simply type EUREKA and press Enter, and Eureka will be loaded into your computer's memory.

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Eureka: The Solver Owner's Handbook

There's one thing you need to remember about loading Etireka and other similar
programs: You must be logged onto the disk and directory where the program is
located in order to load it; otherwise, unless you have set up a DOS path (described
shortly), DOS won't know where to find the program. (In DOS 3.0, however,you
can load a program by giving the full path name.)
For instance, if your distribution disk with the EUREKA.EXE program is in
drive A but the prompt you see on your screen is B >, DOS won't know what
you're talking about if you type EUREKA and press Enter, and will give you the
message "Bad command or file name."
It's as if you were shufHing through the "School Records" :(ile in your file cabinet
looking for information about your home finances. You're in the wrong place. So if
you happen to get that DOS message, simply switch to drive A by typing A: and
then press Enter. Then type EUREKA and press Enter to load Eureka.
You can set up a "path" to the Eureka files so that DOS can find them, using the
DOS path command. See the section on the AUTOEXEC.BAT file for more information.

Directories
A directory is a convenient way to organize your floppy or hard disk files. Directories allow you to subdivide your disk into sections, much the way you. might put
groups of manila file folders into separate file boxes. For instance, you might want
to put all your file folders having to do with finance - for instance, a bank statement
file, an income tax file, or the like - into a box labeled "Finances."
On your computer, it would be convenient to make a directory to hold all your
Eureka files, another for your SideKick files, another for your letters, and so on.
That way, when you type DIR on the DOS command line, you don't have to wade
through hundreds of file names looking for the file you want. You'll get a listing of
only the files on the directory you're currently logged onto.
Although you can make directories on either floppy or hard disks, they are used
most often on hard disks. This is because a hard disk can hold a greater volume of
data, so there is a greater need for organization and compartmentalization.
When you're at the DOS level, rather than in Eureka or ,another program, you
can tell DOS to create directories, move files around between directories, and
display which files are in a particular directory.
In the examples that follow, we assume you are using a hard disk system, and
that you are logged on to the hard disk so that the prompt you see on your screen is
C >. If you want to create directories on your floppy disks, just substitute A or B

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To make a directory for your Eureka files, do the following:
1. At the C > prompt, type MKDIR EUREKA and press Enter. The MKDIR
command tells DOS to make a directory called EUREKA.
2. Type CHDIR EUREKA and press Enter. The CHDIR command tells DOS to
move you into the EUREKA directory.
3. Now, put the Eureka disk you want to copy from into one of your Hoppy
drives -let's say A for this example - and type COPY A:*. * Enter. (The asterisks are wildcards that stand for all files.) The COpy command tells DOS to
copy all files on the A drive to the EUREKA directory on the C drive. As each
file on the disk is copied, you will see it listed on the screen.
That's all there is to it. Treat a directory the same way you would a disk drive: To
load Eureka, you must be in the EUREKA directory before typing Eureka and
pressing Enter, or DOS won't be able to find the program.

Subdirectories
If you are someone who really likes organization, you can further subdivide your
directories into subdirectories. You can create as many directories and subdirectories as you like - just don't forget where you put your filesl
A subdirectory is created the same way as a directory. To create a subdirectory
from the EUREKA directory (for instance, for storing your equation files), do the
following:
1. Be sure you are in the EUREKA directory.
2. Type MKDIR FILES Enter.
3. Type CHDIR FILES. You are now in the FILES subdirectory.
4. Copy your equation files to the new subdirectory.
If you do put your files in a subdirectory, you can let Eureka know where they
are by using the EINST program (see Appendix F).

Where Am I? The $p $g Prampt
You probably have noticed that when you change directories, you still see the C >
prompt; there is no evidence of what directory or subdirectory you are in. This can
be confusing, especially if you leave your computer for a while. It's easy to forget
where you were when you left.

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DOS gives you an easy way to find out. Just type
prompt=$p $g

and from now on (until you turn your computer off or reboot), the prompt will show
you exactly where you are. Try it. If you are still in the FILES subdirectory, your
DOS prompt should look like
C:\EUREKA\FILES >

The AUTOEXEC.BAT Fik
To avoid typing the prompt command (discussed in the previous section) to see
"where you are" every time you turn on your computer, you can set up an
AUTOEXEC.BAT file to do it for you each time you turn on your computer.
The AUTOEXEC.BAT file is a useful tool to set your computer to do things
automatically. There are many more things it can do, but rather than go into great
detail here, we suggest referring to your DOS manual for more information. We
will show you how to create an AUTOEXEC.BAT file that will automatically
change your prompt so you know where you are in your directory structure, set a
path to the Eureka directory, and change to the EUREKA directory.
The DOS path command tells your computer where to look for commands it
doesn't recognize. DOS only recognizes programs in the current (logged) directory, unless there is a path to the directory containing pertinent programs or files.
In the following example, we will set a path to the EUREKA directory.

If you have an AUTOEXEC.BAT file in your root directory, your computer will
do everything in that file when you first turn your computer on. (The root directory
is where you see the C> or C:\ prompt, with no directory names following it.)
Here's how to create an AUTOEXEC.BAT file.

1. Type CHDIR \ Enter to get to the root directory.
2. Type COPY CON AUTOEXEC.BAT Enter. This tells DOS to copy whatever
you type next into a file called AUTOEXEC.BAT.
3. Type
PROMPT=$P $G
PATH=C:\EUREKA
CHOIR EUREKA
Ctrl-Z

The Ctrl-Z sequence saves your commands in the AUTOEXEC.BAT file.
To test your new AUTOEXEC.BAT file, reboot your computer by holding down
the Ctrl and Alt keys and then pressing Del. You should see C:\EUREKA>.

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Clwnging Directories
How do you get from one directory to another? It depends on where you want to
go. The basic DOS command for changing directories is CHDIR. Use it like this:
• To move' from one directory to another: For example, to change from the
EUREKA directory to one called WP, type the following from the EUREKA
directory:
C:\EUREKA> CHOIR \WP Enter

Notice the backslash (\) before the directory name. Whenever you are moving
from one directory to another unrelated directory, type the name of the directory,
preceded by a backslash.
• To move from a directory to its subdirectory: For example, to move from the
EUREKA directory to the FILES subdirectory, type the following from the
EUREKA directory:
C:\EUREKA> CHOIR FILES Enter

In this case, you did not need the backslash, because the FILES directory is a
direct offshoot of the EUREKA directory. In fact, DOS would have misunderstood what you meant if you had used the backslash in this case. If you had
included the backslash, DOS would have thought that FILES was a directory off
the main (root) directory.
• To move from a subdirectory to its parent directory: For example, to move from
the FILES subdirectory to the EUREKA directory, type the following from the
FILES subdirectory:
C:\EUREKA\FILES> CHOIR •. Enter

DOS will move you back to the EUREKA directory. Any time you want to
move back to the parent directory, type two periods after the CHDIR command.
• To move to the root directory: The root directory is the original directory. It is the
parent (or grandparent) of all directories (and subdirectories). When you are in
the root directory, you see this prompt: C:\ >.

To move to the root directory from any other directory, simply type
CHOIR \ Enter

The backs lash without a directory name signals DOS that you want to return
to the root directory.

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This appendix has presented only a quick look at DOS and some of its functions.
Once you're familiar with the information given here, you may want to study your
DOS manual and discover the many things you can do with your computer's operating system. There are many other DOS functions not mentioned here that can
simplify and enhance your computer use.

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Error Messages

The various error messages you may encounter while using Eureka are described
in alphabetical order in this appendix. These messages appear in the Error window. When applicable, a line reference may also appear with the error message.
The line reference specifies the line in the equation file that contains the error.
However, when you use the Calculator, only the word ERROR appears, although
the cause of the problem may be one described in this appendix.

Assignment Must Be to a Variable or Function
Cause: Syntax problem. Attempting to solve an equation file in which an assignment (using: =) has been made to something other than a recognizable variable or
function.
Solution: Edit the file to provide a legitimate variable or function, as required.

Bad Function Call
Cause: Internal error. Generated if Eureka attempts to solve an incorrectly defined
function.
Solution: Quit Eureka, then reactivate it.

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Canrwt Complexify Floor or Frac
Cause: Setting error. The active equation file includes the floor or frac function,
but either the complex setting or complex directive has been set to yes.
Solution: Change the setting or directive to no.

Canrwt Have Complex Derivatives or Integrals
Cause: Setting error. The active equation file includes the deriv or integ function,
but either the complex setting or complex directive has been set to yes.
Solution: Change the setting or directive to no.

Circular Function Definition
Cause: Attempting to solve an equation file that contains a tautological function
definition; that is, one that does not produce an assignable value. An example of
this is:
f(x) := x~2
g(x) := f(x

+ 3 * g(x)
+ 1) - 7 *

Solution: Rewrite the incorrect definition.

Comma Expected
Cause: Syntax problem. In attempting to solve the equation file, Eureka has
encountered an equation that lacks one or more commas between values or constants that are function arguments.
Solution: Edit the equation so that it contains the correct number of commas.

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Command Not Understood
Cause: Syntax error. In attempting to solve an equation file, Eureka has encountered an unintelligible directive (to the left of the equal sign); for example:
$ ccomplex = yes

Solution: Restore the directive to its proper form:
$ complex = yes

Constant Expected
Cause: Attempting to solve an equation file in which something other than a recognizable constant appears where one is syntactically expected.
Solution: Edit the file to provide a legitimate constant.

Dynamic Memory Depleted
Cause: Memory problem. The area allocated to Eureka has been filled.
Solution: Quit Eureka, then reactivate it.

Equations are Inconsistent
Cause: Attempting to solve an equation file in which the equations do not follow
the correct sequence.
Solution: Edit the equations so they follow the correct syntax.

File Already Exists
Cause: In using the Rename command, you have entered a file name that already
exists.
Solution: Select Rename again, but enter a name not currently in use.

Error Messages

219

File Not Found
Cause: In attempting to use Load to bring a file to the screen, no file name has
been found to match the one entered.
Solution: Doublecheck the file name and try again, remembering extensions.

File Too Large
Cause: Attempting to use Save on a file that will not fit in the available disk space.
Solution: Cut down the size of" the file, or if possible, save to another memory
device (such as a different data disk).

Function is Multiply Defined
Cause: Syntax error. Attempting to solve an equation file in which the same function definition is used more than once. For example:
f(a) := pos(x + y)
feb) := pos(z)

Solution: Edit the equation file to eliminate redundant version{s} of the function
definition:
f(a) := pos(x
g(b) : = pos(z)

+ y)

Function is Undefined
Cause: Syntax problem. Attempting to solve an equation file that contains an equation with an undefined function.
Solution: Edit the equation so that it contains an appropriate built-in function or
correctly stated user-defined function.

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Identifier Expected
Cause: Syntax problem. Attempting to solve an equation file that lacks a variable
name where one should syntactically appear.
Solution: Edit the file to include the necessary character.

Identifier Too Long
Cause: A variable name is too long.
Solution: Edit the file to eliminate redundant characters or recast the formula as
necessary.

Incorrect Number of Function Arguments
Cause: Syntax problem. Attempting to solve an equation file that contains a function with a different number of arguments than those indicated in the function
definition.
Solution: Edit the function to contain the correct number of arguments.

Invalid Function Argument
Cause: Attempting to solve an equation file containing a function with an improper
argument. For example, replacing either of the plus signs in this equation with an
equal sign:
frac(x + 2.5 + 0.33).

Solution: Edit the necessary function arguments.

Error Messages

221

Invalid Function Definition
Cause: Syntax problem. Attempting to solve an equation file that contains an
improperly defined function, such as a numeral appearing between the parentheses.
Solution: Edit the function definition so that it is properly formed.

Invalid Number
Cause: In attempting to solve the equation file, Eureka has encountered an illegal
symbolic value. For instance, while e (indicating scientific notation) is allowed, it
may be entered incorrectly.
Solution: Edit the equation file so that it contains a legal symbolic value.

Memory Heap Overflow
Cause: Attempting to solve an equation file that contains an equation requiring
more memory than is available in RAM on your computer.
Solution: Recast the equation or get more RAM for your computer.

Must Differentiate With Respect to a Variable
Cause: Attempting to solve an equation file that uses the deriv function incorrectly.
Solution: Recast the deriv function.

Numeral or Operator Expected
Cause: Attempting to solve an equation file that contains something other than a
numeral or operator where, syntactically, one should appear.
Solution: Edit the file to correct the character(s}.

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Polyrwmial is Invalid
Cause: An internal error is generated if Eureka attempts to solve an invalid polynomial.
Solution: Exit Eureka, then reactivate it.

Printer Not Ready
Cause: Hardware problem. In attempting to use the. Report Go command to print a
report, some difficulty with the printer has been encountered.
Solution: Check that the printer is on, select is on, ribbon and paper are properly
feeding and in place, cables are tight, and so on.

Quote Mark Expected
Cause: Syntax error. Attempting to solve an equation file that lacks a quotation
mark where one should appear around a file name specified by the include directive.
Solution: Edit the file to insert the necessary quotation mark.

Relational Operator Expected
Cause: Syntax error. In attempting to solve the equation file, Eureka has encountered an equation lacking an expected relational operator « or ».
Solution: Edit the incomplete equation so that it contains such a relational operator.

Second Argument Must Be a Variable
Cause: Attempting to solve an equation file containing a function in which the
second argument must be a variable (such as deriv or integ) but is not.
Solution: Recast the function argument as a variable.

Error Messages

223

String Space Overflow
Cause: The memory area allocated to storing strings is full.
Solution: Quit Eureka, then reactivate it.

Sum is Invalid
Cause: Internal error. In attempting to solve the sum function, Eureka has encountered an internal problem.
Solution: Quit Eureka, then reactivate it.

Sum is Too Large
Cause: Attempting to solve for the sum function when the spread between the
initial and final endpoints overtaxes Eureka's internal capacity.
Solution: Recast the function.

Syntax Error for Unit Conversion
Cause: Attempting to solve an equation file that contains an incorrectly formulated
unit conversion.
Solution: Reformulate the incorrect syntax.

Too Few Arguments to Poly
Cause: Syntax problem. Attempting to solve an equation file that includes the poly
function but with only one argument (minimum requirement is two arguments).
Solution: Edit the function to contain the correct number of arguments.

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Too Many Active Variables
Cause: Attempting to solve an equation file having more than twenty active variables.
Solution: Recast the file so that it contains twenty or fewer active variables.

Too Many Constants
Cause: Attempting to solve an equation file that causes Eureka to handle more
than 200 constants. While an equation file may have only a relatively small number
of constants, the solving process itself may sometimes cause Eureka to generate a
large number of internal substitution constants and thus cause this message to
appear.
Solution: Set the substlevel setting to O.

Too Many Formulas
Cause: Attempting to solve an equation file with more than twenty equations.
Solution: Recast the equation file so that it contains no more than twenty equations. Note: Solving in complex mode causes Eureka to actually handle three times
the number of equations that appear in the file: the original equation, one for imaginary numbers, and one for real numbers.

Too Many Functions
Cause: Attempting to solve an equation file with more than ten user-defined functions.
Solution: Recast the file so that it contains ten or fewer functions.

Too Many Unit Conversions
Cause: Attempting to solve an equation file having more than ten unit conversions.
Solution: Recast the file so that it contains ten or fewer such conversions.

Error Messages

225

Too Many Variables
Cause: Attempting to solve an equation file having more than twenty variables.
Solution: Recast the file so that it contains twenty or fewer variables.

Unable to Create File
Cause: Attempting to use Edit to start a new file when, for any of a number of
reasons (for example, the data disk is read-only), this is impossible.
Solution: Ascertain the reason for the failure, then use Edit again.

Unexpected End of File
Cause: Attempting to solve an equation file in which the expected terminating
character (for example, a closing }) is missing.
Solution: Edit the file to supply a recognizable termination character.

Unexpected $ End Directive
Cause: Attempting to solve an equation file that includes an end directive where,
syntactically, one should not appear.
Solution: Edit the file to reposition the end directive.

Unmatched Parentheses
Cause: Syntax problem. Attempting to solve an equation file containing an ex:pression or comment that lacks either an initial or terminal parenthesis or curly
bracket.
Solution: Edit the· file so that all parenthetical expressions are correctly enclosed
by a pair of symbols.

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Unrecognized Character
Cause: Attempting to solve an equation file that contains a character other than a
letter, identifier, operator, or numeral in a position where it might be something
other than part of a variable name or comment.
Solution: Edit the file to supply a recognizable character.

Unrecognized Setting Value
Cause: Attempting to solve an equation file in which a directive appears that
assigns an unknown setting value (for example, $ initval = four, where a numeral
is expected).
Solution: Edit the directive to include a legitimate setting value ($ initval

= 4).

Use of Uninitialized Variable
Cause: Syntax problem. Attempting to solve an equation file in which no assignment of a value (using: =) has been made to a variable that must be initialized.
Solution: Edit the file to provide an appropriate initialization.

Variable Expected
Cause: Attempting to solve an equation file that contains something other than an
identifiable variable where, syntactically, one should appear.
Solution: Edit the file to correct the erroneous variable.

Variable Has Not Been Initialized
Cause: Syntax problem. The active equation file contains a variable that has not
been properly initialized.
Solution: Edit the file so that the variable is initialized.

Error Messages

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Customizing Eureka

The program EINST.COM lets you do four things:
• set up a path to your help and setup files
• customize your keyboard to use with Eureka's editor
• modify the default edit modes
• set the screen mode (default, color, black and white, or monochrome)

If you want to store your help (HELP.EKA) and/or setup (INITIAL.EKAcreated with Write setup on the Options menu) files on a directory other than the
one where you have EUREKA.EXE, or if you are running DOS 2.0, you will need
to use the Eureka directory option to set a path to those files.
If you don't like Eureka's default editor keystrokes, or if you want the Eureka
editor to behave more like your own text editor, you can use the Text editor command option to customize the editor keystrokes to your liking.
Finally, you can use the Default editor mode option to set several defaults for
the editor: insert or overwrite mode, tabs, and autoindenting.

Running EINST
To get started, type EINST at the DOS prompt. The first menu lets you select
Eureka directory, Text editor commands, Default edit modes, or Quit. You can

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either press the highlighted capital letter of the option you want, or use the Up and
Down arrow keys to move to your selection and then press Enter. For instance,
press D to modify the Default edit modes. Pressing Esc will eventually return you
to the main screen.

The Eureka Directory Option
The Eureka directory option is really only useful for hard disk users. You'll use this
option to specifY a path to your help and setup files (HELP.EKA and INITIAL.
EKA), so that they will be accessible from wherever you call up Eureka. (INITIAL.
EKA is the file created when you select Write setup from the Options menu.)
When you select the Eureka directory option, you're prompted to enter the full
path to the directory where you are storing your files. For example, if you want to
keep the files in a subdirectory off a directory called EUREKA, you might type for
your path name
C:\EUREKA\FILES

Mter entering a path, press Enter to accept it and the main menu will redisplay.
When you exit the program, you're prompted whether or not to save the changes.
Once you save the path, the location is written to disk. (Note that the status line at
the bottom of the screen tells you which keystrokes to use when you're in this
screen.)

The Text Editor Command Option
This option allows you to change the default keys that you use while in the Eureka
editor. Press T to modifY the editor commands. The help line at the top of the
screen shows you which keys to use to move around and make changes. Most of
these commands are simply cursor movement commands; however, the R option is
useful if you make a lot of changes, then decide you want to restore the keystrokes
to their factory defaults.
Notice that you can only modifY the secondary, or highlighted, keystrokes; the
other keystrokes cannot be changed.
To change a keystroke, move the selection bar to the key you want to change,
then press Enter. You'll then see a selection bar next to the command you want to
redefine. Press the key(s) you want to use. If you take another look at the top of the
screen, you'll see the help line now lists the available commands:
~

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accept edit literal

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Use the Backspace key to backspace and/or delete something in the keystroke
box. The C option clears, or erases, the whole box. Use R to restore the original
keystrokes before exiting from the screen. Press Enter to accept the keystroke modification you've made. And finally, the < Scroll Lock> is a toggle that lets you
alternate between command and literal modes.
To explain the < Scroll Lock> option, let's take a look at the Enter key, which
is used to modify and accept the editing of a key command. But suppose you
want to change the Find String command from < CtrlQ > < CtrlF > to < CtrlQ >
< Enter> . To do so, you would have to toggle Scroll-Lock to literal mode, so that
when you press the Enter key, it will be interpreted literally as part of the new
keystroke you are entering. Follow these steps:
1. Make sure < Scroll Lock> is toggled to command mode (check the upper
right-hand corner of your screen).
2. Then press Enter at the Find String command line.
3. Press Backspace to delete the < CtrlF > part of the string.
4. Now toggle Scroll-Lock> to literal and press Enter.
5. Again, toggle Scroll-Lock to command mode and then press Enter to accept.
After you've defined a new keystroke(s) for a command, press Enter to accept it.
If you're finished making changes, press Esc to exit. If you still have more changes
to make, use the arrow keys to scroll up and down the list and select your next
command. At this point, if you've accidentally assigned a keystroke sequence that's
been used as a control character sequence in the primary command column, the
message
Command conflicts need to be corrected. Press Esc

will flash across the screen. Any duplicated sequences will be highlighted, so you
can easily search for any disallowed items and reselect a sequence. If you change
your mind, you can use the R option to restore the factory default key definitions.

The Default Edit Mode Option
Press 0 to bring up the Default edit modes menu. There are three editor modes
that can be installed: Insert mode, Autoindent mode, and Tabs.
Use this menu to set the editor's modes to the settings you prefer. You'll still be
able to toggle them ON/OFF from inside the editor; this menu is used to determine whether they start ON or OFF when you first load Eureka.

Customizing Eureka

231

With Insert mode on, anything you enter at the keyboard is inserted at the
cursor position, pushing any text to the right of the cursor further right. Toggling
Insert mode off allows you to overwrite text at the cursor.
With Autoindent mode on, the cursor returns to the starting column of the previous line. When toggled off, the cursor always returns to column one.
Toggle on Tab mode when you want to insert tabs; toggle off and the tab is
automatically set to the beginning of the first word in the previous line.
When you load Eureka, the default values for all three modes are on. You can
change the defaults to suit your preferences and save them back to Eureka. Of
course, you'll still be able to toggle these modes from inside Eureka's editor.
Either use the arrow keys to move the selection bar to the option and then press

Enter or else press the key that corresponds to the highlighted capital letter of the
option.

The Screen Mode Option
Press S to select Screen mode from the installation menu. A pull-down menu will
appear from which you can select the screen mode Eureka will use during operation. Your options include
• Default
• Color
• Black and white
• Monochrome

Default Display Mode
By default, Eureka will always operate in the mode that is active when you load it.

Color Display Mode
Eureka will use color mode with 80 X 25 characters no matter what mode is
active, switching back to the active mode when you exit.

Black and White Display Mode
Eureka will use black and white mode with 80 x 25 characters no matter what
mode is active, switching back to the active mode when you exit.

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Morwchrome Display Mode
Eureka will use monochrome mode no matter what mode is active, switching back
to the active mode when you exit.

Quitting the Program
Once you have finished making all desired changes, select Quit (or press Esc) at the
main menu. The message "Save changes to EUREKA.EXE?" will appear at the
bottom of the screen. If you press Y(for Yes), all of the changes you have made will
be permanently installed into Eureka. (Of course, you can always run this program
again if you want to change them.) If you press N (for No), your changes will be
ignored and you will be returned to the DOS prompt.

If you decide you want to restore the original Eureka factory defaults, simply
copy EUREKA.EXE from your master disk onto your work disk.

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Glossary

absolute value: The value of a positive or negative number when the sign has been
removed. For instance, the absolute value of both - 2 and + 2 is 2.
argument: A variable or expression representing a value in the definition of a builtin or user-defined function.
comment: In Eureka, statements used to help you identify or document various
equation file components. Comments are set off by a semicolon (;) or are enclosed
in curly brackets ({like this}); Eureka ignores comments when solving an equation
file.
confidence level: A rating that appears in the Solution window when Eureka solves
a minimization or maximization problem. The confidence level reveals the degree
of certainty Eureka feels about the found solution.
constraint: A condition that must be met for an equation to be satisfactorily solved.
The accuracy and penalty settings can alter the relative strength of a constraint.
default: The value or instruction Eureka uses until you enter a different value. For
example, Eureka automatically solves a variable to eight decimal places; that is the
default value. You can change selected default values that are listed in the main
menu's Options pull-down menu. You can change nearly all the defaults by including a directive in the equation file.
directive: A command to Eureka, embedded within an equation file, that affects
only that file. For example, $ complex = yes means variables are solved for as
complex numbers.

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equation file: The file that contains the problem to be solved. It must contain an
equation or inequality and, where pertinent, initialization values and user-defined
functions. It may also include directives and comments.
error message: A message that appears in the error message window on the
Eureka screen if you type something that is incorrect or inappropriate. Appendix E
lists Eureka's error messages and describes how to correct or avoid errors.
file name extension: An optional 1 to 3 character extension to a file name. As
shown in these examples, a period separates the file name and the extension:
FINANCE.TOM, FORMULA.l, and FORMULA.165. Extensions are convenient
for identifying different file versions, or for identifying file ownership.
floating-point error: A warning message that may appear when Eureka has solved
for an extremely large number or when a number is divided by zero. It does not
necessarily mean the solution is erroneous.
header: A standardized format for the information appearing at the beginning of a
Eureka report. The Eureka report headers include the date, time, and file name.
help message: A message that appears in the Help window when you press Fl.
Help messages explain the current area of the program, whether it is a highlighted
menu item, window, or other Eureka feature.
identifier: Name for a variable.
initialization: The process of setting initial values for variables or the starting point
of an iterative search. Initializations are flagged with the symbol : = .
list: A table of values for a function, generated with Eureka's Graph/List command.
mask: A way to search selectively for a file. In Eureka, wildcard characters (both
the asterisk (*) and question mark (?)) are used in combination with fragments of
file names as masks. The asterisk finds all character strings starting from the position of the wildcard character; the ? finds only single characters.
math co-processor: A peripheral chip that is used in PCs to speed up mathematical operations.
path name: A listing of the complete DOS path to a particular file, starting with
the root directory and including all directories leading up to the file. For example,
the path name \TOM\FINANCE\PROB.21 refers to a file called PROB.21 that is
located in the finance directory. The finance directory is, in turn, a subdirectory of
TOM, which is a subdirectory of the root directory (\).
Progress window: A window that opens when you select Solve. It displays the
amount of time the program is taking to find the solution, the current amount of
error, and the current solution. When Eureka finds a solution, the Progress window
disappears and the Solution window opens.

236

Eureka: The Solver Owner's Handbook

RAM disk: A software program that causes a definable portion of RAM memory to
be treated as an additional disk drive (usually D:).
report file: A Eureka file that contains a report about an equation file. Such a file
can be edited using the Eureka editor, but cannot be solved.
Solution window: A window that displays the values Eureka has found for the
indicated variables, as well as a rating expressing the degree of confidence Eureka
has in the solution(s).
status line: The line at the bottom of the program screen that provides information
about the special function keys that are currently available.
steepest descent method: A method of minimizing functions that starts at an initial
point and proceeds along a path that is always in the direction of the most rapid
decrease for the function.
user-defined function: A one-variable function defined by the user in an equation
file or with the Graph Function command. Such a function is essential for a plot to
be generated.
verify window: Displays the results of evaluating both sides of the equations in a
file separately, and then comparing the results. Eureka must solve an equation file
before it can evaluate it.

Glossary

237

238

Eureka: The Solver Owner's Handbook

Software
For the dealer nearest you
or to order by phone

Call (800) 543-7543

4585 Scotts Valley Drive
Scotts Valley, CA 95066

.II'.~~1I'fll

IJ'~

r;n...~

rlE
'E'IT"
: ••,AIIIE.
®

Whether you're running WordStar,® Lotus,® dBASE,®
or any other program, SideKick puts al/ these desktop
accessories at your fingertips-Instantly!
A lull-screen WordStar-like Editor to jot
down notes and edit files up to 25 pages
long.
A Phone Directory for names, addresses,
and telephone numbers. Finding a name or a
number is a snap.
An Autodialer for all your phone calls. It will
look up and dial telephone numbers for you.
(A modem is required to use this function.)

All the SideKick windows stacked up over Lotus 1-2-3.e
From bottom to top: SideKick's "Menu Window," ASCII
Table, Notepad, Calculator, Appointment Calendar, Monthly
Calendar, and Phone Dialer.

A Monthly Calendar from 1901 through
2099.
Appointment Calendar to remind you
of important meetings and appointments.
A lull-featured Calculator ideal for
business use. It also performs decimal
to hexadecimal to binary conversions.
An ASCII Table for easy reference.

Here's SideKick running over Lotus 1-2-3. In the
SideKick Notepad you'll notice data that's been imported
directly from the Lotus screen. In the upper right you can
see the Calculator.

The Critics' Choice
"In a simple, beautiful implementation of WordStar's
block copy commands, SideKick can transport all
or any part of the display screen (even an area
overlaid by the notepad display) to the notepad."
-Charles Petzold, PC MAGAZINE
"SideKick deserves a place in every PC."
-Gary Ray, PC WEEK

"SideKick is by far the best we've seen. It is also
the least expensive."
-Ron Mansfield, ENTREPRENEUR
"If you use a PC, get SideKick. You'll soon become
dependent on it."
-Jerry Pournelle, BYTE

Suggested Retail Price: $84.95 (not copy protected)
Minimum system configuration: IBM PC, XT, AT, PCjr and true compatibles. PC-DOS (MS-DOS) 2.0 or greater. 128K RAM. One disk
drive. A Hayes-compatible modem, IBM PCjr internal modem, or AT&T Modem 4000 is required for the autodialer function.
SideKick is a registered trademark of Borland International, Inc. dBASE is a registered trademark of
Ashton-Tate. IBM, XT, AT, and PCjr are registered trademarks of International Business Machines Corp.
AT&T is a registered trademark of American Telephone & Telegraph Company Lotus and 1-2-3 are
registered trademarks of Lotus Development Corp. WordStar is a registered trademark of MicroPro
International Corp. Hayes is a trademark of Hayes Microcomputer Products, Inc.
Copyright 1987 Borland International
BOR0060C

The Organizer For The Computer Age!
Traveling SideKick is BinderWare,@ both a binder you take with you when you travel
and a software program-which includes a Report Generator-that generates and
prints out all the information you'll need to take with you,
Information like your phone list, your client list,
your address book, your calendar, and your
appointments. The appointment or calendar files
you're already using in your SideKickllJ can automatically be used by your Traveling SideKick. You
don't waste time and effort reentering information
that's already there.
One keystroke prints out a form like your address
book. No need to change printer paper;

you simply punch three holes, fold and clip
the form into your Traveling SideKick binder, and
you're on your way. Because Traveling SideKick is
CAD (Computer-Age Designed), you don't fool
around with low-tech tools like scissors, tape, or
staples. And because Traveling SideKick is
electronic, it works this year, next year, and all the
"next years" after that. Old-fashioned daytime
organizers are history in 365 days.

What's inside Traveling SideKick

What the software program and its
Report Generator do for you before
you go-and when you get back
Before you go:
• Prints out your calendar,
appointments, addresses, phone
directory, and whatever other
information you need from your
data files
When you return:
• Lets you quickly and easily enter all
the new names you obtained while
you were away into your
SideKick data files

TRAVELING SIDEKICK SOfTWARE
GENERATES. UPDATES. AND PRINTS YOUR
ADDRESS AND CALENDAR FILES

It can also:
• Sort your address book by contact,
zip code or company name
• Print mailing labels
• Print information selectively
• Search files for existing addresses
or calendar engagements

Suggested Retail Price: $69,95 (not copy protected)
Minimum system configuration: IBM PC, XT, AT, Portable, PCjr, 3270 and true compatibles. PC-DOS (MS-DOS) 2.0 or later.
256K RAM mimimum.

SideKick, BinderWare and Traveling SideKick are registered trademarks of Borland
International, Inc. IBM, AT, XT, and PCjr are registered trademarks of International Business
Machines Corp. MS-DOS is a registered trademark of Microsoft Corp. Copyright 1987
Borland International
BOR 0083A

RAM-resident
Increased productivity lor IBM~PCs or compatibles
SuperKey's simple macros are electronic shortcuts to sueeess.
By lelting you reduce a lengthy paragraph into a single keystroke
of your choice, SuperKey eliminates repetition.
SuperKey turns 1,000 keystrokes into 11
SuperKey can record lengthy keystroke sequences and play them back at the touch of a single key.
Instantly. Like magic.
In fact, with Super Key's simple macros, you can turn "Dear Customer: Thank you for your inquiry.
We are pleased to let you know that shipment will be made within 24 hours. Sincerely," into the
one keystroke of your choice!
SuperKey keeps your confidential files-confidential!
Without encryption, your files are open secrets. Anyone can walk up to your PC and read your
confidential files (tax returns, business plans, customer lists, personal letters, etc.).
With SuperKey you can encrypt any file, even while running another program. As long as you keep
the password secret, only you can decode your file correctly. SuperKey also implements the U.S.
government Data Encryption Standard (DES).
~
~

~
~
~
~

~
~

RAM resident-accepts new macro files
even while running other programs
Pull-down menus
Superfast file encryption
Choice of two encryption schemes
On-line context-sensitive help
One-finger mode reduces key
commands to single keystroke
Screen OFF/ON blanks out and restores
screen to protect against "burn in"
Partial or complete reorganization of
keyboard

Keyboard buffer increases 16 character
keyboard "type-ahead" buffer to 128
characters
~ Real-time delay causes macro playback
to pause for specified interval
~ Transparent display macros allow
creation of menus on top of application
programs
~ Data entry and format control using
"fixed" or "variable" fields
~ Command stack recalls last 256
characters entered

Suggested Retail Price: $99.95 (not copy protected)
Minimum system conllguratlon: iBM PC, XT, AT, PCjr, and true compatibles. PC-DOS (MS-DOS)
2.0 or greater. 128K RAM. One disk drive.
SuperKey is a registered trademark 01 Borland International, Inc. IBM, Xl, AT, and PCjr are
registered trademarks 01 International Business Machines Corp. MS-DOS is a registered
trademark 01 Microsoft Corp.
BOA 0062C

If you use an IBM® PC, you need

Lightning@
Turbo Lightning teams up
with the Random House
Concise Word List to
check your spelling as
you type!
Turbo Lightning, using the
BO,OOO-word Random House
Dictionary, checks your spelling
as you type. If you misspell a
word, it alerts you with a
"beep." At the touch of a key,
Turbo Lightning opens a
window on top of your
application program and
suggests the correct spelling.
Just press one key and the
misspelled word is instantly
replaced with the correct word.
Turbo Lightning works
hand-in-hand with the
Random House Thesaurus
to give you instant access
to synonyms
Turbo Lightning lets you
choose just the right word from
a list of alternates, so you
don't say the same thing the
same way every time. Once
Turbo Lightning opens the
Thesaurus window, you see a
list of alternate words; select
the word you want, press
ENTER and your new word will
instantly replace the original
word. Pure magic!

If you ever write

word, think a. word, or
say a word, you need
Turbo Lightning

You can teach Turbo
Lightning new words
You can teach your new Turbo
Lightning your name, business
associates' names, street
names, addresses, correct
capitalizations, and any
specialized words you use
frequently. Teach Turbo
Lightning once, and it
knows forever.
Turbo Lightning is the
engine that powers
Borland's Turbo Lightning
Library@

The Turbo Lightning Proofreader

Turbo Lightning brings
electronic power to the
Random House Concise Word
List and Random House
Thesaurus. They're at your
fingertips-even while you're
running other programs. Turbo
Lightning will also "drive"
soon-to-be-released
encyclopedias, extended
thesauruses, specialized
dictionaries, and many other
popular reference works. You
get a head start with this
first volume in the Turbo
Lightning Library.

The Turbo Lightning Thesaurus

Suggested Retail Price: $99.95 (not copy protected)
Minimum system configuration: IBM PC, Xl, AT, PCjr, and true compatibles with 2 floppy disk drives. PC-DOS (MS-DOS) 2.0 or greater.
256K RAM. Hard disk recommended.

BORLAND
INTERNATIONAL

Turbo Lightning and Turbo Lightning Library are registered trademarks of Borland International. Inc.
IBM, XT, AT, and PCjr are registered trademarks of International Business Machines Corp Random
House is a registered trademark of Randorn House, Inc
Copyright 1987 Borland International
BOA 0070B

Your Development Toolbox and Technical Reference Manual for Thrbo Lightning 8
l

Lightning Word Wizard includes complete, commented Turbo
Pascal@ source code and all the technical information you'll
need to understand and work with Turbo Lightning's engine."
More than 20 fully documented Turbo Pascal procedures
reveal powerful Turbo Lightning engine calls. Harness the full power
of the complete and authoritative Random House@ Concise
Word List and Random House Thesaurus.
6I

Turbo Lightning's "Reference
Manual"

The ultimate collection of word
games and crossword solvers!

Developers can use the versatile on-line
examples to harness Turbo Lightning's
power to do rapid word searches. Lightning
Word Wizard is the forerunner of the database access systems that will incorporate
and engineer the Turbo Lightning Library®
of electronic reference works.

The excitement, challenge, competition,
and education of four games and three
solver utilities-puzzles, scrambles, spellsearches, synonym-seekings, hidden words,
crossword solutions, and more. You and
your friends (up to four people total) can
set the difficulty level and contest the highspeed smarts of Lightning Word Wizard!

Turbo Lightning-Critics' Choice
"Lightning's good enough to make programmers and users cheer, executives of other
software companies weep."
Jim Seymour, PC Week
"The real future of Lightning clearly lies not with the spelling checker and thesaurus currently
included, but with other uses of its powerful look-up engine."
Ted Silveira, Profiles
"This newest product from Borland has it all."

Don Roy, Computing Now!

Minimum system configuration: IBM PC, Xl, AT, PCjr, Portable, and true compatibles. 256K RAM minimum. PC-DOS (MS-DOS) 2.0
or greater. Turbo Lightning software required. Optional-Turbo Pascal 3.0 or greater to edit and compile Turbo Pascal source code.

Suggested Retail Price: $69.95
(not copy protected)
Turbo Pascal. Turbo Lightning and Turbo Lightning Library are registered trademarks and Lightning Word Wizard is a trademark of Borland International, Inc. Random
House is a registered trademark of Random House. Inc. IBM, XT, AT. and PCjr are registered trademarks of International BUSiness Machines Corp. MS-DOS is a
registered trademark of Microsoft Corp. Copyright 1987 Borland International
BOR0087B

'££1 £1®1IE IA1A'AIE
~r'~~

: IAIAIE.

The high-performance database manager
that's so advanced it's easy to use!

Lets you organize, analyze and report information faster than ever before! If you manage mailing lists,
customer files, or even your company's budgets-Reflex is the database manager for you!
Reflex is the acclaimed, high-performance database manager you've been waiting for. Reflex extends
database management with business graphics. Because a picture is often worth a 1000 words, Reflex
lets you extract critical information buried in mountains of data. With Reflex, when you look, you see.
The REPORT VIEW allows you to generate everything from mailing labels to sophisticated reports.
You can use database files created with Reflex or transferred from Lotus 1-2-3,e dBASE,e PFS: File,e
and other applications.

Reflex: The Critics' Choice
"... if you use a PC, you should know about Reflex ... may be the best bargain in software today."
Jerry Pournelle, BYTE
"Everyone agrees that Reflex is the best-looking database they've ever seen."
Adam B. Green, InloWorld
"The next generation of software has officially arrived."
Peter Norton, PC Week
Reflex: don't use your PC without it!
Join hundreds of thousands of enthusiastic Reflex users and experience the power and ease of use of
Borland's award~winning Reflex.
Suggested Retail Price: $149.95 (not copy protected)
Minimum system configuration: IBM PC, XT, AT, and true compatibles. 384K RAM minimum. IBM Color Graphics Adapter, Hercules
Monochrome Graphics CArd, or equivalent. PC·DDS (MS·DOS) 2.0 or greater. Hard disk and mouse optional. Lotus 1·2·3, dBASE,
or PFS: File optional.
Reflex is a trademark of Borland/Analytica Inc. Lotus 1-2-3 is a registered trademark of Lotus
Development Corporation. dBASE is a registered trademark of Ashton-Tate. PFS: File is a
registered trademark of Software Publishing Corporation. IBM, XT, AT, and IBM Color Graphics
Adapter are registered trademarks of International Business Machines Corporation. Hercules
Graphics Card is a trademark of Hercules Computer Technology. MS-DOS is a registered
trademark of Microsoft Corp
Copyright 1987 Borland International
BOR 0066e

REFLEX

Includes 22 "instant templates" covering a broad range of
business applications (listed below). Also shows you how to
customize databases, graphs, cross tabs, and reports. It's an invaluable
analytical tool and an important addition to another one of
our best sellers, Reflex: The Analyst 1.1.
Fast-start tutorial examples:
Learn Reflex8 as you work with practical business applications. The Reflex Workshop Disk supplies
databases and reports large enough to illustrate the power and variety of Reflex features. Instructions in each
Reflex Workshop chapter take you through a step-by-step analysis of sample data. You then follow simple
steps to adapt the files to your own needs.

22 practical business applications:
Workshop's 22 "instant templates" give you a wide range of analytical tools:
Administration
• Tracking Manufacturing Quality Assurance
•
•
•
•
•

Scheduling Appointments
Planning Conference Facilities
Managing a Project
Creating a Mailing System
Managing Employment Applications

Sales and Marketing
•
•
•
•

Researching Store Check Inventory
Tracking Sales Leads
Summarizing Sales Trends
Analyzing Trends

Production and Operations
• Summarizing Repair Turnaround

• Analyzing Product Costs

Accounting and Financial Planning
•
•
•
•
•
•
•
•
•
•

Tracking Petty Cash
Entering Purchase Orders
Organizing OutgOing Purchase Orders
Analyzing Accounts Receivable
Maintaining Letters of Credit
Reporting Business Expenses
Managing Debits and Credits
Examining Leased Inventory Trend~
Tracking Fixed Assets
Planning Commercial Real Estate Investment

~rether you're a newcomer learning Reflex basics or an experienced "power user" looking for tips, Reflex
Workshop will help you quickly become an expert database analyst.

Minimum system configuration: IBM PC, AT, and Xl, and true compatibles. PC-DOS (MS-DOS) 2.0 or greater. 384K RAM minimum. Requires Rellex: TIle
Anllyst, Ind IBM Color Graphics Adapter, Hercules Monochrome Graphics Card or equivalent.

Suggested Retail Price: $69.95
(not copy protected)
Reflex is a registered trademark and Reflex Workshop is a trademark of Borland/Analytica, Inc. IBM, AT, and XT are registered trademarks of International BusineSS .
Machines Corp. Hercules is a trademark of Hercules Computer Technology. MS-DOS is a registered trademark of Microsoft Corp.
BOROO88A

Version 3.0 with 8087 support and BCD reais

Free MicroCa/c Spreadsheet With Commented Source Code!
FEATURES:

THE CRITICS' CHOICE:

One-Step Compile: No hunting &fishing
expeditions! Turbo finds the errors, takes you
to them, lets you correct them, and instantly
recompiles. You're off and running in
record time.

"Language deal of the century ... Turbo Pascal:
it introduces a new programming environment
and runs like magic."
-Jeff Duntemann, PC Magazine
"Most Pascal compilers barely fit on a disk, but
Turbo Pascal packs an editor, compiler, linker,
and run-time library into just 39K bytes of
random access memory."
-Dave Garland, Popular Computing

Built-in Interactive Editor: WordStar®-like
easy editing lets you debug quickly.
Automatic Overlays: Fits big programs into
small amounts of memory.
MicloCalc: A sample spreadsheet on your disk
with ready-to-compile source code.
IBM® PC Version: Supports Turtle Graphics,
color, sound, full tree directories, window
routines, input/output redirection, and
much more.

"What I think the computer industry is headed
for: well-documented, standard, plenty of
good features, and a reasonable price."
-Jerry Pournelle, BYTE

LOOK AT TURBO NOW!
~ More than 500,000 users worldwide.
~ Turbo Pascal is the de facto industry
standard.
~ Turbo Pascal wins PC MAGAZINE'S
award for technical excellence.

~ Turbo Pascal named "Most
Significant Product of the Year" by
PC WEEK.

5l'

Turbo Pascal 3.0-the fastest Pascal
development environment on the
planet, period.

Suggested Retail Price: $99.95; CPIM®-80 version without 8087 and BCD: $69.95
Features for 16-bit Systems: 8087 math co-processor support for intensive calculations.
Binary Coded Decimals (BCD): eliminates round-off error! A must for any serious business application.
Minimum system configuration: 128K RAM minimum. Includes 8087 & BCD features for 16-bit MS-DOS 2.0 or later and
CP/M-861.1 or later. CP/M-80 version 2.2 or later 48K RAM minimum (8087 and BCD features not available). 8087
version requires 8087 or 80287 co-processor.

BORLAND
INTERNATIONAL

Turbo Pascal is a registered trademark of Borland International. Inc. CP/M is a registered trademark
of Digital Research Inc. IBM is a registered trademark of International Business Machines Corp.
MS-DOS is a registered trademark of Microroft Corp. WordStar is a registered trademark of
MicroPro International. Copyright 1987 Borland International BOR 0061B

VERSION 2.0

Learn Pascal From The Folks Who Created
The Turbo Pascal® Family
Borland International proudly presents Turbo Tutor, the perfect complement
to your Turbo P~scal compiler. Turbo Tutor is really for everyoneeven if you've never programmed before.
And if you're already proficient, Turbo Tutor can sharpen up the fine points.
The manual and program disk focus on the whole spectrum of Turbo
Pascal programming techniques.
• For the Novice: It gives you a concise history of Pascal, tells you how to write a
simple program, and defines the basic programming terms you need to know.
• Programmer's Guide: The heart of Turbo Pascal. The manual covers the fine paints
of every aspect of Turbo Pascal programming: program structure, data types, control
structures, procedures and functions, scalar types, arrays, strings, pointers, sets, files,
and records.
• Advanced Concepts: If you're an expert, you'll love the sections detailing such topics as
linked lists, trees, and graphs. You'll also find sample program examples for PC-DOS and
MS-OOS.®
10,000 lines of commented source code, demonstrations of 20 Turbo Pascal features, multiplechoice quizzes, an interactive on-line tutor, and more!
Turbo Tutor may be the only reference work about Pascal and programming you'll ever need!
Suggested Retail Price: $39.95 (not copy protected)
Minimum system configuration: Turbo Pascal 3.0. PC·DOS (MS·DOS) 2.0 or later. 192K RAM minimum (CP/M·aO
version 2.2 or later: 64K RAM minimum).

Turbo Pascal and Turbo Tutor are registered trademarks of Borland International Inc. CPIM is a
registered trademark of Digital Research Inc. MS-DOS is a· registered trademark of Microsoft Corp.
Copyright 1987 Borland International
BOA 0064C

Is The Perfect Complement To Turbo Pasca/~
It contains a complete library of Pascal procedures that
allows you to sort and search your data and build powerful database
applications. It's another set of tools from Borland that will give
even the beginning programmer the expert's edge.
THE TOOLS YOU NEED!
TURBO ACCESS Using B+ trees: The best way to organize and search your data. Makes it
possible to access records in a file using key words instead of numbers. Now available with
complete source code on disk, ready to be included in your programs.
TURBO SORT: The fastest way to sort data using the QUICKSORT algorithm-the method
preferred by knowledgeable professionals. Includes source code.
GINST (General Installation Program): Gets your programs up and running on other
terminals. This feature alone will save hours of work and research. Adds tremendous value
to all your programs.

GET STARTED RIGHT AWAY.-FREE DATABASE!
Included on every Toolbox diskette is the source code to a working database which
demonstrates the power and simplicity of our Turbo Access search system. Modify it to suit
your individual needs or just compile it and run.

THE CRITICS' CHOICE!
'The tools include a B+ tree search and a sorting system. I've seen stuff like this, but not as
well thought out, sell for hundreds of dollars."
-Jerry Pournelle, BYTE MAGAZINE
"The Turbo Database Toolbox is solid enough and useful enough to come recommended."
-Jell Duntemann, PC TECH JOURNAL
Suggested Retail Price: $69.95 (not copy protected)

Minimum system configuration: 128K RAM and one disk drive (CP/M-80: 48K). 16-bit systems: Turbo Pascal 2.0 or greater for
MS-DOS or PC-DOS 2.0 or greater. Turbo Pascal 2.1 or greater for CP/M-86 1.0 or greater. 8-bit systems: Turbo Pascal 2.0 or
greater for CP/M-80 2.2 or greater.

Turbo Pascal and Turbo Database Toolbox are registered trademarks of Borland International
Inc. CP1M is a registered trademark of Dignal Research. Inc. MS-DOS is a registered
trademark of Microsoft Corp.
Copyright 1987 Borland International BOR 00630

A Library of Graphics Routines for Use with Turbo Pasca/®
High-resolution graphics for your IBM" PC, AT," XT,'"' PC;,., true PC compatibles, and the Heath
Zenith Z-100." Comes complete with graphics window management.
Even if you're new to Turbo Pascal programming, the Turbo Pascal Graphix Toolbox will get you started
right away. It's a collection of tools that will get you right into the fascinating world of high-resolution
business graphics, including graphics window management. You get immediate, satisfying results. And
we keep Royalty out of American business because you don't pay any-even if you distribute your own
compiled programs that include all or part of the Turbo Pascal Graphix Toolbox procedures.

What you get includes:
• Complete commented source code on disk.
• Tools for drawing simple graphics.
• Tools for drawing complex graphics, including
curves with optional smoothing.
• Routines that let you store and restore graphic
images to and from disk.
• Tools allowing you to send screen images to
Epson@l-compatible printers.

•
•
•
•

Full graphics window management.
Two different font styles for graphic labeling.
Choice of line-drawing styles.
Routines that will let you quickly plot functions
and model experimental data.
• And much, much more ...

"While most people only talk about low-cost personal computer software, Borland has been dOing
something about it. And Borland provides good technical support as part of the price."
John Markov & Paul Freiberger, syndicated columnists.

If you ever plan to create Turbo Pascal programs that make use 01 business graphics or scientific
graphics, you need the Turbo Pascal Graphix Toolbox.

Suggested Retail Price: $69.95 (not copy protected)
Minimum system configuration: IBM PC, Xl, AT, PCjr, true compatibles and the Heath Zenith Z·100. Turbo Pascal 3.0 or later. 192K
RAM minimum. Two disk drives and an IBM Color Graphics Adapter (CGA), IBM Enhanced Graphics Adapter (EGA), Hercules Graphics
Card or compatible.

BORLAND
INTERNATIONAL

Turbo Pascal and Turbo Graphix Toolbox are registered trademarks of Borland International.
Inc. IBM, XT, AT, and PCjr are registered trademarks of International Business Machines
Corporation. Hercules Graphics Card is a trademark of Hercules Computer Technology. Heath
Zenith Z-100 is a trademark of Zenith Data Systems Epson is a registered trademark of
Epson Corp. Copyright 1987 Borland International
BOR 0068C

2VR8O PASCAl.

.EDJ1OR 2txltSOX®
It's All You Need To Build Your Own Text Editor
Or Word Processor
Build your own lightning-fast editor and incorporate it into your Turbo Pascal@ programs.
Turbo Editor Toolbox gives you easy-to-install
modules. Now you can integrate a fast and powerful
editor into your own programs. You get the source
code, the manual, and the know-how.

Create your own word processor. We provide all
the editing routines. You plug in the features you want.
You could build aWordStar@-like editor with pull-down
menus like Microsoft's@ Word, and make it work as fast
as WordPerfect.@

To demonstrate the tremendous power of Turbo Editor Toolbox, we give you the source code for
two sample editors:
Simple Editor
MicroStar

A complete editor ready to include in your programs. With windows, block commands, and
memory-mapped screen routines.
A full-blown text editor with a complete pull-down menu user interface, plus a lot more.
Modify MicroStar's pull-down menu system and include it in your Turbo Pascal programs.

The Turbo Editor Toolbox gives you all the
standard features you would expect to find
in any word processor:
•
•
•
•
•
•
•

Wordwrap
UN-delete last line
Auto-indent
Find and Find/Replace with options
Set left and right margin
Block mark, move, and copy
Tab, insert and overstrike modes,
centering, etc.

MicroStar's pull-down menus.

And Turbo Editor Toolbox has features that word processors selling for several hundred dollars can't begin to match.
Just to name a few:

[!]

RAM-based editor. You can edit very large
files and yet editing is lightning fast.
~ Memory-mapped screen routines. Instant paging, scrolling, and text display.
~ Keyboard installation. Change control
keys from WordStar -like commands to any that
you prefer.

~ Multiple windows. See and edit up to eight
documents-or up to eight parts of the same
document-all at the same time.
~ Multitasking. Automatically save your .
text. Plug in a digital clock, an apPOintment
alarm-see how it's done with MicroStar's
"background" printing.

Best of all, source code is included for everything in the Editor Toolbox.

Suggested Retail Price: $69.95 (not copy protected)
Minimum system configuration: IBM PC, XT, AT, 3270, PCjr, and true compatibles. PC-DOS (MS-DOS) 2.0 or greater. 192K RAM.
You must be using Turbo Pascal 3.0 for IBM and compatibles.
Turbo Pascal and Turbo Editor Toolbox are registered trademarks of Borland Inlernational, Inc.
WordStar is a registered trademark of MicroPro International Corp. Word and MS-DOS are
registered trademarks of Microsoft Corp. WordPerfect is a trademark of Satellite Software
International. IBM, XT, AT, and PCjr are registered trademarks of International BUSiness Machines
Cor~

OOROO~B

Secrets And Strategies 01 The Masters Are
Revealed For The First Time
Explore the world of state-of-the-art computer games with Turbo GameWorks. Using
easy-to-understand examples, Turbo GameWorks teaches you techniques to quickly create
your own computer games using Turbo Pascal~ Or, for instant excitement, play the three
great computer games we've included on disk-compiled and ready to run.

TURBO CHESS
Test your chess-playing skills against your computer challenger. With Turbo GameWorks, you're on your way to
becoming a master chess player. Explore the complete Turbo Pascal source code and discover the secrets of
Turbo Chess.
"What impressed me the most was the fact that with this program you can become a computer chess analyst.
You can add new variations to the program at any time and make the program play stronger and stronger chess.
There's no limit to the fun and enjoyment of playing Turbo GameWorks Chess, and most important of all, with this
chess program there's no limit to how it can help you improve your game."
-George Koitanowski, Dean of American Chess, former President of
the United Chess Federation, and syndicated chess columnist.

TURBO BRIDGE
Now play the world's most popular card game-bridge. Play one-on-one with your computer or against up to
three other opponents. With Turbo Pascal source code, you can even program your own bidding or scoring
conventions.
''There has never been a bridge program written which plays at the expert level, and the ambitious user will
enjoy tackling that challenge, with the format already structured in the program: And for the inexperienced player,
the bridge program provides an easy-to-follow format that allows the user to start right out playing. The user can
'play bridge' against real competition without having to gather three other people."
-Kit Woolsey, writer of several articles and books on bridge,
and twice champion of the Blue Ribbon Pairs.

TURBO GO-MOKU
Prepare for battle when you challenge your computer to a game of Go-Moku-the exciting strategy game also
known as Pente.e In this battle of wits, you and the computer take turns placing X's and D's on a grid of 19X19
squares until five pieces are lined up in a row. Vary the game if you like, using the source code available on your
disk.

Suggested Retail Price: $69.95 (not copy protected)
Minimum system configuration: IBM PC, XT, AT, Portable, 3270, PCjr, and true compatibles. PC·DOS (MS·DOS) 2.0 or later. 192K
RAM minimum. To edit and compile the Turbo Pascal source code, you must be using Turbo Pascal 3.0 for IBM PCs and
compatibles.

TURBO PASCAL

IUlfEIICAllfE1111B 1111BI,TN
New .from Borland's Scientific &Engineering Division!

A complete collection of Turbo Pascaf routines and programs
New from Borland's Scientific & Engineering Division, Turbo Pascal Numerical Methods Toolbox
implements the latest high-level mathematical methods to solve common scientific and engineering
problems. Fast.
So every time you need to calculate an integral, work with Fourier Transforms or incorporate any of the
classical numerical analysis tools into your programs, you don't have to reinvent the wheel. Because
the Numerical Methods Toolbox is a complete collection of Turbo Pascal routines and programs that
gives you applied state-of-the-art math tools. It also includes two graphics demo programs, Least
Squares Fit and Fast Fourier Transforms, to give you the picture along with the numbers.
The Numerical Methods Toolbox is a must for you if you're involved with any type of scientific or
engineering computing. Because it comes with complete source code, you have total control of
your application.

What Numerical Methods Toolbox will do for you now:
• Find solutions to equations
• Interpolations
• Calculus: numerical derivatives and
integrals
• Fourier transforms

• Matrix operations: inversions, determinants
and eigenvalues
• Differential equations
• Least squares approximations

5 free ways to look at ilL east Squares Fit"!
As well as a free demo "Fast Fourier Transforms," you also get "Least Squares Fit" in 5
different forms-which gives you 5 different methods of fitting curves to a collection of data pOints.
You instantly get the picture! The 5 different forms are:
1. Power
4. 5-term Fourier
2. Exponential
5. 5-term
3. Logarithm
Polynomial
They're all ready to compile and run "as is." To modify or add graphics to your own programs,
you simply add Turbo Graphix Toolbox@) to your software library. Our Numerical Methods Toolbox is
designed to work hand-in-hand with our Turbo Graphix Toolbox to make professional graphics in
your own programs an instant part of the picture!

Suggested Retail Price: $99.95 (not copy protected)
Minimum system configuration: IBM PC. XT, AT and true compatibles. PC-DOS (MS-DOS) 2.0 or later. 256K. Turbo Pascal 2.0 or later.
The graphics modules require a graphics monitor with an IBM CGA, IBM EGA, or Hercules compatible adapter card, and require the Turbo
Graphix Toolbox. MS-DOS generic version will not support Turbo Graphix Toolbox routines. An 8087 or 80287 numeric co-processor is
not required, but recommended for optimal performance.
Turbo Pascal Numerical Methods Toolbox is a trademark and Turbo Pascal and Turbo Graphix
Toolbox are registered trademarks of Borland International, Inc. IBM, XT, and AT are
registered trademarks of International Business Machines Corp MS-DOS is a registered
trademark of Microsoft Corp. Hercules is a trademark of Hercules Computer Technology.
Apple is a registered trademark of Apple Computer, Inc. Macintosh is a trademark of Mcintosh
Laboratory, Inc. licensed to Apple ComputeLCopyright 1987 Borland International BOR 0219A

TURBO

the natural language of ArtirlCiallntelligence
, Turbo· Prolog brings fifth-generation supercomputer
power to your IBM®PC!

Turbo Prolog takes
Turbo Prolog provides
programming into a new,
a fully integrated pronatural, and logical
gramming environment
environment
like Borland's Turbo
With Turbo Prolog,
Pascal,® the de facto
because of its natural,
worldwide standard.
logical approach, both
You get the
people new to programming
complete Turbo
and professional programmers
Prolog programming
can build powerful applicasystem
tions such as expert systems,
You get the 200-page
customized knowledge
manual you're holding,
bases, natural language
software that includes
interfaces, and smart
the lightning-fast Turbo
information management systems.
Prolog six-pass
Turbo Prolog is a declarative language which
compiler and interactive editor, and the
uses deductive reasoning to solve
\ free GeoBase natural query language
programming problems.
database, which includes commented
source code on disk, ready to compile.
(GeoBase is a complete database de-stgned
Turbo Prolog's development system
and developed around U.S. geography.
includes:
You can modify it or use it "as is.")

o
o
o

A complete Prolog compiler that is a variation
of the Clocksin and Mellish Edinburgh
standard Prolog.
A full-screen interactive editor.
Support for both graphic and text windows.
All the tools that let you build your own
expert systems and AI applications with
unprecedented ease.

BORLAND
INTERNATIONAL

Minimum system configuration: IBM PC, Xl, AT, Portable, 3270, PCjr
and true compatibles. PC-DOS (MS-DOS) 2.0 or later. 384K RAM
minimum.

Suggested Retail Price: $99.95
(not copy protected)
Turbo Prolog is a trademark and Turbo Pascal is a registered trademark of Borland Intemational, Inc.
IBM, AT, XT, and PCjr are registered trademarks of International Business Machines Corp. MS-DOS is a
registered trademark of Microsoft Corp. Copyright 1987 Borland International
BOR oo16D

Enhances Turbo Prolog with more than 80 tools
and over 8,000 lines of source code
Turbo Prolog, the natural language of Artificial Intelligence, is the
most popular AI package in the world with more than 100,000 users.
Our new Turbo Prolog Toolbox extends its possibilities.
The Turbo Prolog Toolbox enhances Turbo Prolog-our 5th-generation computer programming
language that brings supercomputer power to your IBM PC and compatibles-with its more than 80
tools and over 8,000 lines of source code that can be incorporated into your programs, quite easily.

g
g
g
g

g
g

g
g
g

Turbo Prolog Toolbox features include:
Business graphics generation: boxes, circles, ellipses, bar charts, pie charts, scaled graphics
Complete communications package: supports XModem protocol
File transfers from Reflex,8 dBASE 111,8 Lotus 1-2-3,~ Symphony~
A unique parser generator: construct your own compiler or query language
Sophisticated user -interface design tools
40 example programs
Easy-to-use screen editor: design your screen layout and 1/0
Calculated fields definition
Over 8,000 lines of source code you can incorporate into your own programs

Suggested Retail Price: $99.95 (not copy protected)

Minimum system configuration: IBM PC, XT, AT or true compatibles. PC-DOS (MS-DOS) 2.0 or later. Requires Turbo Prolog 1.10
or higher. Dual-floppy disk drive or hard disk. 512K.
Turbo Prolog Toolbox and Turbo Prolog are trademarks of Borland International, Inc. Reflex
is a registered trademark of Borland/Analytica, Inc. dBASE III is a registered trademark of
Ashton-Tate. Lotus 1-2-3 and Symphony are registered trademarks of Lotus Development
Corp. IBM, XT, and AT are registered trademarks 01 International Business Machines Corp.
MS-DOS is a registered trademark of Microsoft Corp.
BOA 0240

lUllI IABIC®
The high-speed BASIC you've been waiting for!
You probably know us for our Turbo Pascal® and Turbo Prolog.® Well, we've done
it again! We've created Turbo Basic, because BASIC doesn't have to be slow.
If BASIC

taught you how to walk, Turbo Basic will teach you how to run!
With Turbo Basic, your only speed is "Full Speed Ahead"! Turbo Basic is a complete development environment with an amazingly fast compiler, an interactive editor and a trace debugging system. And because
Turbo Basic is also compatible with BASICA, chances are that you already know how to use Turbo Basic.

Turbo Basic ends the basic confusion
There's now one standard: Turbo Basic. And because Turbo Basic is a Borland product, the price is right,
the quality is there, and the power is at your fingertips. Turbo Basic is part of the fast-growing Borland
family of programming languages we call the "Turbo Family." And hundreds of thousands of users are
already using Borland's languages. So, welcome to a whole new generation of smart PC users!
Free spreadsheet included with source code!
Yes, we've included MicroCalc,'" our sample spreadsheet, complete with source code. So you can get
started right away with a "real program." You can compile and run it "as is," or modify it.

B
B
B
B
B
B

A technical look at Turbo Basic
Full recursion supported
executable program, with separate windows
for editing, messages, tracing, and execution
Standard IEEE floating-point format
Floating-point support, with full 8087 copro- B Compile and run-time errors place you in
source code where error occurred
cessor integration. Software emulation if no
B Access to local, static and global variables
8087 present
B New long integer (32-bit) data type
Program size limited only by available
memory (no 64K limitation)
B Full 80-bit precision
EGA, CGA, MCGA and VGA support
B PUll-down menus
Full integration of the compiler, editor, and
B Full window management

Suggested Retail Price: $99.95 (not copy protected)
Minimum system configuration: IBM PC, AT, XT, PS/2 or true compatibles. 320K. One floppy drive. PC-DOS (MS-DOS) 2.0 or later.

Turbo Basic, Turbo Prolog and Turbo Pascal are registered trademarks and MicroCalc is a trademark of Borland International, Inc. Other brand and product names are trademarks or registered
trademarks of their respective holders.
Copyright 1987 Borland International
BOR 0265B

.11,rll'''I®
liE IEII1I1'
In'.. :

IJ'~

III.flll Release 2.0

Macintosh

•

The most complete and comprehensive collection 01
desk accessories available lor your Macintosh!
Thousands of users already know that SideKick is the best collection of desk accessories available
for the Macintosh. With our new Release 2.0, the best just got better.
We've just added two powerful high-performance tools to SideKick-OutlookTM: The Outliner
and MacPlan™: The Spreadsheet. They work in perfect harmony with each other and while you
run other programs!
Outlook: The Outliner
• It's the desk accessory with more power than a stand-alone outliner
• A great desktop publishing tool, Outlook lets you incorporate both text and graphics
into your outlines
• Works hand-in-hand with MacPlan
• Allows you to work on several outlines at the same time
•
•
•
•
•

MacPlan: The Spreadsheet
Integrates spreadsheets and graphs
Does both formulas and straight numbers
Graph types include bar charts, stacked bar charts, pie charts and line graphs
Includes 12 example templates free!
Pastes graphics and data right into Outlook creating professional memos and reports, complete
with headers and footers.

1!1
1!1
1!1
1!1
1!1
1!1
1!1
1!1
1!1
1!1

SideKick: The Desktop Organizer,
Release 2.0 now includes
Outlook: The Outliner
MacPlan: The Spreadsheet
Mini word processor
Calendar
Phone Log
Analog clock
Alarm system
Calculator
Report generator
Telecommunications (new version now
supports XModem file transfer protocol)

•

13.67'fi5 5.l1u,",

~ ::::: :~::~venQes I:J!Hl J~H1LI
III

•

em
em

txP«IU'f

Q.ll9i1 Labor

4.66'iii1i Natwia1s

6.219a Ov.rh"ad

I] 11.IB~ Tol~IExpens(l$

El
•

""
leA!!; fletPn:>fil

MacPlan does both spreadsheets and business
graphs. Paste them into your Outlook files and
generate professional reports.

Suggested Retail Price: $99.95 (not copy protected)
Minimum system configurations: Macintosh 512K or Macintosh Plus with one disk drive. One aOOK or two 400K drives are recommended.
With one 400K drive, a limited number of desk accessories will be installable per disk.

BORLAND
INTERNATIONAL

SideKick is a registered trademark and Outlook and MacPlan are trademarks of Borland
International, Inc. Macintosh is a trademark of Mcintosh Laboratory, Inc. licensed to Apple
Computer, Inc. Copyright 1987 Borland International
BOR 0069D

'~~III; 111 IA1AIA"
r;,]
• • AIAIEI
1
The easy-to-use relational database that thinks like a spreadsheet.
Rellex lor the Mac lets you crunch numbers by entering lormulas and link
databases by drawing on-screen lines.

• A client billing application set up for a law
office, but easily customized by any
profeSSional who bills time.
• A parts explosion application that breaks
down an object into its component parts
for cost analysis.

• A sample 1040 tax application with Schedule A, Schedule B, and Schedule 0, each
contained in a separate report document.
• A portfolio analysis application with linked
databases of stock purchases, sales, and
dividend payments.
• A checkbook application.

Reflex for the Mac accomplishes all of these tasks without programming-using
spreadsheet-like formulas. Some other Reflex for the Mac features are:
• Visual database design.
• "What you see is what you get" report and form layout
with pictures.
• Automatic restructuring ·of database files when data
types are changed, or fields are added and deleted.
• Display formats which include General, Decimal,
Scientific, Dollars, Percent.
•

File Edit formet

• Data types which include variable length text, number,
integer, automatically incremented sequence number,
date, time, and logical.
• Up to 255 fields per record.
• Up to 16 files simultaneously open.
• Up to 16 Mac fonts and styles are selectable for
individual fields and labels.

De5tribellDlllll!llWlnclow

~~"~~T ,p.~j--J
.__ .........

a~~~:~-~_~~J~g------.~"__
----.

POtt

-----.-

-'",--

F~~::~:~~,,::··mbi~-'
After opening the "Overview" window. you
draw link lines between databases directly
onto your Macintosh screen.

The link lines you draw establish both
visual and electronic relationships between
your databases.

You can have multiple windows open
simultaneously to view all members of a
linked set-which are interactive and truly
relational.

Critic's Choice
" ... a powerful relational database ... uses a visual approach to information management." InloWorld
"... gives you a lot of freedom in report design; you can even import graphics." A+ Magazine
"... bridges the gap between the pretty programs and the power programs."

Stewart Alsop, PC Letter

Suggested Retail Price: $99.95*
(not copy protected)
Minimum $Y$tem conll,uration: Macirtosh 512K or Macintosh Plus with one disk drive. Second external drive recommended.
Reflex is a registered trademark of BorlandiAnalytica, Inc. Macintosh is a trademark of Mcintosh Laboratory, Inc. and is used with express permiSSion of its owner.
Copyright 1987 Borland International

BOR0149A

The ultimate Pascal development environment
Borland's new Turbo Pascal lor the Mac is so incredibly last that it can
compile 1,420 lines 01 source code in the 7.1 seconds it took you to read this!
And reading the rest of this takes about 5 minutes, which is plenty of time for Turbo Pascal for the Mac
to compile at least 60,000 more lines of source code!
Turbo Pascal for the Mac does both Windows and "Units"
The separate compilation of routines offered by Turbo Pascal for the Mac creates modules called "Units,"
which can be linked to any Turbo Pascal program. This "modular pathway" gives you "pieces" which can
then be integrated into larger programs. You get a more efficient use of memory and a reduction in the
time it takes to develop large programs.
Turbo Pascal for the Mac is so compatible with Lisa~ that they should be living together
Routines from Macintosh Programmer's Workshop Pascal and Inside Macintosh can be compiled and run
with only the subtlest changes. Turbo Pascal for the Mac is also compatible with the Hierarchical File
System of the Macintosh.
The 21-second Guide to Turbo Pascal for the Mac
• Compilation speed of more than 12,000 lines
per minute
• "Unit" structure lets you create programs in
modular form
• Multiple editing windows-up to 8 at once
• Compilation options include compiling to disk or
memory, or compile and run
• No need to switch between programs to compile
or run a program
• Streamlined development and debugging
• Compatibility with Macintosh Programmer's

Workshop Pascal (with minimal changes)
• Compatibility with Hierarchical File System of
your Mac
• Ability to define default volume and folder names
used in compiler directives
• Search and change features in the editor speed up
and simplify alteration of routines
• Ability to use all available Macintosh memory
without limit
• "Units" included to call all the routines provided by
Macintosh Toolbox

Suggested Retail Price: $99.95* (not copy protected)

Minimum system configuration: Macintosh 512K or Macintosh Plus with one disk drive.
Turbo Pascal and SideKick are registered trademarks of Borland Internalional, Inc. and Reflex is a
registered trademark of Borland/Analytica, Inc. Macintosh is a trademark of Mcintosh Laboratories, Inc. licensed
to Apple Computer with its express permission. Lisa is a registered trademark of Apple Computer, Inc. Inside
Macintosh is a copyright of Apple Computer, Inc.
Copyright 1987 Borland International BOA 0167A

',IBB PABCAl®

','111

From the folks who created Turbo Pascal. Borland's new
Turbo Pascal Tutor is everything you need to start programming in Turbo Pascal on the Macintosht It takes
you from the bare basics to advanced programming in a
simple, easy-to-understand fashion.
M

Using the Power of the Macintosh
discusses the revolutionary hardware and software features of this machine. It introduces the
600-plus utility routines in the Apple Toolbox.
Programming the Macintosh in Turbo Pascal
shows you how to create true Macintosh programs that use graphics, pull-down menus, dialog boxes, and so on. Finally, MacTypist, a complete stand-alone application featuring animated
graphics, builds on Turbo Typist and demonstrates what you can do with all the knowledge
you've just acquired.
The disk contains the source code for all the
sample programs, including Turbo Typist, MacTypist, and Turbo Tutor. The Tutor's split screen lets
you run a procedure and view its source code
simultaneously. After running it, you can take a
test on the procedure. If you're stuck for an
answer, a Hint option steers you in the right
direction.

No gimmicks. It's all here.
The manual, the Tutor application, and 30 sample
programs provide a step-by-step tutorial in three
phases: programming in Pascal, programming on
the Macintosh, and programming in Turbo Pascal
on the Macintosh. Here's how the manual is set
up:
Turbo Pascal for the Absolute Novice
delivers the basics-a concise histor~ of Pascal,
key terminology, your first program.
.
A Programmer's Guide to Turbo Pascal
covers Pascal specifics-program structure,
procedures and functions, arrays, strings, and so
on. We've also included Turbo Typist, a textbook
sample program.
Advanced Programming
takes you a step higher into stacks, queues,
binary trees, linked structures, writing large programs, and more.

Macintosh topics included are
5f
5f
5f
g

memory management
resources and resource files
QuickDraw
events
g windows
5f controls

menus

dialogs

debugging

g desk accessory support
g File Manager

Suggested Retail Price: $69.95
Minimum sYltem requirements: Any Macintosh with at least 512K 01 RAM. Requires Turbo PascaL

Turbo Pascal and Turbo TUlor are regislered Irademarks 01 Borland Inlernalional, Inc Other brand and prOduct names
are trademarks or registered trademarks ot their respective hotders Copyright 1987 Borland tnlernational BOR 0381

I"cludes tree
MicroCalc spreadsheet
witll source code

A complete interactive development environment
With Turbo C, you can expect what only Borland delivers:
Quality, Speed, Power and Price. And with its compilation
speed 01 more than 1000 lines a minute, Turbo C makes
everything else look like an exercise in slow motion.
Turbo C: The C compiler for both amateurs and professionals
If you're just beginning and you've "kinda wanted to learn C," now's your chance to do it the easy way.
Turbo C's got everything to get you going. If you're already programming in C, switching to Turbo C will
considerably increase your productivity and help make your programs both smaller and faster.
Turbo C: a complete interactive development environment
Like Turbo Pascale and Turbo Prolog," Turbo C comes with an interactive editor that will show
you syntax errors right in your source code. Developing, debugging, and running a Turbo C
program is a snap!
Technical Specifications
~ Development Environment: A powerful "Make" is
included so that managing Turbo C program
development is easy. Borland's fast liTurbo
Linker" is also included. Also includes pull-down
menus and windows. Can run from the environment or generate an executable file.
[iV' Links with relocatable object modules created
using Borland's Turbo Prolog into a
,-,/ single program..
~ ANSI C compatible.
.
~ Start-up routine ~ource ~ode Included..
~ Both. co~mand line and Integrated environment
versions Included.

[iV' Compiler: One-pass compiler generating native inline code, linkable object modules and assembler.
The object module format is compatible with the
PC-DOS linker. Supports small, medium, compact,
large, and huge memory model libraries. Can mix
models with near and far pOinters. Includes
floating point emulator (utilizes 8087/80287 if
installed).
[iV' Interactive Editor: The system includes a powerful,
interactive full-screen text editor. If the compiler
detects an error, the editor automatically positions
the cursor appropriately in the source code.

"Sieve" benchmark (25 iterations)

Turbo C

Microsoft'" C

Lattice C

Compile time

3.89

16.37

13.90

Compile and link time

9.94

29.06

27.79

Execution time

9.51

13.79

Object code size

5.77
274

297

301

Price

$99.95

$450.00

$500.00

Benchmark run on a 6 Mhz IBM AT using Turbo eversion 1.0 and the Turbo Linker version 1.0; Microsoft eversion 4.0 and the
MS overlay linker version 3.51; Lattice eversion 3.1 and the MS object linker version 3.05.

Suggested Retail Price: $99.95* (not copy protected)
Minimum system configuration: IBM PC, XT, AT and true compatibles. PC-DOS (MS-DOS) 2.0 or later. One floppy drive. 320K.
Turbo C and Turbo Pascal are registered trademarks and Turbo Prolog is a trademark of Borland
International, Inc. Microsoft C and MS-DOS ere registered trademarks of Microsoft Corp. Lattice C
is a registered trademark of Lattice. Inc. IBM. Xl, and AT are registered trademarks of International
Business Machines Corp.
BOA 0243

For the dealer nearest you
or to order by phone
Call (800) 543-7543

INTERNATIONAL

4585 Scotts Valley Drive
Scotts Valley, CA 95066

Index

Accuracy setting, 31-32, 44, 88
ASCII files, importing to Eureka, 43
AUTOEXEC.BAT, 15, 213

Edit command, 56
Editor, 2, 9, 21-23, 28, 35-37, 56, 191-196
customizing, 229-233
key summary, 23, 37, 192-196
EINST.COM, 229-233
.EKA files, 105-186, see also Problem files
Equation file,
capturing, 65
creating, 21, 24
directories for, 53-54
loading, 51, 52
renaming, 54
saving, 23, 34, 52-53
syntax, 9-10, 79-87, 96
comments, 25, 82, 85
directives, 31, 82, 86
functions, 102
initialization, 10, 40-41, 44, 71, 81
order of evaluation, 83
reserved words, 10
symbols, 9, 10, 25, 80, 83-85, 96
units conversion, 84, 97
values, 83
variables, 9, 82
Equations,
accuracy of, 31-32, 43-46, 88
confidence level, Eureka's, 11
constraints on, 31, 41, 44, 93
directives in, 31, 72, 82, 86-98
evaluating, 12, 33, 59
functions in, 27-29, 41, 45, 58, 66-69,
81
graphing, 27
integral (example), 107-109
modeling tips for, 44-46
nonlinear, 127-129, 130-133
report of, 12
roots, finding other, 29, 32, 40, 42, 44,
61
roots, polynomial, 58, 148-151
simultaneous, 127-129, 130-133,
158-160
solving, 11, 26, 31, 40-42, 57, 93-95
batch problems, 42
iterative search, 26, 31, 41, 61
Progress window and, 26
range searched, 94
selected variables, 95
time spent, 93

B
BASIC problem file, 43, 139
Batch file problems, 42
Branch cut, 89
C
Calculator command, 2, 60
Capture command, 65
CARDS.EKA, 165-168
CARLOAN.EKA,123
Change dir command, 54
CHEMBAL.EKA, 172-177
CHEMEQ.EKA,177-186
Chemistry problems, 172-186
CLASSICS.EKA, 108
Close command, 77
Colors, changing, 74, 75
Colors command, 74
Commands menu, 59-62
Comments, 25, 82, 85
COMPLEX.EKA,109
Complex numbers, 89
examples, 107, 109, 148-151
Confidence level, Eureka's, 11
Constants, 45, 71
complex, 89
example, 137-140
Constraints on solving, 31-32, 41, 44, 93
D
DECAY.EKA, 134-136
Decimal places displayed, 31, 89
DERINT.EKA, 108
Deriv function (example), 107, 108
Derivatives, 107, 108
Digits directive, 31, 89
Directives, 31, 72, 82, 86-98, 230
Directories, changing, 53, 54, 74, 211-213
Directories command, 74
Directory command, 53
DOS, 209-215
DOS shell, 55

Eureka: The Solver Owner's Handbook

transcendental (examples), 108,
134-136, 141-l43
trigonometric (example), 108
Verify command and, 12, 33, 59
EQUITY.EKA,161-164
Error messages, 46, 217-227
EUREKA.LOG, 65
EUREKA.MAC, 86
Evaluating equations, see Verify command

F
Fact function (example), 165-168
Factorial, see Fact function
File menu, 51-55
Files on Eureka disk, 4
Financial functions, 90, 99-101
examples, 123-126, 161-164
Finansmooth directive (example), 162
Find other command, 32, 40, 42, 61
Floating point error, 46
Formatted command, 65
Function command, 69
Functions
built-in, 99-10l, 197-208
complex, 89
definite integrals, 107-109, 169-171
financial, 90, 99-10l
examples, 123-l26, 161-164
lists, 68, 92
maximizing/minimizing, 41, 92-93
examples, 107, 144-147, 152-157,
158-160
nonlinear, 152-157
plotting, 27-29, 66-69
polynomial, 58, 148-151
poorly scaled, 45
transcendental, 134-136, 141-143
trigonometric, 89
user-defined,2~ 81, 101-103,107-108
variables and, 101, 152-157
Fval function (example), 161-164
G

Go command, 64
Goto command, 78
Graphs, see Plots
Graph menu, 27-29, 66-69
Graphic card, and plots, 28

Index

Hard copy, see Report
Help, 13
HOUSE.EKA, 152-157
I

Imaginary numbers, 89,107,109,148-151
Include directive (example), 175
Include files, 91
INITIAL.EKA, 75
Initialization values, 81, 91
Installation, 14-16
Floppy disk, 14
Hard disk, 15
RAM disk, 15
INTEG.EKA,169-171
Integ function (examples), 107-109,
169-171
Integrals (examples), 107-109, 169-171
Iterate command, 26, 41
Iterative searches, 31, 41

K
Keys, in Eureka, 8, 18, 20
summary, 20, 23
L

LADDERS.EKA, 127-l29
Least squares fit (example), 137-140
LINECHG.EKA, 130-133
Linear programming (example), 158-160
LINPROG.EKA, 158-160
List command, 68
Load command, 51
Load setup command, 75
Loading J;:ureka, 14-16
Loan problem, 123
LSQRFIT.EKA, 137-140

M
Main menu, 48
MASSCH;G.EKA, 144-147
Math coprocessor chip, 5
Mathematical symbols in Eureka, 4, 9, 25,
83-85,96
Max directive, 41, 92, 107, 108
examples, 152-157, 158-160
Maximization, see
Minimization/maximization problems

MAXMIN.EKA,108
Menus, 8, 17, 19, 47-78
selecting from, 20
summary, 19, 49
Min directive, 41, 93, 107, 108
example, 144-147
Minimization/maximization problems, 11,
26, 40-42, 92-93,
examples, 107, 108, 144-147, 152-157,
158-160
Modeling tips, 44-46

INTEG.EKA,169-171
LADDERS.EKA, 127-129
LINECHG.EKA,130-133
LINPROG.EKA, 158-160
LSQRFIT.EKA, 137-140
MASSCHG.EKA, 144-147
MAXMIN.EKA,108-121
POLYNOM.EKA, 148-151
PROJCTL.EKA, 141-143
Progress window, 26
PROJCTL.EKA, 141-143
Pval function (example), 161-164

N
NOP.COM,5
New command, 52
Next command, 77
Nonlinear equations, simultaneous
(examples), 127-129, 130-133

Q
Quick reference, 187-190
Quit command, 55
Quitting Eureka, 17, 34, 55

Open command, 76
Operators, see Mathematical symbols
Options menu, 31, 70-75, 86
Order of evaluation, 83
OS Shell command, 55
Output command, 64, 67

RAM disk, Eureka with, 15
Rename command, 54
Report, 12,30, 63-65
Report menu, 63-65
Reserved words, 10
Roots, alternate, 29-32, 42, 44
positive/negative, 94

Paymt function (example), 161-164
Penalty setting, 31, 93
Plot command, 66, 94
Plots, 27-29, 66-69
graphics mode, 28
zooming, 28
Poly function (example), 148-151
POLYNOM.EKA, 148-151
Polynomials, roots to, 58, 148-151
Printing, 12, 30, 63-65, 67-68
plots, 67
reports, 63
tables, 68
Problem files (.EKA) 105-186
CARDS.EKA, 165-168
CARLOAN.EKA, 123-126
CHEMBAL.EKA, 172-177
CHEMEQ.EKA,172-186
CLASSICS.EKA, 108-114
COMPLEX.EKA, 108-122
OECAY.EKA,134-136
OERINT.EKA,108-117
EQUITY.EKA, 161-164
HOUSKEKA, 152-157

Save command, 52
Scientific notation, 96-97
Screen mode, installing, 232
Settings command, 31, 47, 95
Setup, saving, 75
Solve command, 11, 26, 57
Solving equations, see Equations, solving
Solving, Eureka's method, 40
Stack command, 78
Substitutions, symbolic, 45, 96
Substlevel setting, 45
example, 137-140
SuperKey macros, 86
Symbols, see Mathematical symbols

T
Tile command, 77
Transcendental, equations (examples), 108,
134-136, 141-143
Trigonometric functions, 89
example, 108
Turbo Basic problem file, 43, 139
Tutorials, 13-34

Eureka: The Solver Owner's Handbook

u
Units conversion, 84, 97
example, 141-143
User-defined functions, 27, 81, 101-103,
107-108
V

Variables,
complex, 89, 107, 109
functions and, 101, 152-157
initializing, 10, 40, 41, 44, 71, 81, 91
~aximizing/minimizing, 92-93,
152-157
modifying, 41
names, 82, 88
polynomial, 58
poorly scaled, 45, 184
solving for selected, 95
Variables command, 32, 41, 70
Verify command, l2, 33, 59
W

Window menu, 33-34, 76-78
Windows
closing, 34, 77
expanding, 37, 77
moving, 34
opening, 34, 76
saving setup of, 75
tiling, 34, 77
zooming, 34, 77
Write setup command, 75
Write to ... command, 53

Z
Zoom command, 77

Index

f you're a scientist, engineer, financial analyst, student,
teacher, or any other professional working with equations,
Eureka: The Solver can do your Algebra, Trigonometry and
Calculus problems in a snap.

Eureka also handles maximization and
minimization problems, plots functions,
generates reports, and saves an incredible
amount of time. Even if you're not a computer specialist. Eureka, can help you solve
your real-world mathematical problems
fast, without having to learn numerical
approximation techniques. Using Borland's
famous pull-down menu design and
context-sensitive help screens, Eureka is
easy to learn and easy to use-as simple
as a hand-held calculator.

X + exp(X) = ID solved Instantly
Instead of e'lentuallyl
Imagine you have to "solve for X,"
where X + exp(X) = 10, and you don't
have Eureka: The Solver. What you do have
is a problem, because it's going to take a
lot of time guessing at "X." With Eureka,
there's no guessing, no dancing in the
dark-you get the right answer, right now.
(PS: X = 2.0705799, and Eureka solved
that one in .4 of a second!)

How to Ule Eureka: The SONer
It's easy.
1. Enter your equation into the
full-screen editor
2. Select the "Solve" command
3. Look at the answer
4. You're done
You can then tell Eureka to
• Evaluate your solution
• Plot a graph

• Generate a report, then send the output
to your printer, disk file or screen
• Or all of the above

. Some of Eureka's key features
You can key in:
~ A formula or formulas
~ A series of equations- and sdlve for
all variables
s' Constraints (like X has to be
< or = 2)
A function to plot
Unit conversions
Maximization and minimization
problems
s' Interesl Rale/Present Value
calculations
s' Variables we call "What happens?,"
like "What happens if I change this
variable to 21 and that variable to 27?"

Eureka: The Sol'ler Includes:
A full-screen editor
Pull-down menus
Context-sensitive Help
On-screen calculator
Automatic 8087 math co-processor
chip support
s' Powerful financial functions
~ Built-in and user-defined math and
financial functions
s' Ability to generate reports complete
with plots and lists
~ Polynomial finder
~ Inequality solutions

Minimum system requirements: IBM PC. AT. XT. Portable. 3270 and true compatibles. PC-DDS

(MS-DOS)

?O and later. 384K.

Eureka: The Solver is a trademark of Borlald Inlernalional. Inc. IBM. AT. and XT are registered trademarks of International
Business Machines Corp. MS· DOS is a registered trademark of Microsoft Corp.
BOR 0236
Copyright 1986 Borland International

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