REDUCEUser’s Manual

Free Version

Anthony C. Hearn and Rainer Schöpf

https://reduce-algebra.sourceforge.io/

September 21, 2019

https://reduce-algebra.sourceforge.io/

Copyright c©2004–2019 Anthony C. Hearn, Rainer Schöpf and contributors to theReduce project. All rights reserved.

Reproduction of this manual is allowed, provided that the source of the material isclearly acknowledged, and the copyright notice is retained.

Contents

Abstract 27

1 Introductory Information 31

2 Structure of Programs 35

2.1 The REDUCE Standard Character Set . . . . . . . . . . . . . . . 35

2.2 Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.3 Identifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.4 Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.5 Strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.6 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.7 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3 Expressions 45

3.1 Scalar Expressions . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.2 Integer Expressions . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.3 Boolean Expressions . . . . . . . . . . . . . . . . . . . . . . . . 47

3.4 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.5 Proper Statements as Expressions . . . . . . . . . . . . . . . . . 49

4 Lists 51

4.1 Operations on Lists . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.1.1 LIST . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.1.2 FIRST . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

1

2 CONTENTS

4.1.3 SECOND . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.1.4 THIRD . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.1.5 REST . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.1.6 . (Cons) Operator . . . . . . . . . . . . . . . . . . . . . . 52

4.1.7 APPEND . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.1.8 REVERSE . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.1.9 List Arguments of Other Operators . . . . . . . . . . . . 53

4.1.10 Caveats and Examples . . . . . . . . . . . . . . . . . . . 53

5 Statements 55

5.1 Assignment Statements . . . . . . . . . . . . . . . . . . . . . . . 56

5.1.1 Set and Unset Statements . . . . . . . . . . . . . . . . . . 56

5.2 Group Statements . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.3 Conditional Statements . . . . . . . . . . . . . . . . . . . . . . . 57

5.4 FOR Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.5 WHILE . . . DO . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.6 REPEAT . . . UNTIL . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.7 Compound Statements . . . . . . . . . . . . . . . . . . . . . . . 62

5.7.1 Compound Statements with GO TO . . . . . . . . . . . . 63

5.7.2 Labels and GO TO Statements . . . . . . . . . . . . . . . 64

5.7.3 RETURN Statements . . . . . . . . . . . . . . . . . . . . 64

6 Commands and Declarations 67

6.1 Array Declarations . . . . . . . . . . . . . . . . . . . . . . . . . 67

6.2 Mode Handling Declarations . . . . . . . . . . . . . . . . . . . . 68

6.3 END . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

6.4 BYE Command . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

6.5 SHOWTIME Command . . . . . . . . . . . . . . . . . . . . . . 69

6.6 DEFINE Command . . . . . . . . . . . . . . . . . . . . . . . . . 69

7 Built-in Prefix Operators 71

7.1 Numerical Operators . . . . . . . . . . . . . . . . . . . . . . . . 71

CONTENTS 3

7.1.1 ABS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

7.1.2 CEILING . . . . . . . . . . . . . . . . . . . . . . . . . . 72

7.1.3 CONJ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

7.1.4 FACTORIAL . . . . . . . . . . . . . . . . . . . . . . . . 73

7.1.5 FIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

7.1.6 FLOOR . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

7.1.7 IMPART . . . . . . . . . . . . . . . . . . . . . . . . . . 73

7.1.8 MAX/MIN . . . . . . . . . . . . . . . . . . . . . . . . . 74

7.1.9 NEXTPRIME . . . . . . . . . . . . . . . . . . . . . . . . 74

7.1.10 RANDOM . . . . . . . . . . . . . . . . . . . . . . . . . 74

7.1.11 RANDOM_NEW_SEED . . . . . . . . . . . . . . . . . . 74

7.1.12 REPART . . . . . . . . . . . . . . . . . . . . . . . . . . 75

7.1.13 ROUND . . . . . . . . . . . . . . . . . . . . . . . . . . 75

7.1.14 SIGN . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

7.2 Mathematical Functions . . . . . . . . . . . . . . . . . . . . . . . 76

7.3 Bernoulli Numbers and Euler Numbers . . . . . . . . . . . . . . 80

7.4 Fibonacci Numbers and Fibonacci Polynomials . . . . . . . . . . 80

7.5 Motzkin numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 81

7.6 CHANGEVAR operator . . . . . . . . . . . . . . . . . . . . . . 81

7.6.1 CHANGEVAR example: The 2-dim. Laplace Equation . . . 83

7.6.2 Another CHANGEVAR example: An Euler Equation . . . . 83

7.7 CONTINUED_FRACTION Operator . . . . . . . . . . . . . . . 84

7.8 DF Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

7.8.1 Switches influencing differentiation . . . . . . . . . . . . 87

7.8.2 Adding Differentiation Rules . . . . . . . . . . . . . . . . 88

7.9 INT Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

7.9.1 Options . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

7.9.2 Advanced Use . . . . . . . . . . . . . . . . . . . . . . . 90

7.9.3 References . . . . . . . . . . . . . . . . . . . . . . . . . 91

7.10 LENGTH Operator . . . . . . . . . . . . . . . . . . . . . . . . . 91

4 CONTENTS

7.11 MAP Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

7.12 MKID Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

7.13 The Pochhammer Notation . . . . . . . . . . . . . . . . . . . . . 93

7.14 PF Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

7.15 SELECT Operator . . . . . . . . . . . . . . . . . . . . . . . . . 94

7.16 SOLVE Operator . . . . . . . . . . . . . . . . . . . . . . . . . . 96

7.16.1 Handling of Undetermined Solutions . . . . . . . . . . . 97

7.16.2 Solutions of Equations Involving Cubics and Quartics . . 98

7.16.3 Other Options . . . . . . . . . . . . . . . . . . . . . . . . 100

7.16.4 Parameters and Variable Dependency . . . . . . . . . . . 101

7.17 Even and Odd Operators . . . . . . . . . . . . . . . . . . . . . . 103

7.18 Linear Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 104

7.19 Non-Commuting Operators . . . . . . . . . . . . . . . . . . . . . 105

7.20 Symmetric and Antisymmetric Operators . . . . . . . . . . . . . 105

7.21 Declaring New Prefix Operators . . . . . . . . . . . . . . . . . . 106

7.22 Declaring New Infix Operators . . . . . . . . . . . . . . . . . . . 107

7.23 Creating/Removing Variable Dependency . . . . . . . . . . . . . 107

8 Display and Structuring of Expressions 109

8.1 Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

8.2 The Expression Workspace . . . . . . . . . . . . . . . . . . . . . 110

8.3 Output of Expressions . . . . . . . . . . . . . . . . . . . . . . . . 111

8.3.1 LINELENGTH Operator . . . . . . . . . . . . . . . . . . 112

8.3.2 Output Declarations . . . . . . . . . . . . . . . . . . . . 112

8.3.3 Output Control Switches . . . . . . . . . . . . . . . . . . 113

8.3.4 WRITE Command . . . . . . . . . . . . . . . . . . . . . 116

8.3.5 Suppression of Zeros . . . . . . . . . . . . . . . . . . . . 119

8.3.6 FORTRAN Style Output Of Expressions . . . . . . . . . 119

8.3.7 Saving Expressions for Later Use as Input . . . . . . . . . 121

8.3.8 Displaying Expression Structure . . . . . . . . . . . . . . 122

8.4 Changing the Internal Order of Variables . . . . . . . . . . . . . . 123

CONTENTS 5

8.5 Obtaining Parts of Algebraic Expressions . . . . . . . . . . . . . 124

8.5.1 COEFF Operator . . . . . . . . . . . . . . . . . . . . . . 124

8.5.2 COEFFN Operator . . . . . . . . . . . . . . . . . . . . . 125

8.5.3 PART Operator . . . . . . . . . . . . . . . . . . . . . . . 125

8.5.4 Substituting for Parts of Expressions . . . . . . . . . . . . 126

9 Polynomials and Rationals 129

9.1 Controlling the Expansion of Expressions . . . . . . . . . . . . . 130

9.2 Factorization of Polynomials . . . . . . . . . . . . . . . . . . . . 130

9.3 Cancellation of Common Factors . . . . . . . . . . . . . . . . . . 132

9.3.1 Determining the GCD of Two Polynomials . . . . . . . . 133

9.4 Working with Least Common Multiples . . . . . . . . . . . . . . 134

9.5 Controlling Use of Common Denominators . . . . . . . . . . . . 134

9.6 divide and mod / remainder Operators . . . . . . . . . . . . 135

9.7 Polynomial Pseudo-Division . . . . . . . . . . . . . . . . . . . . 136

9.8 RESULTANT Operator . . . . . . . . . . . . . . . . . . . . . . . 139

9.9 DECOMPOSE Operator . . . . . . . . . . . . . . . . . . . . . . 140

9.10 INTERPOL operator . . . . . . . . . . . . . . . . . . . . . . . . 141

9.11 Obtaining Parts of Polynomials and Rationals . . . . . . . . . . . 141

9.11.1 DEG Operator . . . . . . . . . . . . . . . . . . . . . . . 141

9.11.2 DEN Operator . . . . . . . . . . . . . . . . . . . . . . . 142

9.11.3 LCOF Operator . . . . . . . . . . . . . . . . . . . . . . . 142

9.11.4 LPOWER Operator . . . . . . . . . . . . . . . . . . . . . 142

9.11.5 LTERM Operator . . . . . . . . . . . . . . . . . . . . . . 143

9.11.6 MAINVAR Operator . . . . . . . . . . . . . . . . . . . . 143

9.11.7 NUM Operator . . . . . . . . . . . . . . . . . . . . . . . 143

9.11.8 REDUCT Operator . . . . . . . . . . . . . . . . . . . . . 144

9.11.9 TOTALDEG Operator . . . . . . . . . . . . . . . . . . . 144

9.12 Polynomial Coefficient Arithmetic . . . . . . . . . . . . . . . . . 145

9.12.1 Rational Coefficients in Polynomials . . . . . . . . . . . . 145

9.12.2 Real Coefficients in Polynomials . . . . . . . . . . . . . . 145

6 CONTENTS

9.12.3 Modular Number Coefficients in Polynomials . . . . . . . 147

9.12.4 Complex Number Coefficients in Polynomials . . . . . . 147

9.13 ROOT_VAL Operator . . . . . . . . . . . . . . . . . . . . . . . . 148

10 Assigning and Testing Algebraic Properties 149

10.1 REALVALUED Declaration and Check . . . . . . . . . . . . . . 149

10.2 SELFCONJUGATE Declaration . . . . . . . . . . . . . . . . . . 150

10.3 Declaring Expressions Positive or Negative . . . . . . . . . . . . 151

11 Substitution Commands 153

11.1 SUB Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

11.2 LET Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

11.2.1 FOR ALL . . . LET . . . . . . . . . . . . . . . . . . . . . 156

11.2.2 FOR ALL . . . SUCH THAT . . . LET . . . . . . . . . . . 157

11.2.3 Removing Assignments and Substitution Rules . . . . . . 157

11.2.4 Overlapping LET Rules . . . . . . . . . . . . . . . . . . 158

11.2.5 Substitutions for General Expressions . . . . . . . . . . . 159

11.3 Rule Lists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

11.4 Asymptotic Commands . . . . . . . . . . . . . . . . . . . . . . . 167

12 File Handling Commands 169

12.1 IN Command . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

12.2 OUT Command . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

12.3 SHUT Command . . . . . . . . . . . . . . . . . . . . . . . . . . 170

12.4 REDUCE startup file . . . . . . . . . . . . . . . . . . . . . . . . 171

13 Commands for Interactive Use 173

13.1 Referencing Previous Results . . . . . . . . . . . . . . . . . . . . 173

13.2 Interactive Editing . . . . . . . . . . . . . . . . . . . . . . . . . . 174

13.3 Interactive File Control . . . . . . . . . . . . . . . . . . . . . . . 175

14 Matrix Calculations 177

14.1 MAT Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

CONTENTS 7

14.2 Matrix Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

14.3 Matrix Expressions . . . . . . . . . . . . . . . . . . . . . . . . . 178

14.4 Operators with Matrix Arguments . . . . . . . . . . . . . . . . . 179

14.4.1 DET Operator . . . . . . . . . . . . . . . . . . . . . . . . 179

14.4.2 MATEIGEN Operator . . . . . . . . . . . . . . . . . . . 180

14.4.3 TP Operator . . . . . . . . . . . . . . . . . . . . . . . . . 181

14.4.4 Trace Operator . . . . . . . . . . . . . . . . . . . . . . . 181

14.4.5 Matrix Cofactors . . . . . . . . . . . . . . . . . . . . . . 181

14.4.6 NULLSPACE Operator . . . . . . . . . . . . . . . . . . . 181

14.4.7 RANK Operator . . . . . . . . . . . . . . . . . . . . . . 182

14.5 Matrix Assignments . . . . . . . . . . . . . . . . . . . . . . . . . 183

14.6 Evaluating Matrix Elements . . . . . . . . . . . . . . . . . . . . 183

15 Procedures 185

15.1 Procedure Heading . . . . . . . . . . . . . . . . . . . . . . . . . 186

15.2 Procedure Body . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

15.3 Matrix-valued Procedures . . . . . . . . . . . . . . . . . . . . . . 188

15.4 Using LET Inside Procedures . . . . . . . . . . . . . . . . . . . . 189

15.5 LET Rules as Procedures . . . . . . . . . . . . . . . . . . . . . . 190

15.6 REMEMBER Statement . . . . . . . . . . . . . . . . . . . . . . 191

16 User Contributed Packages 193

16.1 ALGINT: Integration of square roots . . . . . . . . . . . . . . . . 194

16.2 APPLYSYM: Infinitesimal symmetries of differential equations . 195

16.2.1 Introduction and overview of the symmetry method . . . . 195

16.2.2 Applying symmetries with APPLYSYM . . . . . . . . . . 201

16.2.3 Solving quasilinear PDEs . . . . . . . . . . . . . . . . . 210

16.2.4 Transformation of DEs . . . . . . . . . . . . . . . . . . . 213

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

16.3 ARNUM: An algebraic number package . . . . . . . . . . . . . . 218

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

8 CONTENTS

16.4 ASSERT: Dynamic Verification of Assertions on Function Types . 224

16.4.1 Loading and Using . . . . . . . . . . . . . . . . . . . . . 224

16.4.2 Type Definitions . . . . . . . . . . . . . . . . . . . . . . 224

16.4.3 Assertions . . . . . . . . . . . . . . . . . . . . . . . . . . 225

16.4.4 Dynamic Checking of Assertions . . . . . . . . . . . . . 225

16.4.5 Switches . . . . . . . . . . . . . . . . . . . . . . . . . . 227

16.4.6 Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . 227

16.4.7 Possible Extensions . . . . . . . . . . . . . . . . . . . . . 229

16.5 ASSIST: Useful utilities for various applications . . . . . . . . . . 230

16.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 230

16.5.2 Survey of the Available New Facilities . . . . . . . . . . 230

16.5.3 Control of Switches . . . . . . . . . . . . . . . . . . . . 232

16.5.4 Manipulation of the List Structure . . . . . . . . . . . . . 233

16.5.5 The Bag Structure and its Associated Functions . . . . . 238

16.5.6 Sets and their Manipulation Functions . . . . . . . . . . . 240

16.5.7 General Purpose Utility Functions . . . . . . . . . . . . . 241

16.5.8 Properties and Flags . . . . . . . . . . . . . . . . . . . . 248

16.5.9 Control Functions . . . . . . . . . . . . . . . . . . . . . 249

16.5.10 Handling of Polynomials . . . . . . . . . . . . . . . . . . 252

16.5.11 Handling of Transcendental Functions . . . . . . . . . . . 253

16.5.12 Handling of n–dimensional Vectors . . . . . . . . . . . . 255

16.5.13 Handling of Grassmann Operators . . . . . . . . . . . . . 255

16.5.14 Handling of Matrices . . . . . . . . . . . . . . . . . . . . 256

16.6 AVECTOR: A vector algebra and calculus package . . . . . . . . 260

16.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 260

16.6.2 Vector declaration and initialisation . . . . . . . . . . . . 260

16.6.3 Vector algebra . . . . . . . . . . . . . . . . . . . . . . . 261

16.6.4 Vector calculus . . . . . . . . . . . . . . . . . . . . . . . 262

16.6.5 Volume and Line Integration . . . . . . . . . . . . . . . . 264

16.6.6 Defining new functions and procedures . . . . . . . . . . 266

CONTENTS 9

16.6.7 Acknowledgements . . . . . . . . . . . . . . . . . . . . . 266

16.7 BIBASIS: A Package for Calculating Boolean Involutive Bases . . 267

16.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 267

16.7.2 Boolean Ring . . . . . . . . . . . . . . . . . . . . . . . . 267

16.7.3 Pommaret Involutive Algorithm . . . . . . . . . . . . . . 268

16.7.4 BIBASIS Package . . . . . . . . . . . . . . . . . . . . . 269

16.7.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 270

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273

16.8 BOOLEAN: A package for boolean algebra . . . . . . . . . . . . 274

16.8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 274

16.8.2 Entering boolean expressions . . . . . . . . . . . . . . . 274

16.8.3 Normal forms . . . . . . . . . . . . . . . . . . . . . . . . 275

16.8.4 Evaluation of a boolean expression . . . . . . . . . . . . 276

16.9 CALI: A package for computational commutative algebra . . . . . 278

16.9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 278

16.9.2 The Computational Model . . . . . . . . . . . . . . . . . 285

16.9.3 Basic Data Structures . . . . . . . . . . . . . . . . . . . . 294

16.9.4 About the Algorithms Implemented in CALI . . . . . . . 300

16.9.5 A Short Description of Procedures Available in AlgebraicMode . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316

16.9.6 The CALI Module Structure . . . . . . . . . . . . . . . . 326

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

16.10CAMAL: Calculations in celestial mechanics . . . . . . . . . . . 329

16.10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 329

16.10.2 How CAMAL Worked . . . . . . . . . . . . . . . . . . . 330

16.10.3 Towards a CAMAL Module . . . . . . . . . . . . . . . . 333

16.10.4 Integration with REDUCE . . . . . . . . . . . . . . . . . 335

16.10.5 The Simple Experiments . . . . . . . . . . . . . . . . . . 336

16.10.6 A Medium-Sized Problem . . . . . . . . . . . . . . . . . 337

16.10.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 339

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342

10 CONTENTS

16.11CANTENS: A Package for Manipulations and Simplifications ofIndexed Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . 343

16.11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 343

16.11.2 Handling of space(s) . . . . . . . . . . . . . . . . . . . . 344

16.11.3 Generic tensors and their manipulation . . . . . . . . . . 348

16.11.4 Specific tensors . . . . . . . . . . . . . . . . . . . . . . . 362

16.11.5 The simplification function CANONICAL . . . . . . . . . 378

16.12CDE: A package for integrability of PDEs . . . . . . . . . . . . . 397

16.12.1 Introduction: why CDE? . . . . . . . . . . . . . . . . . . 397

16.12.2 Jet space of even and odd variables, and total derivatives . 398

16.12.3 Differential equations in even and odd variables . . . . . . 402

16.12.4 Calculus of variations . . . . . . . . . . . . . . . . . . . 404

16.12.5 C-differential operators . . . . . . . . . . . . . . . . . . . 404

16.12.6 C-differential operators as superfunctions . . . . . . . . . 407

16.12.7 The Schouten bracket . . . . . . . . . . . . . . . . . . . . 408

16.12.8 Computing linearization and its adjoint . . . . . . . . . . 409

16.12.9 Higher symmetries . . . . . . . . . . . . . . . . . . . . . 412

16.12.10Setting up the jet space and the differential equation. . . . 413

16.12.11Solving the problem via dimensional analysis. . . . . . . 413

16.12.12Solving the problem using CRACK . . . . . . . . . . . . 417

16.12.13Local conservation laws . . . . . . . . . . . . . . . . . . 418

16.12.14Local Hamiltonian operators . . . . . . . . . . . . . . . . 419

16.12.15Korteweg–de Vries equation . . . . . . . . . . . . . . . . 420

16.12.16Boussinesq equation . . . . . . . . . . . . . . . . . . . . 423

16.12.17Kadomtsev–Petviashvili equation . . . . . . . . . . . . . 424

16.12.18Examples of Schouten bracket of local Hamiltonian operators425

16.12.19Bi-Hamiltonian structure of the KdV equation . . . . . . . 426

16.12.20Bi-Hamiltonian structure of the WDVV equation . . . . . 427

16.12.21Schouten bracket of multidimensional operators . . . . . . 431

16.12.22Non-local operators . . . . . . . . . . . . . . . . . . . . . 433

CONTENTS 11

16.12.23Non-local Hamiltonian operators for the Korteweg–de Vriesequation . . . . . . . . . . . . . . . . . . . . . . . . . . . 433

16.12.24Non-local recursion operator for the Korteweg–de Vriesequation . . . . . . . . . . . . . . . . . . . . . . . . . . . 436

16.12.25Non-local Hamiltonian-recursion operators for Plebanskiequation . . . . . . . . . . . . . . . . . . . . . . . . . . . 436

16.12.26Appendix: old versions of CDE . . . . . . . . . . . . . . 438

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439

16.13CDIFF: A package for computations in geometry of DifferentialEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442

16.13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 442

16.13.2 Computing with CDIFF . . . . . . . . . . . . . . . . . . 443

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466

16.14CGB: Computing Comprehensive Gröbner Bases . . . . . . . . . 467

16.14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 467

16.14.2 Using the REDLOG Package . . . . . . . . . . . . . . . . 467

16.14.3 Term Ordering Mode . . . . . . . . . . . . . . . . . . . . 468

16.14.4 CGB: Comprehensive Gröbner Basis . . . . . . . . . . . 468

16.14.5 GSYS: Gröbner System . . . . . . . . . . . . . . . . . . 468

16.14.6 GSYS2CGB: Gröbner System to CGB . . . . . . . . . . 470

16.14.7 Switch CGBREAL: Computing over the Real Numbers . . 470

16.14.8 Switches . . . . . . . . . . . . . . . . . . . . . . . . . . 471

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471

16.15COMPACT: Package for compacting expressions . . . . . . . . . 472

16.16CRACK: Solving overdetermined systems of PDEs or ODEs . . . 473

16.17CVIT: Fast calculation of Dirac gamma matrix traces . . . . . . . 474

16.18DEFINT: A definite integration interface . . . . . . . . . . . . . . 483

16.18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 483

16.18.2 Integration between zero and infinity . . . . . . . . . . . 483

16.18.3 Integration over other ranges . . . . . . . . . . . . . . . . 484

16.18.4 Using the definite integration package . . . . . . . . . . . 485

16.18.5 Integral Transforms . . . . . . . . . . . . . . . . . . . . . 487

12 CONTENTS

16.18.6 Additional Meijer G-function Definitions . . . . . . . . . 489

16.18.7 The print_conditions function . . . . . . . . . . . . . . . 490

16.18.8 Tracing . . . . . . . . . . . . . . . . . . . . . . . . . . . 491

16.18.9 Acknowledgements . . . . . . . . . . . . . . . . . . . . . 491

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491

16.19DESIR: Differential linear homogeneous equation solutions in theneighborhood of irregular and regular singular points . . . . . . . 493

16.19.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . 493

16.19.2 FORMS OF SOLUTIONS . . . . . . . . . . . . . . . . . 494

16.19.3 INTERACTIVE USE . . . . . . . . . . . . . . . . . . . . 495

16.19.4 DIRECT USE . . . . . . . . . . . . . . . . . . . . . . . 495

16.19.5 USEFUL FUNCTIONS . . . . . . . . . . . . . . . . . . 496

16.19.6 LIMITATIONS . . . . . . . . . . . . . . . . . . . . . . . 499

16.20DFPART: Derivatives of generic functions . . . . . . . . . . . . . 500

16.20.1 Generic Functions . . . . . . . . . . . . . . . . . . . . . 500

16.20.2 Partial Derivatives . . . . . . . . . . . . . . . . . . . . . 501

16.20.3 Substitutions . . . . . . . . . . . . . . . . . . . . . . . . 503

16.21DUMMY: Canonical form of expressions with dummy variables . 505

16.21.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 505

16.21.2 Dummy variables and dummy summations . . . . . . . . 506

16.21.3 The Operators and their Properties . . . . . . . . . . . . . 508

16.21.4 The Function CANONICAL . . . . . . . . . . . . . . . . 509

16.21.5 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . 510

16.22EXCALC: A differential geometry package . . . . . . . . . . . . 512

16.22.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 512

16.22.2 Declarations . . . . . . . . . . . . . . . . . . . . . . . . 513

16.22.3 Exterior Multiplication . . . . . . . . . . . . . . . . . . . 514

16.22.4 Partial Differentiation . . . . . . . . . . . . . . . . . . . 515

16.22.5 Exterior Differentiation . . . . . . . . . . . . . . . . . . . 516

16.22.6 Inner Product . . . . . . . . . . . . . . . . . . . . . . . . 518

16.22.7 Lie Derivative . . . . . . . . . . . . . . . . . . . . . . . . 519

CONTENTS 13

16.22.8 Hodge-* Duality Operator . . . . . . . . . . . . . . . . . 519

16.22.9 Variational Derivative . . . . . . . . . . . . . . . . . . . 520

16.22.10Handling of Indices . . . . . . . . . . . . . . . . . . . . . 521

16.22.11Metric Structures . . . . . . . . . . . . . . . . . . . . . . 524

16.22.12Riemannian Connections . . . . . . . . . . . . . . . . . . 528

16.22.13Killing Vectors . . . . . . . . . . . . . . . . . . . . . . . 529

16.22.14Ordering and Structuring . . . . . . . . . . . . . . . . . . 530

16.22.15Summary of Operators and Commands . . . . . . . . . . 532

16.22.16Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 533

16.23FIDE: Finite difference method for partial differential equations . 544

16.23.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . 544

16.23.2 EXPRES . . . . . . . . . . . . . . . . . . . . . . . . . . 545

16.23.3 IIMET . . . . . . . . . . . . . . . . . . . . . . . . . . . 548

16.23.4 APPROX . . . . . . . . . . . . . . . . . . . . . . . . . . 561

16.23.5 CHARPOL . . . . . . . . . . . . . . . . . . . . . . . . . 564

16.23.6 HURWP . . . . . . . . . . . . . . . . . . . . . . . . . . 567

16.23.7 LINBAND . . . . . . . . . . . . . . . . . . . . . . . . . 568

16.24FPS: Automatic calculation of formal power series . . . . . . . . 572

16.24.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 572

16.24.2 REDUCE operator FPS . . . . . . . . . . . . . . . . . . 572

16.24.3 REDUCE operator SimpleDE . . . . . . . . . . . . . . 574

16.24.4 Problems in the current version . . . . . . . . . . . . . . 574

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574

16.25GCREF: A Graph Cross Referencer . . . . . . . . . . . . . . . . 576

16.25.1 Basic Usage . . . . . . . . . . . . . . . . . . . . . . . . . 576

16.25.2 Shell Script "gcref" . . . . . . . . . . . . . . . . . . . . . 576

16.25.3 Redering with yED . . . . . . . . . . . . . . . . . . . . . 576

16.26GENTRAN: A code generation package . . . . . . . . . . . . . . 578

16.27GNUPLOT: Display of functions and surfaces . . . . . . . . . . . 579

16.27.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 579

14 CONTENTS

16.27.2 Command plot . . . . . . . . . . . . . . . . . . . . . . 579

16.27.3 Paper output . . . . . . . . . . . . . . . . . . . . . . . . 583

16.27.4 Mesh generation for implicit curves . . . . . . . . . . . . 583

16.27.5 Mesh generation for surfaces . . . . . . . . . . . . . . . . 584

16.27.6 GNUPLOT operation . . . . . . . . . . . . . . . . . . . . 584

16.27.7 Saving GNUPLOT command sequences . . . . . . . . . . 584

16.27.8 Direct Call of GNUPLOT . . . . . . . . . . . . . . . . . . 585

16.27.9 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 585

16.28GROEBNER: A Gröbner basis package . . . . . . . . . . . . . . 590

16.28.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . 590

16.28.2 Loading of the Package . . . . . . . . . . . . . . . . . . . 593

16.28.3 The Basic Operators . . . . . . . . . . . . . . . . . . . . 593

16.28.4 Ideal Decomposition & Equation System Solving . . . . . 613

16.28.5 Calculations “by Hand” . . . . . . . . . . . . . . . . . . 617

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 620

16.29GUARDIAN: Guarded Expressions in Practice . . . . . . . . . . 622

16.29.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 622

16.29.2 An outline of our method . . . . . . . . . . . . . . . . . . 623

16.29.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 632

16.29.4 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . 634

16.29.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 637

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637

16.30IDEALS: Arithmetic for polynomial ideals . . . . . . . . . . . . 639

16.30.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 639

16.30.2 Initialization . . . . . . . . . . . . . . . . . . . . . . . . 639

16.30.3 Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639

16.30.4 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 640

16.30.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 641

16.31INEQ: Support for solving inequalities . . . . . . . . . . . . . . . 642

16.32INVBASE: A package for computing involutive bases . . . . . . . 644

CONTENTS 15

16.32.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 644

16.32.2 The Basic Operators . . . . . . . . . . . . . . . . . . . . 645

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647

16.33LALR: A parser generator . . . . . . . . . . . . . . . . . . . . . 648

16.33.1 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . 649

16.33.2 An example . . . . . . . . . . . . . . . . . . . . . . . . . 650

16.34LAPLACE: Laplace transforms . . . . . . . . . . . . . . . . . . . 651

16.35LIE: Functions for the classification of real n-dimensional Lie al-gebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 656

16.36LIMITS: A package for finding limits . . . . . . . . . . . . . . . 657

16.36.1 Normal entry points . . . . . . . . . . . . . . . . . . . . 657

16.36.2 Direction-dependent limits . . . . . . . . . . . . . . . . . 657

16.37LINALG: Linear algebra package . . . . . . . . . . . . . . . . . 658

16.37.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 658

16.37.2 Getting started . . . . . . . . . . . . . . . . . . . . . . . 659

16.37.3 What’s available . . . . . . . . . . . . . . . . . . . . . . 660

16.37.4 Fast Linear Algebra . . . . . . . . . . . . . . . . . . . . . 684

16.37.5 Acknowledgments . . . . . . . . . . . . . . . . . . . . . 685

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685

16.38LISTVECOPS: Vector operations on lists . . . . . . . . . . . . . 686

16.39LPDO: Linear Partial Differential Operators . . . . . . . . . . . . 689

16.39.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 689

16.39.2 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 690

16.39.3 Shapes of F-elements . . . . . . . . . . . . . . . . . . . . 691

16.39.4 Commands . . . . . . . . . . . . . . . . . . . . . . . . . 692

16.40MODSR: Modular solve and roots . . . . . . . . . . . . . . . . . 699

16.41MRVLIMIT: A new exp-log limits package . . . . . . . . . . . . 700

16.41.1 The Exp-Log Limits package . . . . . . . . . . . . . . . . 700

16.41.2 The Algorithm . . . . . . . . . . . . . . . . . . . . . . . 701

16.41.3 The tracing facility . . . . . . . . . . . . . . . . . . . . . 703

16 CONTENTS

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706

16.42NCPOLY: Non–commutative polynomial ideals . . . . . . . . . . 706

16.42.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 706

16.42.2 Setup, Cleanup . . . . . . . . . . . . . . . . . . . . . . . 706

16.42.3 Left and right ideals . . . . . . . . . . . . . . . . . . . . 708

16.42.4 Gröbner bases . . . . . . . . . . . . . . . . . . . . . . . . 708

16.42.5 Left or right polynomial division . . . . . . . . . . . . . . 709

16.42.6 Left or right polynomial reduction . . . . . . . . . . . . . 710

16.42.7 Factorization . . . . . . . . . . . . . . . . . . . . . . . . 710

16.42.8 Output of expressions . . . . . . . . . . . . . . . . . . . 711

16.43NORMFORM: Computation of matrix normal forms . . . . . . . 713

16.43.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 713

16.43.2 Smith normal form . . . . . . . . . . . . . . . . . . . . . 714

16.43.3 smithex_int . . . . . . . . . . . . . . . . . . . . . . . . . 715

16.43.4 frobenius . . . . . . . . . . . . . . . . . . . . . . . . . . 716

16.43.5 ratjordan . . . . . . . . . . . . . . . . . . . . . . . . . . 717

16.43.6 jordansymbolic . . . . . . . . . . . . . . . . . . . . . . . 718

16.43.7 jordan . . . . . . . . . . . . . . . . . . . . . . . . . . . . 720

16.43.8 Algebraic extensions: Using the ARNUM package . . . . . 721

16.43.9 Modular arithmetic . . . . . . . . . . . . . . . . . . . . . 722

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723

16.44NUMERIC: Solving numerical problems . . . . . . . . . . . . . 724

16.44.1 Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . 724

16.44.2 Minima . . . . . . . . . . . . . . . . . . . . . . . . . . . 725

16.44.3 Roots of Functions/ Solutions of Equations . . . . . . . . 726

16.44.4 Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 727

16.44.5 Ordinary Differential Equations . . . . . . . . . . . . . . 728

16.44.6 Bounds of a Function . . . . . . . . . . . . . . . . . . . . 730

16.44.7 Chebyshev Curve Fitting . . . . . . . . . . . . . . . . . . 731

16.44.8 General Curve Fitting . . . . . . . . . . . . . . . . . . . 732

CONTENTS 17

16.44.9 Function Bases . . . . . . . . . . . . . . . . . . . . . . . 733

16.45ODESOLVE: Ordinary differential equations solver . . . . . . . . 735

16.45.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 735

16.45.2 Installation . . . . . . . . . . . . . . . . . . . . . . . . . 736

16.45.3 User interface . . . . . . . . . . . . . . . . . . . . . . . . 737

16.45.4 Output syntax . . . . . . . . . . . . . . . . . . . . . . . . 743

16.45.5 Solution techniques . . . . . . . . . . . . . . . . . . . . . 743

16.45.6 Extension interface . . . . . . . . . . . . . . . . . . . . . 748

16.45.7 Change log . . . . . . . . . . . . . . . . . . . . . . . . . 751

16.45.8 Planned developments . . . . . . . . . . . . . . . . . . . 751

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 752

16.46ORTHOVEC: Manipulation of scalars and vectors . . . . . . . . . 754

16.46.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 754

16.46.2 Initialisation . . . . . . . . . . . . . . . . . . . . . . . . 755

16.46.3 Input-Output . . . . . . . . . . . . . . . . . . . . . . . . 755

16.46.4 Algebraic Operations . . . . . . . . . . . . . . . . . . . . 756

16.46.5 Differential Operations . . . . . . . . . . . . . . . . . . . 758

16.46.6 Integral Operations . . . . . . . . . . . . . . . . . . . . . 760

16.46.7 Test Cases . . . . . . . . . . . . . . . . . . . . . . . . . . 760

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763

16.47PHYSOP: Operator calculus in quantum theory . . . . . . . . . . 764

16.47.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 764

16.47.2 The NONCOM2 Package . . . . . . . . . . . . . . . . . 764

16.47.3 The PHYSOP package . . . . . . . . . . . . . . . . . . . 765

16.47.4 Known problems in the current release of PHYSOP . . . . 773

16.47.5 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . 773

16.47.6 Appendix: List of error and warning messages . . . . . . 774

16.48PM: A REDUCE pattern matcher . . . . . . . . . . . . . . . . . . 776

16.48.1 M(exp,temp) . . . . . . . . . . . . . . . . . . . . . . 777

16.48.2 temp _= logical_exp . . . . . . . . . . . . . . . . . . . . 778

18 CONTENTS

16.48.3 S(exp,{temp1 -> sub1, temp2 -> sub2, . . . }, rept, depth) . 779

16.48.4 temp :- exp and temp ::- exp . . . . . . . . . . . . . . . . 780

16.48.5 Arep({rep1,rep2,. . . }) . . . . . . . . . . . . . . . . . . . 781

16.48.6 Drep({rep1,rep2,..}) . . . . . . . . . . . . . . . . . . . . 781

16.48.7 Switches . . . . . . . . . . . . . . . . . . . . . . . . . . 781

16.49QHULL: Compute the complex hull . . . . . . . . . . . . . . . . 783

16.50QSUM: Indefinite and Definite Summation of q-hypergeometricTerms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784

16.50.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 784

16.50.2 Elementary q-Functions . . . . . . . . . . . . . . . . . . 784

16.50.3 q-Gosper Algorithm . . . . . . . . . . . . . . . . . . . . 785

16.50.4 q-Zeilberger Algorithm . . . . . . . . . . . . . . . . . . . 786

16.50.5 REDUCE operator QGOSPER . . . . . . . . . . . . . . . 787

16.50.6 REDUCE operator QSUMRECURSION . . . . . . . . . . 789

16.50.7 Simplification Operators . . . . . . . . . . . . . . . . . . 794

16.50.8 Global Variables and Switches . . . . . . . . . . . . . . . 795

16.50.9 Messages . . . . . . . . . . . . . . . . . . . . . . . . . . 796

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 797

16.51RANDPOLY: A random polynomial generator . . . . . . . . . . . 799

16.51.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 799

16.51.2 Basic use of randpoly . . . . . . . . . . . . . . . . . . 800

16.51.3 Advanced use of randpoly . . . . . . . . . . . . . . . . 801

16.51.4 Subsidiary functions: rand, proc, random . . . . . . . . . 802

16.51.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 804

16.51.6 Appendix: Algorithmic background . . . . . . . . . . . . 805

16.52RATAPRX: Rational Approximations Package for REDUCE . . . 809

16.52.1 Periodic Decimal Representation . . . . . . . . . . . . . . 809

16.52.2 Continued Fractions . . . . . . . . . . . . . . . . . . . . 811

16.52.3 Padé Approximation . . . . . . . . . . . . . . . . . . . . 820

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824

CONTENTS 19

16.53RATINT: Integrate Rational Functions using the Minimal Alge-braic Extension to the Constant Field . . . . . . . . . . . . . . . . 825

16.53.1 Rational Integration . . . . . . . . . . . . . . . . . . . . 825

16.53.2 The Algorithm . . . . . . . . . . . . . . . . . . . . . . . 827

16.53.3 The log_sum operator . . . . . . . . . . . . . . . . . . . 829

16.53.4 Options . . . . . . . . . . . . . . . . . . . . . . . . . . . 830

16.53.5 Hermite’s method . . . . . . . . . . . . . . . . . . . . . . 833

16.53.6 Tracing the ratint program . . . . . . . . . . . . . . . . 834

16.53.7 Bugs, suggestions and comments . . . . . . . . . . . . . 835

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835

16.54REACTEQN: Support for chemical reaction equation systems . . 835

16.55REDLOG: Extend REDUCE to a computer logic system . . . . . 840

16.56RESET: Code to reset REDUCE to its initial state . . . . . . . . . 840

16.57RESIDUE: A residue package . . . . . . . . . . . . . . . . . . . 841

16.58RLFI: REDUCE LATEX formula interface . . . . . . . . . . . . . . 845

16.58.1 APPENDIX: Summary and syntax . . . . . . . . . . . . . 847

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 849

16.59ROOTS: A REDUCE root finding package . . . . . . . . . . . . . 851

16.59.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 851

16.59.2 Root Finding Strategies . . . . . . . . . . . . . . . . . . . 851

16.59.3 Top Level Functions . . . . . . . . . . . . . . . . . . . . 852

16.59.4 Switches Used in Input . . . . . . . . . . . . . . . . . . . 855

16.59.5 Internal and Output Use of Switches . . . . . . . . . . . . 856

16.59.6 Root Package Switches . . . . . . . . . . . . . . . . . . . 856

16.59.7 Operational Parameters and Parameter Setting. . . . . . . 857

16.59.8 Avoiding truncation of polynomials on input . . . . . . . 858

16.60RSOLVE: Rational/integer polynomial solvers . . . . . . . . . . . 859

16.60.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 859

16.60.2 The user interface . . . . . . . . . . . . . . . . . . . . . . 859

16.60.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 860

16.60.4 Tracing . . . . . . . . . . . . . . . . . . . . . . . . . . . 861

20 CONTENTS

16.61RTRACE: Tracing in REDUCE . . . . . . . . . . . . . . . . . . 862

16.61.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 862

16.61.2 RTrace versus RDebug . . . . . . . . . . . . . . . . . . . 862

16.61.3 Procedure tracing: RTR, UNRTR . . . . . . . . . . . . . 863

16.61.4 Assignment tracing: RTRST, UNRTRST . . . . . . . . . 865

16.61.5 Tracing active rules: TRRL, UNTRRL . . . . . . . . . . 867

16.61.6 Tracing inactive rules: TRRLID, UNTRRLID . . . . . . . 868

16.61.7 Output control: RTROUT . . . . . . . . . . . . . . . . . 869

16.62SCOPE: REDUCE source code optimization package . . . . . . . 870

16.63SETS: A basic set theory package . . . . . . . . . . . . . . . . . 871

16.63.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 871

16.63.2 Infix operator precedence . . . . . . . . . . . . . . . . . . 872

16.63.3 Explicit set representation and mkset . . . . . . . . . . . 872

16.63.4 Union and intersection . . . . . . . . . . . . . . . . . . . 873

16.63.5 Symbolic set expressions . . . . . . . . . . . . . . . . . . 873

16.63.6 Set difference . . . . . . . . . . . . . . . . . . . . . . . . 874

16.63.7 Predicates on sets . . . . . . . . . . . . . . . . . . . . . . 875

16.63.8 Possible future developments . . . . . . . . . . . . . . . . 879

16.64SPARSE: Sparse Matrix Calculations . . . . . . . . . . . . . . . 880

16.64.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 880

16.64.2 Sparse Matrix Calculations . . . . . . . . . . . . . . . . . 880

16.64.3 Sparse Matrix Expressions . . . . . . . . . . . . . . . . . 881

16.64.4 Operators with Sparse Matrix Arguments . . . . . . . . . 881

16.64.5 The Linear Algebra Package for Sparse Matrices . . . . . 883

16.64.6 Available Functions . . . . . . . . . . . . . . . . . . . . . 884

16.64.7 Fast Linear Algebra . . . . . . . . . . . . . . . . . . . . . 906

16.64.8 Acknowledgments . . . . . . . . . . . . . . . . . . . . . 906

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 906

16.65SPDE: Finding symmetry groups of PDE’s . . . . . . . . . . . . . 907

16.65.1 Description of the System Functions and Variables . . . . 907

CONTENTS 21

16.65.2 How to Use the Package . . . . . . . . . . . . . . . . . . 910

16.65.3 Test File . . . . . . . . . . . . . . . . . . . . . . . . . . . 916

16.66SPECFN: Package for special functions . . . . . . . . . . . . . . 919

16.66.1 Simplification and Approximation . . . . . . . . . . . . . 920

16.66.2 Constants . . . . . . . . . . . . . . . . . . . . . . . . . . 922

16.66.3 Bernoulli Numbers and Euler Numbers . . . . . . . . . . 922

16.66.4 Fibonacci Numbers and Fibonacci Polynomials . . . . . . 923

16.66.5 Stirling Numbers . . . . . . . . . . . . . . . . . . . . . . 923

16.66.6 The Γ Function, and Related Functions . . . . . . . . . . 923

16.66.7 Bessel Functions . . . . . . . . . . . . . . . . . . . . . . 925

16.66.8 Hypergeometric and Other Functions . . . . . . . . . . . 926

16.66.9 Integral Functions . . . . . . . . . . . . . . . . . . . . . 926

16.66.10Airy Functions . . . . . . . . . . . . . . . . . . . . . . . 927

16.66.11Polynomial Functions . . . . . . . . . . . . . . . . . . . 927

16.66.12Spherical and Solid Harmonics . . . . . . . . . . . . . . . 928

16.66.13Jacobi’s Elliptic Functions . . . . . . . . . . . . . . . . . 929

16.66.14Elliptic Integrals . . . . . . . . . . . . . . . . . . . . . . 931

16.66.15Elliptic Theta Functions . . . . . . . . . . . . . . . . . . 934

16.66.16Lambert’s W function . . . . . . . . . . . . . . . . . . . 936

16.66.173j symbols and Clebsch-Gordan Coefficients . . . . . . . 936

16.66.186j symbols . . . . . . . . . . . . . . . . . . . . . . . . . 936

16.66.19Acknowledgements . . . . . . . . . . . . . . . . . . . . . 936

16.66.20Table of Operators and Constants . . . . . . . . . . . . . 937

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 940

16.67SPECFN2: Package for special special functions . . . . . . . . . 940

16.67.1 REDUCE operator HYPERGEOMETRIC . . . . . . . . . 941

16.67.2 Extending the HYPERGEOMETRIC operator . . . . . . 942

16.67.3 REDUCE operator meijerg . . . . . . . . . . . . . . . 942

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 943

16.68SSTOOLS: Computations with supersymmetric algebraic and dif-ferential expressions . . . . . . . . . . . . . . . . . . . . . . . . 944

22 CONTENTS

16.68.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . 944

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 945

16.69SUM: A package for series summation . . . . . . . . . . . . . . . 946

16.70SYMMETRY: Operations on symmetric matrices . . . . . . . . . 948

16.70.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 948

16.70.2 Operators for linear representations . . . . . . . . . . . . 948

16.70.3 Display Operators . . . . . . . . . . . . . . . . . . . . . 950

16.70.4 Storing a new group . . . . . . . . . . . . . . . . . . . . 950

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 952

16.71TAYLOR: Manipulation of Taylor series . . . . . . . . . . . . . . 953

16.71.1 Basic Use . . . . . . . . . . . . . . . . . . . . . . . . . . 953

16.71.2 Caveats . . . . . . . . . . . . . . . . . . . . . . . . . . . 957

16.71.3 Warning messages . . . . . . . . . . . . . . . . . . . . . 958

16.71.4 Error messages . . . . . . . . . . . . . . . . . . . . . . . 958

16.71.5 Comparison to other packages . . . . . . . . . . . . . . . 960

16.72TPS: A extendible power series package . . . . . . . . . . . . . . 962

16.72.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 962

16.72.2 PS Operator . . . . . . . . . . . . . . . . . . . . . . . . . 962

16.72.3 PSEXPLIM Operator . . . . . . . . . . . . . . . . . . . . 964

16.72.4 PSPRINTORDER Switch . . . . . . . . . . . . . . . . . 964

16.72.5 PSORDLIM Operator . . . . . . . . . . . . . . . . . . . 964

16.72.6 PSTERM Operator . . . . . . . . . . . . . . . . . . . . . 965

16.72.7 PSORDER Operator . . . . . . . . . . . . . . . . . . . . 965

16.72.8 PSSETORDER Operator . . . . . . . . . . . . . . . . . . 965

16.72.9 PSDEPVAR Operator . . . . . . . . . . . . . . . . . . . 966

16.72.10PSEXPANSIONPT operator . . . . . . . . . . . . . . . . 966

16.72.11PSFUNCTION Operator . . . . . . . . . . . . . . . . . . 966

16.72.12PSCHANGEVAR Operator . . . . . . . . . . . . . . . . 967

16.72.13PSREVERSE Operator . . . . . . . . . . . . . . . . . . . 967

16.72.14PSCOMPOSE Operator . . . . . . . . . . . . . . . . . . 968

CONTENTS 23

16.72.15PSSUM Operator . . . . . . . . . . . . . . . . . . . . . . 969

16.72.16PSTAYLOR Operator . . . . . . . . . . . . . . . . . . . 970

16.72.17PSCOPY Operator . . . . . . . . . . . . . . . . . . . . . 970

16.72.18PSTRUNCATE Operator . . . . . . . . . . . . . . . . . . 971

16.72.19Arithmetic Operations . . . . . . . . . . . . . . . . . . . 971

16.72.20Differentiation . . . . . . . . . . . . . . . . . . . . . . . 972

16.72.21Restrictions and Known Bugs . . . . . . . . . . . . . . . 972

16.73TRI: TeX REDUCE interface . . . . . . . . . . . . . . . . . . . . 974

16.74TRIGD: Trigonometrical Functions with Degree Arguments . . . 975

16.74.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 975

16.74.2 Simplification . . . . . . . . . . . . . . . . . . . . . . . . 977

16.74.3 Numerical Evaluation . . . . . . . . . . . . . . . . . . . 978

16.74.4 Bugs, Restrictions and Planned Extensions . . . . . . . . 981

16.75TRIGINT: Weierstrass substitution in REDUCE . . . . . . . . . . 983

16.75.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 983

16.75.2 Statement of the Algorithm . . . . . . . . . . . . . . . . . 984

16.75.3 REDUCE implementation . . . . . . . . . . . . . . . . . 984

16.75.4 Definite Integration . . . . . . . . . . . . . . . . . . . . . 986

16.75.5 Tracing the trigint function . . . . . . . . . . . . . . . . 987

16.75.6 Bugs, comments, suggestions . . . . . . . . . . . . . . . 987

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 988

16.76TRIGSIMP: Simplification and factorization of trigonometric andhyperbolic functions . . . . . . . . . . . . . . . . . . . . . . . . 988

16.76.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 988

16.76.2 Simplifying trigonometric expressions . . . . . . . . . . . 988

16.76.3 Factorizing trigonometric expressions . . . . . . . . . . . 992

16.76.4 GCDs of trigonometric expressions . . . . . . . . . . . . 993

16.76.5 Further Examples . . . . . . . . . . . . . . . . . . . . . . 993

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 997

16.77TURTLE: Turtle Graphics Interface for REDUCE . . . . . . . . . 998

16.77.1 Turtle Graphics . . . . . . . . . . . . . . . . . . . . . . . 998

24 CONTENTS

16.77.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . 998

16.77.3 Turtle Functions . . . . . . . . . . . . . . . . . . . . . . 999

16.77.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 1004

16.77.5 References . . . . . . . . . . . . . . . . . . . . . . . . . 1010

16.78WU: Wu algorithm for polynomial systems . . . . . . . . . . . . 1012

16.79XCOLOR: Color factor in some field theories . . . . . . . . . . . 1014

16.80XIDEAL: Gröbner Bases for exterior algebra . . . . . . . . . . . 1016

16.80.1 Description . . . . . . . . . . . . . . . . . . . . . . . . . 1016

16.80.2 Declarations . . . . . . . . . . . . . . . . . . . . . . . . 1017

16.80.3 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 1018

16.80.4 Switches . . . . . . . . . . . . . . . . . . . . . . . . . . 1020

16.80.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 1020

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1023

16.81ZEILBERG: Indefinite and definite summation . . . . . . . . . . 1024

16.81.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 1024

16.81.2 Gosper Algorithm . . . . . . . . . . . . . . . . . . . . . 1024

16.81.3 Zeilberger Algorithm . . . . . . . . . . . . . . . . . . . . 1025

16.81.4 REDUCE operator GOSPER . . . . . . . . . . . . . . . . 1026

16.81.5 REDUCE operator EXTENDED_GOSPER . . . . . . . . . 1029

16.81.6 REDUCE operator SUMRECURSION . . . . . . . . . . . 1029

16.81.7 REDUCE operator EXTENDED_SUMRECURSION . . . . 1032

16.81.8 REDUCE operator HYPERRECURSION . . . . . . . . . . 1033

16.81.9 REDUCE operator HYPERSUM . . . . . . . . . . . . . . 1035

16.81.10REDUCE operator SUMTOHYPER . . . . . . . . . . . . . 1037

16.81.11Simplification Operators . . . . . . . . . . . . . . . . . . 1037

16.81.12Tracing . . . . . . . . . . . . . . . . . . . . . . . . . . . 1039

16.81.13Global Variables and Switches . . . . . . . . . . . . . . . 1041

16.81.14Messages . . . . . . . . . . . . . . . . . . . . . . . . . . 1042

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1043

16.82ZTRANS: Z-transform package . . . . . . . . . . . . . . . . . . 1045

CONTENTS 25

16.82.1 Z-Transform . . . . . . . . . . . . . . . . . . . . . . . . 1045

16.82.2 Inverse Z-Transform . . . . . . . . . . . . . . . . . . . . 1045

16.82.3 Input for the Z-Transform . . . . . . . . . . . . . . . . . 1045

16.82.4 Input for the Inverse Z-Transform . . . . . . . . . . . . . 1046

16.82.5 Application of the Z-Transform . . . . . . . . . . . . . . 1047

16.82.6 EXAMPLES . . . . . . . . . . . . . . . . . . . . . . . . 1047

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1053

17 Symbolic Mode 1055

17.1 Symbolic Infix Operators . . . . . . . . . . . . . . . . . . . . . . 1057

17.2 Symbolic Expressions . . . . . . . . . . . . . . . . . . . . . . . . 1057

17.3 Quoted Expressions . . . . . . . . . . . . . . . . . . . . . . . . . 1057

17.4 Lambda Expressions . . . . . . . . . . . . . . . . . . . . . . . . 1057

17.5 Symbolic Assignment Statements . . . . . . . . . . . . . . . . . 1058

17.6 FOR EACH Statement . . . . . . . . . . . . . . . . . . . . . . . 1059

17.7 Symbolic Procedures . . . . . . . . . . . . . . . . . . . . . . . . 1059

17.8 Standard Lisp Equivalent of Reduce Input . . . . . . . . . . . . . 1060

17.9 Communicating with Algebraic Mode . . . . . . . . . . . . . . . 1060

17.9.1 Passing Algebraic Mode Values to Symbolic Mode . . . . 1061

17.9.2 Passing Symbolic Mode Values to Algebraic Mode . . . . 1064

17.9.3 Complete Example . . . . . . . . . . . . . . . . . . . . . 1064

17.9.4 Defining Procedures for Intermode Communication . . . . 1065

17.10Rlisp ’88 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1066

17.11References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1066

18 Calculations in High Energy Physics 1067

18.1 High Energy Physics Operators . . . . . . . . . . . . . . . . . . . 1067

18.1.1 . (Cons) Operator . . . . . . . . . . . . . . . . . . . . . . 1067

18.1.2 G Operator for Gamma Matrices . . . . . . . . . . . . . . 1068

18.1.3 EPS Operator . . . . . . . . . . . . . . . . . . . . . . . . 1069

18.2 Vector Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 1069

26 CONTENTS

18.3 Additional Expression Types . . . . . . . . . . . . . . . . . . . . 1070

18.3.1 Vector Expressions . . . . . . . . . . . . . . . . . . . . . 1070

18.3.2 Dirac Expressions . . . . . . . . . . . . . . . . . . . . . 1070

18.4 Trace Calculations . . . . . . . . . . . . . . . . . . . . . . . . . 1071

18.5 Mass Declarations . . . . . . . . . . . . . . . . . . . . . . . . . . 1071

18.6 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1072

18.7 Extensions to More Than Four Dimensions . . . . . . . . . . . . 1073

19 REDUCE and Rlisp Utilities 1075

19.1 The Standard Lisp Compiler . . . . . . . . . . . . . . . . . . . . 1075

19.2 Fast Loading Code Generation Program . . . . . . . . . . . . . . 1076

19.3 The Standard Lisp Cross Reference Program . . . . . . . . . . . . 1077

19.3.1 Restrictions . . . . . . . . . . . . . . . . . . . . . . . . . 1078

19.3.2 Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1078

19.3.3 Options . . . . . . . . . . . . . . . . . . . . . . . . . . . 1078

19.4 Prettyprinting REDUCE Expressions . . . . . . . . . . . . . . . . 1078

19.5 Prettyprinting Standard Lisp S-Expressions . . . . . . . . . . . . 1079

20 Maintaining REDUCE 1081

A Reserved Identifiers 1085

B Changes since Version 3.8 1091

Abstract

This document provides the user with a description of the algebraic programmingsystem REDUCE. The capabilities of this system include:

1. expansion and ordering of polynomials and rational functions,

2. substitutions and pattern matching in a wide variety of forms,

3. automatic and user controlled simplification of expressions,

4. calculations with symbolic matrices,

5. arbitrary precision integer and real arithmetic,

6. facilities for defining new functions and extending program syntax,

7. analytic differentiation and integration,

8. factorization of polynomials,

9. facilities for the solution of a variety of algebraic equations,

10. facilities for the output of expressions in a variety of formats,

11. facilities for generating numerical programs from symbolic input,

12. Dirac matrix calculations of interest to high energy physicists.

27

28 CONTENTS

Acknowledgment

The production of this version of the manual has been the result of the contribu-tions of a large number of individuals who have taken the time and effort to suggestimprovements to previous versions, and to draft new sections. Particular thanksare due to Gerry Rayna, who provided a draft rewrite of most of the first half ofthe manual. Other people who have made significant contributions have includedJohn Fitch, Martin Griss, Stan Kameny, Jed Marti, Herbert Melenk, Don Morri-son, Arthur Norman, Eberhard Schrüfer, Larry Seward and Walter Tietze. Finally,Richard Hitt produced a TEX version of the REDUCE 3.3 manual, which has beena useful guide for the production of the LATEX version of this manual.

29

30 CONTENTS

Chapter 1

Introductory Information

REDUCE is a system for carrying out algebraic operations accurately, no matterhow complicated the expressions become. It can manipulate polynomials in a va-riety of forms, both expanding and factoring them, and extract various parts ofthem as required. REDUCE can also do differentiation and integration, but weshall only show trivial examples of this in this introduction. Other topics not con-sidered include the use of arrays, the definition of procedures and operators, thespecific routines for high energy physics calculations, the use of files to eliminaterepetitious typing and for saving results, and the editing of the input text.

Also not considered in any detail in this introduction are the many options thatare available for varying computational procedures, output forms, number systemsused, and so on.

REDUCE is designed to be an interactive system, so that the user can input an al-gebraic expression and see its value before moving on to the next calculation. Forthose systems that do not support interactive use, or for those calculations, espe-cially long ones, for which a standard script can be defined, REDUCE can also beused in batch mode. In this case, a sequence of commands can be given to RE-DUCE and results obtained without any user interaction during the computation.

In this introduction, we shall limit ourselves to the interactive use of REDUCE,since this illustrates most completely the capabilities of the system. When RE-DUCE is called, it begins by printing a banner message like:

Reduce (Free CSL version), 25-Oct-14 ...

where the version number and the system release date will change from time totime. It proceeds to execute the commands in user’s startup (reducerc) file, ifsuch a file is present, then prompts the user for input by:

1:

31

32 CHAPTER 1. INTRODUCTORY INFORMATION

You can now type a REDUCE statement, terminated by a semicolon to indicate theend of the expression, for example:

(x+y+z)^2;

This expression would normally be followed by another character (a Return onan ASCII keyboard) to “wake up” the system, which would then input the expres-sion, evaluate it, and return the result:

2 2 2X + 2*X*Y + 2*X*Z + Y + 2*Y*Z + Z

Let us review this simple example to learn a little more about the way that RE-DUCE works. First, we note that REDUCE deals with variables, and constantslike other computer languages, but that in evaluating the former, a variable canstand for itself. Expression evaluation normally follows the rules of high schoolalgebra, so the only surprise in the above example might be that the expression wasexpanded. REDUCE normally expands expressions where possible, collecting liketerms and ordering the variables in a specific manner. However, expansion, order-ing of variables, format of output and so on is under control of the user, and variousdeclarations are available to manipulate these.

Another characteristic of the above example is the use of lower case on input andupper case on output. In fact, input may be in either mode, but output is usually inlower case. To make the difference between input and output more distinct in thismanual, all expressions intended for input will be shown in lower case and outputin upper case. However, for stylistic reasons, we represent all single identifiers inthe text in upper case.

Finally, the numerical prompt can be used to reference the result in a later compu-tation.

As a further illustration of the system features, the user should try:

for i:= 1:40 product i;

The result in this case is the value of 40!,

815915283247897734345611269596115894272000000000

You can also get the same result by saying

factorial 40;

Since we want exact results in algebraic calculations, it is essential that integerarithmetic be performed to arbitrary precision, as in the above example. Further-

33

more, the FOR statement in the above is illustrative of a whole range of combiningforms that REDUCE supports for the convenience of the user.

Among the many options in REDUCE is the use of other number systems, such asmultiple precision floating point with any specified number of digits — of use ifroundoff in, say, the 100th digit is all that can be tolerated.

In many cases, it is necessary to use the results of one calculation in succeedingcalculations. One way to do this is via an assignment for a variable, such as

u := (x+y+z)^2;

If we now use U in later calculations, the value of the right-hand side of the abovewill be used.

The results of a given calculation are also saved in the variable WS (for WorkSpace),so this can be used in the next calculation for further processing.

For example, the expression

df(ws,x);

following the previous evaluation will calculate the derivative of (x+y+z)^2withrespect to X. Alternatively,

int(ws,y);

would calculate the integral of the same expression with respect to y.

REDUCE is also capable of handling symbolic matrices. For example,

matrix m(2,2);

declares m to be a two by two matrix, and

m := mat((a,b),(c,d));

gives its elements values. Expressions that include M and make algebraic sensemay now be evaluated, such as 1/m to give the inverse, 2*m - u*m^2 to give usanother matrix and det(m) to give us the determinant of M.

REDUCE has a wide range of substitution capabilities. The system knows aboutelementary functions, but does not automatically invoke many of their well-knownproperties. For example, products of trigonometrical functions are not convertedautomatically into multiple angle expressions, but if the user wants this, he can say,for example:

(sin(a+b)+cos(a+b))*(sin(a-b)-cos(a-b))

34 CHAPTER 1. INTRODUCTORY INFORMATION

where cos(~x)*cos(~y) = (cos(x+y)+cos(x-y))/2,cos(~x)*sin(~y) = (sin(x+y)-sin(x-y))/2,sin(~x)*sin(~y) = (cos(x-y)-cos(x+y))/2;

where the tilde in front of the variables X and Y indicates that the rules apply forall values of those variables. The result of this calculation is

-(COS(2*A) + SIN(2*B))

See also the user-contributed packages ASSIST (chapter 16.5), CAMAL (chap-ter 16.10) and TRIGSIMP (chapter 16.76).

Another very commonly used capability of the system, and an illustration of one ofthe many output modes of REDUCE, is the ability to output results in a FORTRANcompatible form. Such results can then be used in a FORTRAN based numericalcalculation. This is particularly useful as a way of generating algebraic formulasto be used as the basis of extensive numerical calculations.

For example, the statements

on fort;df(log(x)*(sin(x)+cos(x))/sqrt(x),x,2);

will result in the output

ANS=(-4.*LOG(X)*COS(X)*X**2-4.*LOG(X)*COS(X)*X+3.*. LOG(X)*COS(X)-4.*LOG(X)*SIN(X)*X**2+4.*LOG(X)*. SIN(X)*X+3.*LOG(X)*SIN(X)+8.*COS(X)*X-8.*COS(X)-8.. *SIN(X)*X-8.*SIN(X))/(4.*SQRT(X)*X**2)

These algebraic manipulations illustrate the algebraic mode of REDUCE. RE-DUCE is based on Standard Lisp. A symbolic mode is also available for executingLisp statements. These statements follow the syntax of Lisp, e.g.

symbolic car ’(a);

Communication between the two modes is possible.

With this simple introduction, you are now in a position to study the material in thefull REDUCE manual in order to learn just how extensive the range of facilitiesreally is. If further tutorial material is desired, the seven REDUCE InteractiveLessons by David R. Stoutemyer are recommended. These are normally distributedwith the system.

Chapter 2

Structure of Programs

A REDUCE program consists of a set of functional commands which are evaluatedsequentially by the computer. These commands are built up from declarations,statements and expressions. Such entities are composed of sequences of numbers,variables, operators, strings, reserved words and delimiters (such as commas andparentheses), which in turn are sequences of basic characters.

2.1 The REDUCE Standard Character Set

The basic characters which are used to build REDUCE symbols are the following:

1. The 26 letters a through z

2. The 10 decimal digits 0 through 9

3. The special characters _ ! " $ % ’ ( ) * + , - . / : ; <> = { } 〈blank〉

With the exception of strings and characters preceded by an exclamation mark, thecase of characters is ignored: depending of the underlying LISP they will all beconverted internally into lower case or upper case: ALPHA, Alpha and alpharepresent the same symbol. Most implementations allow you to switch this con-version off. The operating instructions for a particular implementation should beconsulted on this point. For portability, we shall limit ourselves to the standardcharacter set in this exposition.

35

36 CHAPTER 2. STRUCTURE OF PROGRAMS

2.2 Numbers

There are several different types of numbers available in REDUCE. Integers consistof a signed or unsigned sequence of decimal digits written without a decimal point,for example:

-2, 5396, +32

In principle, there is no practical limit on the number of digits permitted as exactarithmetic is used in most implementations. (You should however check the spe-cific instructions for your particular system implementation to make sure that thisis true.) For example, if you ask for the value of 22000 you get it displayed as anumber of 603 decimal digits, taking up several lines of output on an interactivedisplay. It should be borne in mind of course that computations with such longnumbers can be quite slow.

Numbers that aren’t integers are usually represented as the quotient of two integers,in lowest terms: that is, as rational numbers.

In essentially all versions of REDUCE it is also possible (but not always desirable!)to ask REDUCE to work with floating point approximations to numbers again, toany precision. Such numbers are called real. They can be input in two ways:

1. as a signed or unsigned sequence of any number of decimal digits with anembedded or trailing decimal point.

2. as in 1. followed by a decimal exponent which is written as the letter Efollowed by a signed or unsigned integer.

e.g. 32. +32.0 0.32E2 and 320.E-1 are all representations of 32.

The declaration SCIENTIFIC_NOTATION controls the output format of float-ing point numbers. At the default settings, any number with five or less dig-its before the decimal point is printed in a fixed-point notation, e.g., 12345.6.Numbers with more than five digits are printed in scientific notation, e.g.,1.234567E+5. Similarly, by default, any number with eleven or more zerosafter the decimal point is printed in scientific notation. To change these defaults,SCIENTIFIC_NOTATION can be used in one of two ways.

SCIENTIFIC_NOTATION m;

where m is a positive integer, sets the printing format so that a number with morethan m digits before the decimal point, or m or more zeros after the decimal point,is printed in scientific notation.

SCIENTIFIC_NOTATION{m,n},

with m and n both positive integers, sets the format so that a number with more

2.3. IDENTIFIERS 37

than m digits before the decimal point, or n or more zeros after the decimal pointis printed in scientific notation.

CAUTION: The unsigned part of any number may not begin with a decimal point,as this causes confusion with the CONS (.) operator, i.e., NOT ALLOWED ARE:.5 -.23 +.12; use 0.5 -0.23 +0.12 instead.

2.3 Identifiers

Identifiers in REDUCE consist of one or more alphanumeric characters (i.e. alpha-betic letters or decimal digits) the first of which must be alphabetic. The maximumnumber of characters allowed is implementation dependent, although twenty-fouris permitted in most implementations. In addition, the underscore character (_) isconsidered a letter if it is within an identifier. For example,

a az p1 q23p a_very_long_variable

are all identifiers, whereas

_a

is not.

A sequence of alphanumeric characters in which the first is a digit is interpreted asa product. For example, 2ab3c is interpreted as 2*ab3c. There is one exceptionto this: If the first letter after a digit is E, the system will try to interpret that part ofthe sequence as a real number, which may fail in some cases. For example, 2E12is the real number 2.0 ∗ 1012, 2e3c is 2000.0*C, and 2ebc gives an error.

Special characters, such as -, *, and blank, may be used in identifiers too, even asthe first character, but each must be preceded by an exclamation mark in input. Forexample:

light!-years d!*!*n good! morning!$sign !5goldrings

CAUTION: Many system identifiers have such special characters in their names(especially * and =). If the user accidentally picks the name of one of them for hisown purposes it may have catastrophic consequences for his REDUCE run. Usersare therefore advised to avoid such names.

Identifiers are used as variables, labels and to name arrays, operators and proce-dures.

38 CHAPTER 2. STRUCTURE OF PROGRAMS

Restrictions

The reserved words listed in section (A may not be used as identifiers. No spacesmay appear within an identifier, and an identifier may not extend over a line of text.

2.4 Variables

Every variable is named by an identifier, and is given a specific type. The type isof no concern to the ordinary user. Most variables are allowed to have the defaulttype, called scalar. These can receive, as values, the representation of any ordinaryalgebraic expression. In the absence of such a value, they stand for themselves.

Reserved Variables

Several variables in REDUCE have particular properties which should not bechanged by the user. These variables include:

CATALAN Catalan’s constant, defined as

∞∑n=0

(−1)n

(2n+ 1)2.

E Intended to represent the base of the natural logarithms. log(e),if it occurs in an expression, is automatically replaced by 1. IfROUNDED is on, E is replaced by the value of E to the current degreeof floating point precision.

EULER_GAMMA Euler’s constant, also available as −ψ(1).

GOLDEN_RATIO The number 1+√

52 .

I Intended to represent the square

root of −1. i^2 is replaced by −1, and appropriately for higherpowers of I. This applies only to the symbol I used on the top level,not as a formal parameter in a procedure, a local variable, nor in thecontext for i:= ....

INFINITY Intended to represent∞in limit and power series calculations for example, as well as in def-inite integration. Note however that the current system does not doproper arithmetic on∞. For example, infinity + infinityis 2*infinity.

2.5. STRINGS 39

KHINCHIN Khinchin’s constant, defined as

∞∏n=1

(1 +

1

n(n+ 2)

)log2 n.

NEGATIVE Used in the Roots package.

NIL In REDUCE (algebraic mode only) taken as a synonym for zero.Therefore NIL cannot be used as a variable.

PI Intended to represent the circular constant. With ROUNDED on, itis replaced by the value of π to the current degree of floating pointprecision.

POSITIVE Used in the Roots package.

T Must not be used as a formal parameter or local variable in proce-dures, since conflict arises with the symbolic mode meaning of T astrue.

Other reserved variables, such as LOW_POW, described in other sections, are listedin Appendix A.

Using these reserved variables inappropriately will lead to errors.

There are also internal variables used by REDUCE that have similar restrictions.These usually have an asterisk in their names, so it is unlikely a casual user woulduse one. An example of such a variable is K!* used in the asymptotic commandpackage.

Certain words are reserved in REDUCE. They may only be used in the mannerintended. A list of these is given in the section “Reserved Identifiers”. There are,of course, an impossibly large number of such names to keep in mind. The readermay therefore want to make himself a copy of the list, deleting the names he doesn’tthink he is likely to use by mistake.

2.5 Strings

Strings are used in WRITE statements, in other output statements (such as errormessages), and to name files. A string consists of any number of characters en-closed in double quotes. For example:

"A String".

40 CHAPTER 2. STRUCTURE OF PROGRAMS

Lower case characters within a string are not converted to upper case.

The string "" represents the empty string. A double quote may be included in astring by preceding it by another double quote. Thus "a""b" is the string a"b,and """" is the string consisting of the single character ".

2.6 Comments

Text can be included in program listings for the convenience of human readers, insuch a way that REDUCE pays no attention to it. There are two ways to do this:

1. Everything from the word COMMENT to the next statement terminator, nor-mally ; or $, is ignored. Such comments can be placed anywhere a blankcould properly appear. (Note that END and >> are not treated as COMMENTdelimiters!)

2. Everything from the symbol % to the end of the line on which it appears isignored. Such comments can be placed as the last part of any line. Statementterminators have no special meaning in such comments. Remember to puta semicolon before the % if the earlier part of the line is intended to be soterminated. Remember also to begin each line of a multi-line % commentwith a % sign.

2.7 Operators

Operators in REDUCE are specified by name and type. There are two types, in-fix and prefix. Operators can be purely abstract, just symbols with no properties;they can have values assigned (using := or simple LET declarations) for specificarguments; they can have properties declared for some collection of arguments(using more general LET declarations); or they can be fully defined (usually by aprocedure declaration).

Infix operators have a definite precedence with respect to one another, and normallyoccur between their arguments. For example:

a + b - c (spaces optional)x

2.7. OPERATORS 41

Prefix operators occur to the left of their arguments, which are written as a listenclosed in parentheses and separated by commas, as with normal mathematicalfunctions, e.g.,

cos(u)df(x^2,x)q(v+w)

Unmatched parentheses, incorrect groupings of infix operators and the like, natu-rally lead to syntax errors. The parentheses can be omitted (replaced by a spacefollowing the operator name) if the operator is unary and the argument is a singlesymbol or begins with a prefix operator name:

cos y means cos(y)cos (-y) – parentheses necessarylog cos y means log(cos(y))log cos (a+b) means log(cos(a+b))

but

cos a*b means (cos a)*bcos -y is erroneous (treated as a variable

“cos” minus the variable y)

A unary prefix operator has a precedence higher than any infix operator, includingunary infix operators. In other words, REDUCE will always interpret cos y +3 as (cos y) + 3 rather than as cos(y + 3).

Infix operators may also be used in a prefix format on input, e.g., +(a,b,c). Onoutput, however, such expressions will always be printed in infix form (i.e., a +b + c for this example).

A number of prefix operators are built into the system with predefined properties.Users may also add new operators and define their rules for simplification. Thebuilt in operators are described in another section.

Built-In Infix Operators

The following infix operators are built into the system. They are all defined inter-nally as procedures.

〈infix operator〉 −→ where | := | or | and | member | memq |= | neq | eq | >= | > |

42 CHAPTER 2. STRUCTURE OF PROGRAMS

These operators may be further divided into the following subclasses:

〈assignment operator〉 −→ :=〈logical operator〉 −→ or | and | member | memq〈relational operator〉 −→ = | neq | eq | >= | > | = geq> greaterp

2.7. OPERATORS 43

Parentheses may be used to specify the order of combination. If parentheses areomitted then this order is by the ordering of the precedence list defined by theright-hand side of the 〈infix operator〉 table at the beginning of this section, fromlowest to highest. In other words, WHERE has the lowest precedence, and . (thedot operator) the highest.

44 CHAPTER 2. STRUCTURE OF PROGRAMS

Chapter 3

Expressions

REDUCE expressions may be of several types and consist of sequences of num-bers, variables, operators, left and right parentheses and commas. The most com-mon types are as follows:

3.1 Scalar Expressions

Using the arithmetic operations + - * / ^ (power) and parentheses, scalarexpressions are composed from numbers, ordinary “scalar” variables (identifiers),array names with subscripts, operator or procedure names with arguments andstatement expressions.

Examples:

xx^3 - 2*y/(2*z^2 - df(x,z))(p^2 + m^2)^(1/2)*log (y/m)a(5) + b(i,q)

The symbol ** may be used as an alternative to the caret symbol (^) for formingpowers, particularly in those systems that do not support a caret symbol.

Statement expressions, usually in parentheses, can also form part of a scalar ex-pression, as in the example

w + (c:=x+y) + z .

When the algebraic value of an expression is needed, REDUCE determines it, start-ing with the algebraic values of the parts, roughly as follows:

Variables and operator symbols with an argument list have the algebraic values

45

46 CHAPTER 3. EXPRESSIONS

they were last assigned, or if never assigned stand for themselves. However, arrayelements have the algebraic values they were last assigned, or, if never assigned,are taken to be 0.

Procedures are evaluated with the values of their actual parameters.

In evaluating expressions, the standard rules of algebra are applied. Unfortunately,this algebraic evaluation of an expression is not as unambiguous as is numericalevaluation. This process is generally referred to as “simplification” in the sense thatthe evaluation usually but not always produces a simplified form for the expression.

There are many options available to the user for carrying out such simplification.If the user doesn’t specify any method, the default method is used. The defaultevaluation of an expression involves expansion of the expression and collectionof like terms, ordering of the terms, evaluation of derivatives and other functionsand substitution for any expressions which have values assigned or declared (seeassignments and LET statements). In many cases, this is all that the user needs.

The declarations by which the user can exercise some control over the way in whichthe evaluation is performed are explained in other sections. For example, if a real(floating point) number is encountered during evaluation, the system will normallyconvert it into a ratio of two integers. If the user wants to use real arithmetic,he can effect this by the command on rounded;. Other modes for coefficientarithmetic are described elsewhere.

If an illegal action occurs during evaluation (such as division by zero) or functionsare called with the wrong number of arguments, and so on, an appropriate errormessage is generated.

3.2 Integer Expressions

These are expressions which, because of the values of the constants and variablesin them, evaluate to whole numbers.

Examples:

2, 37 * 999, (x + 3)^2 - x^2 - 6*x

are obviously integer expressions.

j + k - 2 * j^2

is an integer expression when J and K have values that are integers, or if not integersare such that “the variables and fractions cancel out”, as in

k - 7/3 - j + 2/3 + 2*j^2.

3.3. BOOLEAN EXPRESSIONS 47

3.3 Boolean Expressions

A boolean expression returns a truth value. In the algebraic mode of REDUCE,boolean expressions have the syntactical form:

〈expression〉〈relational operator〉〈expression〉

or

〈boolean operator〉(〈arguments〉)

or

〈boolean expression〉〈logical operator〉〈boolean expression〉.

Parentheses can also be used to control the precedence of expressions.

In addition to the logical and relational operators defined earlier as infix operators,the following boolean operators are also defined:

EVENP(U) determines if the number U is even or not;

FIXP(U) determines if the expression U is integer or not;

FREEOF(U,V) determines if the expression U does not contain the kernelV anywhere in its structure;

NUMBERP(U) determines if U is a number or not;

ORDP(U,V) determines if U is ordered ahead of V by some canonicalordering (based on the expression structure and an internalordering of identifiers);

PRIMEP(U) true if U is a prime object, i.e., any object other than 0 andplus or minus 1 which is only exactly divisible by itself ora unit.

Examples:

j0 or x=-2numberp xfixp x and evenp xnumberp x and x neq 0

48 CHAPTER 3. EXPRESSIONS

Boolean expressions can only appear directly within IF, FOR, WHILE, and UNTILstatements, as described in other sections. Such expressions cannot be used in placeof ordinary algebraic expressions, or assigned to a variable.

NB: For those familiar with symbolic mode, the meaning of some of these oper-ators is different in that mode. For example, NUMBERP is true only for integers andreals in symbolic mode.

When two or more boolean expressions are combined with AND, they are evaluatedone by one until a false expression is found. The rest are not evaluated. Thus

numberp x and numberp y and x>y

does not attempt to make the x>y comparison unless X and Y are both verified tobe numbers.

Similarly, evaluation of a sequence of boolean expressions connected by OR stopsas soon as a true expression is found.

NB: In a boolean expression, and in a place where a boolean expression is expected,the algebraic value 0 is interpreted as false, while all other algebraic values areconverted to true. So in algebraic mode a procedure can be written for direct usagein boolean expressions, returning say 1 or 0 as its value as in

procedure polynomialp(u,x);if den(u)=1 and deg(u,x)>=1 then 1 else 0;

One can then use this in a boolean construct, such as

if polynomialp(q,z) and not polynomialp(q,y) then ...

In addition, any procedure that does not have a defined return value (for example,a block without a RETURN statement in it) has the boolean value false.

3.4 Equations

Equations are a particular type of expression with the syntax

〈expression〉=〈expression〉.

In addition to their role as boolean expressions, they can also be used as argumentsto several operators (e.g., SOLVE), and can be returned as values.

Under normal circumstances, the right-hand-side of the equation is evaluated butnot the left-hand-side. This also applies to any substitutions made by the SUB

3.5. PROPER STATEMENTS AS EXPRESSIONS 49

operator. If both sides are to be evaluated, the switch EVALLHSEQP should beturned on.

To facilitate the handling of equations, two selectors, LHS and RHS, which re-turn the left- and right-hand sides of an equation respectively, are provided. Forexample,

lhs(a+b=c) -> a+band

rhs(a+b=c) -> c.

3.5 Proper Statements as Expressions

Several kinds of proper statements deliver an algebraic or numerical result of somekind, which can in turn be used as an expression or part of an expression. Forexample, an assignment statement itself has a value, namel