Contents

• Abstract
• Acknowledgment
• 1 Introductory Information
• 2 Structure of Programs
  • 2.1 The REDUCE Standard Character Set
  • 2.2 Numbers
  • 2.3 Identifiers
    • 18.3.1 Restrictions
  • 2.4 Variables
    • Reserved Variables
  • 2.5 Strings
  • 2.6 Comments
  • 2.7 Operators
    • Built-In Infix 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 FIRST
    • 4.1.2 SECOND
    • 4.1.3 THIRD
    • 4.1.4 REST
    • 4.1.5 (Cons) Operator
    • 4.1.6 APPEND
    • 4.1.7 REVERSE
    • 4.1.8 List Arguments of Other Operators
• 5 Statements
  • 5.1 Assignment Statements
    • 5.1.1 Set Statement
  • 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 DF Operator
    • 7.3.1 Adding Differentiation Rules
  • 7.4 INT Operator
    • 7.4.1 Options
    • 7.4.2 Advanced Use
    • 7.4.3 References
  • 7.5 LENGTH Operator
  • 7.6 MKID Operator
  • 7.7 PF Operator
  • 7.8 SOLVE Operator
    • 7.8.1 Handling of Undetermined Solutions
    • 7.8.2 Solutions of Equations Involving Cubics and Quartics
    • 7.8.3 Other Options
    • 7.8.4 Parameters and Variable Dependency
  • 7.9 Even and Odd Operators
  • 7.10 Linear Operators
  • 7.11 Non-Commuting Operators
  • 7.12 Symmetric and Antisymmetric Operators
  • 7.13 Declaring New Prefix Operators
  • 7.14 Declaring New Infix Operators
  • 7.15 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
      • ORDER Declaration
      • FACTOR Declaration
    • 8.3.3 Output Control Switches
      • ALLFAC Switch
      • DIV Switch
      • LIST Switch
      • NOSPLIT Switch
      • RAT Switch
      • RATPRI Switch
      • REVPRI Switch
    • 8.3.4 WRITE Command
    • 8.3.5 Suppression of Zeros
    • 8.3.6 FORTRAN Style Output Of Expressions
      • FORTRAN Output Options
    • 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 REMAINDER Operator
  • 9.7 RESULTANT Operator
  • 9.8 DECOMPOSE Operator
  • 9.9 INTERPOL operator
  • 9.10 Obtaining Parts of Polynomials and Rationals
    • 9.10.1 DEG Operator
    • 9.10.2 DEN Operator
    • 9.10.3 LCOF Operator
    • 9.10.4 LPOWER Operator
    • 9.10.5 LTERM Operator
    • 9.10.6 MAINVAR Operator
    • 9.10.7 NUM Operator
    • 9.10.8 REDUCT Operator
  • 9.11 Polynomial Coefficient Arithmetic
    • 9.11.1 Rational Coefficients in Polynomials
    • 9.11.2 Real Coefficients in Polynomials
    • 9.11.3 Modular Number Coefficients in Polynomials
    • 9.11.4 Complex Number Coefficients in Polynomials
• 10 Substitution Commands
  • 10.1 SUB Operator
  • 10.2 LET Rules
    • 10.2.1 FOR ALL ...LET
    • 10.2.2 FOR ALL ...SUCH THAT ...LET
    • 10.2.3 Removing Assignments and Substitution Rules
    • 10.2.4 Overlapping LET Rules
    • 10.2.5 Substitutions for General Expressions
  • 10.3 Rule Lists
    • Displaying Rules Associated with an Operator
    • Order of Application of Rules
  • 10.4 Asymptotic Commands
• 11 File Handling Commands
  • 11.1 IN Command
  • 11.2 OUT Command
  • 11.3 SHUT 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
    • 13.4.1 DET Operator
    • 13.4.2 MATEIGEN Operator
    • 13.4.3 TP Operator
    • 13.4.4 Trace Operator
    • 13.4.5 Matrix Cofactors
    • 13.4.6 NULLSPACE Operator
    • 13.4.7 RANK Operator
  • 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 CONTFR: Approximation of a number by continued fractions
  • 15.12 CRACK: Solving overdetermined systems of PDEs or ODEs
  • 15.13 CVIT: Fast calculation of Dirac gamma matrix traces
  • 15.14 DEFINT: A definite integration interface
  • 15.15 DESIR: Differential linear homogeneous equation solutions in the neighborhood of irregular and regular singular points
  • 15.16 DFPART: Derivatives of generic functions
  • 15.17 DUMMY: Canonical form of expressions with dummy variables
  • 15.18 EXCALC: A differential geometry package
  • 15.19 FPS: Automatic calculation of formal power series
  • 15.20 FIDE: Finite difference method for partial differential equations
  • 15.21 GENTRAN: A code generation package
  • 15.22 GNUPLOT: Display of functions and surfaces
  • 15.23 GROEBNER: A Gröbner basis package
  • 15.24 IDEALS: Arithmetic for polynomial ideals
  • 15.25 INEQ: Support for solving inequalities
  • 15.26 INVBASE: A package for computing involutive bases
  • 15.27 LAPLACE: Laplace transforms
  • 15.28 LIE: Functions for the classification of real n-dimensional Lie algebras
  • 15.29 LIMITS: A package for finding limits
  • 15.30 LINALG: Linear algebra package
  • 15.31 MODSR: Modular solve and roots
  • 15.32 NCPOLY: Non--commutative polynomial ideals
  • 15.33 NORMFORM: Computation of matrix normal forms
  • 15.34 NUMERIC: Solving numerical problems
  • 15.35 ODESOLVE: Ordinary differential equations solver
  • 15.36 ORTHOVEC: Manipulation of scalars and vectors
  • 15.37 PHYSOP: Operator calculus in quantum theory
  • 15.38 PM: A REDUCE pattern matcher
  • 15.39 RANDPOLY: A random polynomial generator
  • 15.40 REACTEQN: Support for chemical reaction equation systems
  • 15.41 RESET: Code to reset REDUCE to its initial state
  • 15.42 RESIDUE: A residue package
  • 15.43 RLFI: REDUCE LaTeX formula interface
  • 15.44 RSOLVE: Rational/integer polynomial solvers
  • 15.45 ROOTS: A REDUCE root finding package
  • 15.46 SCOPE: REDUCE source code optimization package
  • 15.47 SETS: A basic set theory package
  • 15.48 SPDE: Finding symmetry groups of PDE's
  • 15.49 SPECFN: Package for special functions
  • 15.50 SPECFN2: Package for special special functions
  • 15.51 SUM: A package for series summation
  • 15.52 SYMMETRY: Operations on symmetric matrices
  • 15.53 TAYLOR: Manipulation of Taylor series
  • 15.54 TPS: A truncated power series package
  • 15.55 TRI: TeX REDUCE interface
  • 15.56 TRIGSIMP: Simplification and factorization of trigonometric and hyperbolic functions
  • 15.57 XCOLOR: Color factor in some field theories
  • 15.58 XIDEAL: Gröbner Bases for exterior algebra
  • 15.59 WU: Wu algorithm for polynomial systems
  • 15.60 ZEILBERG: A package for indefinite and definite summation
  • 15.61 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
    • 16.9.1 Passing Algebraic Mode Values to Symbolic Mode
    • 16.9.2 Passing Symbolic Mode Values to Algebraic Mode
    • 16.9.3 Complete Example
    • 16.9.4 Defining Procedures for Intermode Communication
  • 16.10 Rlisp '88
  • 16.11 References
• 17 Calculations in High Energy Physics
  • 17.1 High Energy Physics Operators
    • 17.1.1 . (Cons) Operator
    • 17.1.2 G Operator for Gamma Matrices
    • 17.1.3 EPS Operator
  • 17.2 Vector Variables
  • 17.3 Additional Expression Types
    • 17.3.1 Vector Expressions
    • 17.3.2 Dirac Expressions
  • 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
    • Restrictions
    • 18.3.2 Usage
    • 18.3.3 Options
  • 18.4 Prettyprinting Reduce Expressions
  • 18.5 Prettyprinting Standard Lisp S-Expressions
• 19 Maintaining REDUCE
• A Reserved Identifiers
• Index
• About this document ...