REDUCE User's Manual, Free Version March 9, 2019 Manual

User Manual:

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Contents
Abstract
1 Introductory Information
2 Structure of Programs
- 2.1 The REDUCE Standard Character Set
- 2.2 Numbers
- 2.3 Identifiers
- 2.4 Variables
- 2.5 Strings
- 2.6 Comments
- 2.7 Operators
3 Expressions
- 3.1 Scalar Expressions
- 3.2 Integer Expressions
- 3.3 Boolean Expressions
- 3.4 Equations
- 3.5 Proper Statements as Expressions
4 Lists
- 4.1 Operations on Lists
5 Statements
- 5.1 Assignment Statements
  - 5.1.1 Set and Unset Statements
- 5.2 Group Statements
- 5.3 Conditional Statements
- 5.4 FOR Statements
- 5.5 WHILE …DO
- 5.6 REPEAT …UNTIL
- 5.7 Compound Statements
6 Commands and Declarations
- 6.1 Array Declarations
- 6.2 Mode Handling Declarations
- 6.3 END
- 6.4 BYE Command
- 6.5 SHOWTIME Command
- 6.6 DEFINE Command
7 Built-in Prefix Operators
- 7.1 Numerical Operators
  - 7.1.1 ABS
  - 7.1.2 CEILING
  - 7.1.3 CONJ
  - 7.1.4 FACTORIAL
  - 7.1.5 FIX
  - 7.1.6 FLOOR
  - 7.1.7 IMPART
  - 7.1.8 MAX/MIN
  - 7.1.9 NEXTPRIME
  - 7.1.10 RANDOM
  - 7.1.11 RANDOM_NEW_SEED
  - 7.1.12 REPART
  - 7.1.13 ROUND
  - 7.1.14 SIGN
- 7.2 Mathematical Functions
- 7.3 Bernoulli Numbers and Euler Numbers
- 7.4 Fibonacci Numbers and Fibonacci Polynomials
- 7.5 Motzkin numbers
- 7.6 CHANGEVAR operator
  - 7.6.1 CHANGEVAR example: The 2-dim. Laplace Equation
  - 7.6.2 Another CHANGEVAR example: An Euler Equation
- 7.7 CONTINUED_FRACTION Operator
- 7.8 DF Operator
  - 7.8.1 Switches influencing differentiation
  - 7.8.2 Adding Differentiation Rules
- 7.9 INT Operator
  - 7.9.1 Options
  - 7.9.2 Advanced Use
  - 7.9.3 References
- 7.10 LENGTH Operator
- 7.11 MAP Operator
- 7.12 MKID Operator
- 7.13 The Pochhammer Notation
- 7.14 PF Operator
- 7.15 SELECT Operator
- 7.16 SOLVE Operator
  - 7.16.1 Handling of Undetermined Solutions
  - 7.16.2 Solutions of Equations Involving Cubics and Quartics
  - 7.16.3 Other Options
  - 7.16.4 Parameters and Variable Dependency
- 7.17 Even and Odd Operators
- 7.18 Linear Operators
- 7.19 Non-Commuting Operators
- 7.20 Symmetric and Antisymmetric Operators
- 7.21 Declaring New Prefix Operators
- 7.22 Declaring New Infix Operators
- 7.23 Creating/Removing Variable Dependency
8 Display and Structuring of Expressions
- 8.1 Kernels
- 8.2 The Expression Workspace
- 8.3 Output of Expressions
- 8.4 Changing the Internal Order of Variables
- 8.5 Obtaining Parts of Algebraic Expressions
9 Polynomials and Rationals
- 9.1 Controlling the Expansion of Expressions
- 9.2 Factorization of Polynomials
- 9.3 Cancellation of Common Factors
  - 9.3.1 Determining the GCD of Two Polynomials
- 9.4 Working with Least Common Multiples
- 9.5 Controlling Use of Common Denominators
- 9.6 divide and mod / remainder Operators
- 9.7 Polynomial Pseudo-Division
- 9.8 RESULTANT Operator
- 9.9 DECOMPOSE Operator
- 9.10 INTERPOL operator
- 9.11 Obtaining Parts of Polynomials and Rationals
- 9.12 Polynomial Coefficient Arithmetic
- 9.13 ROOT_VAL Operator
10 Assigning and Testing Algebraic Properties
- 10.1 REALVALUED Declaration and Check
- 10.2 SELFCONJUGATE Declaration
- 10.3 Declaring Expressions Positive or Negative
11 Substitution Commands
- 11.1 SUB Operator
- 11.2 LET Rules
- 11.3 Rule Lists
- 11.4 Asymptotic Commands
12 File Handling Commands
- 12.1 IN Command
- 12.2 OUT Command
- 12.3 SHUT Command
- 12.4 REDUCE startup file
13 Commands for Interactive Use
- 13.1 Referencing Previous Results
- 13.2 Interactive Editing
- 13.3 Interactive File Control
14 Matrix Calculations
- 14.1 MAT Operator
- 14.2 Matrix Variables
- 14.3 Matrix Expressions
- 14.4 Operators with Matrix Arguments
- 14.5 Matrix Assignments
- 14.6 Evaluating Matrix Elements
15 Procedures
- 15.1 Procedure Heading
- 15.2 Procedure Body
- 15.3 Matrix-valued Procedures
- 15.4 Using LET Inside Procedures
- 15.5 LET Rules as Procedures
- 15.6 REMEMBER Statement
16 User Contributed Packages
- 16.1 ALGINT: Integration of square roots
- 16.2 APPLYSYM: Infinitesimal symmetries of differential equations
  - 16.2.1 Introduction and overview of the symmetry method
  - 16.2.2 Applying symmetries with APPLYSYM
  - 16.2.3 Solving quasilinear PDEs
  - 16.2.4 Transformation of DEs
  - Bibliography
- 16.3 ARNUM: An algebraic number package
  - Bibliography
- 16.4 ASSERT: Dynamic Verification of Assertions on Function Types
  - 16.4.1 Loading and Using
  - 16.4.2 Type Definitions
  - 16.4.3 Assertions
  - 16.4.4 Dynamic Checking of Assertions
  - 16.4.5 Switches
  - 16.4.6 Efficiency
  - 16.4.7 Possible Extensions
- 16.5 ASSIST: Useful utilities for various applications
  - 16.5.1 Introduction
  - 16.5.2 Survey of the Available New Facilities
  - 16.5.3 Control of Switches
  - 16.5.4 Manipulation of the List Structure
  - 16.5.5 The Bag Structure and its Associated Functions
  - 16.5.6 Sets and their Manipulation Functions
  - 16.5.7 General Purpose Utility Functions
  - 16.5.8 Properties and Flags
  - 16.5.9 Control Functions
  - 16.5.10 Handling of Polynomials
  - 16.5.11 Handling of Transcendental Functions
  - 16.5.12 Handling of n–dimensional Vectors
  - 16.5.13 Handling of Grassmann Operators
  - 16.5.14 Handling of Matrices
- 16.6 AVECTOR: A vector algebra and calculus package
  - 16.6.1 Introduction
  - 16.6.2 Vector declaration and initialisation
  - 16.6.3 Vector algebra
  - 16.6.4 Vector calculus
  - 16.6.5 Volume and Line Integration
  - 16.6.6 Defining new functions and procedures
  - 16.6.7 Acknowledgements
- 16.7 BIBASIS: A Package for Calculating Boolean Involutive Bases
  - 16.7.1 Introduction
  - 16.7.2 Boolean Ring
  - 16.7.3 Pommaret Involutive Algorithm
  - 16.7.4 BIBASIS Package
  - 16.7.5 Examples
  - Bibliography
- 16.8 BOOLEAN: A package for boolean algebra
  - 16.8.1 Introduction
  - 16.8.2 Entering boolean expressions
  - 16.8.3 Normal forms
  - 16.8.4 Evaluation of a boolean expression
- 16.9 CALI: A package for computational commutative algebra
- 16.10 CAMAL: Calculations in celestial mechanics
  - 16.10.1 Introduction
  - 16.10.2 How CAMAL Worked
  - 16.10.3 Towards a CAMAL Module
  - 16.10.4 Integration with REDUCE
  - 16.10.5 The Simple Experiments
  - 16.10.6 A Medium-Sized Problem
  - 16.10.7 Conclusion
  - Bibliography
- 16.11 CANTENS: A Package for Manipulations and Simplifications of Indexed Objects
  - 16.11.1 Introduction
  - 16.11.2 Handling of space(s)
  - 16.11.3 Generic tensors and their manipulation
  - 16.11.4 Specific tensors
  - 16.11.5 The simplification function CANONICAL
- 16.12 CDE: A package for integrability of PDEs
  - 16.12.1 Introduction: why CDE?
  - 16.12.2 Jet space of even and odd variables, and total derivatives
  - 16.12.3 Differential equations in even and odd variables
  - 16.12.4 Calculus of variations
  - 16.12.5 C-differential operators
  - 16.12.6 C-differential operators as superfunctions
  - 16.12.7 The Schouten bracket
  - 16.12.8 Computing linearization and its adjoint
  - 16.12.9 Higher symmetries
  - 16.12.10 Setting up the jet space and the differential equation.
  - 16.12.11 Solving the problem via dimensional analysis.
  - 16.12.12 Solving the problem using CRACK
  - 16.12.13 Local conservation laws
  - 16.12.14 Local Hamiltonian operators
  - 16.12.15 Korteweg–de Vries equation
  - 16.12.16 Boussinesq equation
  - 16.12.17 Kadomtsev–Petviashvili equation
  - 16.12.18 Examples of Schouten bracket of local Hamiltonian operators
  - 16.12.19 Bi-Hamiltonian structure of the KdV equation
  - 16.12.20 Bi-Hamiltonian structure of the WDVV equation
  - 16.12.21 Schouten bracket of multidimensional operators
  - 16.12.22 Non-local operators
  - 16.12.23 Non-local Hamiltonian operators for the Korteweg–de Vries equation
  - 16.12.24 Non-local recursion operator for the Korteweg–de Vries equation
  - 16.12.25 Non-local Hamiltonian-recursion operators for Plebanski equation
  - 16.12.26 Appendix: old versions of CDE
  - Bibliography
- 16.13 CDIFF: A package for computations in geometry of Differential Equations
  - 16.13.1 Introduction
  - 16.13.2 Computing with CDIFF
  - Bibliography
- 16.14 CGB: Computing Comprehensive Gröbner Bases
  - 16.14.1 Introduction
  - 16.14.2 Using the REDLOG Package
  - 16.14.3 Term Ordering Mode
  - 16.14.4 CGB: Comprehensive Gröbner Basis
  - 16.14.5 GSYS: Gröbner System
  - 16.14.6 GSYS2CGB: Gröbner System to CGB
  - 16.14.7 Switch CGBREAL: Computing over the Real Numbers
  - 16.14.8 Switches
  - Bibliography
- 16.15 COMPACT: Package for compacting expressions
- 16.16 CRACK: Solving overdetermined systems of PDEs or ODEs
- 16.17 CVIT: Fast calculation of Dirac gamma matrix traces
- 16.18 DEFINT: A definite integration interface
  - 16.18.1 Introduction
  - 16.18.2 Integration between zero and infinity
  - 16.18.3 Integration over other ranges
  - 16.18.4 Using the definite integration package
  - 16.18.5 Integral Transforms
  - 16.18.6 Additional Meijer G-function Definitions
  - 16.18.7 The print_conditions function
  - 16.18.8 Tracing
  - 16.18.9 Acknowledgements
  - Bibliography
- 16.19 DESIR: Differential linear homogeneous equation solutions in the neighborhood of irregular and regular singular points
  - 16.19.1 INTRODUCTION
  - 16.19.2 FORMS OF SOLUTIONS
  - 16.19.3 INTERACTIVE USE
  - 16.19.4 DIRECT USE
  - 16.19.5 USEFUL FUNCTIONS
  - 16.19.6 LIMITATIONS
- 16.20 DFPART: Derivatives of generic functions
  - 16.20.1 Generic Functions
  - 16.20.2 Partial Derivatives
  - 16.20.3 Substitutions
- 16.21 DUMMY: Canonical form of expressions with dummy variables
  - 16.21.1 Introduction
  - 16.21.2 Dummy variables and dummy summations
  - 16.21.3 The Operators and their Properties
  - 16.21.4 The Function CANONICAL
  - 16.21.5 Bibliography
- 16.22 EXCALC: A differential geometry package
  - 16.22.1 Introduction
  - 16.22.2 Declarations
  - 16.22.3 Exterior Multiplication
  - 16.22.4 Partial Differentiation
  - 16.22.5 Exterior Differentiation
  - 16.22.6 Inner Product
  - 16.22.7 Lie Derivative
  - 16.22.8 Hodge-* Duality Operator
  - 16.22.9 Variational Derivative
  - 16.22.10 Handling of Indices
  - 16.22.11 Metric Structures
  - 16.22.12 Riemannian Connections
  - 16.22.13 Killing Vectors
  - 16.22.14 Ordering and Structuring
  - 16.22.15 Summary of Operators and Commands
  - 16.22.16 Examples
- 16.23 FIDE: Finite difference method for partial differential equations
  - 16.23.1 Abstract
  - 16.23.2 EXPRES
  - 16.23.3 IIMET
  - 16.23.4 APPROX
  - 16.23.5 CHARPOL
  - 16.23.6 HURWP
  - 16.23.7 LINBAND
- 16.24 FPS: Automatic calculation of formal power series
  - 16.24.1 Introduction
  - 16.24.2 REDUCE operator FPS
  - 16.24.3 REDUCE operator SimpleDE
  - 16.24.4 Problems in the current version
  - Bibliography
- 16.25 GCREF: A Graph Cross Referencer
  - 16.25.1 Basic Usage
  - 16.25.2 Shell Script "gcref"
  - 16.25.3 Redering with yED
- 16.26 GENTRAN: A code generation package
- 16.27 GNUPLOT: Display of functions and surfaces
  - 16.27.1 Introduction
  - 16.27.2 Command plot
  - 16.27.3 Paper output
  - 16.27.4 Mesh generation for implicit curves
  - 16.27.5 Mesh generation for surfaces
  - 16.27.6 GnuPlot operation
  - 16.27.7 Saving GnuPlot command sequences
  - 16.27.8 Direct Call of GnuPlot
  - 16.27.9 Examples
- 16.28 GROEBNER: A Gröbner basis package
  - 16.28.1 Background
  - 16.28.2 Loading of the Package
  - 16.28.3 The Basic Operators
  - 16.28.4 Ideal Decomposition & Equation System Solving
  - 16.28.5 Calculations ``by Hand''
  - Bibliography
- 16.29 GUARDIAN: Guarded Expressions in Practice
  - 16.29.1 Introduction
  - 16.29.2 An outline of our method
  - 16.29.3 Examples
  - 16.29.4 Outlook
  - 16.29.5 Conclusions
  - Bibliography
- 16.30 IDEALS: Arithmetic for polynomial ideals
  - 16.30.1 Introduction
  - 16.30.2 Initialization
  - 16.30.3 Bases
  - 16.30.4 Algorithms
  - 16.30.5 Examples
- 16.31 INEQ: Support for solving inequalities
- 16.32 INVBASE: A package for computing involutive bases
  - 16.32.1 Introduction
  - 16.32.2 The Basic Operators
  - Bibliography
- 16.33 LALR: A parser generator
  - 16.33.1 Limitations
  - 16.33.2 An example
- 16.34 LAPLACE: Laplace transforms
- 16.35 LIE: Functions for the classification of real n-dimensional Lie algebras
  - Bibliography
- 16.36 LIMITS: A package for finding limits
  - 16.36.1 Normal entry points
  - 16.36.2 Direction-dependent limits
- 16.37 LINALG: Linear algebra package
  - 16.37.1 Introduction
  - 16.37.2 Getting started
  - 16.37.3 What's available
  - 16.37.4 Fast Linear Algebra
  - 16.37.5 Acknowledgments
  - Bibliography
- 16.38 LISTVECOPS: Vector operations on lists
- 16.39 LPDO: Linear Partial Differential Operators
  - 16.39.1 Introduction
  - 16.39.2 Operators
  - 16.39.3 Shapes of F-elements
  - 16.39.4 Commands
- 16.40 MODSR: Modular solve and roots
- 16.41 MRVLIMIT: A new exp-log limits package
  - 16.41.1 The Exp-Log Limits package
  - 16.41.2 The Algorithm
  - 16.41.3 The tracing facility
  - Bibliography
- 16.42 NCPOLY: Non–commutative polynomial ideals
  - 16.42.1 Introduction
  - 16.42.2 Setup, Cleanup
  - 16.42.3 Left and right ideals
  - 16.42.4 Gröbner bases
  - 16.42.5 Left or right polynomial division
  - 16.42.6 Left or right polynomial reduction
  - 16.42.7 Factorization
  - 16.42.8 Output of expressions
- 16.43 NORMFORM: Computation of matrix normal forms
  - 16.43.1 Introduction
  - 16.43.2 Smith normal form
  - 16.43.3 smithex_int
  - 16.43.4 frobenius
  - 16.43.5 ratjordan
  - 16.43.6 jordansymbolic
  - 16.43.7 jordan
  - 16.43.8 Algebraic extensions: Using the arnum package
  - 16.43.9 Modular arithmetic
  - Bibliography
- 16.44 NUMERIC: Solving numerical problems
  - 16.44.1 Syntax
  - 16.44.2 Minima
  - 16.44.3 Roots of Functions/ Solutions of Equations
  - 16.44.4 Integrals
  - 16.44.5 Ordinary Differential Equations
  - 16.44.6 Bounds of a Function
  - 16.44.7 Chebyshev Curve Fitting
  - 16.44.8 General Curve Fitting
  - 16.44.9 Function Bases
- 16.45 ODESOLVE: Ordinary differential equations solver
  - 16.45.1 Introduction
  - 16.45.2 Installation
  - 16.45.3 User interface
  - 16.45.4 Output syntax
  - 16.45.5 Solution techniques
  - 16.45.6 Extension interface
  - 16.45.7 Change log
  - 16.45.8 Planned developments
  - Bibliography
- 16.46 ORTHOVEC: Manipulation of scalars and vectors
  - 16.46.1 Introduction
  - 16.46.2 Initialisation
  - 16.46.3 Input-Output
  - 16.46.4 Algebraic Operations
  - 16.46.5 Differential Operations
  - 16.46.6 Integral Operations
  - 16.46.7 Test Cases
  - Bibliography
- 16.47 PHYSOP: Operator calculus in quantum theory
  - 16.47.1 Introduction
  - 16.47.2 The NONCOM2 Package
  - 16.47.3 The PHYSOP package
  - 16.47.4 Known problems in the current release of PHYSOP
  - 16.47.5 Final remarks
  - 16.47.6 Appendix: List of error and warning messages
- 16.48 PM: A REDUCE pattern matcher
  - 16.48.1 M(exp,temp)
  - 16.48.2 temp _= logical_exp
  - 16.48.3 S(exp,{temp1 -> sub1, temp2 -> sub2, …}, rept, depth)
  - 16.48.4 temp :- exp and temp ::- exp
  - 16.48.5 Arep({rep1,rep2,…})
  - 16.48.6 Drep({rep1,rep2,..})
  - 16.48.7 Switches
- 16.49 QSUM: Indefinite and Definite Summation of q-hypergeometric Terms
  - 16.49.1 Introduction
  - 16.49.2 Elementary q-Functions
  - 16.49.3 q-Gosper Algorithm
  - 16.49.4 q-Zeilberger Algorithm
  - 16.49.5 REDUCE operator QGOSPER
  - 16.49.6 REDUCE operator QSUMRECURSION
  - 16.49.7 Simplification Operators
  - 16.49.8 Global Variables and Switches
  - 16.49.9 Messages
  - Bibliography
- 16.50 RANDPOLY: A random polynomial generator
  - 16.50.1 Introduction
  - 16.50.2 Basic use of randpoly
  - 16.50.3 Advanced use of randpoly
  - 16.50.4 Subsidiary functions: rand, proc, random
  - 16.50.5 Examples
  - 16.50.6 Appendix: Algorithmic background
- 16.51 RATAPRX: Rational Approximations Package for REDUCE
  - 16.51.1 Periodic Decimal Representation
  - 16.51.2 Continued Fractions
  - 16.51.3 Padé Approximation
  - Bibliography
- 16.52 RATINT: Integrate Rational Functions using the Minimal Algebraic Extension to the Constant Field
  - 16.52.1 Rational Integration
  - 16.52.2 The Algorithm
  - 16.52.3 The log_sum operator
  - 16.52.4 Options
  - 16.52.5 Hermite's method
  - 16.52.6 Tracing the ratint program
  - 16.52.7 Bugs, suggestions and comments
  - Bibliography
- 16.53 REACTEQN: Support for chemical reaction equation systems
- 16.54 REDLOG: Extend REDUCE to a computer logic system
- 16.55 RESET: Code to reset REDUCE to its initial state
- 16.56 RESIDUE: A residue package
- 16.57 RLFI: REDUCE LaTeX formula interface
  - 16.57.1 APPENDIX: Summary and syntax
  - Bibliography
- 16.58 ROOTS: A REDUCE root finding package
  - 16.58.1 Introduction
  - 16.58.2 Root Finding Strategies
  - 16.58.3 Top Level Functions
  - 16.58.4 Switches Used in Input
  - 16.58.5 Internal and Output Use of Switches
  - 16.58.6 Root Package Switches
  - 16.58.7 Operational Parameters and Parameter Setting.
  - 16.58.8 Avoiding truncation of polynomials on input
- 16.59 RSOLVE: Rational/integer polynomial solvers
  - 16.59.1 Introduction
  - 16.59.2 The user interface
  - 16.59.3 Examples
  - 16.59.4 Tracing
- 16.60 RTRACE: Tracing in REDUCE
  - 16.60.1 Introduction
  - 16.60.2 RTrace versus RDebug
  - 16.60.3 Procedure tracing: RTR, UNRTR
  - 16.60.4 Assignment tracing: RTRST, UNRTRST
  - 16.60.5 Tracing active rules: TRRL, UNTRRL
  - 16.60.6 Tracing inactive rules: TRRLID, UNTRRLID
  - 16.60.7 Output control: RTROUT
- 16.61 SCOPE: REDUCE source code optimization package
- 16.62 SETS: A basic set theory package
  - 16.62.1 Introduction
  - 16.62.2 Infix operator precedence
  - 16.62.3 Explicit set representation and mkset
  - 16.62.4 Union and intersection
  - 16.62.5 Symbolic set expressions
  - 16.62.6 Set difference
  - 16.62.7 Predicates on sets
  - 16.62.8 Possible future developments
- 16.63 SPARSE: Sparse Matrix Calculations
  - 16.63.1 Introduction
  - 16.63.2 Sparse Matrix Calculations
  - 16.63.3 Sparse Matrix Expressions
  - 16.63.4 Operators with Sparse Matrix Arguments
  - 16.63.5 The Linear Algebra Package for Sparse Matrices
  - 16.63.6 Available Functions
  - 16.63.7 Fast Linear Algebra
  - 16.63.8 Acknowledgments
  - Bibliography
- 16.64 SPDE: Finding symmetry groups of PDE's
  - 16.64.1 Description of the System Functions and Variables
  - 16.64.2 How to Use the Package
  - 16.64.3 Test File
- 16.65 SPECFN: Package for special functions
- 16.66 SPECFN2: Package for special special functions
  - 16.66.1 REDUCE operator HYPERGEOMETRIC
  - 16.66.2 Extending the HYPERGEOMETRIC operator
  - 16.66.3 REDUCE operator meijerg
  - Bibliography
- 16.67 SSTOOLS: Computations with supersymmetric algebraic and differential expressions
  - 16.67.1 Overview
  - Bibliography
- 16.68 SUM: A package for series summation
- 16.69 SYMMETRY: Operations on symmetric matrices
  - 16.69.1 Introduction
  - 16.69.2 Operators for linear representations
  - 16.69.3 Display Operators
  - 16.69.4 Storing a new group
  - Bibliography
- 16.70 TAYLOR: Manipulation of Taylor series
  - 16.70.1 Basic Use
  - 16.70.2 Caveats
  - 16.70.3 Warning messages
  - 16.70.4 Error messages
  - 16.70.5 Comparison to other packages
- 16.71 TPS: A truncated power series package
  - 16.71.1 Introduction
  - 16.71.2 PS Operator
  - 16.71.3 PSEXPLIM Operator
  - 16.71.4 PSPRINTORDER Switch
  - 16.71.5 PSORDLIM Operator
  - 16.71.6 PSTERM Operator
  - 16.71.7 PSORDER Operator
  - 16.71.8 PSSETORDER Operator
  - 16.71.9 PSDEPVAR Operator
  - 16.71.10 PSEXPANSIONPT operator
  - 16.71.11 PSFUNCTION Operator
  - 16.71.12 PSCHANGEVAR Operator
  - 16.71.13 PSREVERSE Operator
  - 16.71.14 PSCOMPOSE Operator
  - 16.71.15 PSSUM Operator
  - 16.71.16 PSTAYLOR Operator
  - 16.71.17 PSCOPY Operator
  - 16.71.18 PSTRUNCATE Operator
  - 16.71.19 Arithmetic Operations
  - 16.71.20 Differentiation
  - 16.71.21 Restrictions and Known Bugs
- 16.72 TRI: TeX REDUCE interface
- 16.73 TRIGINT: Weierstrass substitution in REDUCE
  - 16.73.1 Introduction
  - 16.73.2 Statement of the Algorithm
  - 16.73.3 REDUCE implementation
  - 16.73.4 Definite Integration
  - 16.73.5 Tracing the trigint function
  - 16.73.6 Bugs, comments, suggestions
  - Bibliography
- 16.74 TRIGSIMP: Simplification and factorization of trigonometric and hyperbolic functions
  - 16.74.1 Introduction
  - 16.74.2 Simplifying trigonometric expressions
  - 16.74.3 Factorizing trigonometric expressions
  - 16.74.4 GCDs of trigonometric expressions
  - 16.74.5 Further Examples
  - Bibliography
- 16.75 TURTLE: Turtle Graphics Interface for REDUCE
  - 16.75.1 Turtle Graphics
  - 16.75.2 Implementation
  - 16.75.3 Turtle Functions
  - 16.75.4 Examples
  - 16.75.5 References
- 16.76 WU: Wu algorithm for polynomial systems
- 16.77 XCOLOR: Color factor in some field theories
- 16.78 XIDEAL: Gröbner Bases for exterior algebra
  - 16.78.1 Description
  - 16.78.2 Declarations
  - 16.78.3 Operators
  - 16.78.4 Switches
  - 16.78.5 Examples
  - Bibliography
- 16.79 ZEILBERG: Indefinite and definite summation
  - 16.79.1 Introduction
  - 16.79.2 Gosper Algorithm
  - 16.79.3 Zeilberger Algorithm
  - 16.79.4 REDUCE operator GOSPER
  - 16.79.5 REDUCE operator EXTENDED_GOSPER
  - 16.79.6 REDUCE operator SUMRECURSION
  - 16.79.7 REDUCE operator EXTENDED_SUMRECURSION
  - 16.79.8 REDUCE operator HYPERRECURSION
  - 16.79.9 REDUCE operator HYPERSUM
  - 16.79.10 REDUCE operator SUMTOHYPER
  - 16.79.11 Simplification Operators
  - 16.79.12 Tracing
  - 16.79.13 Global Variables and Switches
  - 16.79.14 Messages
  - Bibliography
- 16.80 ZTRANS: Z-transform package
  - 16.80.1 Z-Transform
  - 16.80.2 Inverse Z-Transform
  - 16.80.3 Input for the Z-Transform
  - 16.80.4 Input for the Inverse Z-Transform
  - 16.80.5 Application of the Z-Transform
  - 16.80.6 EXAMPLES
  - Bibliography
17 Symbolic Mode
- 17.1 Symbolic Infix Operators
- 17.2 Symbolic Expressions
- 17.3 Quoted Expressions
- 17.4 Lambda Expressions
- 17.5 Symbolic Assignment Statements
- 17.6 FOR EACH Statement
- 17.7 Symbolic Procedures
- 17.8 Standard Lisp Equivalent of Reduce Input
- 17.9 Communicating with Algebraic Mode
- 17.10 Rlisp '88
- 17.11 References
18 Calculations in High Energy Physics
- 18.1 High Energy Physics Operators
- 18.2 Vector Variables
- 18.3 Additional Expression Types
  - 18.3.1 Vector Expressions
  - 18.3.2 Dirac Expressions
- 18.4 Trace Calculations
- 18.5 Mass Declarations
- 18.6 Example
- 18.7 Extensions to More Than Four Dimensions
19 REDUCE and Rlisp Utilities
- 19.1 The Standard Lisp Compiler
- 19.2 Fast Loading Code Generation Program
- 19.3 The Standard Lisp Cross Reference Program
- 19.4 Prettyprinting REDUCE Expressions
- 19.5 Prettyprinting Standard Lisp S-Expressions
20 Maintaining REDUCE
A Reserved Identifiers
B Bibliography
C Changes since Version 3.8
Index

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