
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
4.1.1 LIST
4.1.2 FIRST
4.1.3 SECOND
4.1.4 THIRD
4.1.5 REST
4.1.6 . (Cons) Operator
4.1.7 APPEND
4.1.8 REVERSE
4.1.9 List Arguments of Other Operators
4.1.10 Caveats and Examples
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
5.7.1 Compound Statements with GO TO
5.7.2 Labels and GO TO Statements
5.7.3 RETURN 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
7.8.3 Options controlling display of derivatives
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
8.3.1 LINELENGTH Operator
8.3.2 Output Declarations
8.3.3 Output Control Switches
8.3.4 WRITE Command
8.3.5 Suppression of Zeros
8.3.6 FORTRAN Style Output Of Expressions
8.3.7 Saving Expressions for Later Use as Input
8.3.8 Displaying Expression Structure
8.4 Changing the Internal Order of Variables
8.5 Obtaining Parts of Algebraic Expressions
8.5.1 COEFF Operator
8.5.2 COEFFN Operator
8.5.3 PART Operator
8.5.4 Substituting for Parts of 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
9.11.1 DEG Operator
9.11.2 DEN Operator
9.11.3 LCOF Operator
9.11.4 LPOWER Operator
9.11.5 LTERM Operator
9.11.6 MAINVAR Operator
9.11.7 NUM Operator
9.11.8 REDUCT Operator
9.11.9 TOTALDEG Operator
9.12 Polynomial Coefficient Arithmetic
9.12.1 Rational Coefficients in Polynomials
9.12.2 Real Coefficients in Polynomials
9.12.3 Modular Number Coefficients in Polynomials
9.12.4 Complex Number Coefficients in Polynomials
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
11.2.1 FOR ALL … LET
11.2.2 FOR ALL … SUCH THAT … LET
11.2.3 Removing Assignments and Substitution Rules
11.2.4 Overlapping LET Rules
11.2.5 Substitutions for General Expressions
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
14.4.1 DET Operator
14.4.2 MATEIGEN Operator
14.4.3 TP Operator
14.4.4 Trace Operator
14.4.5 Matrix Cofactors
14.4.6 NULLSPACE Operator
14.4.7 RANK Operator
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
16.3 ARNUM: An algebraic number package
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
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
16.9.1 Introduction
16.9.2 The Computational Model
16.9.3 Basic Data Structures
16.9.4 About the Algorithms Implemented in CALI
16.9.5 A Short Description of Procedures Available in Algebraic Mode
16.9.6 The CALI Module Structure
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
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 Local conservation laws
16.12.11 Local Hamiltonian operators
16.12.12 Examples of Schouten bracket of local Hamiltonian operators
16.12.13 Non-local operators
16.12.14 Non-local recursion operator for the Korteweg–de Vries equation.
16.12.15 Non-local Hamiltonian-recursion operators for Plebanski equation.
16.13 CDIFF: A package for computations in geometry of Differential Equations
16.13.1 Introduction
16.13.2 Computing with CDIFF
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
16.15 COEFF2: Computing Comprehensive Gröbner Bases
16.16 COMPACT: Package for compacting expressions
16.17 CRACK: Solving overdetermined systems of PDEs or ODEs
16.18 CVIT: Fast calculation of Dirac gamma matrix traces
16.19 DEFINT: A definite integration interface
16.19.1 Introduction
16.19.2 Integration between zero and infinity
16.19.3 Integration over other ranges
16.19.4 Using the definite integration package
16.19.5 Integral Transforms
16.19.6 Additional Meijer G-function Definitions
16.19.7 The print_conditions function
16.19.8 Tracing
16.19.9 Acknowledgements
16.20 DESIR: Differential linear homogeneous equation solutions in the
neighborhood of irregular and regular singular points
16.20.1 INTRODUCTION
16.20.2 FORMS OF SOLUTIONS
16.20.3 INTERACTIVE USE
16.20.4 DIRECT USE
16.20.5 USEFUL FUNCTIONS
16.20.6 LIMITATIONS
16.21 DFPART: Derivatives of generic functions
16.21.1 Generic Functions
16.21.2 Partial Derivatives
16.21.3 Substitutions
16.22 DUMMY: Canonical form of expressions with dummy variables
16.22.1 Introduction
16.22.2 Dummy variables and dummy summations
16.22.3 The Operators and their Properties
16.22.4 The Function CANONICAL
16.22.5 Bibliography
16.23 EXCALC: A differential geometry package
16.23.1 Introduction
16.23.2 Declarations
16.23.3 Exterior Multiplication
16.23.4 Partial Differentiation
16.23.5 Exterior Differentiation
16.23.6 Inner Product
16.23.7 Lie Derivative
16.23.8 Hodge-* Duality Operator
16.23.9 Variational Derivative
16.23.10 Handling of Indices
16.23.11 Metric Structures
16.23.12 Riemannian Connections
16.23.13 Killing Vectors
16.23.14 Ordering and Structuring
16.23.15 Summary of Operators and Commands
16.23.16 Examples
16.24 FIDE: Finite difference method for partial differential equations
16.24.1 Abstract
16.24.2 EXPRES
16.24.3 IIMET
16.24.4 APPROX
16.24.5 CHARPOL
16.24.6 HURWP
16.24.7 LINBAND
16.25 FPS: Automatic calculation of formal power series
16.25.1 Introduction
16.25.2 REDUCE operator FPS
16.25.3 REDUCE operator SimpleDE
16.25.4 Problems in the current version
16.26 GCREF: A Graph Cross Referencer
16.26.1 Basic Usage
16.26.2 Shell Script "gcref"
16.26.3 Redering with yED
16.27 GENTRAN: A code generation package
16.28 GNUPLOT: Display of functions and surfaces
16.28.1 Introduction
16.28.2 Command plot
16.28.3 Paper output
16.28.4 Mesh generation for implicit curves
16.28.5 Mesh generation for surfaces
16.28.6 GNUPLOT operation
16.28.7 Saving GNUPLOT command sequences
16.28.8 Direct Call of GNUPLOT
16.28.9 Examples
16.29 GROEBNER: A Gröbner basis package
16.29.1 Background
16.29.2 Loading of the Package
16.29.3 The Basic Operators
16.29.4 Ideal Decomposition & Equation System Solving
16.29.5 Calculations “by Hand”
16.30 GUARDIAN: Guarded Expressions in Practice
16.30.1 Introduction
16.30.2 An outline of our method
16.30.3 Examples
16.30.4 Outlook
16.30.5 Conclusions
16.31 IDEALS: Arithmetic for polynomial ideals
16.31.1 Introduction
16.31.2 Initialization
16.31.3 Bases
16.31.4 Algorithms
16.31.5 Examples
16.32 INEQ: Support for solving inequalities
16.33 INVBASE: A package for computing involutive bases
16.33.1 Introduction
16.33.2 The Basic Operators
16.34 LALR: A parser generator
16.34.1 Limitations
16.34.2 An example
16.35 LAPLACE: Laplace transforms
16.36 LIE: Functions for the classification of real n-dimensional Lie algebras
16.37 LIMITS: A package for finding limits
16.37.1 Normal entry points
16.37.2 Direction-dependent limits
16.38 LINALG: Linear algebra package
16.38.1 Introduction
16.38.2 Getting started
16.38.3 What’s available
16.38.4 Fast Linear Algebra
16.38.5 Acknowledgments
16.39 LISTVECOPS: Vector operations on lists
16.40 LPDO: Linear Partial Differential Operators
16.40.1 Introduction
16.40.2 Operators
16.40.3 Shapes of F-elements
16.40.4 Commands
16.41 MODSR: Modular solve and roots
16.42 MRVLIMIT: A new exp-log limits package
16.42.1 The Exp-Log Limits package
16.42.2 The Algorithm
16.42.3 The tracing facility
16.43 NCPOLY: Non–commutative polynomial ideals
16.43.1 Introduction
16.43.2 Setup, Cleanup
16.43.3 Left and right ideals
16.43.4 Gröbner bases
16.43.5 Left or right polynomial division
16.43.6 Left or right polynomial reduction
16.43.7 Factorization
16.43.8 Output of expressions
16.44 NORMFORM: Computation of matrix normal forms
16.44.1 Introduction
16.44.2 Smith normal form
16.44.3 smithex_int
16.44.4 frobenius
16.44.5 ratjordan
16.44.6 jordansymbolic
16.44.7 jordan
16.44.8 Algebraic extensions: Using the ARNUM package
16.44.9 Modular arithmetic
16.45 NUMERIC: Solving numerical problems
16.45.1 Syntax
16.45.2 Minima
16.45.3 Roots of Functions/ Solutions of Equations
16.45.4 Integrals
16.45.5 Ordinary Differential Equations
16.45.6 Bounds of a Function
16.45.7 Chebyshev Curve Fitting
16.45.8 General Curve Fitting
16.45.9 Function Bases
16.46 ODESOLVE: Ordinary differential equations solver
16.46.1 Introduction
16.46.2 Installation
16.46.3 User interface
16.46.4 Output syntax
16.46.5 Solution techniques
16.46.6 Extension interface
16.46.7 Change log
16.46.8 Planned developments
16.47 ORTHOVEC: Manipulation of scalars and vectors
16.47.1 Introduction
16.47.2 Initialisation
16.47.3 Input-Output
16.47.4 Algebraic Operations
16.47.5 Differential Operations
16.47.6 Integral Operations
16.47.7 Test Cases
16.48 PHYSOP: Operator calculus in quantum theory
16.48.1 Introduction
16.48.2 The NONCOM2 Package
16.48.3 The PHYSOP package
16.48.4 Known problems in the current release of PHYSOP
16.48.5 Final remarks
16.48.6 Appendix: List of error and warning messages
16.49 PM: A REDUCE pattern matcher
16.49.1 M(exp,temp)
16.49.2 temp _= logical_exp
16.49.3 S(exp,{temp1 -> sub1, temp2 -> sub2, …}, rept, depth)
16.49.4 temp :- exp and temp ::- exp
16.49.5 Arep({rep1,rep2,…})
16.49.6 Drep({rep1,rep2,..})
16.49.7 Switches
16.50 QHULL: Compute the complex hull
16.51 QSUM: Indefinite and Definite Summation of q-hypergeometric Terms
16.51.1 Introduction
16.51.2 Elementary q-Functions
16.51.3 q-Gosper Algorithm
16.51.4 q-Zeilberger Algorithm
16.51.5 REDUCE operator QGOSPER
16.51.6 REDUCE operator QSUMRECURSION
16.51.7 Simplification Operators
16.51.8 Global Variables and Switches
16.51.9 Messages
16.52 RANDPOLY: A random polynomial generator
16.52.1 Introduction
16.52.2 Basic use of randpoly
16.52.3 Advanced use of randpoly
16.52.4 Subsidiary functions: rand, proc, random
16.52.5 Examples
16.52.6 Appendix: Algorithmic background
16.53 RATAPRX: Rational Approximations Package for REDUCE
16.53.1 Periodic Decimal Representation
16.53.2 Continued Fractions
16.53.3 Padé Approximation
16.54 RATINT: Integrate Rational Functions using the Minimal Algebraic
Extension to the Constant Field
16.54.1 Rational Integration
16.54.2 The Algorithm
16.54.3 The log_sum operator
16.54.4 Options
16.54.5 Hermite’s method
16.54.6 Tracing the ratint program
16.54.7 Bugs, suggestions and comments
16.55 REACTEQN: Support for chemical reaction equation systems
16.56 REDLOG: Extend REDUCE to a computer logic system
16.57 RESET: Code to reset REDUCE to its initial state
16.58 RESIDUE: A residue package
16.59 RLFI: REDUCE LATEX formula interface
16.59.1 APPENDIX: Summary and syntax
16.60 ROOTS: A REDUCE root finding package
16.60.1 Introduction
16.60.2 Root Finding Strategies
16.60.3 Top Level Functions
16.60.4 Switches Used in Input
16.60.5 Internal and Output Use of Switches
16.60.6 Root Package Switches
16.60.7 Operational Parameters and Parameter Setting.
16.60.8 Avoiding truncation of polynomials on input
16.61 RSOLVE: Rational/integer polynomial solvers
16.61.1 Introduction
16.61.2 The user interface
16.61.3 Examples
16.61.4 Tracing
16.62 RTRACE: Tracing in REDUCE
16.62.1 Introduction
16.62.2 RTrace versus RDebug
16.62.3 Procedure tracing: RTR, UNRTR
16.62.4 Assignment tracing: RTRST, UNRTRST
16.62.5 Tracing active rules: TRRL, UNTRRL
16.62.6 Tracing inactive rules: TRRLID, UNTRRLID
16.62.7 Output control: RTROUT
16.63 SCOPE: REDUCE source code optimization package
16.64 SETS: A basic set theory package
16.64.1 Introduction
16.64.2 Infix operator precedence
16.64.3 Explicit set representation and mkset
16.64.4 Union and intersection
16.64.5 Symbolic set expressions
16.64.6 Set difference
16.64.7 Predicates on sets
16.64.8 Possible future developments
16.65 SPARSE: Sparse Matrix Calculations
16.65.1 Introduction
16.65.2 Sparse Matrix Calculations
16.65.3 Sparse Matrix Expressions
16.65.4 Operators with Sparse Matrix Arguments
16.65.5 The Linear Algebra Package for Sparse Matrices
16.65.6 Available Functions
16.65.7 Fast Linear Algebra
16.65.8 Acknowledgments
16.66 SPDE: Finding symmetry groups of PDE’s
16.66.1 Description of the System Functions and Variables
16.66.2 How to Use the Package
16.66.3 Test File
16.67 SPECFN: Package for special functions
16.67.1 Simplification and Approximation
16.67.2 Constants
16.67.3 Bernoulli Numbers and Euler Numbers
16.67.4 Fibonacci Numbers and Fibonacci Polynomials
16.67.5 Stirling Numbers
16.67.6 The Γ Function, and Related Functions
16.67.7 Bessel Functions
16.67.8 Hypergeometric and Other Functions
16.67.9 Integral Functions
16.67.10 Airy Functions
16.67.11 Polynomial Functions
16.67.12 Spherical and Solid Harmonics
16.67.13 Jacobi’s Elliptic Functions
16.67.14 Elliptic Integrals
16.67.15 Elliptic Theta Functions
16.67.16 Lambert’s W function
16.67.17 3j symbols and Clebsch-Gordan Coefficients
16.67.18 6j symbols
16.67.19 Acknowledgements
16.67.20 Table of Operators and Constants
16.68 SPECFN2: Package for special special functions
16.68.1 REDUCE operator HYPERGEOMETRIC
16.68.2 Extending the HYPERGEOMETRIC operator
16.68.3 REDUCE operator meijerg
16.69 SSTOOLS: Computations with supersymmetric algebraic and differential
expressions
16.69.1 Overview
16.70 SUM: A package for series summation
16.71 SYMMETRY: Operations on symmetric matrices
16.71.1 Introduction
16.71.2 Operators for linear representations
16.71.3 Display Operators
16.71.4 Storing a new group
16.72 TAYLOR: Manipulation of Taylor series
16.72.1 Basic Use
16.72.2 Caveats
16.72.3 Warning messages
16.72.4 Error messages
16.72.5 Comparison to other packages
16.73 TPS: A extendible power series package
16.73.1 Introduction
16.73.2 PS Operator
16.73.3 PSEXPLIM Operator
16.73.4 PSPRINTORDER Switch
16.73.5 PSORDLIM Operator
16.73.6 PSTERM Operator
16.73.7 PSORDER Operator
16.73.8 PSSETORDER Operator
16.73.9 PSDEPVAR Operator
16.73.10 PSEXPANSIONPT operator
16.73.11 PSFUNCTION Operator
16.73.12 PSCHANGEVAR Operator
16.73.13 PSREVERSE Operator
16.73.14 PSCOMPOSE Operator
16.73.15 PSSUM Operator
16.73.16 PSTAYLOR Operator
16.73.17 PSCOPY Operator
16.73.18 PSTRUNCATE Operator
16.73.19 Arithmetic Operations
16.73.20 Differentiation
16.73.21 Restrictions and Known Bugs
16.74 TRI: TeX REDUCE interface
16.75 TRIGD: Trigonometrical Functions with Degree Arguments
16.75.1 Introduction
16.75.2 Simplification
16.75.3 Numerical Evaluation
16.75.4 Bugs, Restrictions and Planned Extensions
16.76 TRIGINT: Weierstrass substitution in REDUCE
16.76.1 Introduction
16.76.2 Statement of the Algorithm
16.76.3 REDUCE implementation
16.76.4 Definite Integration
16.76.5 Tracing the trigint function
16.76.6 Bugs, comments, suggestions
16.77 TRIGSIMP: Simplification and factorization of trigonometric and
hyperbolic functions
16.77.1 Introduction
16.77.2 Simplifying trigonometric expressions
16.77.3 Factorizing trigonometric expressions
16.77.4 GCDs of trigonometric expressions
16.77.5 Further Examples
16.78 TURTLE: Turtle Graphics Interface for REDUCE
16.78.1 Turtle Graphics
16.78.2 Implementation
16.78.3 Turtle Functions
16.78.4 Examples
16.78.5 References
16.79 WU: Wu algorithm for polynomial systems
16.80 XCOLOR: Color factor in some field theories
16.81 XIDEAL: Gröbner Bases for exterior algebra
16.81.1 Description
16.81.2 Declarations
16.81.3 Operators
16.81.4 Switches
16.81.5 Examples
16.82 ZEILBERG: Indefinite and definite summation
16.82.1 Introduction
16.82.2 Gosper Algorithm
16.82.3 Zeilberger Algorithm
16.82.4 REDUCE operator GOSPER
16.82.5 REDUCE operator EXTENDED_GOSPER
16.82.6 REDUCE operator SUMRECURSION
16.82.7 REDUCE operator EXTENDED_SUMRECURSION
16.82.8 REDUCE operator HYPERRECURSION
16.82.9 REDUCE operator HYPERSUM
16.82.10 REDUCE operator SUMTOHYPER
16.82.11 Simplification Operators
16.82.12 Tracing
16.82.13 Global Variables and Switches
16.82.14 Messages
16.83 ZTRANS: Z-transform package
16.83.1 Z-Transform
16.83.2 Inverse Z-Transform
16.83.3 Input for the Z-Transform
16.83.4 Input for the Inverse Z-Transform
16.83.5 Application of the Z-Transform
16.83.6 EXAMPLES
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
17.9.1 Passing Algebraic Mode Values to Symbolic Mode
17.9.2 Passing Symbolic Mode Values to Algebraic Mode
17.9.3 Complete Example
17.9.4 Defining Procedures for Intermode Communication
17.10 Rlisp ’88
17.11 References
18 Calculations in High Energy Physics
18.1 High Energy Physics Operators
18.1.1 . (Cons) Operator
18.1.2 G Operator for Gamma Matrices
18.1.3 EPS Operator
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
19.3.1 Restrictions
19.3.2 Usage
19.3.3 Options
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