REDUCE
User’s Manual
Version 3.8 Anthony C. Hearn
Santa Monica, CA, USA

Email: reduce@rand.org

February 2004

Copyright ©2004 Anthony C. Hearn. All rights reserved.

Registered system holders may reproduce all or any part of this publication for internal purposes, provided that the source of the material is clearly acknowledged, and the copyright notice is retained.

Contents
Abstract
Abstract
Acknowledgment
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
5.2 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
7.2 Mathematical Functions
7.3 DF Operator
7.4 INT Operator
7.5 LENGTH Operator
7.6 MAP Operator
7.7 MKID Operator
7.8 PF Operator
7.9 SELECT Operator
7.10 SOLVE Operator
7.11 Even and Odd Operators
7.12 Linear Operators
7.13 Non-Commuting Operators
7.14 Symmetric and Antisymmetric Operators
7.15 Declaring New Prefix Operators
7.16 Declaring New Infix Operators
7.17 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
9.4 Working with Least Common Multiples
9.5 Controlling Use of Common Denominators
9.6 REMAINDER Operator
9.7 RESULTANT Operator
9.8 DECOMPOSE Operator
9.9 INTERPOL operator
9.10 Obtaining Parts of Polynomials and Rationals
9.11 Polynomial Coefficient Arithmetic
10 Substitution Commands
10.1 SUB Operator
10.2 LET Rules
10.3 Rule Lists
10.4 Asymptotic Commands
11 File Handling Commands
.1IN Command
.2OUT Command
.3SHUT Command
12 Commands for Interactive Use
12.1 Referencing Previous Results
12.2 Interactive Editing
12.3 Interactive File Control
13 Matrix Calculations
13.1 MAT Operator
13.2 Matrix Variables
13.3 Matrix Expressions
13.4 Operators with Matrix Arguments
13.5 Matrix Assignments
13.6 Evaluating Matrix Elements
14 Procedures
14.1 Procedure Heading
14.2 Procedure Body
14.3 Using LET Inside Procedures
14.4 LET Rules as Procedures
15 User Contributed Packages
15.1 ALGINT: Integration of square roots
15.2 APPLYSYM: Infinitesimal symmetries of differential equations
15.3 ARNUM: An algebraic number package
15.4 ASSIST: Useful utilities for various applications
15.5 AVECTOR: A vector algebra and calculus package
15.6 BOOLEAN: A package for boolean algebra
15.7 CALI: A package for computational commutative algebra
15.8 CAMAL: Calculations in celestial mechanics
15.9 CHANGEVR: Change of Independent Variable(s) in DEs
15.10 COMPACT: Package for compacting expressions
15.11 CRACK: Solving overdetermined systems of PDEs or ODEs
15.12 CVIT: Fast calculation of Dirac gamma matrix traces
15.13 DEFINT: A definite integration interface
15.14 DESIR: Differential linear homogeneous equation solutions in the neighborhood of irregular and regular singular points
15.15 DFPART: Derivatives of generic functions
15.16 DUMMY: Canonical form of expressions with dummy variables
15.17 EXCALC: A differential geometry package
15.18 FIDE: Finite difference method for partial differential equations
15.19 FPS: Automatic calculation of formal power series
15.20 GENTRAN: A code generation package
15.21 GNUPLOT: Display of functions and surfaces
15.22 GROEBNER: A Gröbner basis package
15.23 IDEALS: Arithmetic for polynomial ideals
15.24 INEQ: Support for solving inequalities
15.25 INVBASE: A package for computing involutive bases
15.26 LAPLACE: Laplace transforms
15.27 LIE: Functions for the classification of real n-dimensional Lie algebras
15.28 LIMITS: A package for finding limits
15.29 LINALG: Linear algebra package
15.30 MODSR: Modular solve and roots
15.31 NCPOLY: Non–commutative polynomial ideals
15.32 NORMFORM: Computation of matrix normal forms
15.33 NUMERIC: Solving numerical problems
15.34 ODESOLVE: Ordinary differential equations solver
15.35 ORTHOVEC: Manipulation of scalars and vectors
15.36 PHYSOP: Operator calculus in quantum theory
15.37 PM: A REDUCE pattern matcher
15.38 RANDPOLY: A random polynomial generator
15.39 REACTEQN: Support for chemical reaction equation systems
15.40 RESET: Code to reset REDUCE to its initial state
15.41 RESIDUE: A residue package
15.42 RLFI: REDUCE LaTeX formula interface
15.43 ROOTS: A REDUCE root finding package
15.44 RSOLVE: Rational/integer polynomial solvers
15.45 SCOPE: REDUCE source code optimization package
15.46 SETS: A basic set theory package
15.47 SPDE: Finding symmetry groups of PDE’s
15.48 SPECFN: Package for special functions
15.49 SPECFN2: Package for special special functions
15.50 SUM: A package for series summation
15.51 SYMMETRY: Operations on symmetric matrices
15.52 TAYLOR: Manipulation of Taylor series
15.53 TPS: A truncated power series package
15.54 TRI: TeX REDUCE interface
15.55 TRIGSIMP: Simplification and factorization of trigonometric and hyperbolic functions
15.56 WU: Wu algorithm for polynomial systems
15.57 XCOLOR: Color factor in some field theories
15.58 XIDEAL: Gröbner Bases for exterior algebra
15.59 ZEILBERG: A package for indefinite and definite summation
15.60 ZTRANS: Z-transform package
16 Symbolic Mode
16.1 Symbolic Infix Operators
16.2 Symbolic Expressions
16.3 Quoted Expressions
16.4 Lambda Expressions
16.5 Symbolic Assignment Statements
16.6 FOR EACH Statement
16.7 Symbolic Procedures
16.8 Standard Lisp Equivalent of Reduce Input
16.9 Communicating with Algebraic Mode
16.10 Rlisp ’88
16.11 References
17 Calculations in High Energy Physics
17.1 High Energy Physics Operators
17.2 Vector Variables
17.3 Additional Expression Types
17.4 Trace Calculations
17.5 Mass Declarations
17.6 Example
17.7 Extensions to More Than Four Dimensions
18 REDUCE and Rlisp Utilities
18.1 The Standard Lisp Compiler
18.2 Fast Loading Code Generation Program
18.3 The Standard Lisp Cross Reference Program
18.4 Prettyprinting Reduce Expressions
18.5 Prettyprinting Standard Lisp S-Expressions
19 Maintaining REDUCE
A Reserved Identifiers
Index