perm filename MACSYM.MIT[UP,DOC] blob
sn#554488 filedate 1980-03-06 generic text, type C, neo UTF8
COMMENT ⊗ VALID 00271 PAGES
C REC PAGE DESCRIPTION
C00001 00001
C00041 00002
C00042 00003
C00043 00004
C00046 00005 MACSYMA TABLE OF CONTENTS i
C00052 00006 ii TABLE OF CONTENTS MACSYMA
C00059 00007 MACSYMA TABLE OF CONTENTS iii
C00065 00008 iv TABLE OF CONTENTS MACSYMA
C00067 00009 MACSYMA 1
C00072 00010 2 1.1 Logging in and out MACSYMA
C00076 00011 MACSYMA 1.1 Logging in and out 3
C00081 00012 4 1.2 General Information MACSYMA
C00085 00013 MACSYMA 1.3 Levels of Control 5
C00090 00014 6 1.4 Miscellaneous Information MACSYMA
C00091 00015 MACSYMA 7
C00095 00016 8 2.3 Quoted Strings MACSYMA
C00099 00017 MACSYMA 2.4 Atomic Variables and Assignment 9
C00103 00018 10 2.5 Mathematical Operators MACSYMA
C00108 00019 MACSYMA 2.5 Mathematical Operators 11
C00111 00020 12 2.6.1 Functions MACSYMA
C00115 00021 MACSYMA 2.6.2 Arrays 13
C00120 00022 14 2.6.2 Arrays MACSYMA
C00123 00023 MACSYMA 15
C00128 00024 16 2.6.4 Subscripted Functions (Arrays of Functions) MACSYMA
C00131 00025 MACSYMA 2.6.5 Additional Information About Functions 17
C00134 00026 18 MACSYMA
C00138 00027 MACSYMA 2.8 Matrices 19
C00141 00028 20 2.9 Equations MACSYMA
C00145 00029 MACSYMA 2.10 IF Statement 21
C00148 00030 22 2.12 Program Blocks MACSYMA
C00153 00031 MACSYMA 2.12 Program Blocks 23
C00156 00032 24 2.13.1 Commonly Used Forms MACSYMA
C00161 00033 MACSYMA 2.13.1 Commonly Used Forms 25
C00164 00034 26 2.13.1 Commonly Used Forms MACSYMA
C00168 00035 MACSYMA 2.13.2 Additional Forms of the DO Statement 27
C00171 00036 28 MACSYMA
C00176 00037 MACSYMA 3.1 Representation 29
C00181 00038 30 3.2 Evaluation MACSYMA
C00186 00039 MACSYMA 3.2 Evaluation 31
C00190 00040 32 3.2 Evaluation MACSYMA
C00193 00041 MACSYMA 3.2 Evaluation 33
C00197 00042 34 MACSYMA
C00201 00043 MACSYMA 4 Miscellaneous Hints and Facilities 35
C00206 00044 36 4 Miscellaneous Hints and Facilities MACSYMA
C00207 00045 MACSYMA 37
C00209 00046 38 MACSYMA
C00212 00047 MACSYMA 5.2.2 Miscellaneous Functions 39
C00216 00048 40 5.2.2 Miscellaneous Functions MACSYMA
C00219 00049 MACSYMA 5.2.3 Trigonometric Functions 41
C00222 00050 42 5.2.3 Trigonometric Functions MACSYMA
C00224 00051 MACSYMA 43
C00226 00052 44 MACSYMA
C00230 00053 MACSYMA 6.1.1 Evaluation and Simplification Functions 45
C00235 00054 46 6.1.1 Evaluation and Simplification Functions MACSYMA
C00239 00055 MACSYMA 6.1.1 Evaluation and Simplification Functions 47
C00242 00056 48 6.1.1 Evaluation and Simplification Functions MACSYMA
C00246 00057 MACSYMA 6.1.1 Evaluation and Simplification Functions 49
C00250 00058 50 6.1.1 Evaluation and Simplification Functions MACSYMA
C00254 00059 MACSYMA 6.1.1 Evaluation and Simplification Functions 51
C00257 00060 52 6.1.1 Evaluation and Simplification Functions MACSYMA
C00260 00061 MACSYMA 6.1.1 Evaluation and Simplification Functions 53
C00264 00062 54 6.1.1 Evaluation and Simplification Functions MACSYMA
C00268 00063 MACSYMA 6.1.1 Evaluation and Simplification Functions 55
C00272 00064 56 6.1.1 Evaluation and Simplification Functions MACSYMA
C00275 00065 MACSYMA 6.1.1 Evaluation and Simplification Functions 57
C00279 00066 58 6.1.2 Sums and Products MACSYMA
C00282 00067 MACSYMA 59
C00285 00068 60 6.1.3 Differentiation and Integration Functions MACSYMA
C00289 00069 MACSYMA 6.1.3 Differentiation and Integration Functions 61
C00293 00070 62 6.1.3 Differentiation and Integration Functions MACSYMA
C00298 00071 MACSYMA 6.1.3 Differentiation and Integration Functions 63
C00301 00072 64 6.1.3 Differentiation and Integration Functions MACSYMA
C00305 00073 MACSYMA 6.1.3 Differentiation and Integration Functions 65
C00308 00074 66 6.1.3 Differentiation and Integration Functions MACSYMA
C00312 00075 MACSYMA 6.2.1 The Part Functions 67
C00315 00076 68 6.2.1 The Part Functions MACSYMA
C00319 00077 MACSYMA 6.2.1 The Part Functions 69
C00323 00078 70 6.2.2 The Substitution Functions MACSYMA
C00326 00079 MACSYMA 6.2.2 The Substitution Functions 71
C00329 00080 72 6.2.2 The Substitution Functions MACSYMA
C00332 00081 MACSYMA 6.2.2 The Substitution Functions 73
C00336 00082 74 6.2.2 The Substitution Functions MACSYMA
C00340 00083 MACSYMA 6.2.3 More Functions for Part Extraction 75
C00344 00084 76 6.2.3 More Functions for Part Extraction MACSYMA
C00347 00085 MACSYMA 6.2.3 More Functions for Part Extraction 77
C00350 00086 78 6.2.3 More Functions for Part Extraction MACSYMA
C00354 00087 MACSYMA 6.2.3 More Functions for Part Extraction 79
C00357 00088 80 6.2.3 More Functions for Part Extraction MACSYMA
C00361 00089 MACSYMA 6.2.3 More Functions for Part Extraction 81
C00362 00090 82 MACSYMA
C00366 00091 MACSYMA 6.3 SOLVE and Related Functions 83
C00369 00092 84 6.3 SOLVE and Related Functions MACSYMA
C00374 00093 MACSYMA 6.3 SOLVE and Related Functions 85
C00378 00094 86 6.3 SOLVE and Related Functions MACSYMA
C00382 00095 MACSYMA 6.3 SOLVE and Related Functions 87
C00386 00096 88 6.3 SOLVE and Related Functions MACSYMA
C00389 00097 MACSYMA 89
C00392 00098 90 6.4 The Matrix Functions MACSYMA
C00395 00099 MACSYMA 6.4 The Matrix Functions 91
C00398 00100 92 6.4 The Matrix Functions MACSYMA
C00401 00101 MACSYMA 6.4 The Matrix Functions 93
C00405 00102 94 6.4 The Matrix Functions MACSYMA
C00410 00103 MACSYMA 6.4.1 The Dot Operator 95
C00412 00104 96 MACSYMA
C00418 00105 MACSYMA 6.5 Functions for Rational Expressions 97
C00422 00106 98 6.5 Functions for Rational Expressions MACSYMA
C00426 00107 MACSYMA 6.5 Functions for Rational Expressions 99
C00430 00108 100 6.5 Functions for Rational Expressions MACSYMA
C00434 00109 MACSYMA 6.5 Functions for Rational Expressions 101
C00438 00110 102 6.5.1 Algebraic Integers MACSYMA
C00442 00111 MACSYMA 6.5.2 Functions for Extended Rational Expressions 103
C00444 00112 104 6.5.2 Functions for Extended Rational Expressions MACSYMA
C00448 00113 MACSYMA 6.6 Poisson Series Functions 105
C00452 00114 106 6.6 Poisson Series Functions MACSYMA
C00455 00115 MACSYMA 6.6 Poisson Series Functions 107
C00459 00116 108 6.7 Taylor Series MACSYMA
C00462 00117 MACSYMA 6.7 Taylor Series 109
C00466 00118 110 6.7 Taylor Series MACSYMA
C00469 00119 MACSYMA 6.7 Taylor Series 111
C00473 00120 112 6.7 Taylor Series MACSYMA
C00476 00121 MACSYMA 6.7 Taylor Series 113
C00479 00122 114 6.8 Trigonometric Simplification MACSYMA
C00482 00123 MACSYMA 6.8 Trigonometric Simplification 115
C00486 00124 116 6.9 Laplace Transforms MACSYMA
C00489 00125 MACSYMA 6.9 Laplace Transforms 117
C00493 00126 118 6.9 Laplace Transforms MACSYMA
C00497 00127 MACSYMA 6.9 Laplace Transforms 119
C00500 00128 120 6.10 Combinatorial Functions MACSYMA
C00503 00129 MACSYMA 121
C00505 00130 122 MACSYMA
C00510 00131 MACSYMA 7.1 Declaring and Assuming 123
C00514 00132 124 7.1 Declaring and Assuming MACSYMA
C00518 00133 MACSYMA 7.2 Contexts 125
C00522 00134 126 7.2 Contexts MACSYMA
C00524 00135 MACSYMA 127
C00529 00136 128 8 List Handling and LISP-like functions MACSYMA
C00532 00137 MACSYMA 8 List Handling and LISP-like functions 129
C00535 00138 130 8 List Handling and LISP-like functions MACSYMA
C00538 00139 MACSYMA 8.1 Property Specification Functions 131
C00543 00140 132 8.1 Property Specification Functions MACSYMA
C00547 00141 MACSYMA 8.1.1 Functions for Handling MACSYMA Properties 133
C00551 00142 134 8.1.2 Functions for Handling Users' Properties MACSYMA
C00552 00143 MACSYMA 135
C00555 00144 136 9.1 Type Testing Functions MACSYMA
C00559 00145 MACSYMA 9.2 General Pattern Matching Functions 137
C00563 00146 138 9.2 General Pattern Matching Functions MACSYMA
C00568 00147 MACSYMA 9.2 General Pattern Matching Functions 139
C00572 00148 140 9.2 General Pattern Matching Functions MACSYMA
C00577 00149 MACSYMA 9.3 Pattern Matching for Rational Expressions 141
C00581 00150 142 9.3 Pattern Matching for Rational Expressions MACSYMA
C00584 00151 MACSYMA 9.3 Pattern Matching for Rational Expressions 143
C00586 00152 144 MACSYMA
C00589 00153 MACSYMA 10.1 Debugging Functions 145
C00593 00154 146 10.2 Functions for Displaying MACSYMA
C00598 00155 MACSYMA 10.2 Functions for Displaying 147
C00602 00156 148 10.3 Functions for Freeing Storage MACSYMA
C00607 00157 MACSYMA 10.3 Functions for Freeing Storage 149
C00611 00158 150 10.4 Functions Which Reference Disk Files MACSYMA
C00616 00159 MACSYMA 10.4 Functions Which Reference Disk Files 151
C00620 00160 152 10.5 Ordering Functions MACSYMA
C00623 00161 MACSYMA 10.6 Miscellaneous Functions 153
C00628 00162 154 10.6 Miscellaneous Functions MACSYMA
C00632 00163 MACSYMA 10.6 Miscellaneous Functions 155
C00637 00164 156 10.7 Options and Variables for I/O, Status, and Display MACSYMA
C00642 00165 MACSYMA 10.7 Options and Variables for I/O, Status, and Display 157
C00646 00166 158 MACSYMA
C00651 00167 MACSYMA 10.8 Functions for Translation and Compilation 159
C00656 00168 160 10.8 Functions for Translation and Compilation MACSYMA
C00658 00169 MACSYMA 161
C00663 00170 162 11 Tensor Manipulation MACSYMA
C00667 00171 MACSYMA 11.1 Explicit Tensor Manipulation 163
C00671 00172 164 11.1 Explicit Tensor Manipulation MACSYMA
C00675 00173 MACSYMA 11.1 Explicit Tensor Manipulation 165
C00679 00174 166 11.1 Explicit Tensor Manipulation MACSYMA
C00683 00175 MACSYMA 11.1 Explicit Tensor Manipulation 167
C00686 00176 168 11.1 Explicit Tensor Manipulation MACSYMA
C00687 00177 MACSYMA 169
C00693 00178 170 11.2 Indicial Tensor Manipulation MACSYMA
C00698 00179 MACSYMA 11.2 Indicial Tensor Manipulation 171
C00703 00180 172 11.2 Indicial Tensor Manipulation MACSYMA
C00706 00181 MACSYMA 11.2 Indicial Tensor Manipulation 173
C00710 00182 174 11.2 Indicial Tensor Manipulation MACSYMA
C00714 00183 MACSYMA 11.2 Indicial Tensor Manipulation 175
C00718 00184 176 11.2 Indicial Tensor Manipulation MACSYMA
C00720 00185 MACSYMA 177
C00725 00186 178 MACSYMA
C00729 00187 MACSYMA 12.3 Solving Differential Equations by Laplace Transforms 179
C00732 00188 180 MACSYMA
C00733 00189 MACSYMA 12.4 Exterior Calculus of Differential Forms 181
C00737 00190 182 12.5 Vector Analysis MACSYMA
C00742 00191 MACSYMA 12.5 Vector Analysis 183
C00747 00192 184 12.6 Dimensional Analysis MACSYMA
C00752 00193 MACSYMA 12.8 Variational Optimization 185
C00757 00194 186 12.8 Variational Optimization MACSYMA
C00761 00195 MACSYMA 12.10 Units Conversion 187
C00766 00196 188 12.11 The Eigen Package MACSYMA
C00770 00197 MACSYMA 12.12 Elimination by Resultants 189
C00773 00198 190 12.14 Integral Equations MACSYMA
C00777 00199 MACSYMA 12.14 Integral Equations 191
C00781 00200 192 12.15.1 Numerical Integration MACSYMA
C00785 00201 MACSYMA 12.15.1 Numerical Integration 193
C00788 00202 194 12.15.2 Fast Fourier Transforms MACSYMA
C00792 00203 MACSYMA 12.15.3 Roots of Equations by Interpolation 195
C00795 00204 196 12.15.4 Special Functions MACSYMA
C00798 00205 MACSYMA 12.15.5 Polynomial Zeros 197
C00801 00206 198 MACSYMA
C00806 00207 MACSYMA 13.2 Entering the Editor 199
C00810 00208 200 13.3 A Description of the Commands MACSYMA
C00814 00209 MACSYMA 13.3 A Description of the Commands 201
C00819 00210 202 13.3 A Description of the Commands MACSYMA
C00821 00211 MACSYMA 203
C00825 00212 204 14.3 The More Complicated Format MACSYMA
C00830 00213 MACSYMA 14.4 The BATCON Function 205
C00835 00214 206 14.5 Miscellany MACSYMA
C00837 00215 MACSYMA 207
C00842 00216 208 15.2 Automatic Storage of Expressions MACSYMA
C00846 00217 MACSYMA 209
C00851 00218 210 15.3.1 Use of the storage functions MACSYMA
C00856 00219 MACSYMA 15.3.2 Retrieval of expressions stored on disk 211
C00861 00220 212 15.4 Saving a MACSYMA Overnight MACSYMA
C00863 00221 MACSYMA 213
C00868 00222 214 16 Storage Management MACSYMA
C00874 00223 MACSYMA 16 Storage Management 215
C00879 00224 216 16 Storage Management MACSYMA
C00883 00225 MACSYMA 217
C00887 00226 218 17 Simple Plotting Functions MACSYMA
C00891 00227 MACSYMA 17 Simple Plotting Functions 219
C00895 00228 220 17 Simple Plotting Functions MACSYMA
C00896 00229 MACSYMA 17 Simple Plotting Functions 221
C00899 00230 222 MACSYMA
C00903 00231 MACSYMA 18.2 Two-Dimensional Plotting 223
C00908 00232 224 18.2 Two-Dimensional Plotting MACSYMA
C00912 00233 MACSYMA 18.2 Two-Dimensional Plotting 225
C00916 00234 226 18.3 What to Type When PLOT2 has Finished Plotting MACSYMA
C00921 00235 MACSYMA 18.4 Three-Dimensional Plotting 227
C00925 00236 228 18.5 Using the XGP from PLOT2 MACSYMA
C00929 00237 230 MACSYMA
C00934 00238 MACSYMA 19 Debugging in MACSYMA 231
C00939 00239 232 19 Debugging in MACSYMA MACSYMA
C00941 00240 MACSYMA 19 Debugging in MACSYMA 233
C00943 00241 234 MACSYMA
C00948 00242 MACSYMA I Multics MACSYMA 235
C00949 00243 236 MACSYMA
C00954 00244 MACSYMA II A MACSYMA Grammar Primer 237
C00958 00245 238 II A MACSYMA Grammar Primer MACSYMA
C00962 00246 MACSYMA II A MACSYMA Grammar Primer 239
C00965 00247 240 II A MACSYMA Grammar Primer MACSYMA
C00970 00248 MACSYMA II A MACSYMA Grammar Primer 241
C00974 00249 242 II A MACSYMA Grammar Primer MACSYMA
C00977 00250 MACSYMA II A MACSYMA Grammar Primer 243
C00981 00251 244 II A MACSYMA Grammar Primer MACSYMA
C00985 00252 MACSYMA 245
C00988 00253 246 III Illustrative Examples MACSYMA
C00991 00254 MACSYMA III Illustrative Examples 247
C00994 00255 248 III Illustrative Examples MACSYMA
C00997 00256 MACSYMA III Illustrative Examples 249
C01000 00257 250 III Illustrative Examples MACSYMA
C01001 00258 MACSYMA III Illustrative Examples 251
C01005 00259 252 III Illustrative Examples MACSYMA
C01009 00260 MACSYMA III Illustrative Examples 253
C01012 00261 254 MACSYMA
C01016 00262 MACSYMA IV Glossary For The Programming Novice 255
C01020 00263 256 IV Glossary For The Programming Novice MACSYMA
C01023 00264 MACSYMA 257
C01027 00265 258 V Bibliography and References MACSYMA
C01031 00266 MACSYMA V Bibliography and References 259
C01035 00267 260 V Bibliography and References MACSYMA
C01039 00268 MACSYMA V Bibliography and References 261
C01041 00269 262 V Bibliography and References MACSYMA
C01043 00270 MACSYMA 263
C01047 00271 MACSYMA INDEX i
C01074 ENDMK
C⊗;
MACSYMA REFERENCE MANUAL
THE MATHLAB GROUP
LABORATORY FOR COMPUTER SCIENCE
MIT
VERSION NINE
Second Printing
December 1978
Copyright (c) 1977 by Massachusetts Institute of Technology
Cambridge, Mass. 02139
ALL RIGHTS RESERVED
Preface
This manual maintains the format of Version Eight but contains corrections and
updates. It corresponds to Version 267 of MACSYMA which has approximately 221,000
words of compiled code. Any comments, suggestions, or criticisms are welcome and
should be addressed to:
Mathlab Group - Room 828
Laboratory for Computer Science
MIT
545 Technology Square
Cambridge, Mass. 02139
Acknowledgment
This manual was written by Richard Bogen with contributions by Jeffrey Golden
(editor, storage management, debugging, and batch chapters), Michael Genesereth
(relational data base, appendix on the grammar), and Alexander Doohovskoy (explicit
tensor manipulation). We are extremely grateful to Pitts Jarvis for the many hours
he spent in formatting it for the PUB program to use the Xerox Graphics Printer and
to Ellen Lewis for her helpful assistance.
Note
A report, thesis, etc. which contains results that were obtained by using MACSYMA
should acknowledge that the work of the Mathlab group is currently supported, in
part, by the United States Energy Research and Development Administration under
Contract Number E(11-1)-3070 and by the National Aeronautics and Space Administration
under Grant NSG 1323. A copy should also be sent to Prof. J. Moses at the address
given above in the Preface.
MACSYMA TABLE OF CONTENTS i
Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Logging in and out . . . . . . . . . . . . . . . . . . . 1
1.2 General Information . . . . . . . . . . . . . . . . . . . 3
1.3 Levels of Control . . . . . . . . . . . . . . . . . . . 4
1.4 Miscellaneous Information . . . . . . . . . . . . . . . 5
Chapter 2 MACSYMA's Data Types and Statement Types. . . . . . . . . . . 7
2.1 Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Names . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Quoted Strings . . . . . . . . . . . . . . . . . . . . . . . 7
2.4 Atomic Variables and Assignment . . . . . . . . . . . . . . . 8
2.5 Mathematical Operators . . . . . . . . . . . . . . . . . . . 9
2.6 Functions and Arrays . . . . . . . . . . . . . . . . . . .11
2.6.1 Functions . . . . . . . . . . . . . . . . . . .11
2.6.2 Arrays . . . . . . . . . . . . . . . . . . . . . . .12
2.6.3 Lambda Notation . . . . . . . . . . . . . . . . . . .14
2.6.4 Subscripted Functions (Arrays of Functions) . . .15
2.6.5 Additional Information About Functions . . . . . . .16
2.7 Lists . . . . . . . . . . . . . . . . . . . . . . . . . . .17
2.8 Matrices. . . . . . . . . . . . . . . . . . . . . . . . . . .18
2.9 Equations . . . . . . . . . . . . . . . . . . . . . . .19
2.10 IF Statement . . . . . . . . . . . . . . . . . . . . . . .20
2.11 Compound Statements . . . . . . . . . . . . . . . . . . .21
2.12 Program Blocks . . . . . . . . . . . . . . . . . . . . . . .22
2.13 The DO Statement. . . . . . . . . . . . . . . . . . . . . . .23
2.13.1 Commonly Used Forms . . . . . . . . . . . . . . .23
2.13.2 Additional Forms of the DO Statement . . . . . . .26
2.14 Syntax Extension. . . . . . . . . . . . . . . . . . . . . . .27
Chapter 3 What a Serious User Should Know . . . . . . . . . . . . . . .28
3.1 Representation . . . . . . . . . . . . . . . . . . . . . . .28
3.2 Evaluation . . . . . . . . . . . . . . . . . . . . . . .29
3.3 Simplification . . . . . . . . . . . . . . . . . . . . . . .33
Chapter 4 Miscellaneous Hints and Facilities . . . . . . . . . . .34
Chapter 5 Predefined Constants and Functions . . . . . . . . . . .37
5.1 Constants . . . . . . . . . . . . . . . . . . . . . . .37
5.2 Functions . . . . . . . . . . . . . . . . . . . . . . .37
5.2.1 Simple Functions. . . . . . . . . . . . . . . . . . .38
5.2.2 Miscellaneous Functions . . . . . . . . . . . . . . .38
5.2.3 Trigonometric Functions . . . . . . . . . . . . . . .40
5.3 Complex Expressions . . . . . . . . . . . . . . . . . . .43
Chapter 6 MACSYMA Functions and Variables . . . . . . . . . . . . . . .44
6.1 General Purpose Functions . . . . . . . . . . . . . . .44
ii TABLE OF CONTENTS MACSYMA
6.1.1 Evaluation and Simplification Functions . . . . . . .44
6.1.2 Sums and Products . . . . . . . . . . . . . . .57
6.1.3 Differentiation and Integration Functions . . .59
6.2 Part Selection and Substitution . . . . . . . . . . . . . . .66
6.2.1 The Part Functions . . . . . . . . . . . . . . .66
6.2.2 The Substitution Functions . . . . . . . . . . .69
6.2.3 More Functions for Part Extraction . . . . . . .74
6.3 SOLVE and Related Functions . . . . . . . . . . . . . . .82
6.4 The Matrix Functions . . . . . . . . . . . . . . . . . . .89
6.4.1 The Dot Operator. . . . . . . . . . . . . . . . . . .94
6.5 Functions for Rational Expressions . . . . . . . . . . .96
6.5.1 Algebraic Integers . . . . . . . . . . . . . . 101
6.5.2 Functions for Extended Rational Expressions . . 102
6.6 Poisson Series Functions. . . . . . . . . . . . . . . . . . 104
6.7 Taylor Series . . . . . . . . . . . . . . . . . . . . . . 107
6.8 Trigonometric Simplification . . . . . . . . . . . . . . 113
6.9 Laplace Transforms . . . . . . . . . . . . . . . . . . 115
6.10 Combinatorial Functions . . . . . . . . . . . . . . . . . . 119
6.11 Continued Fractions . . . . . . . . . . . . . . . . . . 120
6.12 Number-Theoretic Functions . . . . . . . . . . . . . . 121
Chapter 7 Declaring and Using Mathematical Information . . . . . . 122
7.1 Declaring and Assuming . . . . . . . . . . . . . . . . . . 122
7.2 Contexts. . . . . . . . . . . . . . . . . . . . . . . . . . 125
Chapter 8 List Handling and LISP-like functions . . . . . . . . . . 127
8.1 Property Specification Functions. . . . . . . . . . . . . . 130
8.1.1 Functions for Handling MACSYMA Properties . . 132
8.1.2 Functions for Handling Users' Properties. . . . . . 133
Chapter 9 Pattern Matching and Related Functions . . . . . . . . . . 135
9.1 Type Testing Functions . . . . . . . . . . . . . . . . . . 135
9.2 General Pattern Matching Functions . . . . . . . . . . 136
9.3 Pattern Matching for Rational Expressions . . . . . . 140
Chapter 10 Utility, Input-Output, and Display Functions . . . . . . 144
10.1 Debugging Functions . . . . . . . . . . . . . . . . . . 144
10.2 Functions for Displaying. . . . . . . . . . . . . . . . . . 145
10.3 Functions for Freeing Storage . . . . . . . . . . . . . . 148
10.4 Functions Which Reference Disk Files . . . . . . . . . . 149
10.5 Ordering Functions . . . . . . . . . . . . . . . . . . 151
10.6 Miscellaneous Functions . . . . . . . . . . . . . . . . . . 152
10.7 Options and Variables for I/O, Status, and Display . . 155
10.8 Functions for Translation and Compilation . . . . . . 158
Chapter 11 Tensor Manipulation . . . . . . . . . . . . . . . . . . 161
11.1 Explicit Tensor Manipulation . . . . . . . . . . . . . . 162
MACSYMA TABLE OF CONTENTS iii
11.2 Indicial Tensor Manipulation . . . . . . . . . . . . . . 169
Chapter 12 The SHARE Directory . . . . . . . . . . . . . . . . . . 177
12.1 Simplification for ABS and SIGNUM . . . . . . . . . . 178
10ε2 Array Manipulation . . . . . . . . . . . . . . . . . . 178
12.3 Solving Differential Equations by Laplace Transforms . . 178
12.4 Exterior Calculus of Differential Forms . . . . . . . . . . 180
12.5 Vector Analysis . . . . . . . . . . . . . . . . . . . . . . 181
12.6 Dimensional Analysis . . . . . . . . . . . . . . . . . . 183
12.7 Analytic Optimization . . . . . . . . . . . . . . . . . . 184
12.8 Variational Optimization. . . . . . . . . . . . . . . . . . 185
12.9 Qualitative Analysis . . . . . . . . . . . . . . . . . . 186
12.10 Units Conversion. . . . . . . . . . . . . . . . . . . . . . 186
12.11 The Eigen Package . . . . . . . . . . . . . . . . . . 187
12.12 Elimination by Resultants . . . . . . . . . . . . . . 188
12.13 Integration of Special Forms . . . . . . . . . . . . . . 189
12.14 Integral Equations . . . . . . . . . . . . . . . . . . 189
12.15 Numerical Techniques . . . . . . . . . . . . . . . . . . 191
12.15.1 Numerical Integration . . . . . . . . . . . . . . 191
12.15.2 Fast Fourier Transforms . . . . . . . . . . . . . . 193
12.15.3 Roots of Equations by Interpolation . . . . . . 194
12.15.4 Special Functions . . . . . . . . . . . . . . 195
12.15.5 Polynomial Zeros. . . . . . . . . . . . . . . . . . 196
Chapter 13 The MACSYMA Editor . . . . . . . . . . . . . . . . . . 198
13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 198
13.2 Entering the Editor . . . . . . . . . . . . . . . . . . 198
13.3 A Description of the Commands . . . . . . . . . . . . . . 199
Chapter 14 Batch Functions . . . . . . . . . . . . . . . . . . . . . . 203
14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 203
14.2 The Simple Format . . . . . . . . . . . . . . . . . . 203
14.3 The More Complicated Format . . . . . . . . . . . . . . 203
14.4 The BATCON Function . . . . . . . . . . . . . . . . . . 204
14.5 Miscellany . . . . . . . . . . . . . . . . . . . . . . 205
Chapter 15 Secondary Storage Functions . . . . . . . . . . . . . . 207
15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 207
15.2 Automatic Storage of Expressions. . . . . . . . . . . . . . 207
15.3 Explicit Storage of Expressions . . . . . . . . . . . . . . 209
15.3.1 Use of the storage functions . . . . . . . . . . 209
15.3.2 Retrieval of expressions stored on disk . . . . . . 210
15.4 Saving a MACSYMA Overnight . . . . . . . . . . . . . . 211
Chapter 16 Storage Management . . . . . . . . . . . . . . . . . . 213
Chapter 17 Simple Plotting Functions . . . . . . . . . . . . . . 217
iv TABLE OF CONTENTS MACSYMA
Chapter 18 Complex Plotting and Graphics---PLOT2 . . . . . . . . . . 222
18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 222
18.2 Two-Dimensional Plotting. . . . . . . . . . . . . . . . . . 222
18.3 What to Type When PLOT2 has Finished Plotting . . . . . . 225
18.4 Three-Dimensional Plotting . . . . . . . . . . . . . . 226
18.5 Using the XGP from PLOT2. . . . . . . . . . . . . . . . . . 228
Chapter 19 Debugging in MACSYMA . . . . . . . . . . . . . . . . . . 230
Appendix I Multics MACSYMA . . . . . . . . . . . . . . . . . . . . . . 234
Appendix II A MACSYMA Grammar Primer. . . . . . . . . . . . . . . . . . 236
Appendix III Illustrative Examples . . . . . . . . . . . . . . . . . . 245
Appendix IV Glossary For The Programming Novice . . . . . . . . . . 254
Appendix V Bibliography and References . . . . . . . . . . . . . . 257
Appendix VI MACSYMA Functions and Argument Evaluation . . . . . . 263
FUNCTION,VARIABLE, and OPTION INDEX . . . . . . . . . . . . . . . . . . .i
MACSYMA 1
1 Introduction
MACSYMA (Project MAC's SYmbolic MAnipulation System) is a large computer
programming system written in LISP [Mn1] used for performing symbolic as well
as numerical mathematical manipulations. It is being developed by the Mathlab
Group of the MIT Laboratory for Computer Science (formerly Project MAC).
With MACSYMA the user can differentiate, integrate, take limits, solve
systems of linear or polynomial equations, factor polynomials, expand
functions in Laurent or Taylor series, solve differential equations (using
direct or transform methods), compute Poisson series, plot curves, and
manipulate matrices and tensors. MACSYMA has a language similar to ALGOL-60 to
permit the user to write his own programs for transforming symbolic
expressions.
This manual is intended to be a complete reference for the principal
features of MACSYMA as of the date shown on the cover. It is not meant to be
a tutorial nor does it discuss all of the issues involved in the efficient
manipulation of algebraic expressions. New features under development are, for
the most part, not mentioned.
The user who is unacquainted with certain concepts of computer programming
may find this document difficult on first reading. The novice will benefit by
first reading the MACSYMA Primer [Mo5] and An Introduction to ITS for the
MACSYMA User [Lew1]. This document serves as both a reference manual and as a
user's manual. When it is used as a user's manual, any sections not of
interest should be skipped. Sections which may be passed over on first
reading are indicated by the symbol [] around the section number.
It is recommended that this manual be re-read from time to time after the
user has worked with MACSYMA so that certain parts which were unclear on prior
readings will be better understood in the context of increased familiarity
with the system.
In a programming system such as MACSYMA there are often many ways to go
about solving a given problem as well as many constraints and frustrations
which must be dealt with. Some ways will not succeed due to space or time
constraints and others may work but may be unnecessarily slow. Frequently a
better understanding of the computer facilities will lead to a reformulation
of the problem lending itself to a much improved solution. For some insight
into the subject see [A1, Mo1, Mo2].
1.1 Logging in and out
MACSYMA is implemented on DEC PDP-10 computers which use the ITS operating
system and on Honeywell 6180 computers using the Multics operating system. The
following section concerns the protocol for usage on the PDP-10 at MIT known
as the MC (MACSYMA Consortium) machine. Appendix I of this manual concerns
the use of MACSYMA on Multics.
2 1.1 Logging in and out MACSYMA
A user may gain access to the PDP-10 computer at MIT by (1) finding a
terminal connected to it or (2) finding one on which he can dial to it over
public telephone lines or (3) logging into it over the ARPA network. Once
this is done the user should be communicating with the ITS time sharing
system. To login he should type a control-Z (depressing the control key while
typing the letter Z) which loads in DDT [Lew1] (the top level system program).
Then he should type :LOGIN followed by a space and his user name. (All system
commands, i.e. those which begin with a colon, are terminated by a carriage
return). At this point the user can start up any of several system programs
available (PEEK, TECO, etc.) by typing a : followed by the name of the
program. In particular :MACSYM loads in and begins execution of MACSYMA.
After printing some descriptive information, including the version number,
MACSYMA prints (C1) which means that it is ready to accept input from the
user. The entire sequence is shown below with the computer's output indented.
1
↑char means the control key is to be held down while the next character char
is typed.
2
MC ITS nnnn CONSOLE ii FREE
3
↑Z
MC ITS nnnn DDT mmm
:LOGIN SMITH
:MACSYM
This is MACSYMA vvv
(C1) NROOTS(2*X↑5-X+5,-4,8);
(D1) 1
↑Z
44130) .CALL 44154 (IOT)
:LOGOUT
MC ITS nnnn CONSOLE ii FREE
In line (C1) the user has typed a command which asks for the number of
realroots between -4 and 8 of a quintic polynomial. In line (D1) MACSYMA has
printed the answer. The ↑Z causes an exit from MACSYMA to DDT after which the
←←←←←←←←←←←←←←←
1. The user should note that there is a separate character, ↑, which is often
used for exponentiation (as in line (C1) below). Although the two characters
print in the same way the user should have no difficulty distinguishing
between them. In the example, the "↑5" could not be "control-5" since the
context strongly indicates exponentiation.
2. nnnn represents a version number which changes from time to time; ii
identifies the console.
3. Not printed on console
MACSYMA 1.1 Logging in and out 3
system typed ".CALL 44154 (IOT)" (meaning that MACSYMA was waiting for input
when interrupted). Typing :LOGOUT causes the system to delete all the user's
jobs (in this case MACSYMA was the only job) and to log him out of ITS. When
the user finishes he should always log the console out before he leaves.
1.2 General Information
Command lines to MACSYMA are strings of characters representing
mathematical expressions involving equations, arrays, functions, and programs.
Extra spaces, tabs, and all carriage returns are ignored (except when these
occur in quoted strings).
Command lines are terminated by ";" or "$" (dollar sign). A ";" causes
the command line to be evaluated and the result displayed. The terminator "$"
causes the command line to be evaluated but the result is not displayed.
When typing command lines, depressing the "rubout" or "delete" key deletes
the previous character (on hardcopy devices and displays which cannot
backspace, the deleted character is echoed). By typing "control-K" , the user
obtains a copy of the current command line free of any echoed erasures. The
two characters ?? delete the whole command line, and cause the line number to
be redisplayed.
The command (input) lines are indexed by labels of the form "(Ci)" where i
is incremented with each new command line typed by the user. Similarly, the
results of computations are also indexed. There are two types of output
lines. The ordinary output line is indexed by a label of the form "(Di)" ;
thus, usually the ith input-output pair will be (Ci)-(Di). Sometimes,however,
a computation produces several intermediate results (for example, several
solutions to an equation); it is convenient to be able to reference these
intermediate lines of output. They are indexed by labels of the form "(Ej)"
where j is incremented by one for each intermediate line. For example,
(C1) SOLVE(X↑2 + B*X + C, X);
2
SQRT(B - 4 C) + B
(E1) X = - ------------------
2
2
B - SQRT(B - 4 C)
(E2) X = - ------------------
2
(D2) [E1, E2]
Note that there is no line D1 since intermediate results were produced and
thus the line index was incremented. The general pattern of indexing is of
the form
Ci, Ei,Ei+1,...,Ej, Dj .
4 1.2 General Information MACSYMA
A command line may refer to the results of any previously indexed line
(even if it was not displayed) through the use of the line labels. For
example, the user might say SUBSTITUTE(7,B,E2); which would substitute 7 for B
in the expression E2 above. The immediately preceding D-line is conveniently
referenced by the symbol "%".
If the input line contains a syntax error, it will be reprinted and the
location of the error will be indicated as closely as possible by a special
string, ***$*** .
From a C-line, the user in need of assistance can type:
SEND(message) which will send your message to some MACSYMA system programmer
who is logged in at that time. If no appropriate recipient is available,
the message will be sent to MACSYMA's mail file where it will be seen and
will be answered. If the message consists of more than one word, then
message must begin and end with double quotes.
BUG(message) sends a bug note to MACSYMA. If more than one word, message must
begin and end with double quotes.
MAIL(message) sends mail to MACSYMA using the syntax above.
DESCRIBE(command) takes as argument any MACSYMA command and prints out the
relevant portion of the MACSYMA manual.
EXAMPLE(command) gives examples (in DEMO mode) of the use of selected
functions in MACSYMA.
PRIMER() provides an on-line primer for the novice including an introduction
to MACSYMA syntax, assignment and function definition, and the
simplification commands.
1.3 Levels of Control
When :MACSYM is typed, a LISP system extended with MACSYMA programs is
loaded into main memory from auxiliary storage. Special top level programs
read in, evaluate, simplify, and display the user's expressions. All of the
functions to be presented in subsequent sections are actually LISP programs
which, when they are called, may invoke many other LISP programs in a process
that is invisible to the user.
Switching between DDT, MACSYMA, and the LISP system in which it is
embedded is accomplished by typing the following characters:
MACSYMA 1.3 Levels of Control 5
Control-↑ typed while in MACSYMA causes LISP to be entered. The user can
now type any LISP S-expression and have it evaluated. Typing (CONTINUE) or ↑G
(control-G) causes MACSYMA to be re-entered.
↑Z causes an immediate exit to DDT. If one is already in DDT then ?? is
printed. At this point the user can run some other program like PEEK or TECO
[Lew1]. When in DDT, typing :CONTINUE causes the current job to be resumed.
(:JOB MACSYM should be typed first if the user wishes to re-enter a MACSYMA
which is not the current job).
1.4 Miscellaneous Information
Files in the ITS system have two names each of at most 6 characters. They
are referenced by giving the two names as well as the device (default is DSK)
and the directory name where the file resides (default is the same as the
user's login name). A > sign may be used for the second file name and stands
for the name which is the largest numerically if there exists a file with the
given first name and a numeric second name; otherwise, the > sign represents
the "greatest" name in an alphanumeric sense. A < sign may be similarly used
for the "least" name.
Any of the four descriptors (1st name, 2nd name, device, user) may be
omitted and either the default or the value given in a previous command (if
there was one given) will be used.
For those without a disk directory of their own, the one named USERS is
available. When placing a file on this directory the user should indicate in
some manner (such as by the first file name) the name of the user who created
it.
There is a special mail file on the ITS system for holding comments from
users which are of general interest and for listing changes to MACSYMA which
occur from time to time. The DDT command :MAIL MACSYM followed by a carriage
return and text terminated by a control-C is used to place comments in this
mail file. The user's login name and time of message are added automatically.
(Control-D may be used to cancel this or any other DDT command prematurely).
If the user encounters any bugs in MACSYMA then he should report these in
MACSYMA mail. Mail may also be sent to other users by using the :MAIL command
followed by the user's login name. If the user has received mail the message
--MAIL-- will appear on his console after he logs in. Typing a space will
cause the mail to be printed. The DDT command :PRMAIL MACSYM is used to print
out the MACSYMA mail file. The user should do this occasionally to be
informed of changes to the system and of other users' comments. :PRMAIL may
also be used to print out any user's mail by following it with the user's
login name. Control-S may be used to silence the printout. In addition, the
command :PRINT MACSYM;UPDATE > may be used to print a file describing updates
to MACSYMA since the last version of the manual. The update file should be
checked regularly so the user can be informed of changes to MACSYMA.
6 1.4 Miscellaneous Information MACSYMA
For further information on DDT commands see [Lew1]. Typing :? will list
the commands with a brief description. In particular, one command worth
noting is :KILL which kills the current job.
An Introduction to ITS for the MACSYMA User [Lew1] and the MACSYMA Primer
[Mo5] are very useful to the novice.
MACSYMA 7
2 MACSYMA's Data Types and Statement Types
This section describes the kinds of expressions MACSYMA permits and their
meanings. Chapters 5 and 6 should be referred to where necessary in order to
clarify the examples presented. Default values of options are indicated in
square brackets, [...].
2.1 Numbers
Numbers are integers, rational numbers, floating point numbers, or
"bigfloats". Integers consist of a string of digits not containing a period;
rational numbers are the quotient of two integers and are written as
numerator/denominator; floating point numbers are written as in FORTRAN, i.e.
strings of digits containing a period and optionally followed by an integer
exponent beginning with the letter E; and bigfloats are written exactly like
floating point numbers except using the letter B rather than E (the B must be
included to indicate a bigfloat). Negative numbers begin with a minus sign.
There is no limit on the number of digits in an integer or rational number but
non-zero floating point numbers must have absolute value between .14E-38 and
1.7E38 and are limited to approximately 8 digits precision. This is the
hardware limitation of the computer. Bigfloats may have any number of digits.
The default precision is 16 but the user can change this by setting the
FPPREC[16] to an integer representing the desired precision.
-17253733574534 6.023E23 -1.6E-19
37.567834987250832568B-98 3.14159 227
-3354665557331/66724255465544 -.7B0
2.2 Names
Names are used to designate variables, functions, and arrays. A name
consists of a string of letters (which may include %) and digits. It may also
include other characters but these must be preceded with a \ when typed in.
Names can be of any length and must begin with a letter (unless the leading
character is a \). Lower case letters may be typed, but they are normally
converted into the corresponding upper case letters.
%PI EPSILON X10Y30ISASTRANGENAME \*SPECIAL
2.3 Quoted Strings
A string of characters of any length may be constructed by enclosing the
8 2.3 Quoted Strings MACSYMA
string in quotation marks. To include a quotation mark, semicolon, or dollar
sign in the string it is necessary to precede it with a \ when typed in.
Quoted strings are useful as messages (such as those giving instructions for
entering data) or in as descriptive titles for printed data.
Certain names that are reserved because of their function as keywords
(operators or delimiters) are listed in Appendix II. If these are used out of
their normal context they must be quoted.
"INPUT AMOUNT IN \$" "RIEMANN'S \"ZETA\" FUNCTION"
2.4 Atomic Variables and Assignment
A name which may be assigned an arbitrary value is a variable. A variable
might or might not be subscripted (see 2.6.2). A non-subscripted variable
is designated as an "atomic variable". Atomic variables are assigned values
by writing the name of the variable followed by a : followed by an expression
representing the value to be assigned to the variable. A variable can be
assigned a new value at any time. The value of a variable can be a number, a
matrix, a list, or any MACSYMA expression. If a variable is not assigned a
value then it just represents itself. There are many variables which have
already been assigned values. These are called "MACSYMA" variables or
options. They are provided in order to give the user some control over the way
in which MACSYMA performs its operations. The user should choose names other
than these for his variables. The index to this manual distinguishes these
"MACSYMA" variables and options with a colon (":") between the variable and
its default value. In the text of the manual, the default value is indicated
with square brackets as in the next sentence. MYOPTIONS[[]] is a MACSYMA
variable---an "infolist" (see 8.1)---which is a list of all the MACSYMA
options ever reset by the user.
Some simple examples of assignment follow. (The comments in parentheses
are only for the reader's benefit and are not actually typed to or by
MACSYMA.) Note that MACSYMA automatically assigns labels Ci to the user's
input lines and Di to the output lines. These labels behave as assigned
variables and can be referenced by the user.
MACSYMA 2.4 Atomic Variables and Assignment 9
line label expression comment
←←←← ←←←←← ←←←←←←←←←← ←←←←←←←
(C1) A:16$ (integer)
(C2) LAMBDA: -3/37$ (rational number)
(C3) X:D1; (X is assigned the value of D1)
(D3) 16
(C4) RHO:SIGMA; (since SIGMA has no value at this
time RHO is assigned the symbol SIGMA)
(D4) SIGMA
(C5) SIGMA: .005$ (floating point)
(C6) RHO; (RHO still has its old value since
(D6) SIGMA it hasn't been reassigned a new one)
Since the value assigned may be any expression it may in particular be
another assignment and therefore multiple assignments are permitted. Thus
A:B:C:X+1 assigns X+1 to A, B, and C.
It is important to note that the expression assigned to the variable is
not re-copied. Only a pointer to the expression is assigned. Thus in the
above example, only one copy of X+1 is created.
The MACSYMA variable VALUES (see 8.1) gives a list (see 2.7) of
all the user's atomic variables which have been bound (i.e. have been assigned
values).
The assignment operator :: assigns the value of the expression on its
right to the value of the quantity on its left, which must evaluate to an
atomic variable or subscripted variable (sec. 2.6.2). Thus continuing with
the above examples:
(C7) RHO::LAMBDA$ (Note that the :: causes the value of
(C8) SIGMA; LAMBDA, i.e. -3/37, to be assigned to
(D8) -3/37 the value of RHO, i.e. SIGMA.)
(C9) VALUES;
(D9) [A, LAMBDA, X, RHO, SIGMA]
2.5 Mathematical Operators
Mathematical expressions are constructed by using the following operators
and also functions (see 2.6). The usage and priorities from highest to
lowest are:
10 2.5 Mathematical Operators MACSYMA
Operator Name Symbol Usage
←←←←←←←← ←←←← ←←←←←← ←←←←←
factorials !! ! postfix
exponentiation ** or ↑ infix
non-commutative ↑↑ infix
exponentiation
non-commutative . infix
multiplication
div'sn mult'pn / * infix
negation - prefix
add'n subt'n + - infix
If an operator is referred to out of context it must be enclosed in
quotation marks.
EXPT is used to display exponentiation when the base or exponent become
unwieldy.
! is the factorial which is the product of all the integers from 1 up to
its argument. Thus 5! = 1*2*3*4*5 = 120. The value of the option FACTLIM[-1]
gives the highest factorial which is automatically expanded. If it is -1 then
all integers are expanded.
!! stands for double factorial which is defined as the product of all the
consecutive odd (or even) integers from 1 (or 2) to the odd (or even)
argument. Thus 8!! is 2*4*6*8 = 384.
Period is used for non-commutative product. It must be preceded and
followed by a space when any ambiguity can arise with respect to floating
point numbers. Non-commutative exponentiation is used in the usual sense that
M↑↑2 means M . M.
Operators of equal priority are performed left to right. Parentheses can
be used to change the order of evaluation. Also functional application has
the highest priority. Thus SIN(A*X↑Y/Z!)↑2 means (SIN(A*(X↑Y)/(Z!)))↑2
The operands may be any MACSYMA expressions whose values are the correct
types of data. Note that every statement in MACSYMA yields a value even if
the value is only a trivial one.
MACSYMA has no restriction on the mixing of modes of operands. Integers,
rationals, floating point numbers, and bigfloats may be freely intermixed in
an expression; when conversions are necessary, the priority of conversion is
in the order of the types just mentioned. If floating point numbers or
bigfloats of differing precision are combined in a operation, they will be
converted to floating point or bigfloat numbers of the current precision by
padding with zeroes or by dropping off low order digits and rounding.
Floating point underflow will return 0.0 unless the MACSYMA variable
MACSYMA 2.5 Mathematical Operators 11
1
ZUNDERFLOW[TRUE] is FALSE in which case an error will be signaled.
2.6 Functions and Arrays
2.6.1 Functions
A function is written as a name followed by the arguments to the function
separated by commas and enclosed in parentheses. The arguments may be any
MACSYMA expressions.
A function of a fixed number of arguments can be defined in MACSYMA by
using the := operator. The left side of a function definition consists of the
name of the function followed by the list of formal parameters enclosed in
parentheses. The right side consists of the function body. When a function
is called, the formal parameters will be bound to the actual arguments, any
free variables in the function body will take on the values which they have at
the time of the call, and the function body will be evaluated. It is
permissible to define functions which are recursive to an arbitrary depth.
Care should be taken when passing an expression which contains a variable with
the same name as a formal parameter to a function defined with that formal
parameter as circularity could result when it is evaluated (see 3.2).
The MACSYMA variable FUNCTIONS is a list of all user defined non-
subscripted functions.
The MACSYMA function DISPFUN may be used to display the definition of a
function (see 10).
(C1) F(X):=X↑2+Y$
(C2) F(2);
(D2) Y + 4
(C3) Y:7$
(C4) F(2);
(D4) 11
(C5) G(Y,Z):=F(Z)+3*Y;
(D5) G(Y, Z) := F(Z) + 3 Y
(C6) G(2*Y+Z,-.5);
←←←←←←←←←←←←←←←
1. The square brackets enclose default options
12 2.6.1 Functions MACSYMA
(D6) 3 (Z + 14) + Z + 14.25
(C7) FUNCTIONS;
(D7) [F(X), G(Y, Z)]
The example involving the function G above requires some explanation. In
going from C6 to D6 the following occurs:
(1) The arguments to G are evaluated giving Z+14 and -.5 (Y has the value
7).
(2) G is then invoked and has its formal parameters bound. Y to Z+14 (the
first argument) and Z to -.5 (the second argument). The evaluation of G then
causes F to be invoked on the argument -.5
(3) F has its formal parameter X bound to -.5 and returns the result of
2
the evaluation X +Y with the current bindings which gives Z+14.25
(4) The evaluation of G continues with 3*Y which yields 3*(Z+14). This is
added to the result from (3) and returned.
2.6.2 Arrays
Arrays enable one to refer to a collection of elements by using a single
name. An element of an array is referred to by a subscripted variable which is
a name followed by a list of subscripts enclosed in square brackets. Arrays
1
in MACSYMA are of two types - declared or undeclared. Declared arrays are
similar to FORTRAN arrays. The user declares the number of dimensions and
indicates the maximum value of each subscript. The system then allocates
space for the entire array. To declare an array the user types:
ARRAY(name, dim1, dim2, ..., dimk)
This sets up a k-dimensional array. A maximum of five dimensions may be used.
The subscripts for the ith dimension are the integers running from 0 to dimi.
If the user assigns to a subscripted variable without declaring the
corresponding array, an undeclared array is set up.
Undeclared arrays, otherwise known as hashed arrays (because hash coding
is done on the subscripts), are more general than declared arrays. The user
does not declare their maximum size, and they grow dynamically by hashing as
←←←←←←←←←←←←←←←
1. For efficient translation, the user can also inform MACSYMA of arrays all
of whose elements are of a single type, e.g. INTEGER,BOOLEAN,FLOAT. (see
10.8).
MACSYMA 2.6.2 Arrays 13
more elements are assigned values. The subscripts of undeclared arrays need
not even be numbers. However, unless an array is rather sparse, it is
probably more efficient to declare it when possible than to leave it
undeclared. The ARRAY function can be used to transform an undeclared array
into a declared array.
Array elements can be assigned values explicitly with the : operator or
implicitly by means of an associated function, and the values assigned may be
any MACSYMA expression. To understand implicit assignment we must understand
what MACSYMA does when asked to evaluate a subscripted variable. MACSYMA
first evaluates the subscripts left to right. Then it does an array access to
see if the requested array element already has a value. If it does, the value
is returned. If it does not, MACSYMA checks to see whether the array has an
associated function (see below). If not, the subscripted variable (with the
subscripts evaluated) is returned. (This is standard MACSYMA practice - if
there is no value for a variable, the variable itself is returned when an
evaluation is done.) If there is an associated function, the parameters of the
function are bound to the given subscripts, and the function body is
evaluated. The value of the function call is stored in the appropriate array
element and returned. Note that once an element is computed by the associated
function it is stored so that next time it is needed it will not be
recomputed. A consequence of this is that unless the user uses the KILL,
REMVALUE, or REMARRAY functions (sec. 10.3) to kill an array element or
the entire array, the associated function will never be called a second time
on the same arguments. Thus the user should be aware that even if he
redefines the associated function, those values which already exist will stay
around. Of course individual array elements can be changed by assignment at
any time.
These associated functions are defined with the := operator. Their
definition looks exactly the same as ordinary function definitions, except
that the parameters in the left side of the definition are enclosed in
brackets instead of parentheses.
In order to use a subscripted variable as a single entity without it being
an array and without ever assigning a value to it, it should be prefixed by an
apostrophe to avoid it being confused with a non-subscripted variable of the
same name. For example SUBST(0,W,W+'W[0]).
The MACSYMA variable ARRAYS is a list of all the arrays that have been
allocated, both declared and undeclared.
DISPFUN (see 10.2) may be used to display the definition of an array
associated function.
ARRAYINFO (see 8.1.1) may be used to find out whether an array is
declared or undeclared, how large it is, how many subscripts it has, and which
elements have values in the case of an undeclared array.
(C1) A[N]:=N*A[N-1]$
(C2) A[0]:1$
14 2.6.2 Arrays MACSYMA
(C3) A[5];
(D3) 120
(C4) A[N]:=N$
(C5) A[6];
(D5) 6
(Note that the definition in C4 is being used because A[6] has no value up to
this time.)
(C6) A[4];
(D6) 24
(Since A[4] was assigned a value as a result of A[5] being computed, the new
definition is not used.)
If one is going to define a recursive function which is to be called
several times then if may be more efficient to use an array with an associated
function for initialization. The reason is that once an element is computed
it is stored and thus need not be computed again whereas with a non-
subscripted function, each recursive call may cause a repeat of a past
computation.
2.6.3 Lambda Notation
The LAMBDA notation is used for unnamed functions in order to indicate the
correspondence between the variables of the function and the arguments which
are to be substituted for them. It is useful when one desires to pass
functional arguments to other functions or when one wants to apply a function
just once without having to define it with :=.
(C1) F:LAMBDA([X,Y,Z],X↑2+Y↑2+Z↑2);
2 2 2
(D1) LAMBDA([X, Y, Z], Z + Y + X )
(C2) F(1,2,A);
2
(D2) A + 5
MACSYMA also permits operators to be used in a functional notation;
however, in order not to get a syntax error they must be surrounded by "s.
(C3) "+"(1,2,A);
(D3) A + 3
MACSYMA 15
[2.6.4] Subscripted Functions (Arrays of Functions)
It is possible for the value of an array element to be a lambda
expression. Thus if the assignment F[1]:LAMBDA([X],X↑2+1) were performed, then
F[1] could be used in the ordinary prefix functional sense with its arguments
following in parentheses, e.g. F[1](3) would yield the value 10. There is an
alternative syntax available for assigning a lambda expression to an array
which introduces the notion of a "subscripted function". In the above case
one could also type F[1](X):=X↑2+1 and this would be entirely equivalent.
Other elements of the array could be assigned different lambda expressions (or
any MACSYMA expressions). If there is an algorithm for computing the
different functions to be stored in an array on the basis of the subscripts
alone, then one may use an associated function. For example,
F[K]:=LAMBDA([X],X↑K+1). Again an alternative syntax of F[K](X):=X↑K+1 may be
used. The left side of the definition consists of the function name followed
by the subscripts, enclosed in brackets, followed by the arguments, enclosed
in parentheses. The subscripts (which are not evaluated at definition time)
must be either all numeric or all symbolic. Note that subscripted functions
are treated exactly like arrays so all of the information in sec. 2.6.2
applies. In particular when a subscripted function is referenced, the element
is immediately retrieved and applied to its arguments if it exists; otherwise
it is computed (this time only) and then applied. Consequently, two
evaluations of the definition are performed. Thus consider the definition
F[K](E):=COEFF(E,X,K) and the call F[2](3*X↑2-1). Although the user may have
2 2
thought that this would return the coefficient of X in 3*X -1, i.e. 3, it
will return 0. The reason is that F[2] is first computed by evaluating the
definition yielding 0, since E has not been bound at this time. Note that
F[K](E):= SUBST(K,'J,'(COEFF(E,X,J))) would return the desired result as would
F(K,E):=COEFF(E,X,K). Thus the user should be clear about the distinction
between subscripted functions (a type of array) and ordinary functions. Also a
subscripted function should not be redefined without KILL'ing or REMARRAY'ing
it first; otherwise the elements which have already been stored will be used.
The ARRAYS[[]] list (see 2.6.2) also includes subscripted functions.
The function ARRAYINFO (see 8.1.1) may also be used on subscripted
functions.
(C1) T[N](X):=RATSIMP(2*X*T[N-1](X)-T[N-2](X))$
This generates the Chebyshev polynomials.
(C2) T[0](X):=1$
(C3) T[1](X):=X$
(C4) T[4](Y);
4 2
(D4) 8 Y - 8 Y + 1
16 2.6.4 Subscripted Functions (Arrays of Functions) MACSYMA
(C5) G[N](X):=SUM(EV(X),I,N,N+2)$
(C6) H(N,X):=SUM(EV(X),I,N,N+2)$
(C7) G[2](I↑2);
2
(D7) 3 I
(C8) H(2,I↑2);
(D8) 29
The following illustrates a definition for the Legendre polynomials.
(C9) P[N](X):=RATSIMP(1/(2↑N*N!)*DIFF((X↑2-1)↑N,X,N))$
(C10) Q(N,X):=RATSIMP(1/(2↑N*N!)*DIFF((X↑2-1)↑N,X,N))$
(C11) P[2];
2
3 X - 1
(D11) LAMBDA([X], --------)
2
(C12) P[2](Y+1);
2
3 (Y + 1) - 1
(D12) --------------
2
(C13) Q(2,Y+1);
2
3 Y + 6 Y + 2
(D13) --------------
2
(C14) P[2](5);
(D14) 37
(C15) Q(2,5);
5
attempt to differentiate wrt a number
[2.6.5] Additional Information About Functions
In order to pass a function as an argument to another function you need
only give its name in the argument list of the call. It may then be used in
the called function by following the name of the corresponding formal
parameter with a parenthesized list of arguments. Subscripted functions (see
2.6.4) are passed by giving the name followed by the subscripts in brackets.
Arrays can be passed by giving the name of the array in the argument list and
they can be referenced by subscripting the corresponding formal parameter.
When passing names of functions or arrays one must take care that there is
MACSYMA 2.6.5 Additional Information About Functions 17
no atomic variable with the same name which is bound because then that value
rather than the name will be passed. In this case the name should be preceded
by a ' (see 3.2) to prevent it from being evaluated.
In order to assign to a formal parameter of a function so that the
corresponding actual parameter gets changed (and remains changed) when the
function is exited, then the :: operator rather than the : operator should be
used.
(C7) F[I,J](X,Y):=X↑I + Y↑J;
I J
(D7) F (X, Y) := X + Y
I, J
(C8) G(FUN,ARG1,ARG2):=PRINT(FUN," APPLIED TO ",ARG1," AND ",
ARG2," IS ",FUN(ARG1,ARG2))$
(C9) G(F[2,1],SIN(%PI),2*A);
2
LAMBDA([X,Y],Y+X ) APPLIED TO 0 AND 2 A IS 2 A
(D9) 2 A
2.7 Lists
Lists are ordered sets of elements which can be any MACSYMA expressions.
They are written enclosed in brackets with elements separated by commas. If
the value of a variable is a list, its elements may be obtained or assigned to
by subscripting as with arrays. In certain cases lists are treated like
vectors (row or column matrices). (see 2.8) Lists are sometimes used as
arguments to MACSYMA functions (e.g. MATRIX, SOLVE, etc.). Chapter 8
describes functions for many list operations such as deleting elements,
selecting an element, reversing a list, etc.
(C1) [X↑2,Y/3,-2]$
(C2) %[1]*X;
3
(D2) X
(C3) [A,D1,D2];
2 Y 3
(D3) [A , [X , - , -2] , X ]
3
18 MACSYMA
2.8 Matrices
A matrix is a 2-dimensional ordered set of elements. It is represented
internally using a list of lists all of the same length which stand for the
rows of the matrix. Matrices may be constructed by using the function MATRIX
whose arguments are lists representing the rows of the matrix. (The functions
ENTERMATRIX and GENMATRIX may also be used to construct a matrix (see
6.4).)
The operators + , - , * , and / may be used between two matrices and take
effect elementwise. (A matrix minus itself gives the zero matrix of the same
size.) They may also be used between a scalar and a matrix and the scalar will
1
be operated on with each element of the matrix.
Matrix multiplication is signified by using the dot operator (non-
commutative product). Raising a matrix to a power (multiplying it by itself)
is accomplished by use of the ↑↑ operator. That is, M.M is equivalent to
M↑↑2. The inverse of a matrix may be obtained by using a negative exponent,
i.e. M↑↑-1.
If the switch LISTARITH[TRUE] is TRUE then
1) Lists will behave arithmetically: they can be added to one
another, etc.
2) In matrix operations they can be used as row or column vectors
and will be converted to such when necessary.
An element of a matrix may be referenced by subscripting the matrix as
with arrays but the same name should not be used to stand for both a matrix
and an array.
There are many functions for operating on matrices as well as many options
which can be set to give the user much flexibility and control over matrix
operations (these are described in sec. 6.4). If a matrix is too wide to
be displayed all at once, it is displayed column by column or as a list of
lists.
(C1) M:MATRIX([A,0],[B,1]);
[ A 0 ]
(D1) [ ]
[ B 1 ]
(C2) M[1,1]*%;
←←←←←←←←←←←←←←←
1. In MACSYMA a scalar is an expression free of lists, matrices, and any atoms
declared NONSCALAR.
MACSYMA 2.8 Matrices 19
[ 2 ]
(D2) [ A 0 ]
[ ]
[ A B A ]
(C3) M*M;
[ 2 ]
[ A 0 ]
(D3) [ ]
[ 2 ]
[ B 1 ]
(C4) M.M;
[ 2 ]
(D4) [ A 0 ]
[ ]
[ A B + B 1 ]
(C5) D2-D4+1;
[ 1 1 ]
(D5) [ ]
[ 1 - B A ]
(C6) M↑-1;
DIVISION BY 0
(C7) M↑↑-1;
[ 1 ]
[ - 0 ]
[ A ]
(D7) [ ]
[ B ]
[ - - 1 ]
[ A ]
(C8) [X,Y].M;
(D8) [ B Y + A X Y]
2.9 Equations
An equation is formed in MACSYMA simply by using an equal sign between any
two expressions. Equations may be added or subtracted, and they may be
multiplied or divided by any expression. They may be operated on just as any
MACSYMA expression can be and may serve as arguments to functions.
(C1) X+1=Y↑2$
(C2) X-1=2*Y+1$
20 2.9 Equations MACSYMA
(C3) D1+D2;
2
(D3) 2 X = Y + 2 Y + 1
(C4) D1/Y;
X + 1
(D4) ----- = Y
Y
(C5) 1/%;
Y 1
(D5) ----- = -
X + 1 Y
[2.10] IF Statement
The IF statement is used for conditional execution. The syntax is
IF condition THEN expression1 ELSE expression2.
The result of an IF statement is expression1 if condition is true and
expression2 if it is false. expression1 and expression2 are any MACSYMA
expressions (including nested IF statements), and condition is an expression
which evaluates to TRUE or FALSE and is composed of relational and logical
operators which are as follows:
Operator name Symbol Type
←←←←←←←← ←←←← ←←←←←← ←←←←
greater than > relational infix
equal to = , EQUAL " "
not equal to # " "
less than < " "
greater than >=
or equal to " "
less than <=
or equal to " "
and AND logical infix
or OR " "
not NOT logical prefix
The relational operators all have equal priorities which are less than the
priorities of the arithmetic operators and greater than that of the logical
operators. The priority of NOT is greater than that of AND which is greater
than that of OR. The difference between "=" and EQUAL is discussed in sec.
7.1.
If the ELSE clause is omitted, this will be the same as if ELSE FALSE were
specified. In order to have several expressions evaluated after the THEN or
MACSYMA 2.10 IF Statement 21
ELSE clauses, the expressions may be enclosed in a compound statement (see
2.11) but care should be taken to return the desired value. The switch
PREDERROR[TRUE] determines the action taken if a clause is not universally
true or universally false (see 7.1).
(C1) FIB[N]:= IF N=1 OR N=2 THEN 1
ELSE FIB[N-1]+FIB[N-2]$
(C2) FIB[1]+FIB[2];
(D2) 2
(C3) FIB[3];
(D3) 2
(C4) FIB[5];
(D4) 5
(C5) ETA(MU,NU):= IF MU=NU THEN MU
ELSE IF MU>NU THEN MU-NU
ELSE MU+NU$
(C6) ETA(5,6);
(D6) 11
(C7) ETA(ETA(7,7),ETA(1,2));
(D7) 4
(C8) IF NOT 5>=2 AND 6<=5 OR 4+1>3 THEN A ELSE B;
(D8) A
[2.11] Compound Statements
In order to execute a sequence of statements in a context where a single
statement is permitted then the user may group these statements into a
compound statement by separating them with commas and enclosing the whole
group in parentheses. The value of a compound statement is the value of the
last statement in the group.
Compound statements are also useful for grouping together a sequence of
related calculations when a computation cannot easily be expressed in a single
MACSYMA statement.
(C1) IF X=Y THEN (X:X+1, Y:Y-1)
ELSE (S:0, FOR I:1 THRU X DO (S:S+F(I), Y:Y-G(Y)))$
[2.12] Program Blocks
22 2.12 Program Blocks MACSYMA
Blocks in MACSYMA are somewhat analogous to subroutines in FORTRAN or
procedures in ALGOL or PL/I. Blocks are like compound statements but also
enable the user to tag statements within the block and to assign "dummy"
variables to values which are local to the block. The syntax is:
BLOCK([v1, ... vk], statement1,..., statementj)
where the vi are atomic variables which are local to the BLOCK and the
statements are any MACSYMA expressions. If no variables are to be made local
then the list may be omitted.
A block uses these local variables to avoid conflict with variables having
the same names used outside of the block (i.e. global to the block). In this
case, upon entry to the block, the global values are saved onto a stack and
are inaccessible while the block is being executed. The local variables then
are unbound so that they evaluate to themselves. They may be bound to
arbitrary values within the block but when the block is exited the saved
values are restored to these variables. The values created in the block for
these local variables are lost. Where a variable is used within a block and
is not in the list of local variables for that block it will be the same as
the variable used outside of the block.
In order to save and restore other local properties besides VALUE, namely
ARRAY (except for complete arrays - (see 10.8)), FUNCTION, DEPENDENCIES,
1
ATVALUE, MATCHDECLARE, ATOMGRAD, CONSTANT, and NONSCALAR (see 8.1), the
function LOCAL should be used inside of the block with arguments being the
names of the variables (see 10.6).
The value of the block is the value of the last statement or the value of
the argument to the function RETURN which may be used to exit explicitly from
the block. The function GO may be used to transfer control to the statement of
the block that is tagged with the argument to GO. To tag a statement, precede
it by an atomic argument as another statement in the BLOCK. For example:
BLOCK([X],X:1,LOOP,X:X+1,...,GO(LOOP),...). The argument to GO may be any
expression which evaluates to a tag. For example GO(IF X>Y THEN PLACE1 ELSE
COMPUTEPLACE(X)). One cannot use GO to transfer to a tag in a BLOCK other
than the one containing the GO.
Blocks typically appear on the right side of a function definition but can
be used in other places as well.
(C1) HESSIAN(F):=BLOCK([DFXX,DFXY,DFXZ,DFYY,DFYZ,DFZZ],
DFXX:DIFF(F,X,2),DFXY:DIFF(F,X,1,Y,1),
DFXZ:DIFF(F,X,1,Z,1),DFYY:DIFF(F,Y,2),
DFYZ:DIFF(F,Y,1,Z,1),DFZZ:DIFF(F,Z,2),
DETERMINANT(MATRIX([DFXX,DFXY,DFXZ],[DFXY,DFYY,DFYZ],
[DFXZ,DFYZ,DFZZ])))$
←←←←←←←←←←←←←←←
1. All of these properties except for FUNCTION are related more closely to the
use of the name as a variable rather than as a function
MACSYMA 2.12 Program Blocks 23
(C2) HESSIAN(X↑3-3*A*X*Y*Z+Y↑3);
3 2 3 2 3
(D2) - 54 A X Y Z - 54 A Y - 54 A X
(C3) SUBST(1,Z,QUOTIENT(%,-54*A↑2));
3 3
(D3) X + A Y X + Y
The above example computes the Hessian of a cubic curve (the Folium of
Descartes) which turns out to be invariant under this transformation, i.e. the
result is of the same form.
The example below illustrates the saving and restoring of values described
at the beginning of this section.
(C4) F(X):=BLOCK([Y], LOCAL(A), Y:4, A[Y]:X, DISPLAY(A[Y]))$
(C5) Y:2$
(C6) A[Y+2]:0$
(C7) F(9);
A = 9
4
(D7) DONE
(C8) A[Y+2];
(D8) 0
If LOCAL(A) had not been used, the value on line D8 would have been 9.
[2.13] The DO Statement
The DO statement is used for performing iteration. Due to its great
generality the DO statement will be described in two parts. First the usual
form will be given which is analogous to that used in several other
programming languages (FORTRAN, ALGOL, PL/I, etc.); then the other features
will be mentioned.
[2.13.1] Commonly Used Forms
There are three variants of this form that differ only in their
terminating conditions. They are:
24 2.13.1 Commonly Used Forms MACSYMA
(a) FOR variable : initial-value STEP increment
THRU limit DO body
(b) FOR variable : initial-value STEP increment
WHILE condition DO body
(c) FOR variable : initial-value STEP increment
UNLESS condition DO body
(Alternatively, the STEP may be given after the termination condition or
limit. )
The initial-value, increment, limit, and body can be any expressions. To
iterate over several statements, the body can be made into a compound
statement (see 2.11) or a BLOCK (see 2.12). The condition is as in the IF
statement. If the increment is 1 then "STEP 1" may be omitted.
The execution of the DO statement proceeds by first assigning the initial-
value to the variable (henceforth called the control-variable). Then: (1) If
the control-variable has exceeded the limit of a THRU specification, or if the
condition of the UNLESS is TRUE, or if the condition of the WHILE is FALSE
then the DO terminates. (2) The body is evaluated. (3) The increment is added
to the control-variable. The process from (1) to (3) is performed repeatedly
until the termination condition is satisfied. One may also give several
termination conditions in which case the DO terminates when any of them is
satisfied.
In general the THRU test is satisfied when the control-variable is greater
than the limit if the increment was non-negative, or when the control-variable
is less than the limit if the increment was negative. The increment and limit
may be non-numeric expressions as long as this inequality can be determined.
However, unless the increment is known to be negative (i.e. is a negative
number) at the time the DO statement is input, MACSYMA assumes it will be
positive when the DO is executed. If it is not positive, then the DO may not
terminate properly.
Note that the limit, increment, and termination condition are evaluated
each time through the loop. Thus if any of these involve much computation,
and yield a result that does not change during all the executions of the body,
then it is more efficient to set a variable to their value prior to the DO and
use this variable in the DO form.
The value normally returned by a DO statement is the atom DONE, as every
statement in MACSYMA returns a value. However, the function RETURN (see 2.12)
may be used inside the body to exit the DO prematurely and give it any desired
value. Note however that a RETURN within a DO that occurs in a BLOCK will
exit only the DO and not the BLOCK. Note also that the GO function may not be
used to exit from a DO into a surrounding BLOCK.
The control-variable is always local to the DO and thus any variable may
MACSYMA 2.13.1 Commonly Used Forms 25
be used without affecting the value of a variable with the same name outside
of the DO. The control-variable is unbound after the DO terminates.
(C1) FOR A:-3 THRU 26 STEP 7 DO LDISPLAY(A)$
(E1) A = -3
(E2) A = 4
(E3) A = 11
(E4) A = 18
(E5) A = 25
The function LDISPLAY generates intermediate labels; DISPLAY does not.
(C6) S:0$
(C7) FOR I:1 WHILE I<=10 DO S:S+I;
(D7) DONE
(C8) S;
(D8) 55
Note that the condition in C7 is equivalent to UNLESS I > 10 and also THRU 10
(C9) SERIES:1$
(C10) TERM:EXP(SIN(X))$
(C11) FOR P:1 UNLESS P>7 DO
(TERM:DIFF(TERM,X)/P,
SERIES:SERIES+SUBST(X=0,TERM)*X↑P)$
(C12) SERIES;
7 6 5 4 2
(D12) X X X X X
-- - --- - -- - -- + -- + X + 1
96 240 15 8 2
sin(x)
which gives 8 terms of the Taylor series for e .
(C13) POLY:0$
(C14) FOR I:1 THRU 5 DO
FOR J:I STEP -1 THRU 1 DO
POLY:POLY+I*X↑J$
(C15) POLY;
5 4 3 2
(D15) 5 X + 9 X + 12 X + 14 X + 15 X
(C16) GUESS:-3.0$
(C17) FOR I:1 THRU 10 DO (GUESS:SUBST(GUESS,X,.5*(X+10/X)),
IF ABS(GUESS↑2-10)<.00005 THEN RETURN(GUESS));
(D17) - 3.1622807
26 2.13.1 Commonly Used Forms MACSYMA
This example computes the negative square root of 10 using the Newton-
Raphson iteration a maximum of 10 times. Had the convergence criterion not
been met the value returned would have been "DONE".
[2.13.2] Additional Forms of the DO Statement
Instead of always adding a quantity to the control-variable one may
sometimes wish to change it in some other way for each iteration. In this
case one may use "NEXT expression" instead of "STEP increment". This will
cause the control-variable to be set to the result of evaluating expression
each time through the loop.
(C1) FOR COUNT:2 NEXT 3*COUNT THRU 20
DO DISPLAY(COUNT)$
COUNT = 2
COUNT = 6
COUNT = 18
As an alternative to FOR variable:value ...DO... the syntax FOR variable
FROM value ...DO... may be used. This permits the "FROM value" to be placed
after the step or next value or after the termination condition. If "FROM
value" is omitted then 1 is used as the initial value.
Sometimes one may be interested in performing an iteration where the
control-variable is never actually used. It is thus permissible to give only
the termination conditions omitting the initialization and updating
information as in the following example to compute the square-root of 5 using
a poor initial guess.
(C1) X:1000
(C2) THRU 10 WHILE X#0.0 DO X:.5*(X+5.0/X)$
(C3) X;
(D3) 2.236068
If it is desired one may even omit the termination conditions entirely and
just give "DO body" which will continue to evaluate the body indefinitely. In
this case the function RETURN (see 2.11) should be used to terminate execution
of the DO.
(C1) NEWTON(F,GUESS):=BLOCK([NUMER,Y],LOCAL(DF),NUMER:TRUE,
DEFINE(DF(X),DIFF(F(X),X)),
DO (Y:DF(GUESS), IF Y=0.0 THEN ERROR(
"DERIVATIVE AT",GUESS," IS ZERO"),
GUESS:GUESS-F(GUESS)/Y,
IF ABS(F(GUESS))<5.0E-6 THEN RETURN(GUESS)))$
(C2) SQR(X):=X↑2-5.0$
MACSYMA 2.13.2 Additional Forms of the DO Statement 27
(C3) NEWTON(SQR,1000);
(D3) 2.236068
(Note that RETURN, when executed, causes the current value of GUESS to be
returned as the value of the DO. The BLOCK is exited and this value of the DO
is returned as the value of the BLOCK because the DO is the last statement in
the block.)
One other form of the DO is available in MACSYMA. The syntax is:
FOR variable IN list [end-tests] DO body
The members of the list (see 2.7) are any expressions which will
successively be assigned to the variable on each iteration of the body. The
optional end-tests can be used to terminate execution of the DO; otherwise it
will terminate when the list is exhausted or when a RETURN is executed in the
body. (In fact, list may be any non-atomic expression, and successive parts
are taken.)
(C1) FOR F IN [LOG, RHO, ATAN] DO LDISP(F(1.0))$
(E1) 0
(E2) RHO(1)
%PI
(E3) ---
4
(C4) EV(E3,NUMER);
(D4) 0.78539816
[2.14] Syntax Extension
It is possible to add new operators to MACSYMA (infix, prefix, postfix,
unary, or matchfix with given precedences), to remove existing operators, or
to redefine the precedence of existing operators. Details may be found in
Appendix II.
28 MACSYMA
3 What a Serious User Should Know
Usually the user need not be concerned with the internal workings of
MACSYMA, but some knowledge of the representation of expressions and of the
way in which they are evaluated, simplified, and displayed should be acquired
in order to use MACSYMA more easily, efficiently, and effectively.
3.1 Representation
After an expression is read by MACSYMA it is automatically translated
(i.e. lexically scanned and parsed) to a LISP "internal" form. This is the
form in which MACSYMA's programs deal with expressions. Initially the
translated expression is in "general" form but certain functions convert this
to other forms.
(1) The general form represents non-atomic expressions as LISP lists whose
first element is the main operator of the expression and whose remaining
elements are the operands also represented in this form. Thus, after
1
simplification, 2*X+3/4 is represented essentially as (PLUS (RAT 3 4) (TIMES
2 X)). F(X)-LOG(X) is represented as (PLUS (F X) (TIMES -1 (LOG X))). Any
expression which MACSYMA deals with can be represented in this form.
(2) Canonical Rational Expressions constitute a second kind of
representation which is especially suitable for expanded polynomials and
rational functions (as well as for partially factored polynomials and rational
functions when RATFAC[FALSE] is set to TRUE, (see 6.5)). In this CRE
form an ordering of variables (from most to least main) is assumed for each
expression. Polynomials are represented recursively by a list consisting of
the main variable followed by a series of pairs of expressions, one for each
term of the polynomial. The first member of each pair is the exponent of the
main variable in that term and the second member is the coefficient of that
term which could be a number or a polynomial in another variable again
represented in this form. Thus the principal part of the CRE form of 3*X↑2-1
is (X 2 3 0 -1) and that of 2*X*Y+X-3 is (Y 1 (X 1 2) 0 (X 1 1 0 -3)) assuming
Y is the main variable, and is (X 1 (Y 1 2 0 1) 0 -3) assuming X is the main
variable. "Main"-ness is usually determined by reverse alphabetical order.
The "variables" of a CRE expression needn't be atomic. In fact any
subexpression whose main operator is not + - * / or ↑ with integer power will
be considered a "variable" of the expression (in CRE form) in which it occurs.
For example the CRE variables of the expression X+SIN(X+1)+2*SQRT(X)+1 are X,
SQRT(X), and SIN(X+1). If the user does not specify an ordering of variables
by using the RATVARS function (see 6.5) MACSYMA will choose an alphabetic
one.
←←←←←←←←←←←←←←←
1. Ignoring the flags MACSYMA places on operators
MACSYMA 3.1 Representation 29
In general, CRE's represent rational expressions, that is, ratios of
polynomials, where the numerator and denominator have no common factors, and
the denominator is positive. The internal form is essentially a pair of
polynomials (the numerator and denominator) preceded by the variable ordering
list.
If an expression to be displayed is in CRE form or if it contains any
subexpressions in CRE form, the symbol /R/ will follow the line label.
(3) An extended CRE form is used for the representation of Taylor series.
The notion of a rational expression is extended so that the exponents of the
variables can be positive or negative rational numbers rather than just
positive integers and the coefficients can themselves be rational expressions
as described above in (2) rather than just polynomials. These are represented
internally by a recursive polynomial form which is similar to and is a
generalization of CRE form, but carries additional information such as the
degree of truncation.
As with CRE form, the symbol /T/ follows the line label of such
expressions.
(4) When RATFAC[FALSE] is TRUE,expressions are brought into partially
factored form: numerator and denominator are relatively prime products of
recursively constructed primitive polynomial kernels. Kernels at the same
level within numerator and denominator may not be relatively prime. In the
future, kernels may be further specified to be square-free.
(5) Another internal form is used to represent Poisson series. This
specialized representation of trigonometric series is described in section
6.6.
3.2 Evaluation
After MACSYMA parses a command line the expression is evaluated and
simplified and the result is displayed. Often the two-phase process of
evaluation and simplification is referred to simply as "evaluation." In this
section though, we use the word "evaluation" to refer only to the evaluation
stage proper and not to the simplification stage.
MACSYMA expressions consist of numbers, variables, function calls, and
operators. When an expression is read by MACSYMA the parsing program
translates it into LISP preserving the order and the result is the value of
the current C line. The evaluation phase proceeds by building up an
expression which is similar in form to the input expression, but has certain
substitutions. The evaluator is recursive, and calls itself on all sub-
expressions.
When the evaluator sees a name, it checks to see whether the name has a
value assigned to it. If there is a value, that value is returned by the
30 3.2 Evaluation MACSYMA
evaluator. If there is no value assigned to the name, the evaluator just
returns the name itself. (For the means of assigning values to names refer to
2.4. For a description of the evaluation process as applied to subscripted
names see 2.6.2). Note that problems could arise if a variable is bound to an
expression containing an occurrence of that variable since each time the
variable is evaluated, the entire expression is substituted for each
occurrence of the variable. For example if Y has the value [X,Y,Z] and if the
value of Y is evaluated the result is [X,[X,Y,Z],Z].
MACSYMA distinguishes between two types of functions - nouns and verbs.
Most functions in the system, including all user-defined functions, are
initially considered to be verb-type. Undefined functions and some system
functions are considered to be noun-type. When the evaluator sees a function
call, it evaluates the arguments to the function (unless that function is of a
type which doesn't have its arguments evaluated, e.g. BATCH; for a list of
such functions see Appendix VI). Then, if the function is verb-type, the
evaluator applies the function to the evaluated arguments and returns the
value of the function. For noun-type functions the evaluator returns an
expression identical to the function call, except that the arguments are
replaced by their evaluations.
The user can explicitly declare a name to be noun-type by using the
1
DECLARE function (see 8.1.1). For example, the function INTEGRATE
normally tries to integrate its first argument. After the command
DECLARE(INTEGRATE,NOUN) is given however, INTEGRATE will not perform the
integration. Sometimes the user may give a verb function arguments which it
is not equipped to deal with. In certain cases the verb function will return
the noun form of itself. If this was because of some undefined functions in
the expression, which the user defines at a later time, he can cause the noun-
form to be re-evaluated at that time by giving the label of that line followed
by the name of the unevaluated function separated with a comma as arguments of
the function EV (see 6.1). For example:
(C1) DIFF(X*F(X),X);
d
(D1) X (-- F(X)) + F(X)
dX
(C2) F(X):=SIN(X)$
(C3) EV(D1,DIFF);
(D3) SIN(X) + X COS(X)
Here we see that the expression returned by the evaluator is similar to
←←←←←←←←←←←←←←←
1. The noun-ness applies to both the function use and array use of the name.
Note that nounifying a function name F doesn't affect occurrences of F that
existed before F was nounified.
MACSYMA 3.2 Evaluation 31
the input expression. The basic difference is that names which have values
are replaced by their values and verb-type function calls are replaced by the
result of applying the function to its arguments.
MACSYMA has several special operators which give the user some control
over the evaluation process. The single-quote operator ' has the effect of
preventing evaluation. Thus an expression preceded by a single-quote
evaluates to that expression. A special case is the evaluation of a function
call where the name of the function is preceded by a quote as in 'F(X). In
this instance the quote causes the function to be treated as though it were
noun-type.
To simply prevent evaluation of F(X) without converting F to a noun, use
'(F(X)).
The quote-quote operator, ' ' , causes an extra evaluation to occur. It is
best considered as a macro character. Inputting an expression preceded by a
quote-quote has exactly the same effect as inputting the result of evaluating
and simplifying the expression. In other words when an inputted expression
contains a sub-expression which begins with a quote-quote that sub-expression
is replaced in the input string by the result of evaluating and simplifying
the expression following the quote-quote. This occurs at the time an
expression is parsed. In the case of evaluating a function call with a ''
preceding the name of the function (i.e. ''F(x) ), the '' causes the function
to be treated as if it were verb-type.
(C1) X;
(D1) X
(C2) X:3$
(C3) X;
(D3) 3
(C4) 'X;
(D4) X
(C5) F(X):=X↑2;
2
(D5) F(X) := X
(C6) 'F(2);
(D6) F(2)
(C7) EV(%,F);
(D7) 4
(C8) '(F(2));
(D8) F(2)
(C9) ''%;
(''atom means evaluate the atom's value)
32 3.2 Evaluation MACSYMA
(D9) 4
(C10) DECLARE(INTEGRATE,NOUN)$
(C11) INTEGRATE(Y↑2,Y);
/
[ 2
(D11) I Y DY
]
/
(C12) ''INTEGRATE(Y↑2,Y);
3
Y
(D12) --
3
(C13) F(Y):=DIFF(Y*LOG(Y),Y,2);
(D13) F(Y) := DIFF(Y LOG(Y), Y, 2)
(C14) F(Y):=''(DIFF(Y*LOG(Y),Y,2));
1
(D14) F(Y) := -
Y
(C15) C14;
1
(D15) F(Y) := -
Y
(Notice that the input expression has been changed due to the use of ' '.)
Referring to line (C14) above, suppose one wished to define the function
F(Y) as DIFF(Y*LOG(Y),Y,I) within another function G(I) where the I in the
definition of F(Y) is to be replaced by the argument to G when G is called.
G(I):=BLOCK(...,F(Y):=''(DIFF(Y*LOG(Y),Y,I)),...) will not do the job
because the
'' operator will cause the differentiation to be carried out at parse time and
thus either an error will result (if I is unbound) or the current global value
of I will be used rather than the value of the argument to G when it is
called. Omitting the '' is also not desirable in this example because that
would force the differentiation to be done each time F is called rather than
at the time it is defined. To remedy this one may use the command
DEFINE(function(arguments),body)
which is like function(arguments):=''body but causes the evaluation of body to
MACSYMA 3.2 Evaluation 33
occur at the time DEFINE is evaluated. Thus
G(I):=BLOCK(...,DEFINE(F(Y),DIFF(Y*LOG(Y),Y,I)),...) will work properly.
DEFINE may also be used for subscripted functions.
3.3 Simplification
The simplifier takes the output of the evaluator and tries to make it
smaller and more manageable, using some built-in algebra. Unless the user
takes some special action (like setting the special variable SIMP to FALSE
(see 6.1)), MACSYMA will never output an unsimplified expression. The
simplifier re-orders expressions in order to obtain a standard form and the
result is the value of the current D line. Thus A+B+C or C+A+B or C+B+A if
input, will all result in the same internal form, (PLUS A B C) which displays
as C + B + A . The simplifier also changes the SQRT function to exponentiation
to the 1/2 power and removes the difference and quotient operators from the
-1
expression by converting X-Y to X+(-1)*Y and X/Y to X*Y .
Roughly speaking, the simplifier orders expressions on the basis of their
subexpressions being ordered first. Variables are ordered alphabetically
i.e., from A to Z. Constants (%E, %PI, %I and any atoms DECLAREd CONSTANT)
come before variables and numbers come before constants. Finally, functions
1
are ordered according to their arguments , and according to their names in
case their arguments are the same. Thus Y+2*A*X-%PI would become (PLUS (TIMES
-1 %PI) (TIMES 2 A X) Y)).
The user should be aware that the line between evaluation and
simplification is not clear-cut. For instance, SIN is a noun-function. When
the evaluator sees SIN(0), it returns SIN(0). However, the simplifier notices
this special case and changes this expression to 0. So simplification will
sometimes obscure the difference between noun and verb functions.
[Mo1] mentions these and many other matters dealing with simplification.
←←←←←←←←←←←←←←←
1. Comparing first arguments first, second arguments second, etc.
34 MACSYMA
4 Miscellaneous Hints and Facilities
Care should be taken in cases where an expression containing % is re-
evaluated since the value of % changes each time a new line is computed. This
is shown in the following example.
(C1) (X+Y)↑3$
(C2) DIFF(%,X);
2
(D2) 3 (Y + X)
(C3) Y:X↑2+1$
(C4) ''C2;
(D4) 2 X
In line C4 the user may have intended to re-evaluate C2 thinking that the
% still referred to D1 while it actually referred to D3. Note the use of the
' ' operator to re-evaluate a previous expression. (see 3.2)
The following interrupt characters typed while holding down the control
key have special functions. They may be typed at any time--- even in the
middle of a command line---and take effect immediately.
↑ (control-shift-N on some terminals) - enters top-level LISP after
resetting all locally bound variables and breaking out of all functions. It
is not possible to continue an interrupted calculation after a control-↑, but
typing (CONTINUE) will return to MACSYMA.
A - makes a breakpoint in MACSYMA and suspends the computation. At this
point the user is in a MACSYMA break loop. If a user function was being
executed at the time of the break, its values may be printed or changed.
Aside from this, it is almost like being at top-level MACSYMA. To exit and
resume the computation type EXIT; (see 10.1).
X - quits a computation started while in a control-A break without
quitting the top-level computation.
] - (control-shift-M on some terminals) prints the time used so far in a
computation (without interrupting it).
K - reprints the current input line. This is useful when many rubouts
have obscured the line (on hardcopy devices).
Y - gets the last command string.
L - clears the screen on display consoles and reprints the current line.
W - stops printout at the console while the computation continues. (If the
user is connected to MACSYMA via the ARPA network, printout will not stop
until the Arpanet buffer is emptied.) The switch TTYOFF[FALSE] if set to TRUE
MACSYMA 4 Miscellaneous Hints and Facilities 35
also stops the printout. This is useful for temporarily turning off the
display for functions which might generate a lot of printing like BATCH.
Setting the switch to FALSE causes printing to be resumed.
V - resumes printout at the console turned off by control-W.
G - aborts a computation and returns control to top-level MACSYMA. This
is like control-↑ immediately followed by (CONTINUE) and is useful for
breaking out of infinite loops or for terminating a computation prematurely.
H - (backspace on some consoles) makes a "breakpoint" in MACSYMA, enters
LISP, and prints the time used in the current computation. Control-H does not
reset any values. Altmode (or Escape) P (for proceed) followed by a space
will return to MACSYMA and resume the computation. Control-B also performs
this function in NEWIO MACSYMA.
D - causes garbage collection statistics to be printed out each time a
garbage collection takes place [Mn1]. See Chapter 16.
C - stops printout of garbage collection statistics turned on by control-
D.
Two of the many MACSYMA variables or options mentioned later on are of
special interest and will be described here.
(1) The value of LINEL gives the number of characters which are printed on
a line. It is initially set by MACSYMA to the line length of the type of
terminal being used (as far as is known) but may be reset at any time by the
user. The user may have to reset it in DDT with :TCTYP as well. See [Lew1].
(2) If the variable SHOWTIME[FALSE] is TRUE then the computation time will
be printed automatically with each output expression.
Sometimes when a user gives a command line the message "... being loaded"
will be printed. This means that a function being used in the command line
and/or the associated programs are not in the initially loaded MACSYMA but are
being loaded in now via the dynamic loader. Infrequently used or inessential
functions are not initially loaded into MACSYMA in an effort to save space.
When in LISP typing (CONTINUE) or control-G will return to MACSYMA.
MACSYMA provides the facility for the user to have an initialization file
which gets loaded automatically before line (C1) is printed. If the user has
a directory then the file should be named MACSYM (INIT). Otherwise he may
place the file whose first file name is his login name and whose second file
name is MACSYM on the directory called (INIT). This file must be in the
format for the LOADFILE function (see 10.4), i.e. it must contain LISP
code. It may be created via the SAVE function (see 10.4) or by
translating a BATCH file (see 10.8).
36 4 Miscellaneous Hints and Facilities MACSYMA
A user who knows LISP should note that preceding a name with a ? causes
the corresponding LISP atom to be invoked. For example, ?FIXP(4.2); returns
FALSE, where FIXP is the name of a LISP system function.
MACSYMA 37
5 Predefined Constants and Functions
5.1 Constants
A number of common mathematical constants have special names in MACSYMA;
%E - the base of the natural logarithms.
%PI - the transcendental number π.
%I - the square root of -1.
INF - real positive infinity.
MINF - real minus infinity.
INFINITY - complex infinity, an infinite magnitude of arbitrary phase
1
angle.
TRUE - the Boolean constant, true. (T in LISP)
FALSE - the Boolean constant, false. (NIL in LISP)
5.2 Functions
All of the functions mentioned below take one argument (shown as X) unless
stated otherwise. The default values of MACSYMA variables which affect
certain functions are given in brackets with the function.
←←←←←←←←←←←←←←←
1. The infinity symbols have meaning only for certain functions, for example,
LIMIT, INTEGRATE, SUM.
38 MACSYMA
5.2.1 Simple Functions
ABS(X) - absolute value of X
ABSBOXCHAR[!] is the character used to draw absolute value signs
around expressions which are more than a single line high.
FLOAT(exp) - converts integers, rational numbers and bigfloats in exp to
floating point numbers.
BFLOAT(X) - converts all numbers and functions of numbers to bigfloat numbers.
Setting FPPREC[16] to N, sets the bigfloat precision to N digits. If
FLOAT2BF[FALSE] is FALSE a warning message is printed when a floating
point number is converted into a bigfloat number (since this may lead to
loss of precision).
ENTIER(X) - largest integer < X.
SIGNUM(X) - if X<0 then -1 else if X>0 then 1 else 0. If X is not numeric
then a simplified but equivalent form is returned. For example, SIGNUM(-X)
gives -SIGNUM(X).
POLYSIGN(X) - same as SIGNUM but always returns a numerical result by looking
at the numerical factor of the highest degree term in X.
MIN(X1, X2, ...) yields the minimum of its arguments (or returns a simplified
form if some of its arguments are non-numeric).
MAX(X1, X2, ...) yields the maximum of its arguments (or returns a simplified
form if some of its arguments are non-numeric).
5.2.2 Miscellaneous Functions
SQRT(X) - the square root of X. It is represented internally by X↑(1/2). Also
see ROOTSCONTRACT in section 6.1.1.
RADPRODEXPAND[TRUE] - if TRUE will cause nth roots of factors of a
product which are powers of n to be pulled outside of the radical, e.g.
SQRT(16*X↑2) will become 4*X only if RADPRODEXPAND is TRUE.
MACSYMA 5.2.2 Miscellaneous Functions 39
ISQRT(X) - takes one integer arg and returns the "integer SQRT" of its
absolute value.
EXP(X) - the exponential function. It is represented internally as %E↑X.
DEMOIVRE[FALSE] - if TRUE will cause %E↑[A+B*%I] to become
%E↑A*(COS(B)+%I*SIN(B)) if B is free of %I. A and B are not expanded.
%EMODE[TRUE] - when TRUE %E↑[%PI*%I*X] will be simplified as follows:
it will become COS(%PI*X)+%I*SIN(%PI*X) if X is an integer or a multiple
of 1/2, 1/3, 1/4, or 1/6 and thus will simplify further. For other
numerical X it will become %E↑[%PI*%I*Y] where Y is X-2*k for some integer
k such that ABS(Y)<1. If %EMODE is FALSE no simplification of
%E↑[%PI*%I*X] will take place.
%ENUMER[FALSE] - when TRUE will cause %E to be converted into 2.718...
whenever NUMER is TRUE. The default is that this conversion will take
place only if the exponent in %E↑X evaluates to a number.
LOG(X) - the natural logarithm.
LOGEXPAND[FALSE] - if TRUE will cause LOG(A/B) to become LOG(A)-LOG(B)
B
and LOG(A*B) to become LOG(A)+LOG(B). This does not effect LOG(A ) which
always becomes B*LOG(A).
LOGSIMP[TRUE] - if FALSE then no simplification of %E to a power
containing LOG's is done.
LOGNUMER[FALSE] - if TRUE then negative floating point arguments to
LOG will always be converted to their absolute value before the log is
taken. If NUMER is also TRUE, then negative integer arguments to LOG will
also be converted to their absolute value.
The LOGCONTRACT command (see 6.1.1) "contracts" expressions containing LOG.
PLOG(X) - the principal branch of the complex-valued natural logarithm with
-%PI < X < +%PI .
GLOG(X) - the generalized logarithm, i.e. all branches. This is sometimes
used by the definite integration package.
BINOMIAL(X, Y) - the binomial coefficient X*(X-1)*...*(X-Y+1)/Y!. If X and Y
are integers, the binomial coefficient is actually computed. If Y or X-Y
is an integer, the binomial coefficient is simplified to a polynomial.
40 5.2.2 Miscellaneous Functions MACSYMA
RANDOM(X) - returns a random integer between 0 and X-1. If no argument is
35 35
given then a random integer between -2 and 2 -1 is returned. If X is
FALSE then the random sequence is restarted from the beginning.
FIB(X) - the Xth Fibonacci number with FIB(0)=0, FIB(1)=1, and FIB(-N)=(-
(N+1)
1) *FIB(N). PREVFIB is FIB(X-1), the Fibonacci number preceding the
last one computed.
GENFACT(X, Y, Z) is the generalized factorial of X which is: X*(X-Z)*(X-
2*Z)*...*(X-(Y-1)*Z). Thus, for integral X, GENFACT(X,X,1)=X! and
GENFACT(X,X/2,2)=X!!
GAMMA(X) - the gamma function. GAMMA(I)=(I-1)! for I a positive integer.
GAMMALIM[1000000] controls simplification of the gamma function for
integral and rational number arguments. If the absolute value of the
argument is not greater than GAMMALIM, then simplification will occur.
Note that the FACTLIM switch (see 2.5) controls simplification of the
result of GAMMA of an integer argument as well.
BETA(X, Y) - same as GAMMA(X)*GAMMA(Y)/GAMMA(X+Y)
ERF(X) - the error function, whose derivative is: 2*EXP(-X↑2)/SQRT(%PI).
EULER(X) - gives the Xth Euler number for integer X.
BERN(X) - gives the Xth Bernoulli number for integer X.
ZEROBERN[TRUE] if set to FALSE excludes the zero BERNOULLI numbers.
ZETA(X) - gives the Riemann zeta function for certain integer values of X.
PSI(X) - derivative of LOG(GAMMA(X)).
5.2.3 Trigonometric Functions
This section outlines the way in which trigonometric functions are called
in MACSYMA; for more information on the simplification of trigonometric
MACSYMA 5.2.3 Trigonometric Functions 41
functions and expressions, the user should read Section 2 of the MACSYMA
Primer [Mo5].
Circular Functions
SIN, COS, TAN, COT, SEC, CSC
Inverse Circular Functions
ASIN, ACOS , ATAN , ACOT , ASEC , ACSC
ATAN2(Y,X) - yields the value of ATAN(Y/X) in the interval -%PI to %PI.
Hyperbolic Functions
SINH , COSH , TANH , COTH, SECH , CSCH
Inverse Hyperbolic Functions
ASINH , ACOSH ,ATANH , ACOTH , ASECH, ACSCH
TRIGSIGN[TRUE] - if TRUE permits simplification of negative arguments to
trigonometric functions. E.g., SIN(-X) will become -SIN(X) only if TRIGSIGN is
TRUE.
EXPONENTIALIZE[FALSE] - if TRUE will cause all circular and hyperbolic
functions to be converted to exponential form.
LOGARC[FALSE] - if TRUE will cause the inverse circular and hyperbolic
functions to be converted into logarithmic form
Examples
(C1) SIN(%PI/12)+TAN(%PI/6);
%PI 1
(D1) SIN(---) + -------
12 SQRT(3)
(C2) EV(%,NUMER);
(D2) 0.8361693
(C3) BETA(1/2,2/5);
2
SQRT(%PI) GAMMA(-)
5
(D3) ------------------
9
GAMMA(--)
10
42 5.2.3 Trigonometric Functions MACSYMA
(C4) EV(%,NUMER);
(D4) 3.6790924
(C5) DIFF(ATANH(SQRT(X)),X);
1
(D5) -----------------
2 SQRT(X) (1 - X)
(C6) SOLVE(X↑2+10↑5*X+1);
SOLUTION
(E6) X = - SQRT(2499999999) - 50000
(E7) X = SQRT(2499999999) - 50000
(D7) [E6, E7]
(C8) E7,NUMER;
(D8) X = 2.9296875E-3
(C9) BFLOAT(E7);
(D9) X = - .9999999747378752B-5
(C10) FPPREC:25$
(C11) SIN(.5B0);
(D11) .4794255386042030002732879B0
The trigonometric simplification routines use declared information in some
simple cases. Declarations about variables are used as follows, e.g.
(C5) DECLARE(J, INTEGER, E, EVEN, O, ODD)$
(C6) SIN(X + (J*E + 1/2)*%PI)$
(D6) COS(X)
(C7) SIN(X + (O + 1/2)*%PI);
(D7) - COS(X)
MACSYMA 43
5.3 Complex Expressions
A complex expression is specified in MACSYMA by adding the real part of
the expression to %I times the imaginary part. Thus the roots of the equation
X↑2-4*X+13=0 are 2+3*%I and 2-3*%I.
Examples
(C1) (SQRT(-4)+SQRT(2.25))↑2;
2
(D1) (2 %I + 1.5)
(C2) EXPAND(%);
6.0 %I - 1.75
(C3) EXPAND(SQRT(2*%I));
(D3) %I + 1
Note that simplification of products of complex expressions can be
effected by expanding the product. Simplification of quotients, roots, and
other functions of complex expressions can usually be accomplished by using
the REALPART, IMAGPART, RECTFORM, POLARFORM, ABS, CARG functions (see
6.2.3).
44 MACSYMA
6 MACSYMA Functions and Variables
Following is a list of all MACSYMA functions divided into functional
classes. MACSYMA variables which affect the operation of some functions are
described under the appropriate function with their default value in brackets.
These are sometimes referred to as MACSYMA options.
6.1 General Purpose Functions
6.1.1 Evaluation and Simplification Functions
EV(exp, arg1, ..., argn) is one of MACSYMA's most powerful and versatile
commands. It evaluates the expression exp in the environment specified by
the argi. This is done in steps, as follows:
(1) First the environment is set up by scanning the argi which may be
as follows:
SIMP causes exp to be simplified regardless of the setting of the
switch SIMP which inhibits simplification if FALSE.
NOEVAL suppresses the evaluation phase of EV (see step (4) below).
This is useful in conjunction with the other switches and in causing exp
to be resimplified without being reevaluated.
EVAL causes an extra post-evaluation of exp to occur. (See step (5)
below.)
INFEVAL leads to an "infinite evaluation" mode. EV repeatedly
evaluates an expression until it stops changing. To prevent a variable,
say X, from being evaluated away in this mode, simply include X='X as an
argument to EV. Of course expressions such as EV(X,X=X+1,INFEVAL); will
generate an infinite loop. CAVEAT EVALUATOR.
EXPAND causes expansion.
EXPAND(m,n) causes expansion, setting the values of MAXPOSEX and
MAXNEGEX to m and n respectively. (see the EXPAND function below)
DETOUT causes any matrix inverses computed in exp to have their
determinant kept outside of the inverse rather than dividing through each
element.
DIFF causes all differentiations indicated in exp to be performed.
(see the DIFF function below.)
MACSYMA 6.1.1 Evaluation and Simplification Functions 45
DERIVLIST(var1,...,vark) causes only differentiations with respect to
the indicated variables.
FLOAT causes non-integral rational numbers to be converted to floating
point.
NUMER causes some mathematical functions (including exponentiation)
with numerical arguments to be evaluated in floating point (see 5.2.1). It
causes variables in exp which have been given numervals (see 8.1.2) to
be replaced by their values. It also sets the FLOAT switch on.
PRED causes predicates (expressions which evaluate to TRUE or FALSE)
to be evaluated.
NOUNS converts all nouns occurring in exp to verbs.
E where E is an atom declared to be an EVFLAG (see 8.1.1) causes E
to be bound to TRUE during the evaluation of exp.
V:expression (or alternatively V=expression) causes V to be bound to
the value of expression during the evaluation of exp. Note that if V is a
MACSYMA option, then expression is used for its value during the
evaluation of exp. If more than one argument to EV is of this type then
the binding is done in parallel. If V is a non-atomic expression then a
substitution rather than a binding is performed.
E where E is a function name declared to be an EVFUN (see 8.1)
causes E to be applied to exp.
Any other function names (e.g. SUM) cause evaluation of occurrences of
those names in exp as though they were verbs (see 3.2).
In addition a function occurring in exp (say F(args)) may be defined
locally for the purpose of this evaluation of exp by giving F(args):=body
as an argument to EV.
If an atom not mentioned above or a subscripted variable or
subscripted expression was given as an argument, it is evaluated and if
the result is an equation or assignment then the indicated binding or
substitution is performed. If the result is a list then the members of
the list are treated as if they were additional arguments given to EV.
This permits a list of equations to be given (e.g. [X=1, Y=A**2] ) or a
list of names of equations (e.g. [E1,E2] where E1 and E2 are equations)
such as that returned by SOLVE. (see 6.3)
The argi of EV usually may be given in any order but since they are
picked up left to right the order may influence the result. This is
strictly true of substitution equations which are handled in sequence,
left to right, and EVFUNS which are composed, e.g.
EV(exp,RATSIMP,RECTFORM) is handled as RECTFORM(RATSIMP(exp)). The SIMP,
NUMER, FLOAT, PRED, and INFEVAL switches may also be set locally in a
46 6.1.1 Evaluation and Simplification Functions MACSYMA
block, or globally at the "top level" in MACSYMA so that they will remain
in effect until being reset. Setting INFEVAL:TRUE locally will cause all
evaluations occurring via explicit calls to EV to be done "infinitely".
If exp is in CRE form (see 3.1) then EV will return a result in CRE
form provided the NUMER and FLOAT switches are both FALSE.
(2) During step (1), a list is made of the non-subscripted variables
appearing on the left side of equations in the argi or in the value of
some argi if the value is an equation. The variables (including
subscripted variables) in the expression exp are replaced by their global
values, except for those appearing in this list. Usually, exp is just a
label or % (as in (C2) below), so this step simply retrieves the
expression named by the label, so that EV may work on it.
(3) If any substitutions are indicated by the argi, they are carried
out now.
(4) The resulting expression is then re-evaluated (unless one of the
argi was NOEVAL) and simplified according the the argi. Note that any
function calls in exp will be carried out after the variables in it are
←←←←←
evaluated and that EV(F(X)) thus may behave like F(EV(X)).
(5) If one of the argi was EVAL, steps (3) and (4) are repeated.
Examples
(C1) SIN(X)+COS(Y)+(W+1)**2+'DIFF(SIN(W),W);
d 2
(D1) COS(Y) + SIN(X) + --SIN(W) + (W + 1)
dW
(C2) EV(%,SIN,EXPAND,DIFF,X=2,Y=1);
2
(D2) COS(W) + W + 2 W + COS(1) + 1.90929742
An alternate "top level" syntax has been provided for EV, whereby one
may just type in its arguments, without the EV(). That is, one may write
simply exp,arg1,...,argn. (This is not permitted as part of another
expression, i.e. in functions, blocks, etc.). exp,RESCAN is equivalent to
EV(exp).
(C4) X+Y,X:A+Y,Y:2;
(D4) Y + A + 2
(Notice the parallel binding process)
(C5) 2*X-3*Y=3$
(C6) -3*X+2*Y=-4$
(C7) SOLVE([D5,D6]);
solution
MACSYMA 6.1.1 Evaluation and Simplification Functions 47
1
(E7) Y = - -
5
6
(E8) X = -
5
(D8) [E7, E8]
(C9) D6,D8;
(D9) - 4 = - 4
(C10) X+1/X > GAMMA(1/2);
1
(D10) X + - > SQRT(%PI)
X
(C11) %,NUMER,X=1/2;
(D11) 2.5 > 1.7724539
(C12) %,PRED;
(D12) TRUE
UNKNOWN(exp) returns TRUE iff exp contains an operator or function not known
to the built-in simplifier.
EXPAND(exp) causes products of sums and exponentiated sums to be multiplied
out, numerators of rational expressions which are sums to be split into
their respective terms, and multiplication (commutative and non-
commutative) to be distributed over addition at all levels of exp. For
polynomials one may wish use RATEXPAND which uses a more efficient
algorithm (see below).
Terms in exp whose exponent is less than MAXNEGEX[1000] or greater
than MAXPOSEX[1000] will not be EXPANDed. However,
EXPAND(exp,p,n) expands exp, using p for MAXPOSEX and n for MAXNEGEX.
This helps the user control how much and what kinds of expansion are to
take place.
EXPON[0] - the exponent of the largest negative power which is
automatically expanded (independent of calls to EXPAND). For example if
EXPON is 4 then (X+1)**(-5) will not be automatically expanded.
EXPOP[0] - the highest positive exponent which is automatically
expanded. Thus (X+1)**3, when typed, will be automatically expanded only
48 6.1.1 Evaluation and Simplification Functions MACSYMA
if EXPOP is greater than or equal to 3. If it is desired to have (X+1)**n
expanded where n is greater than EXPOP then executing EXPAND((X+1)**n)
will work only if MAXPOSEX is not less than n.
(C1) (1/(X+Y)**4-3/(Y+Z)**3)**2;
1 3 2
(D1) (-------- - --------)
4 3
(Y + X) (Z + Y)
(C2) EXPAND(%,2,0);
6 9 1
(D2) - ----------------- + -------- + --------
4 3 6 8
(Y + X) (Z + Y) (Z + Y) (Y + X)
(C3) EXPAND(A.(B+C.(D+E)+F));
(D3) A . F + A . C . E + A . C . D + A . B
RATEXPAND(exp) expands exp by multiplying out products of sums and
exponentiated sums, combining fractions over a common denominator,
cancelling the greatest common divisor of the numerator and denominator,
then splitting the numerator (if a sum) into its respective terms divided
by the denominator. This is accomplished by converting exp to CRE form
(see 3.1) and then back to general form.
RATEXPAND[FALSE] - if TRUE will cause CRE expressions to be fully
expanded when they are converted back to general form or displayed, while
if it is FALSE then they will be put into a recursive form. (see RATSIMP
below)
RATDENOMDIVIDE[TRUE] - if FALSE will stop the splitting up of the
terms of the numerator of RATEXPANDed expressions from occurring.
KEEPFLOAT[FALSE] if set to TRUE will prevent floating point numbers
from being rationalized when expressions which contain them are converted
to CRE form.
GCD[EZ] if FALSE will prevent the greatest common divisor from being
taken when expressions are converted to CRE form. This will sometimes
speed the calculation if gcds are not required. (cf. the function GCD in
6.5)
(C1) RATEXPAND((2*X-3*Y)**3);
MACSYMA 6.1.1 Evaluation and Simplification Functions 49
3 2 2 3
(D1) - 27 Y + 54 X Y - 36 X Y + 8 X
(C2) (X-1)/(X+1)**2+1/(X-1);
X - 1 1
(D2) -------- + -----
2 X - 1
(X + 1)
(C3) EXPAND(D2);
X 1 1
(D3) ------------ - ------------ + -----
2 2 X - 1
X + 2 X + 1 X + 2 X + 1
(C4) RATEXPAND(D2);
2
2 X 2
(D4) --------------- + ---------------
3 2 3 2
X + X - X - 1 X + X - X - 1
RATSIMP(exp) "rationally" simplifies (similar to RATEXPAND) the expression exp
and all of its subexpressions including the arguments to non-rational
functions. The result is returned as the quotient of two polynomials in a
recursive form, i.e. the coefficients of the main variable are polynomials
in the other variables. Variables may, as in RATEXPAND, include non-
rational functions (e.g. SIN(X**2+1) ) but with RATSIMP, the arguments to
non-rational functions are rationally simplified. Note that RATSIMP is
affected by some of the variables which affect RATEXPAND.
RATSIMPEXPONS[FALSE] - if TRUE will cause exponents of expressions to
be RATSIMPed automatically during simplification.
RATSIMP(exp,v1,...,vn) enables rational simplification with the specification
of variable ordering as in RATVARS.
(C1) SIN(X/(X↑2+X))=%E↑((LOG(X)+1)**2-LOG(X)**2);
2 2
X - LOG (X) + (LOG(X) + 1)
(D1) SIN(------) = %E
2
X + X
50 6.1.1 Evaluation and Simplification Functions MACSYMA
(C2) RATSIMP(%);
1 2
(D2) SIN(-----) = %E X
X + 1
(C3) ((X-1)**(3/2)-(X+1)*SQRT(X-1))/SQRT((X-1)*(X+1));
3/2
(X - 1) - SQRT(X - 1) (X + 1)
(D3) --------------------------------
SQRT(X - 1) SQRT(X + 1)
(C4) RATSIMP(%);
2
(D4) - -----------
SQRT(X + 1)
(C5) X**(A+1/A),RATSIMPEXPONS:TRUE;
2
A + 1
------
A
(D5) X
RADCAN(exp) simplifies exp, which can contain logs, exponentials, and
radicals, by converting it into a form which is canonical over a large
class of expressions and a given ordering of variables; that is, all
functionally equivalent forms are mapped into a unique form. For a
somewhat larger class of expressions, RADCAN produces a regular form
[Fa2]. Two equivalent expressions in this class will not necessarily have
the same appearance, but their difference will be simplified by RADCAN to
zero. For some expressions RADCAN can be quite time consuming. This is
the cost of exploring certain relationships among the components of the
expression for simplifications based on factoring and partial-fraction
expansions of exponents.
RADPRODEXPAND[TRUE] when set to FALSE will inhibit certain
transformations: RADCAN(SQRT(1-X)) will remain SQRT(1-X) and will not
2 2
become %I SQRT(X-1). RADCAN(SQRT(X -2*X+1)) will remain SQRT(X -2*X + 1)
and will not be transformed to X-1.
(C1) (LOG(X**2+X)-LOG(X))**A/LOG(X+1)**(A/2);
2 A
(LOG(X + X) - LOG(X))
(D1) -----------------------
A/2
LOG(X + 1)
MACSYMA 6.1.1 Evaluation and Simplification Functions 51
(C2) RADCAN(%);
A/2
(D2) LOG(X + 1)
(C3) LOG(A**(2*X)+2*A**X+1)/LOG(A**X+1);
2 X X
LOG(A + 2 A + 1)
(D3) --------------------
X
LOG(A + 1)
(C4) RADCAN(%);
(D4) 2
(C5) (%E**X-1)/(%E**(X/2)+1);
X
%E - 1
(D5) ---------
X/2
%E + 1
(C6) RADCAN(%);
X/2
(D6) %E - 1
COMBINE(exp) simplifies the sum exp by combining terms with the same
denominator into a single term.
MULTTHRU(exp) multiplies a factor (which should be a sum) of exp by the other
factors of exp. That is exp is f1*f2*...*fn where at least one factor, say
fi, is a sum of terms. Each term in that sum is multiplied by the other
factors in the product. (Namely all the factors except fi). MULTTHRU
does not expand exponentiated sums. This function is the fastest way to
distribute products (commutative or noncommutative) over sums. Since
quotients are represented as products (see 3.3) MULTTHRU can be used to
divide sums by products as well.
MULTTHRU(exp1, exp2) multiplies each term in exp2 (which should be a sum or an
equation) by exp1. If exp1 is not itself a sum then this form is
equivalent to MULTTHRU(exp1*exp2).
(C1) X/(X-Y)**2-1/(X-Y)-F(X)/(X-Y)**3;
52 6.1.1 Evaluation and Simplification Functions MACSYMA
1 X F(X)
(D1) - ----- + -------- - --------
X - Y 2 3
(X - Y) (X - Y)
(C2) MULTTHRU((X-Y)**3,%);
2
(D2) - (X - Y) + X (X - Y) - F(X)
(C3) RATEXPAND(D2);
2
(D3) - Y + X Y - F(X)
(C4) ((A+B)**10*S**2+2*A*B*S+(A*B)**2)/(A*B*S**2);
10 2 2 2
(B + A ) S + 2 A B S + A B
(D4) --------------------------------
2
A B S
(C5) MULTTHRU(%);
10
2 A B (B + A)
(D5) - + --- + -------
S 2 A B
S
(notice that (B+A)**10 is not expanded)
(C6) MULTTHRU(A.(B+C.(D+E)+F));
(D6) A . F + A . (C . (E + D)) + A . B
(compare with similar example under EXPAND)
XTHRU(exp) combines all terms of exp (which should be a sum) over a common
denominator without expanding products and exponentiated sums as RATSIMP
does. XTHRU cancels common factors in the numerator and denominator of
rational expressions but only if the factors are explicit. Sometimes it
is better to use XTHRU before RATSIMPing an expression in order to cause
explicit factors of the gcd of the numerator and denominator to be
canceled thus simplifying the expression to be RATSIMPed.
(C1) ((X+2)**20-2*Y)/(X+Y)**20+(X+Y)**-19-X/(X+Y)**20;
MACSYMA 6.1.1 Evaluation and Simplification Functions 53
20
1 X (X + 2) - 2 Y
(D1) --------- - --------- + ---------------
19 20 20
(Y + X) (Y + X) (Y + X)
(C2) XTHRU(%);
20
(X + 2) - Y
(D2) -------------
20
(Y + X)
PARTFRAC(exp, var) expands the expression exp in partial fractions with
respect to the main variable, var. Each power of a different denominator
will be represented by only a single term (i.e. the decomposition is not
"complete"). The algorithm employed is based on the fact that the
denominators of the partial fraction expansion (the factors of the
original denominator) are relatively prime. The numerators can be written
as linear combinations of denominators, and the expansion falls out.
(C1) 2/(X+2)-1/(X+1)-X/(X+1)**2$
(C2) RATSIMP(%);
X
(D2) - -------------------
3 2
X + 4 X + 5 X + 2
(C3) PARTFRAC(%,X);
- 2 X - 1 2
(D3) ---------- + -----
2 X + 2
(X + 1)
FACTOR(exp) factors the expression exp, containing any number of variables or
functions, into factors irreducible over the integers.
FACTOR(exp, p) factors exp over the field of integers with an element adjoined
whose minimum polynomial is p.
FACTORFLAG[FALSE] if FALSE suppresses the factoring of integer factors
of rational expressions.
DONTFACTOR may be set to a list of variables with respect to which
factoring is not to occur. (It is initially empty). Factoring also will
54 6.1.1 Evaluation and Simplification Functions MACSYMA
not take place with respect to any variables which are less important
(using the variable ordering assumed for CRE form) than those on the
DONTFACTOR list. (see 6.5)
SAVEFACTORS[FALSE] if TRUE causes the factors of an expression which
is a product of factors to be saved by certain functions in order to speed
up later factorizations of expressions containing some of the same
factors.
BERLEFACT[TRUE] if FALSE then the Kronecker factoring algorithm will
be used otherwise the Berlekamp algorithm, which is the default, will be
used. (see [Be1, Wa4])
INTFACLIM[1000] is the largest divisor which will be tried when
factoring a bignum integer. If set to FALSE (this is the case when the
user calls FACTOR explicitly), or if the integer is a fixnum (i.e. fits in
one machine word), complete factorization of the integer will be
attempted. The user's setting of INTFACLIM is used for internal calls to
FACTOR. Thus, INTFACLIM may be reset to prevent MACSYMA from taking an
inordinately long time factoring large integers.
GCFACTOR(n) factors the gaussian integer n over the gaussians, i.e. numbers of
the form a + b i where a and b are rational integers. Factors are
normalized by making a and b non-negative.
(C1) FACTOR(2**63-1);
2
(D1) 73 127 337 92737 649657 7
(C2) FACTOR(Z**2*(X+2*Y)-4*X-8*Y);
(D2) (2 Y + X) (Z - 2) (Z + 2)
(C3) X**2*Y**2+2*X*Y**2+Y**2-X**2-2*X-1;
2 2 2 2 2
(D3) X Y + 2 X Y + Y - X - 2 X - 1
(C4) DONTFACTOR:[X]$
(C5) FACTOR(D3/36/(Y**2+2*Y+1));
2
(X + 2 X + 1) (Y - 1)
(D5) ----------------------
36 (Y + 1)
(C6) FACTOR(%E**(3*X)+1);
X 2 X X
(D6) (%E + 1) (%E - %E + 1)
MACSYMA 6.1.1 Evaluation and Simplification Functions 55
(C7) FACTOR(X**4+1,A**2-2);
2 2
(D7) (X + A X + 1) (X - A X + 1)
When FACTOR is applied to integers, note that the value returned by
FACTOR when used in other computations may not lead to a simplified
result. Using D1 above, the user can check that D1 + 1; will not return
6
2 3.
FACTORSUM(exp) tries to group terms in factors of exp which are sums into
groups of terms such that their sum is factorable. It can recover the
2 2 2 2
result of EXPAND((X+Y) +(Z+W) ) but it can't recover EXPAND((X+1) +(X+Y) )
because the terms have variables in common.
(C1) (X+1)*((U+V)↑2+A*(W+Z)↑2),EXPAND;
2 2 2 2
(D1) A X Z + A Z + 2 A W X Z + 2 A W Z + A W X + V X
2 2 2 2
+ 2 U V X + U X + A W + V + 2 U V + U
(C2) FACTORSUM(%);
2 2
(D2) (X + 1) (A (Z + W) + (V + U) )
FACTOROUT(exp,var1,var2,...) rearranges the sum exp into a sum of terms of the
form f(var1,var2,...)*g where g is a product of expressions not containing
the vari's and f is factored.
Another technique of factoring complex expressions uses the function
SCANMAP (see Chapter 8).
SQFR(exp) is similar to FACTOR except that the polynomial factors are "square-
free." That is, they have factors only of degree one. This algorithm,
which is also used by the first stage of FACTOR, utilizes the fact that a
polynomial has in common with its nth derivative all its factors of degree
> n. Thus by taking gcds with the polynomial of the derivatives with
respect to each variable in the polynomial, all factors of degree > 1 can
be found.
56 6.1.1 Evaluation and Simplification Functions MACSYMA
(C1) SQFR(4*X**4+4*X**3-3*X**2-4*X-1);
2 2
(D1) (X - 1) (2 X + 1)
GFACTOR(exp) factors the polynomial exp over the Gaussian integers (i. e. with
SQRT(-1) = %I adjoined). This is like FACTOR(exp,A**2+1) where A is %I.
(C1) GFACTOR(X**4-1);
(D1) (X - 1) (X + 1) (X + %I) (X - %I)
GFACTORSUM(exp) is similar to FACTORSUM but applies GFACTOR instead of FACTOR.
IRREDUCIBLE(exp) returns exp flagged as being irreducible, i.e. it doesn't
factor. Exp must be a sum. If FACTOR is ever called on an expression
marked as irreducible it returns immediately. For example, if the value
of H is a large expression which the user knows to be irreducible and the
expression G*H is to be factored (where the value of G is arbitrary) then
FACTOR(G*IRREDUCIBLE(H)) is faster than FACTOR(G*H).
PARTITION(exp, var) returns a list of two expressions. They are (1) the
factors of exp (if it is a product) or the terms of exp (if it is a sum)
which don't contain var and, (2) the factors or terms which do.
(C1) PARTITION(2*A*X*F(X),X);
(D1) [ 2 A , X F(X) ]
(C2) PARTITION(A+B,X);
(D2) [ A + B , 0 ]
LOGCONTRACT(exp) recursively scans an exp, transforming subexpressions of the
form a1*LOG(b1) + a2*LOG(b2) + c into LOG(RATSIMP(b1↑a1 * b2↑a2)) + c
(C1) 2*(A*LOG(X) + 2*A*LOG(Y))$
(C2) LOGCONTRACT(%);
2 4
(D3) A LOG(X Y )
MACSYMA 6.1.1 Evaluation and Simplification Functions 57
ROOTSCONTRACT(exp) converts products of roots into roots of products. For
example, ROOTSCONTRACT(SQRT(X)*Y↑(3/2)); gives SQRT(X*Y↑3). Currently it
only knows about rational number exponents whose denominators are 2, but
extension to other roots will follow.
6.1.2 Sums and Products
SUM(exp, ind, lo, hi) performs a summation of the values of exp as the index
ind varies from lo to hi. If the upper and lower limits differ by an
integer then each term in the sum is evaluated and added together.
Otherwise the summand is evaluated with the index of summation unbound and
(if SIMPSUM [FALSE] is TRUE) the result is simplified. This
simplification may sometimes be able to produce a closed form. If SIMPSUM
is FALSE or if 'SUM is used, the value is a sum noun form which is a
representation of the sigma notation used in mathematics.
Sums may be differentiated, added, subtracted, or multiplied with some
automatic simplification being performed.
CAUCHYSUM[FALSE] when TRUE causes the Cauchy product to be used when
multiplying sums together rather than the usual product. In the Cauchy
product the index of the inner summation is a function of the index of the
outer one rather than varying independently.
GENINDEX[I] is the alphabetic prefix used to generate the next
variable of summation when necessary.
GENSUMNUM[0] is the numeric suffix used to generate the next variable
of summation. If it is set to FALSE then the index will consist only of
GENINDEX with no numeric suffix.
(C1) SIMPSUM:TRUE$
(C2) SUM(I**2+2**I,I,0,N);
3 2
N + 1 2 N + 3 N + N
(D2) 2 + --------------- - 1
6
(C3) SUM(3**(-I),I,1,INF);
1
(D3) -
2
(C4) SUM(I↑2,I,1,4)*SUM(1/I↑2,I,1,INF);
58 6.1.2 Sums and Products MACSYMA
2
(D5) 5 %PI
NUSUM(exp,var,low,high) performs indefinite summation of exp with respect to
var using a decision procedure due to R.W. Gosper. exp and the potential
th
answer must be expressible as products of n powers, factorials,
binomials, and rational functions.
UNSUM(fun,n) is the first backward difference fun(n) - fun(n-1).
(C1) G(P):=P*4↑N/BINOMIAL(2*N,N);
N
P 4
(D1) G(P) := ----------------
BINOMIAL(2 N, N)
(C2) G(N↑4);
4 N
N 4
(D2) ----------------
BINOMIAL(2 N, N)
(C3) NUSUM(D2,N,0,N);
4 3 2 N
2 (N + 1) (63 N + 112 N + 18 N - 22 N + 3) 4 2
(D3) ------------------------------------------------ - ------
693 BINOMIAL(2 N, N) 3 11 7
(C4) UNSUM(%,N);
4 N
N 4
(D4) ----------------
BINOMIAL(2 N, N)
PRODUCT(exp, ind, lo, hi) gives the product of the values of exp as the index
ind varies from lo to hi. The evaluation is similar to that of SUM. No
simplification of products is available at this time.
(C1) PRODUCT(X+I*(I+1)/2,I,1,4);
(D1) (X + 1) (X + 3) (X + 6) (X + 10)
MACSYMA 59
6.1.3 Differentiation and Integration Functions
DIFF(exp, v1, n1, v2, n2, ...) differentiates exp with respect to each vi, ni
times. If just the first derivative with respect to one variable is
desired then the form DIFF(exp,v) may be used. If the noun form of the
function is required (as, for example, when writing a differential
equation), 'DIFF should be used and this will display in a two dimensional
format.
DERIVABBREV[FALSE] if TRUE will cause derivatives to display as
subscripts.
DIFF(exp) gives the "total differential", that is, the sum of the
derivatives of exp with respect to each of its variables times the
function DEL of the variable. No further simplification of DEL is
offered.
(C1) DIFF(EXP(F(X)),X,2);
2
F(X) d F(X) d 2
(D1) %E (--- F(X)) + %E (-- F(X))
2 dX
dX
(C2) DERIVABBREV:TRUE$
(C3) 'INTEGRATE(F(X,Y),Y,G(X),H(X));
H(X)
/
[
(D3) I F(X, Y) dY
]
/
G(X)
(C4) DIFF(%,X);
H(X)
/
[
(D4) I F(X, Y) dY + F(X, H(X)) H(X) - F(X, G(X)) G(X)
] X X X
/
G(X)
60 6.1.3 Differentiation and Integration Functions MACSYMA
DEPENDS(funlist1,varlist1,funlist2,varlist2,...) declares functional
dependencies for variables to be used by DIFF.
DEPENDS([F,G],[X,Y],[R,S],[U,V,W],U,T) informs DIFF that F and G depend on
X and Y, that R and S depend on U,V, and W, and that U depends on T. The
arguments to DEPENDS are evaluated. The variables in each funlist are
←←← ←←←←←←←←←
1
declared to depend on all the variables in the next varlist. A funlist
can contain the name of an atomic variable or array. In the latter case,
it is assumed that all the elements of the array depend on all the
variables in the succeeding varlist. Initially, DIFF(F,X) is 0; executing
DEPENDS(F,X) causes future differentiations of F with respect to X to give
DF/DX or Y (if DERIVABBREV:TRUE).
X
(C1) DEPENDS([F,G],[X,Y],[R,S],[U,V,W],U,T);
(D1) [F(X, Y), G(X, Y), R(U, V, W), S(U, V, W), U(T)]
(C2) DEPENDENCIES;
(D2) [F(X, Y), G(X, Y), R(U, V, W), S(U, V, W), U(T)]
(C3) DIFF(R.S,U);
dR dS
(D3) -- . S + R . --
dU dU
Since MACSYMA knows the chain rule for symbolic derivatives, it takes
advantage of the given dependencies as follows:
(C4) DIFF(R.S,T);
dR dU dS dU
(D4) (-- --) . S + R . (-- --)
dU dT dU dT
If we set
(C5) DERIVABBREV:TRUE;
(D5) TRUE
then re-executing the command C4, we obtain
(C6) ''C4;
(D6) (R U ) . S + R . (S U )
U T U T
To eliminate a previously declared dependency, the REMOVE command can
be used. For example, to say that R no longer depends on U as declared in
C1, the user can type REMOVE(R,DEPENDENCY). This will eliminate all
dependencies that may have been declared for R.
←←←←←←←←←←←←←←←
1. In this command, lists of length one can be typed in directly as atoms.
MACSYMA 6.1.3 Differentiation and Integration Functions 61
(C7) REMOVE(R,DEPENDENCY);
(D7) DONE
(C8) ''C4;
(D8) R . (S U )
U T
CAVEAT: DIFF is the only MACSYMA command which uses DEPENDENCIES
information. The arguments to INTEGRATE,LAPLACE,etc. must be given their
dependencies explicitly in the command, e.g., INTEGRATE(F(X),X).
GRADEF(f(x1, ..., xn), g1, ..., gn) defines the derivatives of the function f
with respect to its n arguments. That is, df/dxi = gi, etc. If fewer
than n gradients, say i, are given, then they refer to the first i
arguments of f. The xi are merely dummy variables as in function
definition headers and are used to indicate the ith argument of f. All
arguments to GRADEF except the first are evaluated so that if g is a
defined function then it is invoked and the result is used.
Gradients are needed when, for example, a function is not known explicitly
but its first derivatives are and it is desired to obtain higher order
derivatives. GRADEF may also be used to redefine the derivatives of
MACSYMA's predefined functions (e.g. GRADEF(SIN(X),SQRT(1-SIN(X)**2)) ).
It is not permissible to use GRADEF on subscripted functions.
GRADEFS is a list of the functions which have been given gradients by
use of the GRADEF command.
PRINTPROPS([f1,f2,...],GRADEF) (see 8.1.1) may be used to display
the gradefs of the functions f1,f2,..
REMOVE([f1,f2,...],GRADEF) may be used to eliminate the GRADEF
property from the functions f1,f2,....
(C1) DEPENDS(Y,X)$
(C2) GRADEF(F(X,Y),X**2,G(X,Y))$
(C3) DIFF(F(X,Y),X);
dY 2
(D3) G(X, Y) -- + X
dX
(C4) GRADEF(J(N,Z), 'DIFF(J(N,Z),N),
RATSIMP(J(N-1,Z)-N/Z*J(N,Z)))$
(C5) DIFF(J(2,X),X,2);
62 6.1.3 Differentiation and Integration Functions MACSYMA
2
J(0, X) X - 3 J(1, X) X + 6 J(2, X)
(D5) ------------------------------------
2
X
(The example above computes the second derivative of a Bessel function
of order two. A subscripted function e.g. J[N], could not have been used
because a gradient for it cannot be defined using GRADEF.)
GRADEF(a,v,exp) may be used to state that the derivative of the atomic
variable a with respect to v is exp. This automatically does a
DEPENDS(a,v). For examples, see example 2 of Appendix III.
PRINTPROPS([a1,a2,...],ATOMGRAD) (see 8.1.1) may be used to display
the atomic gradient properties of a1,a2,...
REMOVE([a1,a2,...],ATOMGRAD) may be used to eliminate the ATOMGRAD
property from a1,a2,....
INTEGRATE(exp, var) integrates exp with respect to var or returns an integral
expression (the noun form) if it cannot perform the integration. Roughly
speaking three stages are used:
(1) INTEGRATE sees if the integrand is of the form
F(G(X))*DIFF(G(X),X) by testing whether the derivative of some
subexpression (i.e. G(X) in the above case) divides the integrand. If so
it looks up F in a table of integrals and substitutes G(X) for X in the
integral of F. This may make use of gradients in taking the derivative.
(If an unknown function appears in the integrand it must be eliminated in
this stage or else INTEGRATE will return the noun form of the integrand.)
(2) INTEGRATE tries to match the integrand to a form for which a
specific method can be used, e.g. trigonometric substitutions.
(3) If the first two stages fail it uses the Risch algorithm. (see
[Mo2, Mo4])
CAVEAT: INTEGRATE knows only about explicit dependencies.
INTEGRATE(exp, var, low, high) finds the definite integral of exp with respect
to var from low to high. Several methods are used,including direct
substitution in the indefinite integral and contour integration (see
[Wa3]). Improper integrals may use the names INF for positive infinity
and MINF for negative infinity. If an integral "form" is desired for
manipulation (for example, an integral which cannot be computed until some
numbers are substituted for some parameters), the noun form 'INTEGRATE may
be used and this will display with an integral sign.
MACSYMA 6.1.3 Differentiation and Integration Functions 63
ABCONVTEST[FALSE] when TRUE causes INTEGRATE to test for absolute
convergence.
The function LDEFINT uses LIMIT (see 6.1.3) to evaluate the indefinite
integral at the lower and upper limits.
Sometimes during integration the user may be asked what the sign of an
expression is. Suitable responses are POS; , ZERO; , or NEG; . (see
7.1)
(C1) INTEGRATE(SIN(X)**3,X);
3
COS (X)
(D1) ------- - COS(X)
3
(C2) INTEGRATE(X**A/(X+1)**(5/2),X,0,INF);
IS A + 1 POSITIVE, NEGATIVE, OR ZERO?
POS;
IS 2 A - 3 POSITIVE, NEGATIVE, OR ZERO?
NEG;
3
(D2) BETA(A + 1, - - A)
2
(C3) GRADEF(Q(X),SIN(X**2));
(D3) Q(X)
(C4) DIFF(LOG(Q(R(X))),X);
d 2
(-- R(X)) SIN(R (X))
dX
(D4) --------------------
Q(R(X))
(C5) INTEGRATE(%,X);
(D5) LOG(Q(R(X)))
RISCH(exp, var) integrates exp with respect to var using the Risch algorithm.
This currently handles the cases of nested exponentials and logarithms
which the main part of INTEGRATE can't do. INTEGRATE will automatically
apply RISCH if given these cases.
ERFFLAG[TRUE] - if FALSE prevents RISCH from introducing the ERF
function in the answer if there were none in the integrand to begin with.
64 6.1.3 Differentiation and Integration Functions MACSYMA
(C1) RISCH(X↑2*ERF(X),X);
2 2
- X X 3 2
%E (%E SQRT(%PI) X ERF(X) + X + 1)
(D1) ------------------------------------------
3 SQRT(%PI)
(C2) DIFF(%,X),RATSIMP;
2
(D2) X ERF(X)
CHANGEVAR(exp,f(x,y),y,x) makes the change of variable given by f(x,y) = 0 in
all integrals occurring in exp with integration with respect to x; y is
the new variable.
(C1) 'INTEGRATE(%E**SQRT(A*Y),Y,0,4);
4
/
[ SQRT(A) SQRT(Y)
(D1) I (%E ) DY
]
/
0
(C2) CHANGEVAR(D1,Y-Z↑2/A,Z,Y);
2 SQRT(A)
/
[ Z
2 I Z %E dZ
]
/
0
(D4) ---------------------
A
LIMIT(exp, var, val, dir) finds the limit of exp as the real variable var
approaches the value val from the direction dir. Dir may have the value
PLUS for a limit from above, MINUS for a limit from below, or may be
omitted (implying a two-sided limit is to be computed). For the method
see [Wa3]. LIMIT uses the following special symbols: INF (positive
infinity) and MINF (negative infinity). On output it may also use UND
(undefined), IND (indefinite but bounded) and INFINITY (complex infinity).
LHOSPITALLIM[4] is the maximum number of times L'Hospital's rule is
MACSYMA 6.1.3 Differentiation and Integration Functions 65
used in LIMIT. This prevents infinite looping in cases like
LIMIT(COT(X)/CSC(X),X,0).
TLIMSWITCH[FALSE] when true will cause the limit package to use Taylor
series when possible.
(C1) LIMIT(X*LOG(X),X,0,PLUS);
(D1) 0
(C2) LIMIT((1+X)**(1/X),X,0);
(D2) %E
(C3) LIMIT(%E**X/X,X,INF);
(D3) INF
(C4) LIMIT(SIN(1/X),X,0);
(D4) IND
TLIMIT(exp,var,val,dir) is just the function LIMIT with TLIMSWITCH set to
TRUE.
LDEFINT(exp,var,low,high) yields the definite integral of exp by using LIMIT
to evaluate the indefinite integral of exp with respect to var at the
upper limit high and at the lower limit low.
TLDEFINT(exp,var,low,high) is just LDEFINT with TLIMSWITCH set to TRUE.
RESIDUE(exp, var, val) computes the residue in the complex plane of the
expression exp when the variable var assumes the value val. The residue
is the coefficient of (var-val)**(-1) in the Laurent series for exp.
(C1) RESIDUE(S/(S**2+A**2),S,A*%I);
1
(D1) -
2
(C2) RESIDUE(SIN(A*X)/X**4,X,0);
3
A
(D2) - --
6
66 6.1.3 Differentiation and Integration Functions MACSYMA
ODE2(diffeq,depvar,indvar) solves ordinary differential equations,diffeq, of
first or second order. The dependent and independent variables are
specified as the second and third arguments. When successful ODE2 returns
either an explicit or implicit solution for the dependent variable. The
symbol %C is used to represent the constant in the case of first order
equations and %K1,%K2 represent the constants for second order equations.
If for some reason ODE2 cannot obtain a solution, it returns FALSE,
sometimes printing an error message to the user.
(C1) X↑2*'DIFF(Y,X) + 3*X*Y = SIN(X)/X;
2 dY SIN(X)
(D1) X -- + 3 X Y = ------
dX X
(C2) ODE2(%,Y,X);
%C - COS(X)
(D2) Y = -----------
3
X
6.2 Part Selection and Substitution
The functions in this section are used to extract or replace parts of
expressions.
6.2.1 The Part Functions
The Part functions make it possible to reference or replace any part of
any MACSYMA expression. A part of a displayed expression is referred to by a
set of indices which are non-negative integers. For example, in
exponentiation the base is considered part 1 and the exponent part 2. In a
quotient the numerator is part 1 and the denominator part 2. In a sum or
product the ith term or factor is part i. In any expression the main operator
is part 0. For -X the 0th part is -, for A↑B it is ↑, for DIFF(F(X),X) it is
DIFF, etc. Note that unary minus is considered an operator.
In MACSYMA the user has some control of the way in which expressions are
displayed. The ordering of factors in a product or terms in a sum may be
changed by the user (see 10.5, 6.5). The ordering of parts in the
displayed form of an expression may differ from the ordering in the internal
representation of the expression.
MACSYMA 6.2.1 The Part Functions 67
PART(exp, n1, ..., nk) deals with the displayed form of exp. It obtains the
part of exp as specified by the indices n1,...,nk. First part n1 of exp
is obtained, then part n2 of that, etc. The result is part nk of ... part
n2 of part n1 of exp. Thus PART(Z+2*Y,2,1) yields 2. PART can be used to
obtain an element of a list, a row of a matrix, etc.
(C1) X+Y/Z**2;
Y
(D1) -- + X
2
Z
(C2) PART(D1,1,2,2);
(D2) 2
(C3) 'INTEGRATE(F(X),X,A,B)+X;
B
/
[
(D3) I F(X)dX + X
]
/
A
(C4) PART(%,1,1);
(D4) F(X)
INPART(exp,n1,...,nk) is similar to PART but works on the internal
representation of the expression (see 3.3) rather than the displayed form
and thus may be faster since no formatting is done. Care should be taken
with respect to the order of subexpressions in sums and products (since
the order of variables in the internal form is often different from that
in the displayed form) and in dealing with unary minus, subtraction, and
division (since these operators are removed from the expression).
PART(X+Y,0) or INPART(X+Y,0) yield +, though in order to refer to the
operator it must be enclosed in "s. For example ...IF INPART(D9,0)="+"
THEN ...
(C1) X+Y+W*Z;
(D1) W Z + Y + X
(C2) INPART(D1,3,2);
(D2) Z
(C3) PART(D1,1,2);
(D3) Z
68 6.2.1 The Part Functions MACSYMA
(C4) 'LIMIT(F(X)**G(X+1),X,0,MINUS);
G(X + 1)
(D4) LIMIT F(X)
X ->0-
(C5) INPART(%,1,2);
(D5) G(X + 1)
DISPFORM(exp) returns the external representation of exp (wrt its main
operator). This should be useful in conjunction with PART which also
deals with the external representation. Suppose EXP is -A . Then the
internal representation of EXP is "*"(-1,A), while the external
representation is "-"(A). LENGTH(EXP) gives 2, while LENGTH(DISPFORM(EXP))
gives 1. MAP(F,EXP) gives F(-1)*F(A), while MAP(F,DISPFORM(EXP)) gives
-F(A).
DISPFORM(exp,ALL) converts the entire expression (not just the top-level)
to external format. For example, if EXP:SIN(SQRT(X)), then
FREEOF(SQRT,EXP) and FREEOF(SQRT,DISPFORM(EXP)) give TRUE, while
FREEOF(SQRT,DISPFORM(EXP,ALL)) gives FALSE.
NOUNIFY(f) returns the noun form of the function name f. This is needed if
one wishes to refer to the name of a verb function as if it were a noun.
Note that some verb functions will return their noun forms if they can't
be evaluated for certain arguments. This is also the form returned if a
function call is preceded by a quote.
(C6) IS(INPART(D4,0)=NOUNIFY(LIMIT));
(D6) TRUE
VERBIFY(f) returns the function name f in its verb form.
BOX(exp) returns exp enclosed in a box. The box is actually part of the
expression. BOX(exp,label) encloses exp in a labeled box. label is a name
which will be truncated in display if it is too long. Simplification will
occur within and outside of a BOXed expression but simplifications which
require interactions across the box boundary will not take place.
BOXCHAR["] - is the character used to draw the box in this and in the
DPART and LPART functions.
DPART(exp, n1, ..., nk) selects the same subexpression as PART, but instead of
just returning that subexpression as its value, it returns the whole
expression with the selected subexpression displayed inside a box. The
box is actually part of the expression.
MACSYMA 6.2.1 The Part Functions 69
(C1) DPART(X+Y/Z**2,1,2,1);
Y
(D1) ---- + X
2
"""""
" Z "
"""""
LPART(label, exp, n1, ..., nk) is similar to DPART but uses a labeled box. A
labeled box is similar to the one produced by DPART but it has a name in
the top line.
REMBOX(exp, arg) removes boxes from exp according to arg. If arg is UNLABELED
then all unlabeled boxes are removed. If arg is the name of some label
then only boxes with that label are removed. If arg is omitted then all
boxes labeled and unlabeled are removed.
6.2.2 The Substitution Functions
SUBST(a, b, c) substitutes a for b in c. b must be an atom or a complete
subexpression of c. For example, X+Y+Z is a complete subexpression of
2*(X+Y+Z)/W while X+Y is not. When b does not have these characteristics,
one may sometimes use SUBSTPART or RATSUBST (see below). Alternatively,
if b is of the form e/f then one could use SUBST(a*f,e,c) while if b is of
the form e**(1/f) then one could use SUBST(a**f,e,c). The SUBST command
Y
also discerns the X in X↑(-Y) so that SUBST(A,SQRT(X),1/SQRT(X)) yields
1/A.
a and b may also be operators of an expression (enclosed in "s) or they
may be function names. If one wishes to substitute for the independent
variable in derivative forms then the AT function (see below) should be
used.
SUBST(eq1,exp) or SUBST([eq1,...,eqk],exp) are other permissible
forms. The eqi are equations indicating substitutions to be made. For
each equation, the right side will be substituted for the left in the
expression exp.
For expressions in CRE representation (see 3.1), SUBST, like many of
MACSYMA's general simplification commands, works on the RATDISREPed form
of the expression.
EXPTSUBST[FALSE] if TRUE permits substitutions such as Y for %E**X in
%E**(A*X) to take place.
70 6.2.2 The Substitution Functions MACSYMA
(C1) SUBST(A,X+Y,X+(X+Y)**2+Y);
2
(D1) Y + X + A
(C2) SUBST(-%I,%I,A+B*%I);
(D2) A - %I B
(Note that C2 is one way of obtaining the complex conjugate of an
expression.) The following examples illustrate the difference between
substitution (as performed by SUBST) and binding (as performed by EV).
(C3) %PI*R,%PI:-%I;
%PI improper value assignment
(C4) SUBST(X=0,DIFF(SIN(X),X));
(D4) 1
(C5) DIFF(SIN(X),X),X=0;
0
attempt to differentiate wrt a number
(C6) MATRIX([A,B],[C,D]);
[ A B ]
(D6) [ ]
[ C D ]
(C8) SUBST("[",MATRIX,%);
(D8) [[A, B], [C, D]]
RATSUBST(a, b, c) substitutes a for b in c. b may be a sum, product, power,
etc. RATSUBST knows something of the meaning of expressions whereas SUBST
does a purely syntactic substitution. Thus SUBST(A,X+Y,X+Y+Z) returns
X+Y+Z whereas RATSUBST would return Z+A.
RADSUBSTFLAG[FALSE] if TRUE permits substitutions such as U for
SQRT(X) in X.
(C1) RATSUBST(A,X*Y↑2,X↑4*Y↑8+X↑4*Y↑3);
3 4
(D1) A X Y + A
(C2) 1 + COS(X) + COS(X)↑2 + COS(X)↑3 + COS(X)↑4;
4 3 2
(D2) COS (X) + COS (X) + COS (X) + COS(X) + 1
(C3) RATSUBST(1-SIN(X)↑2,COS(X)↑2,%);
MACSYMA 6.2.2 The Substitution Functions 71
4 2 2
(D3) SIN (X) + COS(X) (2 - SIN (X)) - 3 SIN (X) + 3
SUBSTPART(x, exp, n1, ..., nk) substitutes x for the subexpression picked out
by the rest of the arguments as in PART. It returns the new value of exp.
x may be some operator to be substituted for an operator of exp. In this
case it is enclosed in "s.
(C1) 1/(X↑2+2);
1
(D1) ------
2
X + 2
(C2) SUBSTPART(3/2,%,2,1,2);
1
(D2) --------
3/2
X + 2
(C3) A*X+F(B,Y);
(D3) A X + F(B, Y)
(C4) SUBSTPART("+",%,1,0);
(D4) X + F(B, Y) + A
(C5) X↑2 + X + 1$
(C6) SUBSTPART("[",%,0);
2
(D6) [X , X, 1]
SUBSTINPART(x, exp, n1, ...) is like SUBSTPART but works on the internal
representation of exp.
(C1) X.'DIFF(F(X),X,2);
2
d
(D1) X . (--- F(X))
2
dX
(C2) SUBSTINPART(D↑2,%,2);
2
(D2) X . D
(C3) SUBSTINPART(F1,F[1](X+1),0);
72 6.2.2 The Substitution Functions MACSYMA
(D3) F1(X + 1)
Additional Information
If the last argument to a Part function is a list of indices then
several subexpressions are picked out, each one corresponding to an index
of the list. Thus PART(X+Y+Z,[1,3]) is Z+X.
PIECE holds the last expression selected when using the Part
functions. It is set during the execution of the function and thus may be
referred to in the function itself as shown below.
If PARTSWITCH[FALSE] is set to TRUE then END is returned when a
selected part of an expression doesn't exist, otherwise an error message
is given.
(C1) 27*Y**3+54*X*Y**2+36*X**2*Y+Y+8*X**3+X+1;
3 2 2 3
(D1) 27 Y + 54 X Y + 36 X Y + Y + 8 X + X + 1
(C2) PART(D1,2,[1,3]);
2
(D2) 54 Y
(C3) SQRT(PIECE/54);
(D3) Y
(C4) SUBSTPART(FACTOR(PIECE),D1,[1,2,3,5]);
3
(D4) (3 Y + 2 X) + Y + X + 1
(C5) 1/X+Y/X-1/Z;
1 Y 1
(D5) - - + - + -
Z X X
(C6) SUBSTPART(XTHRU(PIECE),%,[2,3]);
Y + 1 1
(D6) ----- - -
X Z
MACSYMA 6.2.2 The Substitution Functions 73
ATVALUE(form, list, value) enables the user to assign the boundary value value
to form at the points specified by list.
(C1) ATVALUE(F(X,Y),[X=0,Y=1],A**2)$
The form must be a function, f(v1,v2,...) , or a derivative,
DIFF(f(v1,v2,...),vi,ni,vj,nj,...) in which the functional arguments
explicitly appear (ni is the order of differentiation with respect vi).
The list of equations determine the "boundary" at which the value is
given; list may be a list of equations, as above, or a single equation, vi
= exp.
The symbols @1, @2,... will be used to represent the functional
variables v1,v2,... when atvalues are displayed.
PRINTPROPS([f1, f2,...], ATVALUE) will display the atvalues of the
functions f1,f2,... as specified in previously given uses of the ATVALUE
function. (see 8.1.1) If the list contains just one element then the
element can be given without being in a list. If a first argument of ALL
is given then atvalues for all functions which have them will be
displayed.
AT(exp, list) will evaluate exp (which may be any expression) with the
variables assuming the values as specified for them in the list of
equations or the single equation similar to that given to the ATVALUE
function. If a subexpression depends on any of the variables in list but
it hasn't had an atvalue specified and it can't be evaluated then a noun
form of the AT will be returned which will display in a two-dimensional
form.
(C1) ATVALUE(F(X,Y),[X=0,Y=1],A**2);
2
(D1) A
(C2) ATVALUE('DIFF(F(X,Y),X),X=0,Y+1);
(D2) @2 + 1
(C3) PRINTPROPS(ALL,ATVALUE);
!
D !
--- F(@1, @2)! = @2 + 1
D@1 !
!@1 = 0
2
F(0, 1) = A
(D3) DONE
74 6.2.2 The Substitution Functions MACSYMA
(C4) DIFF(4*F(X,Y)**2-U(X,Y)**2,X);
d d
(D4) 8 F(X, Y) (-- F(X, Y)) - 2 U(X, Y) (-- U(X, Y))
dX dX
(C5) AT(%,[X=0,Y=1]);
!
2 d !
(D5) 16 A - 2 U(0, 1) ( -- U(X, Y)! )
dX !
!X = 0, Y = 1
6.2.3 More Functions for Part Extraction
LISTOFVARS(exp) yields a list of the variables in exp.
LISTCONSTVARS[FALSE] if TRUE will cause LISTOFVARS to include %E, %PI,
%I, and any variables declared constant (see 8.1) in the list it
returns if they appear in exp. The default is to omit these.
(C1) LISTOFVARS(F(X[1]+Y)/G**(2+A));
(D1) [X[1], Y, A, G]
COEFF(exp, v, n) obtains the coefficient of v**n in exp. n may be omitted if
it is 1. v may be an atom, or complete subexpression of exp e.g., X,
SIN(X), A[I+1], X+Y, etc. (In the last case the expression (X+Y) should
occur in exp). Sometimes it may be necessary to expand or factor exp in
n
order to make v explicit. This is not done automatically by COEFF.
(C1) COEFF(2*A*TAN(X)+TAN(X)+B=5*TAN(X)+3,TAN(X));
(D1) 2 A + 1 = 5
(C2) COEFF(Y+X*%E**X+1,X,0);
(D2) Y + 1
RATCOEF(exp, v, n) returns the coefficient, C, of the expression v**n in the
expression exp. n may be omitted if it is 1. C will be free (except
possibly in a non-rational sense) of the variables in v. If no
coefficient of this type exists, zero will be returned. RATCOEF expands
and rationally simplifies its first argument and thus it may produce
MACSYMA 6.2.3 More Functions for Part Extraction 75
answers different from those of COEFF which is purely syntactic. Thus
RATCOEF((X+1)/Y+X,X) returns (Y+1)/Y whereas COEFF returns 1.
RATCOEF(exp,v,0) is currently the same as RATSUBST(0,v,exp). Therefore if
v occurs to any negative powers, RATCOEF should not be used. Since exp is
rationally simplified before it is examined, coefficients may not appear
quite the way they were envisioned.
(C1) S:A*X+B*X+5$
(C2) RATCOEF(S,A+B);
(D2) X
BOTHCOEF(exp, var) returns a list whose first member is the coefficient of var
in exp (as found by RATCOEF if exp is in CRE form otherwise by COEFF) and
whose second member is the remaining part of exp. That is, [A,B] where
exp=A*var+B.
(C1) ISLINEAR(EXP,VAR):=BLOCK([C],
C:BOTHCOEF(RAT(EXP,VAR),VAR),
IS(FREEOF(VAR,C) AND C[1]#0))$
(C2) ISLINEAR((R**2-(X-R)**2)/X,X);
(D2) TRUE
ISOLATE(exp, var) returns exp with subexpressions which are sums and which do
not contain var replaced by intermediate expression labels (these being
atomic symbols like E1, E2, ...). This is often useful to avoid
unnecessary expansion of subexpressions which don't contain the variable
of interest. Since the intermediate labels are bound to the
subexpressions they can all be substituted back by evaluating the
expression in which they occur.
EXPTISOLATE[FALSE] if TRUE will cause ISOLATE to examine exponents of
atoms (like %E) which contain var.
(C1) (A+B)↑4*(1+X*(2*X+(C+D)↑2));
4 2
(D1) (B + A) (X (2 X + (D + C) ) + 1)
(C2) ISOLATE(%,X);
2
(E2) (D + C)
76 6.2.3 More Functions for Part Extraction MACSYMA
4
(E3) (B + A)
(D3) E3 (X (2 X + E2) + 1)
(C4) RATEXPAND(D3)$
(C5) EV(%);
4 2 4 2 4
(D5) 2 (B + A) X + (B + A) (D + C) X + (B + A)
(C6) (A+B)*(X+A+B)↑2*EXP(X↑2+A*X+B);
2
2 X + A X + B
(D6) (B + A) (X + B + A) %E
(C7) ISOLATE(%,X),EXPTISOLATE:TRUE;
(E7) B + A
B
(E8) %E
2
2 X + A X
(D8) E7 E8 (X + E7) %E
PICKAPART(exp,depth) will assign E labels to all subexpressions of exp down to
the specified integer depth. This is useful for dealing with large
expressions and for automatically assigning parts of an expression to a
variable without having to use the Part functions.
(C1) INTEGRATE(1/(X↑3+2),X)$
(C2) PICKAPART(D1,1);
1/3
LOG(X + 2 )
(E2) -------------
2/3
3 2
MACSYMA 6.2.3 More Functions for Part Extraction 77
1/3
2 X - 2
ATAN(------------)
1/3
2 SQRT(3)
(E3) ------------------
2/3
2 SQRT(3)
2 1/3 2/3
LOG(X - 2 X + 2 )
(E4) - -----------------------
2/3
6 2
(D4) E4 + E3 + E2
REVEAL(exp,depth) will display exp to the specified integer depth with the
length of each part indicated. Sums will be displayed as SUM(n) and
products as PRODUCT(n) where n is the number of subparts of the sum or
product. Exponentials will be displayed as EXPT.
(C1) INTEGRATE(1/(X↑3+2),X)$
(C2) REVEAL(%,2);
(D2) PRODUCT(3) + PRODUCT(3) + PRODUCT(3)
(C3) REVEAL(D1,3);
EXPT LOG EXPT LOG
(D3) - -------- + EXPT EXPT ATAN + --------
6 3
NUMFACTOR(exp) gives the numerical factor multiplying the expression exp which
should be a single term. If the gcd of all the term coefficients in a sum
is desired the CONTENT function (see 6.5) may be used.
(C1) GAMMA(7/2);
(D1) 15 SQRT(%PI)
------------
8
(C2) NUMFACTOR(%)
15
(D2) --
8
78 6.2.3 More Functions for Part Extraction MACSYMA
HIPOW(exp, v) gives the highest explicit exponent of v in exp. Sometimes it
may be necessary to expand exp since this is not done automatically by
HIPOW. Thus HIPOW(Y**3*X**2+X*Y**4,X) is 2.
LOPOW(exp, v) gives the lowest exponent of v which explicitly appears in exp.
Thus LOPOW((X+Y)**2+(X+Y)**A,X+Y) is MIN(A,2).
DERIVDEGREE(exp, dv, iv) finds the highest degree of the derivative of the
dependent variable dv with respect to the independent variable iv
occurring in exp.
(C1) 'DIFF(Y,X,2)+'DIFF(Y,Z,3)*2+'DIFF(Y,X)*X**2$
(C2) DERIVDEGREE(%,Y,X);
(D2) 2
The next several functions deal with complex variables. The user should
note the following conventions.
1) all variables are assumed to take on real values
exclusively;
2) all functions are assumed to be real-valued;
3) the complex argument is maintained in the half-open
interval (-p,p] whenever possible;
4) the argument of 0 is (arbitrarily) assumed to be 0,
although normally the user need not worry about this, since
0*%E↑(%I*0) is simplified to 0;
5) trigonometric functions are normally assumed to take on
their principal values.
REALPART(exp) gives the real part of exp. REALPART and IMAGPART will work on
expressions involving trigonometric and hyperbolic functions, as well as
SQRT, LOG, and exponentiation.
IMAGPART(exp) returns the imaginary part of the expression exp.
N
The real or imaginary part of an expression of the form Z , where Z is not
purely real, will be algebraic if n <= MAXPOSEX; otherwise, for compactness,it
N N
will be expressed as ABS(Z) * COS(N*ARG Z) or ABS(Z) * SIN(N*ARG Z).
MACSYMA 6.2.3 More Functions for Part Extraction 79
RECTFORM(exp) returns an expression of the form A + B*%I, where A and B are
purely real.
POLARFORM(exp) returns R*%E↑(%I*THETA) where R and THETA are purely real.
CAVEAT: Simplification of algebraic and transcendental functions of a
complex variable may give rise to apparent factors, %I. For example, SQRT(-
C+D) may be transformed to %I*SQRT(C-D).
CABS(exp) returns the complex absolute value (the complex modulus) of exp.
CARG(exp) returns the argument (phase angle) of exp. Due to the conventions
and restrictions (described above), principal value cannot be guaranteed.
(C1) RECTFORM(SIN(2*%I+X));
(D1) COSH(2) SIN(X) + %I SINH(2) COS(X)
(C2) POLARFORM(%);
2 2 2 2
(D2) SQRT(COSH (2) SIN (X) + SINH (2) COS (X))
%I ATAN2(SINH(2) COS(X), COSH(2) SIN(X))
%E
(C3) RECTFORM(LOG(3+4*%I));
(D3) LOG(5) + %I ATAN2(4,3)
(C4) POLARFORM(%);
2 2 %I ATAN2(ATAN2(4, 3), LOG(5))
(D4) SQRT(LOG (5) + ATAN2 (4, 3)) %E
(C5) RECTFORM((2+3.5*%I)↑.25),NUMER;
(D5) 0.36825881 %I + 1.36826627
(C6) POLARFORM(D5);
0.26291253 %I
(D6) 1.416957 %E
LHS(eqn) returns the left side of the equation eqn. If eqn is not an
equation, then LHS(eqn) = eqn.
80 6.2.3 More Functions for Part Extraction MACSYMA
RHS(eqn) returns the right side of the equation eqn. If eqn is not an
equation, then RHS(eqn) = 0.
NUM(exp) obtains the numerator, exp1, of the rational expression exp =
exp1/exp2.
DENOM(exp) returns the denominator, exp2, of the rational expression exp =
exp1/exp2.
The above two commands do not alter the internal representations of
expressions and have the desirable property that for all expressions
NUM(exp)/DENOM(exp) is the same as exp.
FIRST(exp) yields the first part of exp which may result in the first element
of a list, the first row of a matrix, the first term of a sum, etc. Note
that FIRST and the following two functions work on the form of exp which
is displayed not the form which is typed on input. If the variable
INFLAG[FALSE] is set to TRUE however, these functions will look at the
internal form of exp. Note that the simplifier re-orders expressions (see
3.3). Thus FIRST(X+Y) will be X if INFLAG is TRUE and Y if INFLAG is
FALSE. (FIRST(Y+X) gives the same results).
REST(exp, n) yields exp with its first n elements removed if n is positive and
its last -n elements removed if n is negative. If n is 1 it may be
omitted. Exp may be a list, matrix, or other expression. If INFLAG:TRUE
the internal form of exp will be used.
LAST(exp) yields the last part (term, row, element, etc.) of the exp. If
INFLAG:TRUE the internal form of exp will be used.
DELETE(exp1, exp2) removes all occurrences of exp1 from exp2. Exp1 may be a
term of exp2 (if it is a sum) or a factor of exp2 (if it is a product).
(C1) DELETE(SIN(X),X+SIN(X)+Y);
(D1) Y + X
LENGTH(exp) gives the number of parts in the internal form of exp. For lists
this is the number of elements, for matrices it is the number of rows, and
for sums it is the number of terms. However for products it may not
always yield the number of factors that would be displayed because of the
fact that -E is represented internally as -1*E and A/B is represented
internally by A*B↑(-1). (cf. DISPFORM)
MACSYMA 6.2.3 More Functions for Part Extraction 81
NTERMS(exp) gives the number of terms that exp would have if it were fully
expanded out and no cancellations or combinations of terms occurred. Note
that expressions like SIN(E), SQRT(E), EXP(E), etc. count as just one term
regardless of how many terms E has (if it is a sum).
82 MACSYMA
6.3 SOLVE and Related Functions
The following functions obtain the roots of equations or yield information
concerning the roots.
NROOTS(poly, low, high) finds the number of real roots of the real univariate
polynomial poly in the half-open interval (low,high]. The endpoints of
the interval may also be MINF,INF respectively for minus infinity and plus
infinity. NROOTS(poly) is equivalent to NROOTS(poly,MINF,INF). The
method of Sturm sequences is used. (see Heindel in [A1].)
(C1) POLY1:X**10-2*X**4+1/2$
(C2) NROOTS(POLY1,-6,9.1);
RAT REPLACED 0.5 BY 1/2 = 0.5
(D2) 4
REALROOTS(poly, bound) finds all of the real roots of the real univariate
polynomial poly within a tolerance of bound which, if less than 1, causes
all integral roots to be found exactly. The parameter bound may be
arbitrarily small in order to achieve any desired accuracy. The first
argument may also be an equation. REALROOTS(poly) is equivalent to
REALROOTS(poly,ROOTSEPSILON). ROOTSEPSILON[1.0E-7] is a real number used
to establish the confidence interval for the roots.
(C1) REALROOTS(X**5+X+1,5.0E-6);
395773
(E1) X = - ------
524288
(D1) [E1]
(C2) E1,FLOAT;
(D2) X = - 0.75487709
(C3) PART(C1,1);
5
(D3) X + X + 1
(C4) %,D2;
(D4) 1.50687992E-6
ALLROOTS(poly) finds all the real and complex roots of the real polynomial
poly which must be univariate and may be an equation. For complex
MACSYMA 6.3 SOLVE and Related Functions 83
1
polynomials an algorithm by Jenkins and Traub is used; for real
2
polynomials the algorithm used is due to Jenkins. The flag
POLYFACTOR[FALSE] when true causes ALLROOTS to factor the polynomial over
the real numbers if the polynomial is real, or over the complex numbers,
if the polynomial is complex.
(C1) (2*X+1)**3=13.5*(X**5+1);
3 5
(D1) (2 X + 1) = 13.5 (X + 1)
(C2) ALLROOTS(%);
(E2) X = - 1.0157555
(E3) X = 0.829675
(E4) X = - 0.96596254 %I - 0.40695972
(E5) X = 0.96596254 %I - 0.40695972
(E6) X = 1.0
(D6) [E2, E3, E4, E5, E6]
LINSOLVE([exp1, exp2, ...], [var1, var2, ...]) solves the list of simultaneous
linear equations for the list of variables. The expi must each be linear
in the variables and may be equations. LINSOLVE does no error checking to
assure linearity.
If GLOBALSOLVE[FALSE] is set to TRUE then variables which are SOLVEd
for will be set to the solution of the set of simultaneous equations.
BACKSUBST[TRUE] if set to FALSE will prevent back substitution after
the equations have been triangularized. This may be necessary in very big
problems where back substitution would cause the storage capacity to be
exceeded.
(C1) X+Z=Y$
(C2) 2*A*X-Y=2*A**2$
(C3) Y-2*Z=2$
←←←←←←←←←←←←←←←
1. Algorithm 419, Comm. ACM, vol. 15, (1972), p. 97
2. Algorithm 493, TOMS, vol. 1, (1975), p.178.
84 6.3 SOLVE and Related Functions MACSYMA
(C4) LINSOLVE([D1,D2,D3],[X,Y,Z]),GLOBALSOLVE:TRUE;
SOLUTION
(E4) X : A + 1
(E5) Y : 2 A
(E6) Z : A - 1
(D6) [E4, E5, E6]
ALGSYS([exp1, exp2, ...], [var1, var2, ...]) solves the list of simultaneous
polynomials or polynomial equations (which can be non-linear) for the list
of variables. The symbols %R1, %R2, etc. will be used to represent
arbitrary parameters when needed for the solution. In the process
described below, ALGSYS is entered recursively if necessary.
The method is as follows:
(1) First the equations are FACTORed and split into subsystems.
(2) For each subsystem S↓i, an equation E and a variable var are
selected (the var is chosen to have lowest nonzero degree). Then the
resultant of E and E↓j with respect to var is computed for each of the
remaining equations E↓j in the subsystem S↓i. This yields a new subsystem
S'↓i in one fewer variables (var has been eliminated). The process now
returns to (1).
(3) Eventually, a subsystem consisting of a single equation is
obtained. If the equation is multivariate and no approximations in the
form of floating point numbers have been introduced, then SOLVE is called
3
to find an exact solution.
If the equation is univariate and is either linear, quadratic, or bi-
quadratic, then again SOLVE is called if no approximations have been
introduced. If approximations have been introduced then if the switch
REALROOTS[TRUE] is TRUE, the function REALROOTS is called to find the
real-valued solutions. If REALROOTS:FALSE then ALLROOTS is called which
looks for real and complex-valued solutions. If ALGSYS produces a
solution which has fewer significant digits than required, the user can
8
change the value of ALGEPSILON[10 ] to a higher value.
(4) Finally, the solutions obtained in step (3) are re-inserted into
previous levels and the solution process returns to (1).
The user should be aware of several caveats.
←←←←←←←←←←←←←←←
3. The user should realize that SOLVE may not be able to produce a
solution or if it does the solution may be a very large expression.
MACSYMA 6.3 SOLVE and Related Functions 85
When ALGSYS encounters a multivariate equation which already contains
floating point approximations, then, presently, it does not attempt to
apply exact methods to such equations and prints the message: ALGSYS
CANNOT SOLVE---SYSTEM TOO COMPLICATED.
Interactions with RADCAN can produce large or complicated expressions.
In that case, the user may use PICKAPART or REVEAL to analyze the
solution. Occasionally, RADCAN may introduce an apparent %I into a
solution which is actually real-valued. To prevent the omission of
possible solutions, the user may prefer to set REALONLY[TRUE] to FALSE.
(C1) X+Z-Y$
(C2) X+B*Y*Z -A$
(C3) X↑2+C*Z -D;
(C4) ALGSYS([D1,D2,D3],[X,Y,Z]);
(D4) []
(C5) X+Z-Y↑2$
(C6) ALGSYS([D1,D2,D5],[X,Y,Z]);
B - A A - 1
(D6) [[X = -----, Y = 1, Z = -----], [X = A, Y = 0, Z = - A]]
B - 1 B - 1
(C7) X+B*Y*Z*X -A$
(C8) ALGSYS([D1,D5,D7],[X,Y,Z]);
2
SQRT(B - 4 A B + 2 B + 1) + B + 1
(D8) [[X = ----------------------------------, Y = 1,
2 B
2
SQRT(B + (2 - 4 A) B + 1) + (1 - 2 A) B + 1
Z = - --------------------------------------------],
2 2
B SQRT(B + (2 - 4 A) B + 1) + B + B
2
- SQRT(B - 4 A B + 2 B + 1) + B + 1
[X = ------------------------------------, Y = 1,
2 B
2
SQRT(B + (2 - 4 A) B + 1) + (2 A - 1) B - 1
Z = - --------------------------------------------],
2 2
B SQRT(B + (2 - 4 A) B + 1) - B - B
86 6.3 SOLVE and Related Functions MACSYMA
[X = A, Y = 0, Z = - A]]
SOLVE(exp, var) solves the algebraic equation exp for the variable var and
returns a list of solution equations in var. If exp is not an equation,
it is assumed to be an expression to be set equal to zero. Var may be a
function (e.g. F(X)), or other non-atomic expression except a sum or
product. It may be omitted if exp contains only one variable. Exp may be
a rational expression, and may contain trigonometric functions,
exponentials, etc.
The following method is used:
Let E be the expression and X be the variable. If E is linear in X
then it is trivially solved for X. Otherwise if E is of the form A*X**N+B
then the result is (-B/A)**(1/N) times the Nth roots of unity.
If E is not linear in X then the gcd of the exponents of X in E (say
N) is divided into the exponents and the multiplicity of the roots is
multiplied by N. Then SOLVE is called again on the result.
If E factors then SOLVE is called on each of the factors. Finally
SOLVE will use the quadratic, cubic, or quartic formulas where necessary.
In the case where E is a polynomial in some function of the variable
to be solved for, say F(X), then it is first solved for F(X) (call the
result C), then the equation F(X)=C can be solved for X provided the
inverse of the function F is known.
BREAKUP[TRUE] if FALSE will cause SOLVE to express the solutions of
cubic or quartic equations as single expressions rather than as made up of
several common subexpressions which is the default.
MULTIPLICITIES[NOT%SET%YET] - will be set to a list of the
multiplicities of the individual solutions returned by SOLVE, REALROOTS,
or ALLROOTS.
SOLVEFACTORS[TRUE] - if FALSE then SOLVE will not try to factor the
expression. The FALSE setting may be desired in some cases where
factoring is not necessary.
SOLVERADCAN[FALSE] - if TRUE then SOLVE will use RADCAN which will
make SOLVE slower but will allow certain problems containing exponentials
and logs to be solved.
When SOLVing polynomials with large integer coefficients, it may be
useful to reset INTFACLIM.
MACSYMA 6.3 SOLVE and Related Functions 87
SOLVE([eq1, ..., eqn], [v1, ..., vn]) solves a system of simultaneous (linear
or non-linear) polynomial equations by calling LINSOLVE or ALGSYS and
returns a list of the solution lists in the variables. In the case of
LINSOLVE this list would contain a single list of solutions. It takes two
lists as arguments. The first list (eqi, i=1,...,n) represents the
equations to be solved; the second list is a list of the unknowns to be
determined. If the total number of variables in the equations is equal to
the number of equations, the second argument-list may be omitted. For
linear systems if the given equations are not compatible, the message
INCONSISTENT will be displayed; if no unique solution exists, then
SINGULAR will be displayed. DISPFLAG[TRUE] when set to FALSE within a
BLOCK will inhibit the display of output generated by the Solve functions
called from within the BLOCK.
SOLVETRIGWARN[TRUE] if set to FALSE will inhibit printing by SOLVE of the
warning message saying that it is using inverse trigonometric functions to
solve the equation, and thereby losing solutions.
SOLVEDECOMPOSES[TRUE] if TRUE, will induce SOLVE to use POLYDECOMP
4
(see below) in attempting to solve polynomials.
PROGRAMMODE[FALSE] when TRUE will inhibit SOLVE from printing E-labels
and will force SOLVE to return its answers explicitly as elements in a
list.
SOLVEEXPLICIT[FALSE] if FALSE, inhibits SOLVE from returning implicit
solutions i.e. of the form F(x)=0.
(C1) SOLVE(ASIN(COS(3*X))*(F(X)-1),X);
Solution
%PI
(E1) X = ---
6
The roots of
(E2) F(X) = 1
(D2) [E1,E2]
(C3) SOLVERADCAN:TRUE$
(C4) SOLVE(5**F(X)=125,F(X));
(D4) F(X) = 3
(C5) [4*X**2-Y**2=12,X*Y-X=2]
2 2
(D5) [4 X - Y = 12, X Y - X = 2]
←←←←←←←←←←←←←←←
4. Under certain circumstances (e.g. if there is a variable in the
exponent), the implicit "solution" may quite complex
88 6.3 SOLVE and Related Functions MACSYMA
(C6) SOLVE(D5,[X,Y]);
(D6) [[Y = - 0.15356758,
X = - 1.733752], [Y = 2.0, X = 2.0]]
(C7) SOLVE(X↑3+A*X+1,X);
3
SQRT(4 A + 27)
(E7) ---------------
6 SQRT(3)
1 1/3
(E8) (E7 - -)
2
Solution
%I SQRT(3) 1
(---------- - -) A
%I SQRT(3) 1 2 2
(E9) X = ( - ---------- - -) E8 - ------------------
2 2 3 E8
%I SQRT(3) 1
( - ---------- - -) A
%I SQRT(3) 1 2 2
(E10) X = (---------- - -) E8 - ---------------------
2 2 3 E8
A
(E11) X = E8 - ----
3 E8
(D11) [E9, E10, E11]
POLYDECOMP(poly,var) returns a list of polynomials
[f1(var),f2(var),...fn(var)] such that poly = f1(f2(...fn(var)...)).
There is no other decomposition which involves more polynomials excepting
linear fi.
MACSYMA 89
6.4 The Matrix Functions
Matrix multiplication is effected by using the dot operator, ".", which is
also convenient if the user wishes to represent other non-commutative
algebraic operations (see 6.4.1). The exponential of the . operation is
"↑↑" .
Thus, for a matrix A, A.A = A↑↑2 and, if it exists, A↑↑-1 is the inverse
of A.
The operations +,-,*,** are all element-by-element operations; all
operations are normally carried out in full, including the . (dot) operation.
Many switches exist for controlling simplification rules involving dot and
matrix-list operations (see below).
ENTERMATRIX(m, n) allows one to enter a matrix element by element with MACSYMA
requesting values for each of the m*n entries.
(C1) ENTERMATRIX(2,1);
ROW 1 COLUMN 1 X+Y/2;
ROW 2 COLUMN 1 34;
MATRIX-ENTERED
[ Y ]
[ - + X ]
(D1) [ 2 ]
[ ]
[ 34 ]
MATRIX(row1, ..., rown) defines a rectangular matrix with the indicated rows.
Each row has the form of a list of expressions, e.g. [A, X**2, Y, 0] is a
list of 4 elements.
GENMATRIX(array, i2, j2, i1, j1) generates a matrix from the array using
array(i1,j1) for the first (upper-left) element and array(i2,j2) for the
last (lower-right) element of the matrix. If j1=i1 then j1 may be
omitted. If j1=i1=1 then i1 and j1 may both be omitted. If a selected
element of the array doesn't exist a symbolic one will be used.
(C1) H[I,J]:=1/(I+J-1)$
(C2) GENMATRIX(H,3,3);
90 6.4 The Matrix Functions MACSYMA
[ 1 1]
[1 - -]
[ 2 3]
[ ]
[1 1 1]
(D2) [- - -]
[2 3 4]
[ ]
[1 1 1]
[- - -]
[3 4 5]
COPYMATRIX(M) creates a copy of the matrix M. This is the only way to make a
copy aside from recreating M elementwise. Copying a matrix may be useful
when SETELMX is used (see below).
COPYLIST(L) creates a copy of the list L.
ADDROW(M,l) appends the row given by the list l onto the matrix M.
IDENT(n) produces an n by n identity matrix.
DIAGMATRIX(n, x) returns a diagonal matrix of size n by n with the diagonal
elements all x. An identity matrix is created by DIAGMATRIX(n,1), or one
may use IDENT(n).
EMATRIX(m, n, x, i, j) will create an m by n matrix all of whose elements are
zero except for the i,j element which is x.
MATRIXMAP(fn, M) will map the function fn onto each element of the matrix M.
SETELMX(x, i, j, M) changes the i,j element of M to x. The altered matrix is
returned as the value. The notation M[i,j]:x may also be used, altering M
in a similar manner, but returning x as the value.
COEFMATRIX([eq1, ...], [var1, ...]) the coefficient matrix for the variables
var1,... of the system of linear equations eq1,...
MACSYMA 6.4 The Matrix Functions 91
AUGCOEFMATRIX([eq1, ...], [var1, ...]) the augmented coefficient matrix for
the variables var1,... of the system of linear equations eq1,.... This is
the coefficient matrix with a column adjoined for the constant terms in
each equation (i.e. those not dependent upon var1,...).
(C1) [2*X-(A-1)*Y=5*B,A*X+B*Y+C=0]$
(C2) AUGCOEFMATRIX(%,[X,Y]);
[2 1 - A -5 B ]
(D2) [ ]
[A B C ]
COL(M,i) gives a matrix of the ith column of the matrix M.
ROW(M, i) gives a matrix of the ith row of matrix M.
SUBMATRIX(m1, ..., M, n1, ...) creates a new matrix composed of the matrix M
with rows mi deleted, and columns ni deleted.
MINOR(M, i, j) computes the i,j minor of the matrix M. That is, M with row i
and column j removed.
TRANSPOSE(M) produces the transpose of M.
ECHELON(M) produces the echelon form of M. That is, M with elementary row
operations performed on it such that the first non-zero element in each
row in the resulting matrix is a one and the column elements under the
first one in each row are all zero.
(C3) ECHELON(D2); (D2 is as above)
[ A - 1 5 B ]
[1 - ----- --- ]
[ 2 2 ]
(D3) [ ]
[ 2 C + 5 A B ]
[0 1 - ------------]
[ 2 ]
[ 2 B + A - A]
92 6.4 The Matrix Functions MACSYMA
TRIANGULARIZE(M) produces the upper triangular form of the matrix M which
needn't be square.
(C4) TRIANGULARIZE(D2);
[2 1 + A 5 B ]
(D4) [ ]
[ 2 ]
[0 2 B + A - A - 2 C - 5 A B]
RANK(M) computes the rank of the matrix M. That is, the order of the largest
non-singular subdeterminant of M.
(C5) RANK(D2);
(D5) 2
DETERMINANT(M) computes the determinant of the square matrix M by a method
similar to Gaussian elimination. The method and the form of the result
depend upon the setting of the switch RATMX (see below). There is a
special routine for dealing with sparse determininants which can be used
by setting the switches RATMX:TRUE and SPARSE:TRUE.
NEWDET(M,n) also computes the determinant of M but uses the Johnson-Gentleman
tree minor algorithm [Ge1]. M may be the name of a matrix or array. The
argument n is the order; it is optional if M is a matrix.
CHARPOLY(M, var) computes the characteristic polynomial for M with respect to
var. That is, DETERMINANT(M - DIAGMATRIX(LENGTH(M),var)).
(C1) A:MATRIX([3,1],[2,4]);
[3 1]
(D1) [ ]
[2 4]
(C2) CHARPOLY(A,LAMBDA);
2
(D2) LAMBDA - 7 LAMBDA + 10
(C3) SOLVE(%);
Solution
(E3) LAMBDA = 2
(E4) LAMBDA = 5
(D4) [E4, E5]
MACSYMA 6.4 The Matrix Functions 93
(C5) X:MATRIX([X1],[X2]);
[X1]
(D5) [ ]
[X2]
(C6) A.X-LAMBDA*X,E4;
[ X2 - 2 X1 ]
(D6) [ ]
[ - X2 + 2 X1]
(C7) D6[1,1]=0;
(D7) X2 - 2 X1 = 0
(C8) X1**2+X2**2=1;
2 2
(D8) X2 + X1 = 1
(C9) SOLVE([D7,D8],[X1,X2]);
(D9) [[X2 = - 0.89442714, X1 = - 0.44721357],
[X2 = 0.89442714, X1 = 0.44721357]]
Options Relating to Matrices
Note: MX stands for Matrix and SC stands for Scalar
By resetting the options LMXCHAR and RMXCHAR (with the defaults [ and ]
respectively), the user can specify the delimiters used in the display of
matrices.
RATMX[FALSE] - if FALSE will cause determinant and matrix addition,
subtraction, and multiplication to be performed in the representation of the
1
matrix elements and will cause the result of matrix inversion to be left in
general representation. If it is TRUE, the 4 operations mentioned above will
be performed in CRE form and the result of matrix inverse will be in CRE form.
Note that this may cause the elements to be expanded (depending on the setting
of RATFAC) which might not always be desired.
LISTARITH[TRUE] - if FALSE causes any arithmetic operations with lists to
be suppressed; when TRUE, list-matrix operations are contagious causing lists
to be converted to matrices yielding a result which is always a matrix.
However, list-list operations should return lists.
←←←←←←←←←←←←←←←
1. Matrix inversion is always performed in CRE form, except when done via the
INVERT command.
94 6.4 The Matrix Functions MACSYMA
DETOUT[FALSE] - if TRUE will cause the determinant of a matrix whose
inverse is computed to be kept outside of the inverse. For this switch to
have an effect DOALLMXOPS and DOSCMXOPS should be FALSE (see below).
Alternatively this switch can be given to EV which causes the other two to be
set correctly.
DOALLMXOPS[TRUE] - if TRUE all operations relating to matrices are carried
out. If it is FALSE then the setting of the following switches govern which
operations are performed.
DOMXMXOPS[FALSE] - if TRUE then all matrix-matrix or matrix-list
operations are carried out (but not scalar-matrix operations); if this switch
is FALSE they are not.
DOSCMXOPS[FALSE] - if TRUE then scalar-matrix operations are performed.
DOSCMXPLUS[FALSE] - if TRUE will cause SCALAR + MATRIX to give a matrix
answer. This switch is not subsumed under DOALLMXOPS.
SCALARMATRIXP[TRUE] - if TRUE causes a square matrix of dimension one
(when produced as a result of a computation) to be converted to a scalar (i.e.
its only element).
SPARSE[FALSE] - if TRUE and if RATMX:TRUE then DETERMINANT will use
special routines for computing sparse determinants.
6.4.1 The Dot Operator
In some applications one would like to work with expressions containing
non-commuting variables. In order to allow other variables to play the role
of scalars and constants, the non-commuting variables may be declared
NONSCALAR by using the DECLARE function (see 8.1.1). Then by setting the
appropriate flags below and using the pattern-matching functions (see
9.2), the user can control simplification and interaction between scalars
and nonscalars.
After the expressions are manipulated into a particular form then perhaps
actual matrices or operators will be substituted for them. Several options
are provided in order to control how MACSYMA treats such expressions. (The
options are checked for and utilized by MACSYMA in the order in which they are
presented here).
In the following discussion A, B, and C are any expressions, and SC is a
scalar expression (i.e. one free of lists, matrices, and any atoms declared
non-scalar).
DOTASSOC[TRUE] - when TRUE causes (A.B).C to simplify to A.(B.C)
DOTSCRULES[FALSE] - when TRUE will cause A.SC or SC.A to simplify to SC*A
and A.(SC*B) to simplify to SC*(A.B)
MACSYMA 6.4.1 The Dot Operator 95
DOTCONSTRULES[TRUE] - is similar to DOTSCRULES but with constants instead
of scalars.
DOTEXPTSIMP[TRUE] - when TRUE causes A.A to simplify to A↑↑2
DOTDISTRIB[FALSE] - if TRUE will cause A.(B+C) to simplify to A.B+A.C
(C1) DECLARE([M1,M2,M3],NONSCALAR)$
(C2) (1-L*M1).(1-L*M2).(1-L*M3),DOTCONSTRULES:TRUE,EXPAND;
(D2) - L M3 + L M2 . L M3 - L M2 + L M1 . L M3
- L M1 . (L M2 . L M3) + L M1 . L M2 - L M1 + 1
(C3) %,DOTSCRULES:TRUE;
2 2
(D3) - L M3 + L (M2 . M3) - L M2 + L (M1 . M3)
3 2
- L (M1 . (M2 . M3)) + L (M1 . M2) - L M1 + 1
(C4) RAT(%,L);
3
(D4)/R/ - (M1 . (M2 . M3)) L + (M2 . M3 + M1 . M3
2
+ M1 . M2) L + ( - M3 - M2 - M1) L + 1
96 MACSYMA
6.5 Functions for Rational Expressions
A rational expression is the quotient of two polynomials. MACSYMA
provides a special internal representation (called CRE for canonical rational
expression form - (see 3.1) ) which can be used for rational expressions (and
polynomials as special cases) and which requires less storage than the general
representation. In addition CRE manipulations are usually faster. Therefore
it may be desirable to use these whenever the problem of interest can be
expressed largely in terms of polynomials or rational expressions. The symbol
/R/ following the line label in the display of an expression indicates that
either the expression is in CRE form or that some subexpression of it is.
CRE form is "contagious" in that any time a CRE expression is added to or
multiplied by another compatible expression, the result is in CRE form. Thus
by initially multiplying by RAT(1) one can force his entire calculation to be
done in CRE form. However, if CRE are mixed into an expression containing
general forms e.g. SIN(RAT(X**2)), such that the result is not totally in CRE
form, then the result is automatically converted into general representation.
Some functions (e.g. RATSIMP, FACTOR, etc.) use CRE form internally in the
implementation of their algorithms. This fact however is usually transparent
to the user.
RATVARS(var1, var2, ..., varn) forms its n arguments into a list in which the
rightmost variable varn will be the main variable of future rational
expressions in which it occurs, and the other variables will follow in
sequence. If a variable is missing from the RATVARS list, it will be
given lower priority than the leftmost variable var1. The arguments to
RATVARS can be either variables or non-rational functions (e.g. SIN(X)).
The variable RATVARS is a list of the arguments which have been given to
this function.
RAT(exp, v1, ..., vn) converts exp to CRE form by expanding and combining all
terms over a common denominator and cancelling out the greatest common
divisor of the numerator and denominator as well as converting floating
point numbers to rational numbers within a tolerance of RATEPSILON[2.0E-
8]. The variables are ordered according to the v1,...,vn as in RATVARS,
if these are specified. RAT does not generally simplify functions other
than + , - , * , / , and exponentiation to an integer power and it does
not deal with equations whereas RATSIMP does handle these cases. Note
that atoms (numbers and names) in CRE form are not the same as they are in
the general form. Thus RAT(X)-X results in RAT(0) which has a different
internal representation than 0.
RATFAC[FALSE] when TRUE invokes a partially factored form for CRE
rational expressions. During rational operations the expression is
maintained as fully factored as possible without an actual call to the
factor package. This should always save space and may save some time in
MACSYMA 6.5 Functions for Rational Expressions 97
some computations. The numerator and denominator are still made
relatively prime, for example
RAT((X↑2 -1)↑4/(X+1)↑2); yields (X-1)↑4*(X+1)↑2),
but the factors within each part may not be relatively prime.
RATPRINT[TRUE] if FALSE suppresses the printout of the message
informing the user of the conversion of floating point numbers to rational
numbers.
KEEPFLOAT[FALSE] if TRUE prevents floating point numbers from being
1
converted to rational numbers.
BFTORAT[FALSE] controls the conversion of bfloats to rational numbers.
If BFTORAT:FALSE, RATEPSILON will be used to control the conversion (this
results in relatively small rational numbers). If BFTORAT:TRUE, the
rational number generated will accurately represent the bfloat.
(Also see the RATEXPAND and RATSIMP functions. sec. 6.1.1)
(C1) ((X-2*Y)**4/(X**2-4*Y**2)**2+1)*(Y+A)*(2*Y+X)
/(4*Y**2+X**2);
4
(X - 2 Y)
(Y + A) (2 Y + X) (------------ + 1)
2 2 2
(X - 4 Y )
(D1) ------------------------------------
2 2
4 Y + X
(C2) RAT(%,Y,A,X);
2 A + 2 Y
(D2)/R/ ---------
X + 2 Y
RATDISREP(exp) changes its argument from CRE form to general form. This is
sometimes convenient if one wishes to stop the "contagion", or use
rational functions in non-rational contexts (see the example at the
beginning of this section). Most CRE functions will work on either CRE or
non-CRE expressions, but the answers may take different forms. If
RATDISREP is given a non-CRE for an argument, it returns its argument
unchanged.
←←←←←←←←←←←←←←←
1. As with all other switches, various MACSYMA algorithms may override the
setting of this switch if they are unable to operate in that mode.
98 6.5 Functions for Rational Expressions MACSYMA
TOTALDISREP(exp) converts every subexpression of exp from CRE to general form.
If exp is itself in CRE form then this is identical to RATDISREP but if
not then RATDISREP would return exp unchanged while TOTALDISREP would
"totally disrep" it. This is useful for ratdisrepping expressions e.g.,
equations, lists, matrices, etc. which have some subexpressions in CRE
form.
NUM(exp) obtains the numerator of the rational expression exp.
DENOM(exp) returns the denominator of the rational expression exp.
The above two commands do not alter the internal representations of
expressions and have the desirable property that for all expressions
NUM(exp)/DENOM(exp) is the same as exp. This is not true of the following two
commands which return expressions in CRE form.
RATNUMER(exp) obtains the numerator of the rational expression exp. If exp is
in general form then the NUM function should be used instead, unless one
wishes to get a CRE result.
RATDENOM(exp) obtains the denominator of the rational expression exp. If exp
is in general form then the DENOM function should be used instead, unless
one wishes to get a CRE result.
RATWEIGHT(v1, w1, ..., vn, wn) assigns a weight of wi to the variable vi.
This causes a monomial to be replaced by 0 if its weight exceeds the value
of the variable RATWTLVL [FALSE] (for the default value FALSE no
truncation occurs). The weight of a monomial is the sum of the products
of the weight of a variable in the term times its power. Thus the weight
of 3*v1**2*v2 is 2*w1+w2. This truncation occurs only when multiplying or
exponentiating CRE forms of expressions.
RATWEIGHTS[[]] returns a list of weight assignments, as does RATWEIGHT();
KILL(...,RATWEIGHTS), SAVE(...,RATWEIGHTS) eliminate and save weight
assignments ((see 10.3),(see 15.3)).
(C5) RATWEIGHT(A,1,B,1);
(D5) [[B, 1], [A, 1]]
(C6) EXP1:RAT(A+B+1)$
(C7) %**2;
2 2
(D7)/R/ B + (2 A + 2) B + A + 2 A + 1
(C8) RATWTLVL:1$
MACSYMA 6.5 Functions for Rational Expressions 99
(C9) EXP1**2;
(D9)/R/ 2 B + 2 A + 1
HORNER(exp, var) will convert exp into a rearranged representation as in
Horner's rule, using var as the main variable if it is specified. Var may
also be omitted in which case the main variable of the CRE form of exp is
used. HORNER sometimes improves stability if expr is to be numerically
evaluated. It is also useful if MACSYMA is used to generate programs to
be run in FORTRAN (see STRINGOUT - 10.4)
(C1) 1.0E-20*X↑2-5.5*X+5.2E20;
2
(D1) 1.0E-20 X - 5.5 X + 5.2E+20
(C2) HORNER(%,X),KEEPFLOAT:TRUE;
(D2) X (1.0E-20 X - 5.5) + 5.2E+20
(C3) D1,X=1.0E20;
ARITHMETIC OVERFLOW
(C4) D2,X=1.0E20;
(D4) 6.9999999E+19
FASTTIMES(p1, p2) multiplies the polynomials p1 and p2 by using a special
algorithm for multiplication of polynomials. They should be multivariate,
dense, and nearly the same size. Classical multiplication is of order N*M
where N and M are the degrees. FASTTIMES is of order MAX(N,M)**1.585.
The rest of the functions in this section return their results in
general representation only if all of their principal arguments are in
that form. If any of their principal arguments are in CRE form then the
result is returned in CRE form.
DIVIDE(p1, p2, var1, ..., varn) computes the quotient and remainder of the
polynomial p1 divided by the polynomial p2, in a main polynomial variable,
varn. The other variables are as in the RATVARS function. The result is
a list whose first element is the quotient and whose second element is the
remainder.
(C1) DIVIDE(X+Y,X-Y,X);
(D1) [1, 2 Y]
(C2) DIVIDE(X+Y,X-Y);
(D2) [ - 1, 2 X]
(Note that Y is the main variable in C2)
100 6.5 Functions for Rational Expressions MACSYMA
QUOTIENT(p1, p2, var1, ...) computes the quotient of the polynomial p1 divided
by the polynomial p2.
REMAINDER(p1, p2, var1, ...) computes the remainder of the polynomial p1
divided by the polynomial p2.
CONTENT(p1, var1, ..., varn) returns a list whose first element is the
greatest common divisor of the coefficients of the terms of the polynomial
p1 in the variable varn (this is the content) and whose second element is
the polynomial p1 divided by the content.
(C1) CONTENT(2*X*Y+4*X**2*Y**2,Y);
(D1) [2*X, 2*X*Y**2+Y].
GCD(p1, p2, var1, ...) computes the greatest common divisor of p1 and p2. The
flag GCD[EZ] determines which algorithm is employed. Setting GCD to
EZ,RED, or MOD selects the EZGCD [Mo6], reduced, or modular [Br1]
algorithm, respectively. If GCD:FALSE then GCD(p1,p2,var) will always
return 1 for all x. Many functions (e.g. RATSIMP, FACTOR, etc.) cause
gcd's to be taken implicitly. For homogeneous polynomials it is
recommended that GCD:RED be used. To take the gcd when an algebraic is
present, e.g. GCD(X↑2-2*SQRT(2)*X+2,X-SQRT(2)); , ALGEBRAIC must be TRUE
and GCD must not be EZ.
EZGCD(p1, p2, ...) gives a list whose first element is the g.c.d of the
polynomials p1,p2,... and whose remaining elements are the polynomials
divided by the g.c.d. This always uses the EZGCD algorithm (not
recommended for homogeneous polynomials).
MOD(p) converts the polynomial p to a modular representation with respect to
the current modulus which is the value of the variable MODULUS.
If MODULUS[FALSE] is set to a positive prime p, then all arithmetic in
the rational function routines will be done modulo p. That is all
integers will be reduced to less than p/2 in absolute value (if p=2 then
all integers are reduced to 1 or 0). This is the so called "balanced"
modulus system, e.g. N MOD 5 = -2, -1, 0, 1, or 2.
RESULTANT(p1, p2, var) computes the resultant of the two polynomials p1 and
p2, eliminating the variable var. The resultant is a determinant of the
coefficients of var in p1 and p2 which equals zero if and only if p1 and
p2 have a non-constant factor in common.
MACSYMA 6.5 Functions for Rational Expressions 101
MODRESULT[FALSE] if TRUE causes the modular resultant algorithm to be
used, otherwise the reduced (which is the default) will be used (see
[Co1]).
(C1) RESULTANT(A*Y+X**2+1,Y**2+X*Y+B,X);
4 3 2 2
(D1) Y + A Y + (2 B + 1) Y + B
RATDIFF(exp, var) differentiates the rational expression exp (which must be a
ratio of polynomials or a polynomial in the variable var) with respect to
var. For rational expressions this is much faster than DIFF. The result
is left in CRE form. However, RATDIFF should not be used on factored CRE
forms; use DIFF instead for such expressions.
(C1) (4*X**3+10*X-11)/(X**5+5);
3
4 X + 10 X - 11
(D1) ----------------
5
X + 5
(C2) MODULUS:3$
(C3) MOD(D1);
2
X + X - 1
(D3) --------------------
4 3 2
X + X + X + X + 1
(C4) RATDIFF(D1,X);
5 4 3
X - X - X + X - 1
(D4) ------------------------------
8 7 5 4 3
X - X + X - X + X - X + 1
6.5.1 Algebraic Integers
An algebraic integer is a solution of a univariate monic polynomial
equation with integer coefficients. Examples of algebraic integers are
2+3*%I, SQRT(7), and 6↑(1/3)-5↑(1/7). In addition to the factorization of
polynomials over the ring of integers with an algebraic integer adjoined,
MACSYMA provides simplification of expressions involving algebraic integers by
representing them in a canonically simplified form, in which there are no
radicals in the denominators of fractions.
102 6.5.1 Algebraic Integers MACSYMA
TELLRAT(poly1,...,polyn) adds to the ring of algebraic integers known to
MACSYMA, the elements which are the solutions of the univariate, monic
polynomials polyj (integer coefficients). MACSYMA initially knows about
%I and all roots of integers. TELLRAT() returns a list of the polynomials
given to TELLRAT. To SAVE or KILL all of one's TELLRATs, just do SAVE
(...,TELLRATS,...) or KILL(...,TELLRATS,...). To undo a TELLRAT(p(X)),
simply do TELLRAT(X).
ALGEBRAIC[FALSE] must be set to TRUE in order for the simplification
of algebraic integers to take effect.
RATALGDENOM[TRUE] if TRUE allows rationalization of denominators wrt.
radicals to take effect. To do this one must use CRE form in algebraic
mode.
(C1) ALGEBRAIC:RATALGDENOM:TRUE$
(C2) RATDIS(E):=RATDISREP(RAT(E))$
(C3) 10*(1+%I)/(3↑(1/3)+%I);
10 (%I + 1)
(D3) -----------
1/3
3 + %I
(C4) RATDIS(%);
2/3 1/3
(D4) (4 %I + 2) 3 + (4 - 2 %I) 3 - 4 %I - 2
(C5) TELLRAT(A↑2+A+1)$
(C6) A/(SQRT(2)+SQRT(3))+1/(A*SQRT(2)-1);
1 A
(D6) ------------- + -----------------
SQRT(2) A - 1 SQRT(3) + SQRT(2)
(C7) RATDIS(%);
(7 SQRT(3) - 10 SQRT(2) + 2) A - 2 SQRT(2) - 1
(D7) ----------------------------------------------
7
6.5.2 Functions for Extended Rational Expressions
An extended rational expression is a truncated power series with rational
functions for coefficients ( as generated by TAYLOR). The truncation
capability (RATWEIGHT,RATWTLVL) described above (see 6.5) is utilized by
extended CRE forms as well as by CRE forms.
MACSYMA 6.5.2 Functions for Extended Rational Expressions 103
TAYLOR(exp,[var1,pt1,ord1],[var2,pt2,ord2],...) returns a truncated power
series in the variables vari about the points pti, truncated at ordi. For
further details see 6.7.
PSEXPAND[FALSE] if TRUE will cause extended rational function
expressions to display fully expanded. (RATEXPAND will also cause this.)
If FALSE, multivariate expressions will be displayed just as in the
rational function package. If PSEXPAND:MULTI, then terms with the same
total degree in the variables are grouped together.
SRRAT(exp) converts exp from extended rational form to CRE form, i.e. it is
like RAT(RATDISREP(exp)) although much faster.
(C1) TAYLOR(1 + X, [X, 0, 3]);
(D1)/T/ 1 + X + . . .
(C2) 1/%;
2 3
(D2)/T/ 1 - X + X - X + . . .
(C3) TAYLOR(1 + X + Y + Z, [X, 0, 3], [Y, 1, 2],
[Z, 2, 1]);
(D3)/T/ 4 + (Z - 2) + (Y - 1) + X + . . .
(C4) 1/%;
104 6.5.2 Functions for Extended Rational Expressions MACSYMA
1 Z - 2 1 Z - 2
(D4)/T/ - - ----- + (- -- + ----- + . . .) (Y - 1)
4 16 16 32
1 3 (Z - 2) 2
+ (-- - --------- + . . .) (Y - 1)
64 256
1 Z - 2 1 3 (Z - 2)
+ (- -- + ----- + (-- - --------- + . . .) (Y - 1)
16 32 32 128
3 3 (Z - 2) 2
+ (- --- + --------- + . . .) (Y - 1) + . . .) X
256 256
1 3 (Z - 2) 3 3 (Z - 2)
+ (-- - --------- + (- --- + --------- + . . .) (Y - 1)
64 256 256 256
3 15 (Z - 2) 2 2
+ (--- - ---------- + . . .) (Y - 1) + . . .) X
512 2048
1 Z - 2 1 5 (Z - 2)
+ (- --- + ----- + (--- - --------- + . . .) (Y - 1)
256 256 256 1024
5 15 (Z - 2) 2 3
+ (- ---- + ---------- + . . .) (Y - 1) + . . .) X
2048 4096
+ . . .
[6.6] Poisson Series Functions
A Poisson series is a finite sum where each term has the form p*trig(q)
where "trig" is either SIN or COS . Usually, p is a polynomial with rational
number or floating point coefficients, or a general MACSYMA expression. The
argument q is a linear combination of no more than 6 variables, whose names
are literally U, V, W, X, Y, and Z. (These restrictions are not vital, but
apparently present no difficulty in usual applications. They could be altered
easily).
Conversion to a Poisson series expands all products or powers of sines
and/or cosines into sums. In order to display the result, it is usually
necessary to convert an expression in Poisson encoding into general MACSYMA
representation using the OUTOFPOIS function, or to print it using the
PRINTPOIS function.
MACSYMA 6.6 Poisson Series Functions 105
POISSIMP(A) converts A into a Poisson series for A in general representation.
INTOPOIS(A) converts A into a Poisson encoding.
OUTOFPOIS(A) converts A from Poisson encoding to general representation. If A
is not in Poisson form, it will make the conversion, i.e. it will look
like the result of OUTOFPOIS(INTOPOIS(A)). This function is thus a
canonical simplifier for sums of powers of SIN's and COS's of a particular
type.
PRINTPOIS(A) prints a Poisson series in a readable format. In common with
OUTOFPOIS, it will convert A into a Poisson encoding first, if necessary.
POISTIMES(A, B) is functionally identical to INTOPOIS(A*B).
POISTRIM() is a reserved function name which (if the user has defined it) gets
applied during Poisson multiplication. It is a predicate function of 6
arguments which are the coefficients of the U, V,..., Z in a term. Terms
for which POISTRIM is TRUE (for the coefficients of that term) are
eliminated during multiplication.
POISPLUS(A, B) is functionally identical to INTOPOIS(A+B).
POISEXPT(A, B) (B a positive integer) is functionally identical to
INTOPOIS(A**B).
POISDIFF(A, B) differentiates A with respect to B. B must occur only in the
trig arguments or only in the coefficients.
POISINT(A, B) integrates in a similarly restricted sense (to POISDIFF). Non-
periodic terms in B are dropped if B is in the trig arguments.
POISSUBST(A, B, C) substitutes A for B in C. C is a Poisson series.
(1) Where B is a variable U, V, W, X, Y, or Z then A must be an expression
linear in those variables (e.g. 6*U+4*V).
(2) Where B is other than those variables, then A must also be free of
those variables, and furthermore, free of sines or cosines.
106 6.6 Poisson Series Functions MACSYMA
POISSUBST(A, B, C, D, N) is a special type of substitution which operates on A
and B as in type (1) above, but where D is a Poisson series, expands
COS(D) and SIN(D) to order N so as to provide the result of substituting
A+D for B in C. The idea is that D is an expansion in terms of a small
parameter. For example, POISSUBST(U,V,COS(V),E,3) results in COS(U)*(1-
2 3
E /2) - SIN(U)*(E-E /6).
POISMAP(series, sinfn, cosfn) will map the functions sinfn on the sine terms
and cosfn on the cosine terms of the poisson series given. sinfn and
cosfn are functions of two arguments which are a coefficient and a
trigonometric part of a term in series respectively.
(C1) PFEFORMAT:TRUE$
(C2) (2*A↑2-B)*COS(X+2*Y)-(A*B+5)*SIN(U-4*X);
(D2) - (A B + 5) SIN(U - 4 X)
2
+ (2 A - B) COS(2 Y + X)
(C3) POISEXPT(%,2)$
(C4) PRINTPOIS(D3);
2
(2 A - B) ( - A B - 5) SIN( - 2 Y - 5 X + U)
2
(2 A - B) ( - A B - 5) SIN(2 Y - 3 X + U)
2
- 1/2 ( - A B - 5) COS(2 U - 8 X)
2 2 2
1/2 (2 A - B) + 1/2 ( - A B - 5)
2 2
1/2 (2 A - B) COS(4 Y + 2 X)
(D4) DONE
(C5) POISINT(D3,Y)$
(C6) POISSIMP(%);
MACSYMA 6.6 Poisson Series Functions 107
2 2
(D6) 1/8 (2 A - B)
2
SIN(4 Y + 2 X) - 1/2 (2 A - B) ( - A B - 5)
2
COS(2 Y - 3 X + U) + 1/2 (2 A - B) ( - A B - 5)
COS( - 2 Y - 5 X + U)
(C7) OUTOFPOIS(SIN(X)↑5+COS(X)↑5);
(D7) 1/16 SIN(5 X) + 1/16 COS(5 X) - 5/16 SIN(3 X)
+ 5/16 COS(3 X) + 5/8 SIN(X) + 5/8 COS(X)
One or two final points: the coefficients in the arguments of the trig
functions must fit in a pre-arranged domain. Initially this is set to [-
n-1 n-1
15,16] but can be set to [-2 +1, 2 ] by the variable POISLIM[5]. This
should not be done in the middle of a calculation. Also, it is possible
to define the coefficient arithmetic to be almost anything. The user
(probably in conjunction with a LISP-MACSYMA programmer) must define the
programs needed to add, multiply, substitute, encode and decode the
coefficients. The encoding for +1 and -1 and a program to test for 0
(zero), completes each package. These packages are available for
coefficients being CRE form, polynomials with floating point coefficients,
and polynomials with rational number coefficients, in addition to the
default general MACSYMA form.
If all coefficients of trig terms are desired in CRE form, the user
should LOADFILE(POIS3,FASL,DSK,MACSYM) and
LOADFILE(RATPOI,FASL,DSK,MACSYM). Only those variables on the RATVARS
list can be used in the coefficients. In many instances this is a much
more efficient technique in terms of speed.
6.7 Taylor Series
TAYLOR(exp, var, pt, pow) expands the expression exp in a truncated Taylor
series (or Laurent series, if required) in the variable var around the
point pt. The terms through (var-pt)**pow are generated. If exp is of the
form f(var)/g(var) and g(var) has no terms up to degree pow then TAYLOR
will try to expand g(var) up to degree 2*pow. If there are still no non-
zero terms TAYLOR will keep doubling the degree of the expansion of g(var)
until reaching pow*2**n where n is the value of the variable
TAYLORDEPTH[3].
108 6.7 Taylor Series MACSYMA
If MAXTAYORDER[TRUE] is TRUE, then during algebraic manipulation of
(truncated) Taylor series, TAYLOR will try to retain as many terms as are
certain to be correct.
(C1) TAYLOR(SQRT(1+A*X+SIN(X)),X,0,3);
2 2
(A + 1) X (A + 2 A + 1) X
(D1)/R/ 1 + --------- - -----------------
2 8
3 2 3
(3 A + 9 A + 9 A - 1) X
+ -------------------------- + . . .
48
(C2) %**2;
3
X
(D2)/R/ 1 + (A + 1) X - -- + . . .
6
(C3) PRODUCT((X**I+1)**2.5,I,1,INF)/(X**2+1);
INF
/===\
! ! I 2.5
! ! (X + 1)
! !
! !
I = 1
(D3) -----------------
2
X + 1
(C4) TAYLOR(%,X,0,3),KEEPFLOAT:TRUE;
2 3
(D4)/R/ 1.0 + 2.5 X + 3.375 X + 6.5625 X + . . .
(C5) TAYLOR(1/LOG(1+X),X,0,3);
2 3
1 1 X X 19 X
(D5)/R/ - + - - -- + -- - ----- + . . .
X 2 12 24 720
Multivariate Taylor Series Expansions
MACSYMA 6.7 Taylor Series 109
For multivariate functions, there are several ways of obtaining Taylor
series expansions. If the variables are truly independent and all
singularities involve only one variable at a time then the expansion may be
done as follows:
TAYLOR(exp, var1, pt1, ord1, var2, pt2, ord2,...)
or
TAYLOR(exp,[var1,pt1,ord1],[var2,pt2,ord2],...)
Naturally the two techniques may be intermixed.
However, if the variables are interdependent or singularities involving
some of the variables together can occur then the following scheme is to be
used:
TAYLOR(exp, [var1, var2, . . .], pt, ord) where each of pt and ord may be
replaced by a list which will correspond to the list of variables. that
is, the nth items on each of the lists will be associated together.
The user should be warned that this scheme uses the RATWTLEVEL scheme
implicitly whenever the variables are expanded to different orders. In this
case the user must not be trying to use RATWTLEVEL simultaneously.
Internally this is done in the following manner; for each X substitute
i
n
i
X -----> T W .
i i
2 3
Then a term like X Y Z would become
2 n + 3 n + n
1 2 3 2 3
T W W W
1 2 3
and truncation is done on T. The W variables as well as T are not seen by the
user. The following are examples of the various modes of Taylor expansions.
(C5) TAYLOR(SIN(X+Y),X,0,3,Y,0,3);
110 6.7 Taylor Series MACSYMA
3 2 2 3 2 3
Y Y X Y X Y X X
(D5)/R/ Y - -- + X - ---- - ---- + ----- - --
6 2 2 12 6
2 3
Y X
+ ----- + . . .
12
(C6) TAYLOR(SIN(X+Y),[X,Y],0,3);
3 2 2 3
X + 3 Y X + 3 Y X + Y
(D6)/R/ Y + X - ------------------------- + . . .
6
(C7) TAYLOR(1/SIN(X+Y),X,0,3,Y,0,3);
2 3
1 Y X X X X
(D7)/R/ - + - - -- + - + -- - -- + . . .
Y 6 2 6 3 4
Y Y Y
(C8) TAYLOR(1/SIN(X+Y),[X,Y],0,3);
1 X + Y
(D8)/R/ ----- + -----
X + Y 6
3 2 2 3
7 X + 21 Y X + 21 Y X + 7 Y
+ -------------------------------- + . . .
360
If one wants to handle asymptotic expansions a facility exists to some
extent. It may be invoked as follows.
TAYLOR(exp, [x,pt,ord,ASYMP]) will give an expansion of exp in negative powers
of (x-pt). The highest order term will be
-ord
(x - pt)
The ASYMP is a syntactic device and not to be assigned to; for example, one
types TAYLOR(F(X),[X,0,4,ASYMP]).
If the user is expanding polynomials he may specify a truncation level of INF
in which case the expansion will never truncate.
MACSYMA 6.7 Taylor Series 111
DEFTAYLOR(function, exp) allows the user to define the Taylor series (about 0)
of an arbitrary function of one variable as exp which may be a polynomial
in that variable or which may be given implicitly as a power series using
the SUM function.
In order to display the information given to DEFTAYLOR one can use
POWERSERIES(F(X),X,0). (see below).
(C1) DEFTAYLOR(F(X),X**2+SUM(X**I/(2**I*I!**2),
I,4,INF));
(D1) [F]
(C2) TAYLOR(%E**SQRT(F(X)),X,0,4);
2 3 4
X 3073 X 12817 X
(D2)/R/ 1 + X + -- + ------- + -------- + . . .
2 18432 307200
TAYLORINFO(exp) returns FALSE if exp is not a Taylor series. Otherwise, a
list of lists is returned describing the particulars of the Taylor
expansion. For example,
(C3) TAYLOR((1-Y↑2)/(1-X),X,0,3,[Y,A,INF]);
2 2
(D3)/R/ 1 - A - 2 A (Y - A) - (Y - A)
2 2
+ (1 - A - 2 A (Y - A) - (Y - A) ) X
2 2 2
+ (1 - A - 2 A (Y - A) - (Y - A) ) X
2 2 3
+ (1 - A - 2 A (Y - A) - (Y - A) ) X
+ . . .
(C4) TAYLORINFO(D3);
(D4) [[Y, A, INF], [X, 0, 3]]
POWERSERIES(exp, var, pt) generates the general form of the power series
expansion for exp in the variable var about the point pt (which may be INF
for infinity). In cases in which POWERSERIES is unable to expand exp the
TAYLOR function may give the first several terms of the series.
VERBOSE[FALSE] - if TRUE will cause comments about the progress of
POWERSERIES to be printed as the execution of it proceeds.
112 6.7 Taylor Series MACSYMA
(C2) VERBOSE:TRUE$
(C3) POWERSERIES(LOG(SIN(X)/(1-X↑2)),X,0);
CAN'T EXPAND
LOG(X - 1)
SO WE WILL TRY AGAIN AFTER APPLYING THE RULE:
d
/ -- (X - 1)
[ dX
LOG(X - 1) = I -------- dX
] X - 1
/
CAN'T EXPAND
LOG(SIN(X))
SO WE WILL TRY AGAIN AFTER APPLYING THE RULE:
d
/ -- SIN(X)
[ dX
LOG(SIN(X)) = I ------- dX
] SIN(X)
/
IN FIRST SIMPLIFICATION WE HAVE RETURNED:
/ /
[ [ 1
I COT(X) dX - I ----- dX - LOG(X + 1) + LOG(- 1)
] ] X - 1
/ /
IS I2 ZERO OR NONZERO?
ZERO;
MACSYMA 6.7 Taylor Series 113
TRYING TO DO A RATIONAL FUNCTION EXPANSION OF
1
-----
X - 1
USING A SPECIAL RULE FOR EXPRESSIONS OF FORM
M - N
(A + C VAR )
HERE WE HAVE
[N = 1, A = - 1, C = 1, M = 1]
INF
==== I3 2 I3 I3 + 1
\ (- 1) 2 BERN(2 I3) LOG(X) X
(D4) ( > ------------------------------- + -------)
/ (2 I3)! I3 + 1
====
I3 = 0
INF
==== I2 I2
\ (- 1) X
+ ( > -----------) + LOG(-1)
/ I2
====
I2 = 1
6.8 Trigonometric Simplification
TRIGEXPAND(exp) expands trigonometric and hyperbolic functions of sums of
angles and of multiple angles occurring in exp. For best results, exp
should be expanded. To enhance user control of simplification, this
function expands only one level at a time, expanding sums of angles or
multiple angles. To obtain full expansion into sines and cosines
immediately, set the switch TRIGEXPAND:TRUE.
TRIGEXPAND[FALSE] if TRUE causes expansion of all expressions
containing SINs and COSs occurring subsequently.
HALFANGLES[FALSE] - if TRUE causes half-angles to be simplified away.
(C1) X+SIN(3*X)/SIN(X),TRIGEXPAND=TRUE,EXPAND;
114 6.8 Trigonometric Simplification MACSYMA
2 2
(D1) - SIN (X) + 3 COS (X) + X
(C2) TRIGEXPAND(SIN(10*X+Y));
(D2) COS(10 X) SIN(Y) + SIN(10 X) COS(Y)
TRIGREDUCE(exp, var) combines products and powers of trigonometric and
hyperbolic SINs and COSs of var into those of multiples of var. It also
tries to eliminate these functions when they occur in denominators. If
var is omitted then all variables in exp are used. Also see the POISSIMP
function (6.6).
(C4) TRIGREDUCE(D1);
(D4) 2 COS(2 X) + X + 1
The trigonometric simplification routines will use declared
information in some simple cases. Declarations about variables (see
10.8) are used as follows, e.g.
(C5) DECLARE(J, INTEGER, E, EVEN, O, ODD)$
(C6) SIN(X + (E + 1/2)*%PI)$
(D6) COS(X)
(C7) SIN(X + (O + 1/2) %PI);
(D7) - COS(X)
There are a number of ways the user may also explicitly invoke identities
2 2
such as sin (x) + cos (x) = 1. The simplest method is substitution.
(C8) D1;
2 2
(D8) - SIN (X) + 3 COS (X) + X
(C9) %,SIN(X)↑2=1-COS(X)↑2;
2
(D9) 4 COS (X) + X - 1
4
Often one wishes to recognize that sin (x) can be transformed using the
same rule. For this one needs the added power of RATSUBST.
MACSYMA 6.8 Trigonometric Simplification 115
(C10) RATSUBST(1-COS(X)↑2,SIN(X)↑2,SIN(X)↑4);
4 2
(D10) COS (X) - 2 COS (X) + 1
In general RATSUBST will perform a RATSIMP (and thus an expansion) as well
as apply the substitution. One can similarly use the LETSIMP and DEFRULE
commands together with additional declarations to define more intricate rules.
Although not as powerful as RATSUBST, the TELLSIMP command enables the
automatic application of a rule.
(C11) TELLSIMP(SIN(X)↑2,1-COS(X)↑2)$
(C12) (SIN(X)+1)↑2;
2
(D12) (SIN(X) + 1)
(C13) EXPAND(%);
2
(D13) 2 SIN(X) - COS (X) + 2
(C14) SIN(X)↑2;
2
(D14) 1 - COS (X)
6.9 Laplace Transforms
LAPLACE(exp, ovar, lvar) takes the Laplace transform of exp with respect to
the variable ovar and transform parameter lvar. Exp may only involve the
functions EXP, LOG, SIN, COS, SINH, COSH, and ERF. It may also be a
linear, constant coefficient differential equation in which case ATVALUE
(see 6.2.2) of the dependent variable will be used. These may be supplied
either before or after the transform is taken. Since the initial
conditions must be specified at zero, if one has boundary conditions
imposed elsewhere he can impose these on the general solution and
eliminate the constants by solving the general solution for them and
substituting their values back. exp may also involve convolution
integrals. Functional relationships must be explicitly represented in
order for LAPLACE to work properly. That is, if F depends on X and Y it
must be written as F(X,Y) wherever F occurs as in
LAPLACE('DIFF(F(X,Y),X),X,S).
116 6.9 Laplace Transforms MACSYMA
(C1) LAPLACE(%E**(2*T+A)*SIN(T)*T,T,S);
A
2 %E (S - 2)
(D1) ---------------
2 2
((S - 2) + 1)
(C2) LAPLACE(DELTA(T-A)*SIN(B*T),T,S);
Is A positive,negative or zero?
POS;
- A S
(D2) SIN(A B) %E
ILT(exp, lvar, ovar) takes the inverse Laplace transform of exp with respect
to lvar and parameter ovar. exp must be a ratio of polynomials whose
denominator has only linear and quadratic factors. By using the functions
LAPLACE and ILT together with the SOLVE or LINSOLVE functions the user can
solve a single differential or convolution integral equation or a set of
them.
(C1) 'INTEGRATE(SINH(A*X)*F(T-X),X,0,T)+B*F(T)=T**2;
T
/
[ 2
(D1) I (SINH(A X) F(T - X)) DX + B F(T) = T
]
/
0
(C2) LAPLACE(%,T,S);
A LAPLACE(F(T), T, S) 2
(D2) --------------------- + B LAPLACE(F(T), T, S) = --
2 2 3
S - A S
(C3) LINSOLVE([%],['LAPLACE(F(T),T,S)]);
SOLUTION
MACSYMA 6.9 Laplace Transforms 117
2 2
2 S - 2 A
(E3) LAPLACE(F(T), T, S) = --------------------
5 2 3
B S + (A - A B) S
(D3) [E3]
(C4) ILT(E3,S,T);
Is A B (A B - 1) positive, negative, or zero?
POS;
2
SQRT(A) SQRT(A B - B) T
2 COSH(------------------------)
B
(D4) F(T) = - --------------------------------
A
2
A T 2
+ ------- + ------------------
A B - 1 3 2 2
A B - 2 A B + A
Laplace Transforms of Special Functions
The following function for taking Laplace transforms of Special Functions
is available in MACSYMA. The user must type LOADFILE(SPECFN,LISP,DSK,SHARE);
to load in the special routines. Since the latter take up a great deal of
space, it is recommended that these computations be carried out in a fresh
MACSYMA.
LAPINT(exp,ovar) takes the Laplace transform of exp with respect to the
variable ovar. exp may involve
1) Special Functions of linear or quadratic argument multiplied by
a) arbitrary powers of the argument, or
b) trigonometric and exponential functions of linear argument
2) Products of two Special Functions of linear or quadratic argument taken
from only one of the following groups:
a) Any kind of Bessel, Modified Bessel, or Hankel functions,
b) Orthogonal polynomials,
c) Confluent Hypergeometric Functions
In this second category, factors of type 1a or 1b are also
permitted.
118 6.9 Laplace Transforms MACSYMA
The basic method is to rewrite the expression in terms of Generalized
Hypergeometric Functions (GHF), apply a general formula for taking the
Laplace transform of GHF's, and then, if possible, present the result in
terms of elementary functions or "common" Special Functions. For further
details, see Symbolic Laplace Transforms of Special Functions [Av].
(C1) T↑(1/2)*GAMMAINCOMPLETE(1/2,A*T)*%E↑(-P*T);
1 - P T
(D1) GAMMAINCOMPLETE(-, A T) SQRT(T) %E
2
(C2) LAPINT(%,T);
%PI 2
(D2) --------------------------- - -------------------------
3/2 A 3/2 3/2 A 3/2
2 (P + A) (1 - -----) (P + A) (1 - -----)
P + A P + A
(C3) T↑(1/2)*J[1](2*A↑(1/2)*T↑(1/2))*%E↑(-P*T);
- P T
(D3) J (2 SQRT(A) SQRT(T)) SQRT(T) %E
1
(C4) LAPINT(%,T);
- A/P
SQRT(A) %E
(D4) ---------------
2
P
(C5) T↑2*J[1](A*T)*%E↑(-P*T);
2 - P T
(D5) J (A T) T %E
1
(C6) LAPINT(%,T);
3 A
(D6) --------------
2
A 5/2 4
(-- + 1) P
2
P
(C7) T↑(3/2)*Y[1](A*T)*%E↑(-T);
3/2 - T
(D7) Y (A T) T %E
MACSYMA 6.9 Laplace Transforms 119
1
(C8) LAPINT(%,T);
%I 1 3/4
15 %I SQRT(2) P (- --) (------ - 1)
- 2, 1/2 A 2
A + 1
(D8) ---------------------------------------------
2 2 2 2 1/4
8 SQRT(%PI) (A + 1) ((A + 1) - 1)
6.10 Combinatorial Functions
MINFACTORIAL(exp) examines exp for occurrences of two factorials which differ
by an integer. It then turns one into a polynomial times the other.
(C1) N!/(N+1)!;
N!
(D1) --------
(N + 1)!
(C2) MINFACTORIAL(%);
1
(D2) -----
N + 1
FACTCOMB(exp) tries to combine the coefficients of factorials in exp with the
factorials themselves by converting, for example, (N+1)*N! into (N+1)!.
SUMSPLITFACT[TRUE] if set to FALSE will cause MINFACTORIAL to be
applied after a FACTCOMB.
(C1) (N+1)↑B*N!↑B;
B B
(D1) (N + 1) N!
(C2) FACTCOMB(%);
B
(D1) (N + 1)!
MAKEFACT(exp) transforms occurrences of binomial,gamma, and beta functions in
exp to factorials.
120 6.10 Combinatorial Functions MACSYMA
MAKEGAMMA(exp) transforms occurrences of binomial,factorial, and beta
functions in exp to gamma functions.
BERNPOLY(v, n) generates the nth Bernoulli polynomial in the variable v.
6.11 Continued Fractions
CF(exp) converts exp into a continued fraction. exp is an expression composed
of arithmetic operators and lists which represent continued fractions. A
continued fraction a+1/(b+1/(c+...)) is represented by the list
[a,b,c,...]. a,b,c,.. must be integers. Exp may also involve SQRT(n)
where n is an integer. In this case CF will give as many terms of the
continued fraction as the value of the variable CFLENGTH[1] times the
period. Thus the default is to give one period.
CFDISREP(list) converts the continued fraction represented by list into
general representation.
(C1) CF([1,2,-3]+[1,-2,1]);
(D1) [1, 1, 1, 2]
(C2) CFDISREP(%);
1
(D2) 1 + ---------
1
1 + -----
1
1 + -
2
CFEXPAND(x) gives a matrix of the numerators and denominators of the next-to-
last and last convergents of the continued fraction x.
(C1) (CFLENGTH:4, CF(SQRT(3));
(D1) [1, 1, 2, 1, 2, 1, 2, 1, 2]
(C2) CFEXPAND(%);
[265 97]
(D2) [ ]
[153 56]
(C3) D2[1,2]/D2[2,2],NUMER;
(D3) 1.73214285
MACSYMA 121
6.12 Number-Theoretic Functions
PRIME(n) gives the nth prime. MAXPRIME[489318] is the largest number accepted
as argument.
DIVSUM(n,k) adds up all the factors of n raised to the kth power. If only one
argument is given then k is assumed to be 1.
TOTIENT(n) is the number of integers less than n which are relatively prime to
n. Also n - DIVSUM(n,0) + 1.
JACOBI(p,q) is the Jacobi symbol of p and q.
QUNIT(n) gives the principal unit of the real quadratic number field SQRT(n)
where n is an integer, i.e. the element whose norm is unity. This amounts
to solving Pell's equation A**2-n*B**2=1.
(C1) QUNIT(17);
(D1) SQRT(17)+4
(C2) EXPAND(%*(SQRT(17)-4));
(D2) 1
122 MACSYMA
7 Declaring and Using Mathematical Information
The commands in this chapter deal with the communication, use, and
manipulation of mathematical information about objects and functions in
MACSYMA. Taken as a whole, this information comprises MACSYMA's relational
data base. Facts take the form of either "predicates" or "features". A
predicate is a well-formed formula consisting of a relation and its arguments,
e.g. A>B, or is a composition of predicates using the logical operators NOT,
AND, and OR. Alternatively, certain facts about mathematical objects and
functions can be expressed more naturally as "features", i.e. unary
predicates. For example, one can say that a certain constant is an INTEGER or
that a function is INCREASING. Any feature, e.g. the linearity of F, can also
be expressed as a predicate via the relation KIND, as in KIND(F,LINEAR).
7.1 Declaring and Assuming
The predicates and features communicated to MACSYMA with the ASSUME and
DECLARE commands may be tested with IS and FEATUREP and removed with FORGET
and REMOVE. The facts in the relational data base are used by the simplifier
and several commands, like SIGN, the IF statement, and INTEGRATE (certain
integrations require sign information). MACSYMA has a rudimentary inference
capability enabling limited deductions from the data base. It excels at
taxonomic deductions, e.g. KIND(N,EVEN) implies KIND(N,INTEGER), and simple
2 N
expression comparisons, e.g. X<0 and KIND(N,EVEN) imply Y +X >0. The only
sort of inequality information used by the inference mechanism at the moment
are relations between variables and other variables and numbers.
The operator "=" is a total relation that holds between two expressions if
and only if the expressions are syntactically identical. It is not a
mathematical comparison. Thus, IS((X+1)↑2=X↑2+2*X+1) would return FALSE. The
relation EQUAL, on the other hand, is a mathematical comparison of its two
arguments. A predicate involving EQUAL is true if and only if its arguments
are mathematically equivalent in light of the current data base. Thus,
IS(EQUAL((X+1)↑2,X↑2+2*X+1)) would return TRUE. The operators ">", ">=", "<",
and "<=" are also mathematical comparisons.
MACSYMA currently recognizes and uses the following features of objects:
EVEN, ODD, INTEGER, RATIONAL, IRRATIONAL, REAL, IMAGINARY, and COMPLEX. The
useful features of functions include: INCREASING, DECREASING, ODDFUN (odd
function), EVENFUN (even function), COMMUTATIVE (or SYMMETRIC), ANTISYMMETRIC,
ASSOCIATIVE.
ASSUME(pred1, pred2, ...) first checks the specified predicates for redundancy
and consistency with the current data base. If the predicates are
consistent and non-redundant, they are added to the data base; if
inconsistent or redundant, no action is taken. ASSUME returns a list
MACSYMA 7.1 Declaring and Assuming 123
whose entries are the predicates added to the data base and the atoms
REDUNDANT or INCONSISTENT where applicable. ASSUME also accepts lists of
predicates as arguments.
FORGET(pred1, pred2, ...) removes the specified predicates from the data base.
Note that it does not guarantee that equivalent facts are removed. FORGET
also accepts lists of predicates as arguments.
DECLARE(a1, f1, a2, f2, ...) declares each ai to have the corresponding
feature fi. DECLARE(F, INCREASING) is in all respects equivalent to
ASSUME(KIND(F,INCREASING)). The ai and fi may also be lists of objects or
features.
REMOVE(a1, f1, a2, f2, ...) removes each feature fi from the corresponding
object ai. The ai and fi may also be lists of objects or features.
IS(pred) attempts to determine whether the specified predicate is provable
from the facts in the current data base. IS returns TRUE if the predicate
is true for all values of its variables consistent with the data base and
returns FALSE if it is false for all such values. Otherwise, its action
depends on the setting of the switch PREDERROR[TRUE]. IS errs out if the
value of PREDERROR is TRUE and returns UNKNOWN if PREDERROR is FALSE.
FEATUREP(a,f ) attempts to determine whether the object a has the feature f on
the basis of the facts in the current data base. If so, it returns TRUE,
else FALSE.
(C1) DECLARE(J,EVEN)$
(C2) FEATUREP(J,INTEGER);
(D2) TRUE
SIGN(exp) attempts to determine the sign of its specified expression on the
basis of the facts in the current data base. It returns one of the
following answers: POS (positive), NEG (negative), ZERO, PZ (positive or
zero), NZ (negative or zero), PN (positive or negative), or PNZ (positive,
negative, or zero, i.e. nothing known).
(C3) ASSUME(A>=B,B>=C,C>=D,D>=A);
(D3) [A >= B, B >= C, C >= D, D >= A]
(C4) SIGN(B-C);
(D4) ZERO
(C5) DECLARE(K,INTEGER,L,ODD,F,INCREASING)$
124 7.1 Declaring and Assuming MACSYMA
(C6) ASSUME(X>0);
(D6) [X > 0]
(C7) F(X+3*Y↑(L+24*K+1))-F(0);
L + 24 K + 1
(D7) F(3 Y + X) - F(0)
(C8) SIGN(%);
(D8) POS
ASKSIGN(exp) first attempts to determine whether the specified expression is
positive, negative, or zero. If it cannot, it asks the user the necessary
1
questions to complete its deduction. The user's answer is recorded in the
data base for the duration of the current computation (one "C-line"). The
value of ASKSIGN is one of POS, NEG, or ZERO.
The following function, when applicable, gives the user relational
information. However, it does NOT use the data base.
ZEROEQUIV(exp,var) tests whether the expression exp in the single variable var
is equivalent to zero. It returns either TRUE, FALSE, or DONTKNOW. For
example ZEROEQUIV(SIN(2*X) - 2*SIN(X)*COS(X),X) returns TRUE and
ZEROEQUIV(%E↑X+X,X) returns FALSE. On the other hand ZEROEQUIV(LOG(A*B) -
LOG(A) - LOG(B),A) will return DONTKNOW because of the presence of an
extra variable. The restrictions are:
(1) Do not use functions that MACSYMA does not know how to differentiate
and evaluate.
(2) If the expression has poles on the real line, there may be errors in
the result (but this is unlikely to occur).
(3) If the expression contains functions which are not solutions to first
order differential equations there may be incorrect results.
(4) The algorithm uses floating-point evaluation at randomly chosen points
using a corresponding "epsilon" for carefully selected subexpressions.
This is always somewhat hazardous, although the algorithm tries to
minimize the potential for error.
7.2 Contexts
←←←←←←←←←←←←←←←
1. If the user wishes to look at the expression more closely before replying,
the variable ASKEXP is set to it. Typing control-A results in a MACSYMA break
(see 4); the user may now analyze the expression in order to give an
appropriate answer.
MACSYMA 7.2 Contexts 125
The context mechanism makes it possible for a user to bind together and
name a selected portion of his data base, called a context. Once this is
done, the user can have MACSYMA assume or forget large numbers of facts merely
by activating or deactivating their context. Any atom can be a context, and
the facts contained in that context will be retained in storage until the user
destroys them individually by using FORGET or destroys them as a whole by
using KILL to destroy the context to which they belong.
FACTS(context) returns a list of the facts in the specified context.
ACTIVATE(cont1, cont2, ...) causes the facts in the specified contexts contk
to be added to the current data base.
DEACTIVATE(cont1, cont2, ...) causes the facts in the specified contexts contk
to be removed from the current data base, unless specified by some other
active context.
CONTEXT[GLOBAL]. Whenever a user assumes a new fact, it is placed in the
context named as the current value of the variable CONTEXT. Similarly,
FORGET references the current value of CONTEXT. To add or delete a fact
from a different context, one must bind CONTEXT to the intended context
and then perform the desired additions or deletions. The context
specified by the value of CONTEXT is automatically activated. All of
MACSYMA's built-in relational knowledge is contained in the default
context GLOBAL.
CONTEXTS[[]] is a list of the currently active contexts, not including
the value of CONTEXT.
(C9) CONTEXT:CON1$
(C10) DECLARE(M,INTEGER)$
(C11) FEATUREP(M,INTEGER);
(D11) TRUE
(C12) CONTEXT:CON2$
(C13) FEATUREP(M,INTEGER);
(D13) FALSE
(C14) DECLARE(N,INTEGER);
(D14) DONE
(C15) CONTEXT:CON1$
(C16) FEATUREP(M,INTEGER);
(D16) TRUE
126 7.2 Contexts MACSYMA
(C17) FEATUREP(N,INTEGER);
(D17) FALSE
(C18) ACTIVATE(CON2)$
(C19) FEATUREP(N,INTEGER);
(D19) TRUE
(C13) CONTEXTS;
(D13) [CON2]
LOCAL(a1, a2, ...) causes the external facts about the objects a1, a2, ... to
be forgotten for the duration of the enclosing BLOCK, independent of
context. Any facts assumed about a1, a2, ... within the BLOCK containing
the LOCAL will be forgotten upon exit, again independent of context.
(C14) DECLARE(P,INTEGER)$
(C15) BLOCK(LOCAL(P),PRINT(FEATUREP(P,INTEGER)),ASSUME(P,IRRATIONAL))$
FALSE
(C16) FEATUREP(P,INTEGER);
(D16) TRUE
(C17) FEATUREP(P,IRRATIONAL);
(D17) FALSE
MACSYMA 127
8 List Handling and LISP-like functions
APPLY(function, list) gives the result of applying the function to the list of
its arguments. This is useful when it is desired to compute the arguments
to a function before applying that function. For example, if L is the
list [1, 5, -10.2, 4, 3], then APPLY(MIN,L) gives -10.2. APPLY is also
useful when calling functions which do not have their arguments evaluated
if it is desired to cause evaluation of them. For example, if FILESPEC is
a variable bound to the list [TEST, CASE] then APPLY(CLOSEFILE,FILESPEC)
is equivalent to CLOSEFILE(TEST,CASE). In general the first argument to
APPLY should be preceded by a ' to to make it evaluate to itself. Since
some atomic variables have the same name as certain functions the values
of the variable would be used rather than the function because APPLY has
its first argument evaluated as well as its second.
FUNMAKE(name,[arg1,...,argn]) returns name(arg1,...,argn) without calling the
function name.
ARRAYMAKE(name,[i1,i2,...]) returns name[i1,i2,...].
MAP(fn, exp1, exp2, ...) returns an expression whose leading operator is the
same as that of the expi but whose subparts are the results of applying fn
to the corresponding subparts of the expi. Fn is either the name of a
function of n arguments (where n is the number of expi) or is a LAMBDA
form of n arguments.
MAPERROR[TRUE] - if FALSE will cause all of the mapping functions to
(1) stop when they finish going down the shortest expi if not all of the
expi are of the same length and (2) apply fn to [exp1, exp2,...] if the
expi are not all the same type of object. If MAPERROR is TRUE then an
error message will be given in the above two instances.
One of the uses of this function is to MAP a function (e.g. PARTFRAC)
onto each term of a very large expression where it ordinarily wouldn't be
possible to use the function on the entire expression due to an exhaustion
of list storage space in the course of the computation.
(C1) MAP(F,X+A*Y+B*Z);
(D1) F(B Z) + F(A Y) + F(X)
(C2) MAP(LAMBDA([U],PARTFRAC(U,X)),X/(X↑3+4*X↑2+5*X+2));
1 X
(D2) X (----- - --------)
X + 2 2
(X + 1)
(C3) MAP(RATSIMP, X/(X**2+X)+(Y**2+Y)/Y);
128 8 List Handling and LISP-like functions MACSYMA
1
(D3) Y + ----- + 1
X + 1
(C4) MAP("=",[A,B],[-.5, 3,2.5]);
MAP IS TRUNCATING
(D4) [A = -.5, B = 3]
MAPATOM(expr) is TRUE if and only if expr is treated by the MAPping routines a
as an "atom", a unit. "Mapatoms" are atoms, numbers (including rational
numbers), and subscripted variables.
MAPLIST(fn, exp1, exp2, ...) yields a list of the applications of fn to the
parts of the expi. This differs from MAP(fn,exp1,exp2,...) which returns
an expression with the same main operator as expi has (except for
simplifications and the case where MAP does an APPLY). Fn is of the same
form as in MAP.
FULLMAP(fn, exp1, ...) is similar to MAP but it will keep mapping down all
subexpressions until the main operators are no longer the same.
The user should be aware that FULLMAP is used by the MACSYMA simplifier
for certain matrix manipulations; thus, the user might see an error
message concerning FULLMAP even though FULLMAP was not explicitly called
by the user.
(C1) A+B*C$
(C2) FULLMAP(G,%);
(D2) G(B) G(C) + G(A)
(C3) MAP(G,D1);
(D3) G(B C) + G(A)
FULLMAPL(fn, list1, ...) is similar to FULLMAP but it only maps onto lists and
matrices
(C1) FULLMAPL("+",[3,[4,5]],[[A,1],[0,-1.5]]);
(D1) [[A + 3, 4], [4, 3.5]]
MACSYMA 8 List Handling and LISP-like functions 129
SCANMAP(function,exp) recursively applies function to exp. This is most
useful when "complete" factorization is desired, for example:
(C1) EXP:(A↑2+2*A+1)*Y + X↑2$
(C2) SCANMAP(FACTOR,EXP);
2 2
(D2) (A + 1) Y + X
Note the way in which SCANMAP applies the given function FACTOR to the
constituent subexpressions of exp; if another form of exp is presented to
SCANMAP then the result may be different. Thus, D2 is not recovered when
SCANMAP is applied to the expanded form of exp:
(C3) SCANMAP(FACTOR,EXPAND(EXP));
2 2
(D3) A Y + 2 A Y + Y + X
Here is another example of the way in which SCANMAP recursively
applies a given function to all subexpressions, including exponents:
(C4) EXPR : U*V↑(A*X+B) + C$
(C5) SCANMAP('F, EXPR);
F(F(F(A) F(X)) + F(B))
(D5) F(F(F(U) F(F(V) )) + F(C))
APPEND(list1, list2, ...) returns a single list of the elements of list1
followed by the elements of list2,...
(C1) APPEND([Y+X,0,-3.2],[2.5E20,X]);
(D1) [Y+X, 0, -3.2, 2.5E20, X]
CONS(exp, list) returns a new list constructed of the element exp as its first
element, followed by the elements of list.
ENDCONS(exp, list) returns a new list consisting of the elements of list
followed by exp.
130 8 List Handling and LISP-like functions MACSYMA
MEMBER(exp, list) returns TRUE if exp occurs as a member of list (not within a
member). Otherwise FALSE is returned.
REVERSE(list) reverses the order of the members of list (not the members
themselves).
The functions FIRST, REST, LAST, DELETE, LENGTH (6.2.3) also work on
lists.
Examples
(C1) UNION(X,Y):=IF X = [] THEN Y ELSE
IF MEMBER(T:FIRST(X),Y) THEN UNION(REST(X),Y)
ELSE CONS(T,UNION(REST(X),Y)$
(C2) UNION([A,B,1,1/2,X**2],[-X**2,A,Y,1/2]);
2 2 1
(D2) [X , 1, B, - X , A, Y, -]
2
In this example T is assigned the value of FIRST(X) in the call to
MEMBER and is referenced later in CONS(T,UNION(...)).
(C3) BERNPOLY(X,5);
4 3
5 5 X 5 X X
(D3) X - ---- + ---- - -
2 3 6
(C4) MAPLIST(NUMFACTOR,%);
5 5 1
(D4) [1, - -, -, - -]
2 3 6
(C5) APPLY(MIN,%);
5
(D5) - -
2
[8.1] Property Specification Functions
The functions in this section are used to specify properties for atomic
variables. A property is a piece of information which may be utilized during
the user's session with MACSYMA. MODEDECLARE (sec. 10.8) is one example
MACSYMA 8.1 Property Specification Functions 131
of a property specification function. Also, the subsequent section deals with
functions for pattern matching and includes the function MATCHDECLARE. As
these are somewhat complicated they are described in other sections. However,
along with DECLARE (see below) and other functions, they all perform the task
of setting up information which is used later.
For most types of properties there exists a name which is an indicator of
that property. For example the command GRADEF(F(X),SIN(X**2)); causes F to
receive a "gradef" property of LAMBDA([X],SIN(X**2)). (The indicator is
GRADEF and the property is the lambda expression).
The presence of some properties is signified merely by the presence of the
indicator and requires no additional information. These indicators are known
as flags. For example %I has associated with it the flag CONSTANT. We can
also speak of "constant" as being a property of %I.
There are three principal operations which are needed by the user in
dealing with properties. He must be able to set up the property, to display
it, and to remove it. Also he may sometimes want to know what properties he
set up in his MACSYMA. There are several lists (known as information lists)
which contain all of the atoms that possess a particular property. Moreover,
the value of the variable INFOLISTS is a list of the names of all of the
information lists in MACSYMA. These are:
LABELS - all bound C,D, and E labels.
VALUES - all bound atoms (set up by : , :: , or functional binding).
FUNCTIONS - all user defined functions (set up by f(x):=...).
ARRAYS - declared and undeclared arrays (set up by : , :: , or a[x]:=...)
MYOPTIONS - all options ever reset by the user (whether or not they get
reset to their default value).
RULES - user defined pattern matching and simplification rules (set up by
TELLSIMP, TELLSIMPAFTER, DEFMATCH, or, DEFRULE.)
ALIASES - atoms which have a user defined alias (set up by the ALIAS,
ORDERGREAT, ORDERLESS functions or by DECLAREing the atom a NOUN).
DEPENDENCIES - atoms which have functional dependencies (set up by the
DEPENDS or GRADEF functions).
GRADEFS - functions which have user defined derivatives (set up by the
GRADEF function).
PROPS - atoms which have any property other than those mentioned above,
such as atvalues, matchdeclares, etc. as well as properties specified in the
DECLARE function.
132 8.1 Property Specification Functions MACSYMA
In addition to these information lists similar lists may be generated by
the function PROPVARS(prop) which yields a list of those atoms on the PROPS
list which have the property indicated by prop. Thus PROPVARS(ATVALUE) will
yield a list of atoms which have atvalues.
8.1.1 Functions for Handling MACSYMA Properties
DECLARE(a1, f1, a2, f2, ...) gives the atom ai the flag fi. The ai's and fi's
may also be lists of atoms and flags respectively in which case each of
the atoms gets all of the properties. The possible flags and their
meanings are:
CONSTANT - makes ai a constant as is %PI.
NONSCALAR - makes ai behave as does a list or matrix with respect to
the dot operator. (see 6.4.1)
NOUN - makes the function ai a noun so that it won't be evaluated
automatically. (see 3.2)
ALPHABETIC - adds ai to MACSYMA's alphabet (initially A-Z,%). Thus,
DECLARE("←",ALPHABETIC) enables NEW←VALUE to be used as a name.
EVFUN - makes ai known to the EV function so that it will get applied
if its name is mentioned. Initial evfuns are FACTOR, TRIGEXPAND,
TRIGREDUCE, BFLOAT, RATSIMP, RATEXPAND, RADCAN, and LOGCONTRACT.
EVFLAG - makes ai known to the EV function so that it will be bound to
TRUE during the execution of EV if it is mentioned. Initial evflags are
FLOAT, PRED, SIMP, NUMER, DETOUT, EXPONENTIALIZE, DEMOIVRE, KEEPFLOAT,
LISTARITH, TRIGEXPAND, SIMPSUM, ALGEBRAIC, RATALGDENOM, FACTORFLAG, and
LOGEXPAND.
BINDTEST - causes ai to signal an error if it ever is used in a
computation unbound (see Chapter 19).
ARRAYINFO(a) returns a list of information about the array a. For hashed
arrays it returns a list of "HASHED", the number of subscripts, and the
subscripts of every element which has a value. For declared arrays it
returns a list of "DECLARED", the number of subscripts, and the bounds
that were given the the ARRAY function when it was called on a.
(C1) B[1,X]:1$
(C2) ARRAY(F,2,3)$
(C3) ARRAYINFO(B);
(D3) [HASHED, 2, [1, X]]
(C4) ARRAYINFO(F);
MACSYMA 8.1.1 Functions for Handling MACSYMA Properties 133
(D4) [DECLARED, 2, [2, 3]]
PROPERTIES(a) will yield a list showing the names of all the properties
associated with the atom a.
PRINTPROPS(a, i) will display the property with the indicator i associated
with the atom a. a may also be a list of atoms or the atom ALL in which
case all of the atoms with the given property will be used. For example,
PRINTPROPS([F,G],ATVALUE). PRINTPROPS is for properties that cannot
otherwise be displayed, i.e. for ATVALUE,ATOMGRAD,GRADEF, and
MATCHDECLARE.
REMOVE(a1, p1, a2, p2, ...) removes the property pi from the atom ai. Ai and
pi may also be lists as with DECLARE. Pi may be any property e.g.
FUNCTION, MODEDECLARE, etc. It may also be TRANSFUN implying that the
translated LISP version of the function is to be removed. This is useful
if one wishes to have the MACSYMA version of the function executed rather
than the translated version. Pi may also be OP or OPERATOR to remove a
syntax extension given to ai (see Appendix II). If ai is "ALL" then the
property indicated by pi is removed from all atoms which have it. Unlike
the more specific remove functions (REMVALUE, REMARRAY, REMFUNCTION, and
REMRULE) REMOVE does not indicate when a given property is non-existent;
it always returns "DONE".
8.1.2 Functions for Handling Users' Properties
PUT(a, p, i) associates with the atom a the property p with the indicator i.
This enables the user to give an atom any arbitrary property.
QPUT(a, p, i) is similar to PUT but it doesn't have its arguments evaluated.
GET(a, i) retrieves the user property indicated by i associated with atom a or
returns FALSE if a doesn't have property i.
(C1) PUT(%E,TRANSCENDENTAL,TYPE);
(D1) TRANSCENDENTAL
(C2) PUT(%PI,TRANSCENDENTAL,TYPE)$
(C3) PUT(%I,ALGEBRAIC,TYPE)$
(C4) TYPEOF(X) := BLOCK([Q], IF NUMBERP(X) THEN RETURN(ALGEBRAIC),
IF NOT ATOM(X) THEN RETURN(MAPLIST(TYPEOF, X)),
Q : GET(X, TYPE),
IF Q=FALSE THEN ERROR("NOT NUMERIC") ELSE Q)$
134 8.1.2 Functions for Handling Users' Properties MACSYMA
(C5) TYPEOF(2*%E+X*%PI);
NOT NUMERIC
QUIT
(C6) TYPEOF(2*%E+%PI);
(D6) [[ALGEBRAIC, TRANSCENDENTAL], TRANSCENDENTAL]
REM(a, i) removes the property indicated by i from the atom a.
NUMERVAL(var1, exp1, var2, exp2, ...) declares vari to have a numerical value
of exp1 which is evaluated and substituted for the variable in any
expressions in which the variable occurs if the NUMER flag is TRUE. (see
the EV function, sec. 6.1.1).
MACSYMA 135
9 Pattern Matching and Related Functions
[9.1] Type Testing Functions
ATOM(exp) is TRUE if exp is atomic (i.e. a number or name) else FALSE. Thus
ATOM(5) is TRUE while ATOM(A[1]) and ATOM(SIN(X)) are FALSE. (Assuming
A[1] and X are unbound.)
SUBVARP(exp) is TRUE if exp is a subscripted variable, for example A[I].
CONSTANTP(exp) is TRUE if exp is a constant (i.e. composed of numbers and %PI,
%E, %I or any variables bound to a constant or DECLAREd constant (see
8.1.1) else FALSE. Any function whose arguments are constant is also
considered to be a constant.
NONSCALARP(exp) is TRUE if exp is a non-scalar, i.e. it contains atoms
declared as non-scalars (see 8.1), lists, or matrices.
INTEGERP(exp) is TRUE if exp is an integer else FALSE.
EVENP(exp) is TRUE if exp is an even integer. FALSE is returned in all other
cases.
ODDP(exp) is TRUE if exp is an odd integer. FALSE is returned in all other
cases.
FLOATNUMP(exp) is TRUE if exp is a floating point number else FALSE.
BFLOATP(exp) is TRUE is exp is a bigfloat number else FALSE.
NUMBERP(exp) is TRUE if exp is an integer, a rational number, a floating point
number or a bigfloat else FALSE.
RATNUMP(exp) is TRUE if exp is a rational number (includes integers) else
FALSE.
136 9.1 Type Testing Functions MACSYMA
LISTP(exp) is TRUE if exp is a list else FALSE.
MATRIXP(exp) is TRUE if exp is a matrix else FALSE.
RATP(exp) is TRUE if exp is in CRE or extended CRE form else FALSE.
FREEOF(x1, x2, ..., exp) yields TRUE if the xi do not occur in exp and FALSE
otherwise. The xi are atoms or they may be subscripted names, functions
(e.g. SIN(X) ), or operators enclosed in "s. They may also be lists of
these objects.
(C1) FREEOF(Y,SIN(X+2*Y));
(D1) FALSE
(C2) FREEOF(COS(Y),"*",SIN(Y)+COS(X));
(D2) TRUE
[9.2] General Pattern Matching Functions
The pattern matching functions permit the user to test expressions for
combinations of syntactic and semantic patterns and to automatically have
variables set to parts of expressions which fit the patterns. This enables
one to transform an expression as well as to see if it fits a pattern.
It is also possible to add simplification rules which apply to user or
system defined functions or operators. In order to choose the particular rule
which applies, a pattern match must usually be performed on the operands of
the expression which is to be simplified.
MATCHDECLARE(patternvar, predicate, ...) associates a predicate with a pattern
variable so that the variable will only match expressions for which the
predicate is not FALSE. (The matching is accomplished by one of the
functions described below). For example after MATCHDECLARE(Q,FREEOF(X,%E))
is executed, Q will match any expression not containing X or %E. If the
match succeeds then the variable is set to the matched expression. The
predicate (in this case FREEOF) is written without the last argument which
should be the one against which the pattern variable is to be tested.
Note that the patternvar and the arguments to the predicate are evaluated
at the time the match is performed.
The odd numbered argument may also be a list of pattern variables all
of which are to have the associated predicate. Any even number of
arguments may be given.
MACSYMA 9.2 General Pattern Matching Functions 137
For pattern matching, predicates refer to functions which are either
FALSE or not FALSE (any non FALSE value acts like TRUE).
MATCHDECLARE(var,TRUE) will permit var to match any expression.
PRINTPROPS([v1,v2,...],MATCHDECLARE) (see 8.1.1) will display the
matchdeclare properties of the variables v1,v2,...
TELLSIMPAFTER(pattern, replacement) defines a replacement for pattern which
the MACSYMA simplifier uses after it applies the built-in simplification
rules. The pattern may be anything but a single variable or a number.
TELLSIMP(pattern, replacement) is similar to TELLSIMPAFTER but places new
information before old so that it is applied before the built-in
simplification rules. The pattern may not be a sum, product, single
variable, or number.
RULES is a list of names having simplification rules added to them by
DEFRULE, DEFMATCH, TELLSIMP, or TELLSIMPAFTER.
(C1) MATCHDECLARE([XX,A,B],TRUE);
(D1) DONE
(C2)
TELLSIMP(D[XX](A,B),B(XX)*DIFF(A(XX),X)-A(XX)*DIFF(B(XX),XX));
RULE PLACED ON SUBVAR
(D2) [SUBVARRULE1, FALSE]
SUBVARRULE1 is the name assigned to the TELLSIMP rule from (C34).
(C3) D[Z](X,Y);
(D3) Y(Z) X(Z) - X(Z) Y(Z)
X Z
Another example of the use of TELLSIMP is shown in the following:
(C4) 0↑0;
0↑0 HAS BEEN GENERATED
To override such default simplification, the user can use the following
paradigm:
(C5) BLOCK([SIMP],SIMP:FALSE,TELLSIMP(0↑0,1));
RULE PLACED ON **
(D5) [**RULE1, SIMPEXPT]
(C6) 0↑0;
(D6) 1
138 9.2 General Pattern Matching Functions MACSYMA
(C7) REMRULE("**","**RULE1");
(D7) [ SIMPEXPT ]
DEFMATCH(progname, pattern, parm1, ..., parmn) creates a function of n+1
arguments with the name progname which tests an expression to see if it
can match a particular pattern. The pattern is some expression containing
pattern variables and parameters. The parms are given explicitly as
arguments to DEFMATCH while the pattern variables (if supplied) were given
implicitly in a previous MATCHDECLARE function. The first argument to the
created function progname, is an expression to be matched against the
"pattern" and the other n arguments are the actual variables occurring in
the expression which are to take the place of dummy variables occurring in
the "pattern". Thus the parms in the DEFMATCH are like the dummy
arguments to the SUBROUTINE statement in FORTRAN. When the function is
"called" the actual arguments are substituted. For example:
(C1) NONZEROANDFREEOF(X,E):= IF E#0 AND FREEOF(X,E)
THEN TRUE ELSE FALSE$
(IS(E#0 AND FREEOF(X,E)) is an equivalent function definition - see sec.
8.1.1.
(C2) MATCHDECLARE(A,NONZEROANDFREEOF(X),B,FREEOF(X))$
(C3) DEFMATCH(LINEAR,A*X+B,X)$
This has caused the function LINEAR(exp,var1) to be defined. It tests
exp to see if it is of the form A*var1+B where A and B do not contain var1
and A is not zero. DEFMATCHed functions return (if the match is
successful) a list of equations whose left sides are the pattern variables
and parms and whose right sides are the expressions which the pattern
variables and parameters matched. The pattern variables, but not the
parameters, are set to the matched expressions. If the match fails, the
function returns FALSE. Thus LINEAR(3*Z+(Y+1)*Z+Y**2,Z) would return
[B=Y**2, A=Y+4, X=Z]. Any variables not declared as pattern variables in
MATCHDECLARE or as parameters in DEFMATCH which occur in pattern will
match only themselves so that if the third argument to the DEFMATCH in
(C4) had been omitted, then LINEAR would only match expressions linear in
X, not in any other variable.
A pattern which contains no parameters or pattern variables returns
TRUE if the match succeeds.
(C1) MATCHDECLARE([A,F],TRUE)$
(C2) CONSTINTERVAL(L,H):=CONSTANTP(H-L)$
(C3) MATCHDECLARE(B,CONSTINTERVAL(A))$
(C4) MATCHDECLARE(X,ATOM)$
MACSYMA 9.2 General Pattern Matching Functions 139
(C5) BLOCK(REMOVE(INTEGRATE,LINEAR),
DEFMATCH(CHECKLIMITS,'INTEGRATE(F,X,A,B),
DECLARE(INTEGRATE,LINEAR))$
(C6) 'INTEGRATE(SIN(T),T,X+%PI,X+2*%PI)$
(C7) CHECKLIMITS(%);
(D7) [B = X + 2 %PI, A = X + %PI, X = T,
F = SIN(T)]
(C8) 'INTEGRATE(SIN(T),T,0,X)$
(C9) CHECKLIMITS(%);
(D9) FALSE
DEFRULE(rulename, pattern, replacement) defines and names a replacement rule
for the given pattern. If the rule named rulename is applied to an
expression (by one of the APPLY functions below), every subexpression
matching the pattern will be replaced by the replacement. All variables
in the replacement which have been assigned values by the pattern match
are assigned those values in the replacement which is then simplified. The
rules themselves can be treated as functions which will transform an
expression by one operation of the pattern match and replacement. If the
pattern fails, the original expression is returned.
APPLY1(exp, rule1, ..., rulen) repeatedly applies the first rule to exp until
it fails, then repeatedly applies the same rule to all subexpressions of
exp, left-to-right, until the first rule has failed on all subexpressions.
Call the result of transforming exp in this manner exp'. Then the second
rule is applied in the same fashion starting at the top of exp'. When the
final rule fails on the final subexpression, the application is finished.
APPLY2(exp, rule1, ..., rulen) differs from APPLY1 in that if the first rule
fails on a given subexpression, then the second rule is repeatedly
applied, etc. Only if they all fail on a given subexpression is the whole
set of rules repeatedly applied to the next subexpression. If one of the
rules succeeds, then the same subexpression is reprocessed, starting with
the first rule.
MAXAPPLYDEPTH[10000] is the maximum depth to which APPLY1 and APPLY2 will
delve.
140 9.2 General Pattern Matching Functions MACSYMA
APPLYB1(exp, rule1, ..., rulen) is similar to APPLY1 but works from the
"bottom up" instead of from the "top down". That is, it processes the
smallest subexpression of exp, then the next smallest, etc.
MAXAPPLYHEIGHT[10000] - is the maximum height to which APPLYB1 will reach
before giving up.
[9.3] Pattern Matching for Rational Expressions
LETSIMP(exp) will continually apply the substitution rules previously defined
by the function LET (see below) until no further change is made to exp.
LET(prod, repl, predname, arg1, arg2, ..., argn) defines a substitution rule
for LETSIMP such that prod gets replaced by repl. prod is a product of
positive or negative powers of the following types of terms:
(1) Atoms which LETSIMP will search for literally unless previous to
calling LETSIMP the MATCHDECLARE function is used to associate a predicate
with the atom. In this case LETSIMP will match the atom to any term of a
product satisfying the predicate.
(2) Kernels such as SIN(X), N!, F(X,Y), etc. As with atoms above
LETSIMP will look for a literal match unless MATCHDECLARE is used to
associate a predicate with the argument of the kernel.
A term to a positive power will only match a term having at least that
power in the expression being LETSIMPed. A term to a negative power on
the other hand will only match a term with a power at least as negative.
In the case of negative powers in "product" the switch LETRAT must be set
to TRUE (see below).
If a predicate is included in the LET function followed by a list of
arguments, a tentative match (i.e. one that would be accepted if the
predicate were omitted) will be accepted only if predname(arg1',...,argn')
evaluates to TRUE where argi' is the value matched to argi. The argi may
be the name of any atom or the argument of any kernel appearing in prod.
repl may be any rational expression. If any of the atoms or arguments
from prod appear in repl the appropriate substitutions will be made.
LETRAT[FALSE] when FALSE, LETSIMP will simplify the numerator and
denominator of expr independently and return the result. Substitutions
such as N!/N goes to (N-1)! will fail. To handle such situations LETRAT
should be set to TRUE, then the numerator, denominator, and their quotient
will be simplified in that order.
These substitution functions allow you to work with several rule
packages at once. Each rule package can contain any number of LETed rules
and is refered to by a user supplied name. To insert a rule into the rule
MACSYMA 9.3 Pattern Matching for Rational Expressions 141
package name, do LET([prod,repl,pred,arg1,...],name). To apply the rules
in rule package name, do LETSIMP(expr, name). The function
LETSIMP(expr,name1,name2,...) is equivalent to doing LETSIMP(expr,name1)
followed by LETSIMP(%,name2) etc.
There is a default rule package name which is assumed when no other
name is supplied to any of the functions. Whenever a LET includes a rule
package name the default rule package is made to look like that rule
package.
REMLET(prod, name) deletes the substitution rule, prod --> repl, most recently
defined by the LET function. If name is supplied the rule is deleted from
the rule package name. REMLET() and REMLET(ALL,name) delete all
substitution rules from the default rule package. If name is supplied the
rule package, name, is also deleted.
If a substitution is to be changed using the same product, REMLET need not
be called, just redefine the substitution using the same product
(literally) with the LET function and the new replacement and/or predicate
name. Should REMLET(product) now be called the original substitution rule
will be revived.
LETRULES(name) and LETRULES() display the rules in the default rule package
and the rule package, name, respectively. Note that the function
LETRULES(name) will set the default rule package to the rule package,
name.
(C1) MATCHDECLARE([A1,A2],TRUE)$
(C2) ONELESS(X,Y):=IS(EQUAL(X,Y-1))$
(C3) LET(A1*A2!,A1!,ONELESS,A2,A1);
(D3) A1 A2! --> A1! WHERE ONELESS(A2, A1)
(C4) LETRAT:TRUE$
(C5) LET(A1!/A1,(A1-1)!);
A1!
(D5) --- --> (A1 - 1)!
A1
(C6) LETSIMP(N*M!*(N-1)!/M);
(D6) (M - 1)! N!
The user should be aware that simplification rules for differential
operators can be specified using MACSYMA's pattern-matching commands.
142 9.3 Pattern Matching for Rational Expressions MACSYMA
Consider a function F(X). To inform MACSYMA that F depends on X, the
user must type DEPENDS(F,X); (otherwise, DIFF(F,X) will return 0). We
will assume that this has been done and that DERIVABBREV has been set to
TRUE in the following example.
Now suppose that the function F(X) satisfies some constraint, say that
the d'Alembertian of f(x) is zero:
2
1 f(x) = 0.
In a curved space, this may take the form:
(C4) -2*(DIFF(F,X)*X +2)
*(L*(DIFF(F,X)↑3)*X↑2
+(4*DIFF(F,X,2)+4*(DIFF(F,X)↑2))*X+8*DIFF(F,X));
3 2 2
(D4) - 2 (F X + 2) ((F ) L X + (4 F + 4 (F ) ) X + 8 F )
X X X X X X
One can solve for the second-order term:
(C5) SOLVE(%,DIFF(F,X,2));
Solution
3 2 2
(F ) L X + 4 (F ) X + 8 F
X X X
(E5) F = - -----------------------------
X X 4 X
(D5) [E5]
which can be restated as a simplification rule:
(C6) LET(DIFF(F,X,2),RHS(E5));
(D6) (F --> RHS(E5))
X X
Then a relatively complicated expression such as
MACSYMA 9.3 Pattern Matching for Rational Expressions 143
4 2 3 3 2
(D10) ((F ) L X + (8 F F L + 8 (F ) L) X
X X X X X
2
+ ((F ) (12 L + 16) + 16 F ) X + 32 F )
X X X X
2 2
/(X ((F ) L X + 4 F X + 4))
X X
can be simplified using the LETSIMP command:
(C11) FACTOR(LETSIMP(%));
2
(D11) - (F ) L
X
144 MACSYMA
10 Utility, Input-Output, and Display Functions
[10.1] Debugging Functions
The functions in this section permit the user to examine his MACSYMA
environment and to obtain debugging information. Further detail is given in
section 12.0.
TRACE(name1, name2, ...) gives a trace printout whenever the functions
mentioned are called. TRACE() prints a list of the functions currently
under TRACE.
UNTRACE(name1, ...) removes tracing incurred by the TRACE function. UNTRACE()
removes tracing from all functions.
REMTRACE() removes the tracing facilities from MACSYMA thus freeing up some
storage. They will be reloaded when TRACE is used again.
DECLARE([var1, var2, ...], BINDTEST) causes MACSYMA to give an error message
whenever any of the vari occur unbound in a computation.
BREAK(arg1, ...) evaluates and prints its arguments then enters a MACSYMA
break loop.
Options and Variables
%% is the value of the last computation performed while in a (MACSYMA-
BREAK).
DEBUGMODE[FALSE] if TRUE causes MACSYMA to enter a MACSYMA break loop
whenever a MACSYMA error occurs. If DEBUGMODE:ALL then the user may examine
BACKTRACE for the list of functions currently entered.
REFCHECK[FALSE] if TRUE causes a message to be printed each time a bound
variable is used for the first time in a computation.
PREDERROR[TRUE] - if TRUE causes a message to be printed whenever the
predicate of an IF statement or an IS function fails to evaluate to either
TRUE or FALSE.
SETCHECK[FALSE] - if set to a list of variables (which can be subscripted)
MACSYMA 10.1 Debugging Functions 145
will cause a printout whenever the variables, or subscripted occurrences of
them, are bound (with : or :: or function argument binding). The printout
consists of the variable and the value it is bound to. SETCHECK may be set to
ALL or TRUE thereby including all variables.
SETCHECKBREAK[FALSE] - if set to TRUE will cause a (MACSYMA-BREAK) to
occur whenever the variables on the SETCHECK list are bound.
BACKTRACE (when DEBUGMODE:ALL has been done) has as value a list of all
functions currently entered. (see Chapter 19).
10.2 Functions for Displaying
DISPFUN(f1, f2, ...) displays the definition of the user defined functions f1,
f2, ... which may also be the names of array associated functions,
subscripted functions, or functions with constant subscripts which are the
same as those used when the functions were defined. DISPFUN(ALL) will
display all user defined functions as given on the FUNCTIONS and ARRAYS
lists except subscripted functions with constant subscripts.
DISPRULE(rulename) will display a rule with the name rulename as was given by
DEFRULE, TELLSIMP, or TELLSIMPAFTER or a pattern defined by DEFMATCH. For
example, the first rule modifying SIN will be called SINRULE1. (see 9.2)
DISPLAY(exp1, exp2, ...) displays equations whose left side is expi
unevaluated, and whose right side is the value of the expression centered
on the line. This function is useful in blocks and FOR statements in
order to have intermediate results displayed. The arguments to DISPLAY
are usually atoms, subscripted variables, or function calls. (see the
DISP function below.)
(C1) DISPLAY(B[1,2]);
2
B = X - X
1, 2
(D1) DONE
LDISPLAY(exp1,exp2,...) is like DISPLAY but also generates intermediate
labels.
146 10.2 Functions for Displaying MACSYMA
DISP(exp1,exp2, ...) is like DISPLAY but only the value of the arguments are
displayed rather than equations. This is useful for complicated arguments
which don't have names or where only the value of the argument is of
interest and not the name.
LDISP(exp1,exp2,...) is like DISP but also generates intermediate labels.
PRINT(exp1, exp2, ...) evaluates and displays its arguments one after the
other "on a line" starting at the leftmost position. If expi is unbound
or is preceded by a single quote or is enclosed in "s then it is printed
literally. For example, PRINT("THE VALUE OF X IS ",X). The value
returned by PRINT is the value of its last argument. No intermediate
lines are generated.
DISPTERMS(exp) displays its argument in parts one below the other. That is,
each term in a sum or factor in a product is displayed separately. This
is useful if exp is too large to be otherwise displayed. For example if
P1, P2, ... are very large expressions then the display program may run
out of storage space in trying to display P1+P2+... all at once. However,
DISPTERMS(P1+P2+...) will display P1, then below it P2, etc. When not
using DISPTERMS, if an exponential expression is too wide to be displayed
as A**B it will appear as EXPT(A,B) (or as NCEXPT(A,B) in the case of
A↑↑B).
REVEAL(exp,depth) will display exp to the specified integer depth with the
length of each part indicated. Sums will be displayed as SUM(n) and
products as PRODUCT(n) where n is the number of subparts of the sum or
product. Exponentials will be displayed as EXP.
(C1) INTEGRATE(1/(X↑3+2),X)$
(C2) REVEAL(%,2);
(D2) PRODUCT(3) + PRODUCT(3) + PRODUCT(3)
(C3) REVEAL(D1,3);
EXPT LOG EXPT LOG
(D3) - -------- + EXPT EXPT ATAN + --------
6 3
PLAYBACK(arg) "plays back" input and output lines. If arg=n (a number) the
last n expressions (Ci, Di, and Ei count as 1 each) are "played-back",
while if arg is omitted, all lines are. If arg=INPUT then only input
lines are played back. If arg=[m,n] then all lines with numbers from m to
n inclusive are played-back. If m=n then [m] is sufficient for arg.
MACSYMA 10.2 Functions for Displaying 147
Arg=SLOW places PLAYBACK in a slow-mode similar to DEMO's (as opposed to
the "fast" BATCH). This is useful in conjunction with SAVE or STRINGOUT
(see below) when creating a secondary-storage file in order to pick out
useful expressions. If arg=TIME then the computation times are displayed
as well as the expressions. arg=NOSTRING displays all input lines when
playing back rather than STRINGing them. If arg=GRIND then the display
will be in a more readable format. One may include any number of options
as in PLAYBACK([5,10],20,TIME,SLOW).
STRING(exp) converts exp to MACSYMA's linear notation (similar to FORTRAN's)
just as if it had been typed in and puts exp into the buffer for possible
editing (in which case exp is usually Ci) (see sec. 14.2). The
STRING'ed expression should not be used in a computation.
STRINGOUT(args) will output an expression to a file in a linear format.
STRINGOUT([filespec],...,FUNCTIONS,...) puts all the user's function
definitions in the specified file. (see 10.4)
GRIND[FALSE] if TRUE will cause the STRING, STRINGOUT, and PLAYBACK
commands to use "grind" mode instead of "string" mode. For PLAYBACK,
"grind" mode can also be turned on (for processing input lines) by
specifying GRIND as an option.
GRIND(arg) prints out arg in a more readable format than the STRING command.
It returns a D-line as value.
FORTRAN(exp) converts exp into a FORTRAN linear expression in legal FORTRAN
with 6 spaces inserted at the beginning of each line,continuation
lines,and ** rather than ↑ for exponentiation. When the option
FORTSPACES[FALSE] is TRUE, the FORTRAN command fills out to 80 columns
using spaces.
FORTMX(name,matrix) converts a MACSYMA matrix into a sequence of FORTRAN
assignment statements of the form
name(i,j)= <the ij matrix element>
DESCRIBE(function) prints out the portion of the MACSYMA manual describing the
function.
EXAMPLE(function) does a DEMO of relevant examples involving function.
10.3 Functions for Freeing Storage
148 10.3 Functions for Freeing Storage MACSYMA
REMOVE(args) will remove some or all of the properties associated with
variables or functions. (see 8.1)
REMFUNCTION(f1, f2, ...) removes the user defined functions f1,f2,... from
MACSYMA. If there is only one argument of ALL then all functions are
removed.
REMVALUE(name1, name2, ...) removes the values of user variables (which can be
subscripted) from the system. If name is ALL then the values of all user
variables are removed. Values are those items given names by the user as
opposed to those which are automatically labeled by MACSYMA as Ci, Di, or
Ei.
REMARRAY(name1, name2, ...) removes arrays and array associated functions and
frees the storage occupied. If name is ALL then all arrays are removed.
It may be necessary to use this function if it is desired to redefine the
values in a hashed array.
REMRULE(function, rulename) will remove a rule with the name rulename from the
function which was placed there by DEFRULE, DEFMATCH, TELLSIMP, or
TELLSIMPAFTER. If rule-name is ALL, then all rules will be removed. (see
example in 9.2)
KILL(arg1, arg2, ...) eliminates its arguments from the MACSYMA system. If
argi is a variable (including a single array element), function, or array,
the designated item with all of its properties is removed from core. If
argi=LABELS then all input, intermediate, and output lines to date (but
not other named items) are eliminated. If argi=CLABELS then only input
lines will be eliminated; if argi=ELABELS then only intermediate E-lines
will be eliminated; if argi=DLABELS only the output lines will be
eliminated. If argi is the name of any of the other information lists
(the elements of the MACSYMA variable INFOLISTS), then every item in that
class (and its properties) is KILLed and if argi=ALL then every item on
every information list previously defined as well as LABELS is KILLed. If
argi=a number (say n), then the last n lines (i.e. the lines with the last
n line numbers) are deleted. If argi is of the form [m,n] then all lines
with numbers between m and n inclusive are killed. Note that KILL(VALUES)
or KILL(variable) will not freeup the storage occupied unless the labels
which are pointing to the same expressions are also KILLed. Thus if a
large expression was assigned to X on line C7 one should do KILL(D7) as
well as KILL(X) to release the storage occupied.
KILL(ALLBUT(name1,...,namek)) will do a KILL(ALL) except it will not
KILL the names specified.
KILL removes all properties from the given argument; thus KILL(VALUES)
MACSYMA 10.3 Functions for Freeing Storage 149
will kill all properties associated with every item on the VALUES list.
←←← ←←←←←
KILL always returns the value "DONE" even if the named item doesn't exist
(see 8.1).
The "REMOVE" functions (REMVALUE,REMFUNCTION, REMARRAY,REMRULE) remove
a specific property. These functions return a list of names or FALSE (if
←←←←←←←←
the specific argument doesn't exist).
MACSYMA options may not be KILLed. The user may do RESET() (see
10.6) to reset all options to their default values.
The error message "NO CORE - FASLOAD" results when either too many
FASL files have been loaded in or when allocation level has gotten too
high. Note that once this occurs, KILLing expressions will not help. In
either of these cases, no amount of killing will cause the size of these
spaces to decrease. Killing expressions only causes some spaces to get
emptied out but not made smaller.
10.4 Functions Which Reference Disk Files
LOADFILE(fn1, fn2, DSK, directory) loads a file as designated by its
arguments. This function may be used to bring back quantities that were
stored from a prior MACSYMA session by use of the SAVE or STORE functions.
If DSK and directory are omitted then the last directory seen (initially
the same as the user's login name or USERS if the user has no file
directory) will be used. If DSK and directory are omitted, fn2 may also
be omitted if fn1 > is to be loaded in (where > follows the conventions of
ITS's file system). Fn1 fn2 must be a file of LISP functions and
expressions, not of MACSYMA command lines, in which case BATCH or DEMO is
to be used. (See Chapter 14).
DELFILE(file-specification) will delete the file given by the file-
specification.
BATCH(file-specification) reads in and evaluates MACSYMA command lines from a
file. (see Chapter 14).
DEMO(file-specification) same as BATCH but pauses after each command line and
continues when a space is typed. (see Chapter 14).
BATCON(argument) continues BATCHing in a file which was interrupted (see
14.4).
150 10.4 Functions Which Reference Disk Files MACSYMA
WRITEFILE(DSK, directory) opens a file for writing.
APPENDFILE(filename1, filename2, DSK, directory) is like
WRITEFILE(DSK,directory) but appends to the file whose name is specified
by the first two arguments. A subsequent CLOSEFILE will delete the
original file and rename the appended file.
CLOSEFILE(filename1, filename2) closes a file opened by WRITEFILE and gives it
the name filename1 filename2. Thus to save a file consisting of the
display of all input and output during some part of a session with MACSYMA
the user issues a WRITEFILE, transacts with MACSYMA, then issues a
CLOSEFILE. The user can also issue the PLAYBACK function after a
WRITEFILE to save the display of previous transactions. (Note that what
is saved this way is a copy of the display of expressions not the
expressions themselves). To save the actual expression in internal form
the SAVE function may be used. The expression can then be brought back
into MACSYMA via the LOADFILE function. To save the expression in a
linear form which may then be BATCHed in later, the STRINGOUT function is
used. (see below)
STRINGOUT(file-specification, A1, A2, ...) outputs to a file given by file-
specification ([filename1,filename2,DSK, directory]) the values given by
A1,A2,.. in a MACSYMA readable format. The file-specification may be
omitted, in which case the default values will be used. (see 15.2 - C)
The Ai are usually C labels or may be INPUT meaning the value of all C
labels. Another option is to make Ai FUNCTIONS which will cause all of
the user's function definitions to be strungout (i.e. all those retrieved
by DISPFUN(ALL)). Ai may also be a list [m,n] which means to stringout all
labels in the range m through n inclusive. This function may be used to
create a file of FORTRAN statements by doing some simple editing on the
strungout expressions. The FORTRAN[FALSE] should be set to TRUE, however,
to cause exponentiation to be strung as ** rather than as ↑, as well as to
effect other FORTRAN-like changes. Alternatively, the function FORTRAN
can be used (see 10.2)
SAVE(args) saves quantities described by its arguments on disk and keeps them
in core also. (see 15.3).
STORE(args) same as SAVE but doesn't retain quantities in core. (see
15.3).
FASSAVE(args) is similar to SAVE but produces a FASL file in which the sharing
of subexpressions which are shared in core is preserved in the file
created. Hence, expressions which have common subexpressions will consume
less space when loaded back from a file created by FASSAVE rather than by
SAVE.
MACSYMA 10.4 Functions Which Reference Disk Files 151
UNSTORE(name1, ...) brings the named expressions into core that were stored
away by use of the STORE function in the current MACSYMA. (see 15.3).
RESTORE(file-specification) reinitializes all quantities filed away by a use
of the SAVE or STORE functions, in a prior MACSYMA session, from the file
given by file-specification without bringing them into core. (see
15.4).
REMFILE() removes files created by the secondary storage scheme in the MACSYMA
under use (see 15.2). REMFILE(ALL) does what REMFILE() does and in
addition deletes any files which have been created by the SAVE or STORE
functions but which have not been assigned names by the user.
[10.5] Ordering Functions
Aside from declaring a variable to be constant or using options like
POWERDISP (see below), the only other way in which a user can alter the
ordering of parts of an expression is to set up special aliases for variables
which cause them to be alphabetically less than or greater than any other
variables. Functions which do this are described below. This technique
requires care because although the names have been aliased, they display with
their original name. Aside from the input/output phase the two names
represent two different symbols and thus expressions which contain both the
original name and the alias will not be simplified as the user desires. This
is shown in the examples below.
ORDERGREAT(V1, ..., Vn) sets up aliases for the variables V1, ..., Vn such
that
V1 > V2 > ... > Vn > any other variable not mentioned as an argument.
ORDERLESS(V1, ..., Vn) sets up aliases for the variables V1, ..., Vn such that
V1 < V2 < ... < Vn < any other variable not mentioned as an argument.
Thus the complete ordering scale is:
numerical constants < declared constants <
< first argument to ORDERLESS < ... < last argument to ORDERLESS <
< variables which begin with A < ... < variables which begin with Z <
< last argument to ORDERGREAT < ... < first argument to ORDERGREAT.
152 10.5 Ordering Functions MACSYMA
ORDERGREATP(exp1,exp2) returns TRUE if exp2 precedes exp1 in the ordering
induced by the variable ordering described above.
ORDERLESSP(exp1,exp2) returns TRUE if exp1 precedes exp2 in the ordering
induced by the variable ordering described above.
UNORDER() removes the aliases created by the last use of the above ordering
commands. ORDERGREAT and ORDERLESS may not be used more than one time
each without calling UNORDER.
(C1) A**2+B*X;
2
(D1) B X + A
(C2) ORDERGREAT(A);
(D2) DONE
(C3) A**2+B*X;
2
(D3) A + B X
(C4) %-D1;
2 2
(D4) A - A
(C5) UNORDER();
(D5) [A]
SORT(list,optional-predicate) sorts the list using a suitable optional-
predicate of two arguments (such as "<" or ORDERLESSP). If the optional-
predicate is not given, then MACSYMA's built-in ordering predicate is
used.
10.6 Miscellaneous Functions
TIME(Di1, Di2, ...) gives a list of the times in milliseconds taken to compute
the Di.
LOGOUT() causes the user to be logged out and all jobs deleted. This is
useful when it is desired to BATCH in a file and have the terminal logged
out automatically when the computations are finished. (Equivalent to ↑Z
and :LOGOUT)
MACSYMA 10.6 Miscellaneous Functions 153
QUIT() kills the current MACSYMA but doesn't affect the user's other jobs.
(Equivalent to ↑Z and :KILL).
READ(string1, ...) prints its arguments, then reads in and evaluates one
expression. For example: A:READ("ENTER THE NUMBER OF VALUES").
READONLY(string1,...) prints its arguments, then reads in an expression (which
in contrast to READ is not evaluated).
DEFINE(f(x1, ...), body) is equivalent to f(x1,...):=''body but when used
inside functions it happens at execution time rather than at the time of
definition of the function which contains it. (see 3.2)
LOCAL(v1, v2, ...) causes the variables v1,v2,... to be local with respect to
all the properties in the statement in which this function is used (see
2.12). LOCAL may only be used in BLOCKs, in the body of function
definitions or LAMBDA expressions, or in the EV function and only one
occurrence is permitted in each.
ERROR(arg1, arg2, ...) will evaluate and print its arguments and then will
cause an error return to top level MACSYMA or to the nearest enclosing
ERRCATCH. This is useful for breaking out of nested functions if an error
condition is detected, or wherever one can't type control-G.
ERRORFUN[FALSE] - if set to the name of a function of no arguments will
cause that function to be executed whenever an error occurs. This is
useful in BATCH files where the user may want his MACSYMA killed or his
terminal logged out if an error occurs. In these cases ERRORFUN would be
set to QUIT or LOGOUT.
ERRCATCH(exp1, exp2, ...) evaluates its arguments one by one and returns a
list of the value of the last one if no error occurs. If an error occurs
in the evaluation of any arguments, ERRCATCH "catches" the error and
immediately returns [] (the empty list). This function is useful in BATCH
files where one suspects an error might occur which would otherwise have
terminated the BATCH if the error weren't caught.
CATCH(exp1,...,expn) evaluates its arguments one by one; if the structure of
the expi leads to the evaluation of an expression of the form THROW(arg),
then the value of the CATCH is the value of THROW(arg). This "non-local
return" thus goes through any depth of nesting to the nearest enclosing
CATCH. There must be a CATCH corresponding to a THROW, else an error is
generated. If the evaluation of the expi does not lead to the evaluation
of any THROW then the value of the CATCH is the value of expn.
154 10.6 Miscellaneous Functions MACSYMA
(C1) G(L):=CATCH(MAP(LAMBDA([X],
IF X<0 THEN THROW(X) ELSE F(X)),L))$
(C2) G([1,2,3,7]);
(D2) [F(1), F(2), F(3), F(7)]
(C3) G([1,2,-3,7]);
(D3) - 3
The function G returns a list of F of each element of L if L consists
only of non-negative numbers; otherwise, G "catches" the first negative
element of L and "throws" it up.
THROW(exp) evaluates exp and throws the value back to the most recent CATCH.
THROW is used with CATCH as a structured nonlocal exit mechanism.
BREAK(arg1, ...) will evaluate and print its arguments and will then cause a
(MACSYMA-BREAK) at which point the user can examine and change his
environment. Upon typing EXIT; the computation resumes. (see Chapter
19)
RESET() causes all MACSYMA options to be set to their default values.
%TH(i) is the ith previous computation. That is, if the next expression to be
computed is D(j) this is D(j-i). This is useful in BATCH files or for
referring to a group of D expressions. For example, if SUM is initialized
to 0 then FOR I:1 THRU 10 DO SUM:SUM+%TH(I) will set SUM to the sum of the
last ten D expressions.
CONCAT(arg1, arg2, ...) evaluates its arguments and returns the concatenation
of their values resulting in a name or a quoted string (see 2.2 and 2.3)
the type being given by that of the first argument. Thus if X is bound to
1 and D is unbound then CONCAT(X,2)="12" and CONCAT(D,X+1)=D2.
GETCHAR(a, i) returns the ith character of the quoted string or atomic name a.
This function is useful in manipulating the LABELS list.
STATUS(arg) will return miscellaneous status information about the user's
MACSYMA depending upon the arg given. Permissible arguments and results
are as follows:
TIME - the time used so far in the computation.
DAY - the day of the week.
DATE - a list of the year, month, and day.
MACSYMA 10.6 Miscellaneous Functions 155
DAYTIME - a list of the hour, minute, and second.
RUNTIME - accumulated cpu time times the atom "MSEC".
REALTIME -the real time (in sec) elapsed since the user started up his
MACSYMA.
WRITEFILE - a list of the device and username for the current
writefile or an empty list if no WRITEFILE has been done.
LOADFILE - a list of the first file name, second file name, device,
and username for the current BATCH, DEMO, or LOADFILE function.
FILE - a list of the current first file name and second file name.
UNIT - a list of the current device and username.
ALARMCLOCK(arg1, arg2, arg3) will execute the function of no arguments whose
name is arg3 when the time specified by arg1 and arg2 elapses. If arg1 is
the atom "TIME" then arg3 will be executed after arg2 seconds of real-time
has elapsed while if arg1 is the atom "RUNTIME" then arg3 will be executed
after arg2 milliseconds of cpu time. If arg2 is negative then the arg1
timer is shut off.
LABELS(char) takes a char C,D,or E as arg and generates a list of all C-
labels,D-labels, or E-labels, respectively. (If you've generated many E-
labels via SOLVE, then FIRST(REST(LABELS(C))); reminds you what the last
C-label was.)
ALIAS(newname1, oldname1, newname2, oldname2, ...) provides an alternate name
for a (user or system) function,variable,array,etc. Any even number of
arguments may be used.
[10.7] Options and Variables for I/O, Status, and Display
GRIND[FALSE] if TRUE will cause the STRING, STRINGOUT, and PLAYBACK
commands to use "grind" mode instead of "string" mode. For PLAYBACK, "grind"
mode can also be turned on (for processing input lines) by specifying GRIND as
an option.
SHOWTIME[FALSE] - if TRUE causes MACSYMA to print the cpu time taken by
each computation. This figure does not include I/O time except in the case of
the time given at the end of running a batch file. By setting SHOWTIME:ALL,
in addition to the cpu time MACSYMA now also prints out (when not zero) the
amount of time spent in garbage collection (gc) in the course of a
computation. This time is of course included in the time printed out as
"time=" .
(It should be noted that since the "time=" time only includes computation
time and not any intermediate display time, and since it is difficult to
ascribe "responsibility" for gc's, the gctime printed will include all gctime
156 10.7 Options and Variables for I/O, Status, and Display MACSYMA
incurred in the course of the computation and hence may in rare cases even be
larger than "time=").
LASTTIME - the time to compute the last expression in milliseconds
presented as a list of "time" and "gctime" .
OPTIONSET[FALSE] - if TRUE, MACSYMA will print out a message whenever a
MACSYMA option is reset. This is useful if the user is doubtful of the
spelling of some option and wants to make sure that the variable he assigned a
value to was truly an option variable.
NOLABELS[FALSE] - if TRUE then no labels will be bound except for E lines
generated by the solve functions (sect. 6.3). This is most useful in the
"BATCH" mode where it eliminates the need to do KILL(LABELS) in order to free
up storage.
BFTRUNC[TRUE] causes trailing zeroes in non-zero bigfloat numbers not to
be displayed. Thus, if BFTRUNC:FALSE, BFLOAT(1); displays as
1.000000000000000B0. Otherwise, this is displayed as 1.0B0.
EXPTDISPFLAG[TRUE] - if TRUE, MACSYMA displays expressions with negative
exponents using quotients e.g., X**(-1) as 1/X.
%EDISPFLAG[FALSE] - if TRUE, MACSYMA displays %E to a negative exponent as
a quotient, i.e. %E↑-X as 1/%E↑X.
SQRTDISPFLAG[TRUE] - if FALSE causes SQRT to display with exponent 1/2.
PFEFORMAT[FALSE] - if TRUE will cause rational numbers to display in a
linear form and denominators which are integers to display as rational number
multipliers.
DISPFLAG[TRUE] - if set to FALSE within a BLOCK (see 2.12) will inhibit
the display of output generated by the solve functions (see 6.3) called from
within the BLOCK. Termination of the BLOCK with a dollar sign, $, sets
DISPFLAG to FALSE.
LOADPRINT[TRUE] - governs the printing of messages accompanying loading of
files. The following options are available: TRUE means always print the
message; 'LOADFILE means print only when the LOADFILE command is used;
'AUTOLOAD means print only when a file is automatically loaded in (e.g. the
integration file SIN FASL); FALSE means never print the loading message.
NOUNDISP[FALSE] - if TRUE will cause NOUNs to display with a single quote.
This switch is always TRUE when displaying function definitions.
POWERDISP[FALSE] - if TRUE will cause sums to be displayed with their
terms in the reverse order. Thus polynomials would display as truncated power
series, i.e., with the lowest power first.
BOTHCASES[FALSE] - if TRUE will cause MACSYMA to retain lower case text as
MACSYMA 10.7 Options and Variables for I/O, Status, and Display 157
well as upper case. Note, however, that the names of any MACSYMA special
variables or functions must be typed in upper case.
STARDISP[FALSE] - if TRUE will cause multiplication to be displayed
explicitly with an * between operands.
DSKGC[FALSE] - if TRUE will cause user defined values, functions, arrays,
and line labelled expressions to be automatically stored on disk whenever the
system determines that the available in-core space is getting low (see also
15.2).
LABELS - a list of C, D, and E lines which are bound.
INCHAR[C] - the alphabetic prefix of the names of expressions typed by the
user.
LINECHAR[E] - the alphabetic prefix of the names of intermediate displayed
expressions.
OUTCHAR[D] - the alphabetic prefix of the names of outputted expressions.
LINENUM - the line number of the last expression.
CURSOR[←] is the prompt symbol of the MACSYMA editor, DEMO function,
PLAYBACK(SLOW) mode, and (MACSYMA-BREAK). (see chapters 13 and
19).
GENINDEX[I] -the alphabetic prefix of the index of summation for generated
sums. (The values of GENINDEX and of the above four variables may be any
number of characters though the default is a single character.)
IBASE[10] - the base for inputting numbers.
BASE[10] - the base for display of numbers.
LINEL - the length of the printed line on the terminal. Also used for
plotting (see Chapters 17 and 18).
PLOTHEIGHT - the height of the area used for plotting (see Chapters
17 and 18).
VERSION[267] - is the version number of MACSYMA. This could be useful if
the user wants to label his output.
INFOLISTS is the list of all the information lists which are in MACSYMA:
[LABELS,VALUES,FUNCTIONS,ARRAYS,MYOPTIONS,PROPS,ALIASES,RULES,
GRADEFS,DEPENDENCIES]
Initially, all these information lists are empty. As the user proceeds
with his computation, he may examine these lists when necessary.
158 MACSYMA
[10.8] Functions for Translation and Compilation
MODEDECLARE(y1, mode1, y2, mode2, ...) is used to declare the modes of
variables and functions for subsequent translation or compilation of
functions. Its arguments are pairs consisting of a variable yi, and a mode
which is one of BOOLEAN, INTEGER, NUMBER, RATIONAL, FLOAT, POLY (for
polynomial), or CRE (for expression in CRE form). Each yi may also be a
list of variables all of which are declared to have modei.
If yi is an array, and if every element of the array which is referenced
in the function has a value then ARRAY(yi, COMPLETE, d1, d2, ...) rather
than ARRAY(yi, d1, d2, ...) should be used when first declaring the bounds
of the array. If all the elements of the array are of mode INTEGER
(FLOAT), use INTEGER (FLOAT) instead of COMPLETE. Also if every element of
the array is of the same mode, say m, then
MODEDECLARE(COMPLETEARRAY(yi),m)) should be used for efficient
translation. Also numeric code using arrays can be made to run faster by
declaring the expected size of the array, as in:
MODEDECLARE(COMPLETEARRAY(A[10,10]),FLOAT)
for a floating point number array which is 10 x 10.
Additionally one may declare the mode of the result of a function by
using FUNCTION(F1,F2,...) as an argument; here F1,F2,... are the names of
functions. For example the expression,
MODEDECLARE([FUNCTION(F1,F2,...),X],POLY,Q,COMPLETEARRAY(Q),FLOAT)
declares that X and the value returned by F1,F2,... are polynomials and
that Q is an array of floating point numbers. MODEDECLARE is used either
immediately inside of a function definition (see below) or at top-level
for global variables.
OPTIMIZE(exp) returns an expression that produces the same value and side
effects as exp but does so more efficiently by avoiding the recomputation
of common subexpressions. OPTIMIZE also has the side effect of
"collapsing" its argument so that all common subexpressions are shared.
←←←←←←←←
(C1) DIFF(%,X,2);
2 2 2 2
2 Y + X Y + X Y + X Y + X
4 X %E 2 %E 4 X %E 2 %E
(D1) ------------- + ---------- - ------------ + ----------
Y + X Y + X 2 3
(Y + X) (Y + X)
(C2) OPTIMIZE(%);
MACSYMA 10.8 Functions for Translation and Compilation 159
2 Y + T2
(D2) BLOCK([T1, T2, T3, T4], T1 : Y + X, T2 : X , T3 : %E ,
1 4 X T3 2 T3
T4: --, 4 T2 T4 T3 + 2 T4 T3 - ------ + ----)
T1 2 3
T1 T1
TRANSLATE(f1, f2, ...) translates the user defined functions f1,f2,... from
the MACSYMA language to LISP (i.e. it makes them EXPRs). This results in
a gain in speed when they are called. The functions should include a call
to MODEDECLARE at the beginning when possible in order to produce more
efficient code. For example:
F(X1,X2,...):=BLOCK([v1,v2,...],MODEDECLARE(v1,mode1,v2,mode2,...),...)
where the X1,X2,... are the parameters to the function and the v1,v2,...
are the local variables. The names of translated functions are added to
the PROPS lists (see 8.1). Functions should not be translated unless they
are fully debugged.
TRANSLATE(FUNCTIONS) or TRANSLATE(ALL) means translate all functions.
TRANSLATE[FALSE] - If TRUE, causes automatic translation of a user's
function to LISP. Note that translated functions may not run identically
to the way they did before translation as certain incompatabilities may
exist between the LISP and MACSYMA versions. Principally, the RAT
function with more than one argument and the RATVARS function (see 6.5)
should not be used if any variables are MODEDECLAREd CRE.
SAVEDEF[TRUE] - if TRUE will cause the MACSYMA version of a user
function to remain when the function is TRANSLATEd. This permits the
definition to be displayed by DISPFUN and allows the function to be
edited. If SAVEDEF is FALSE, the names of translated functions are
removed from the FUNCTIONS list.
TRANSRUN[TRUE] - if FALSE will cause the interpreted version of all
functions to be run (provided they are still around) rather than the
translated version.
TRANSBIND[TRUE] - if TRUE removes global declarations in the local
context. This applies to variables which are formal parameters to
functions.
One can translate functions stored in a file by giving TRANSLATE an
argument which is a file specification. This is a list of the form
[fn1,fn2,DSK,dir] where fn1 fn2 is the name of the file of MACSYMA
functions, and dir is the name of a file directory.
160 10.8 Functions for Translation and Compilation MACSYMA
Such a file may contain declarations involving DECLARE,MODEDECLARE, or
MATCHDECLARE in addition to the function definitions. The file should not
use %, since % is not maintained when the translated file is loaded.
TRANSLATE will return DONE when translation is accomplished. In the
case of a file translation, the file containing the translated code is
named fn1 LISP on the disk directory dir. This file of LISP code may be
read into MACSYMA by using the LOADFILE function (see 10.4).
COMPFILE([filespec],f1,f2,...) will translate (if necessary) and write out
MACSYMA function definitions and other expressions into a disk file which
can be read into the compiler. The filespec (optional) specifies the file
to be written. The default for users with a directory is CMPFIL > and the
standard MACSYMA default file for other users. The file written contains
declarations used by the compiler. When COMPGRIND[FALSE] is TRUE the
function definitions are pretty-printed. The remaining arguments are
atomic function names.
MACSYMA 161
11 Tensor Manipulation
MACSYMA implements symbolic tensor manipulation of two distinct types:
explicit tensor manipulation and indicial tensor manipulation.
Explicit tensor manipulation means that tensors are represented as arrays
or matrices; tensor operations such as contraction or covariant
differentiation are carried out by actually summing over repeated (dummy)
indices---by explicitly performing operations on the appropriate tensor
components stored in an array or matrix.
Indicial tensor manipulation is implemented by representing tensors simply
as functions of their covariant and contravariant indices; tensor operations
such as contraction or covariant differentiation are represented by
manipulating the indices themselves rather than the components which they
refer to.
These two approaches to the question of treating differential, algebraic
and analytic processes in the context of Riemannian geometry have various
advantages and disadvantages which reveal themselves only through the
particular nature and difficulty of the user's problem. However, one should
keep in mind the following characteristics of the two implementations:
Explicit Tensor Manipulation (ETENSR)
i) The standard representation of tensors and tensor operations
explicitly in terms of their components makes ETENSR easy to use:
specification of the metric and the computation of the induced
tensors and invariants is straightforward.
ii) Although all of MACSYMA's powerful simplification capacity is
at hand, a complex metric with intricate functional and coordinate
dependencies can easily lead to expressions whose size is excessive
and whose structure is hidden.
Indicial Tensor Manipulation (ITENSR)
i) Because of the special way in which tensors and tensor
operations are represented in terms of symbolic operations on their
indices, expressions which in the explicit representation would be
unmanageable can be greatly simplified by contraction and reduction
to a "canonical" form (for symmetric tensors). In this way the
structure of a large expression may be more transparent.
ii) On the other hand, because of the the special indicial
representation in ITENSR, in some cases the user must be careful
about the specification of the metric, function definition, and the
evaluation of differentiated "indexed" objects.
These two tensor manipulation packages, ETENSR and ITENSR, are available
162 11 Tensor Manipulation MACSYMA
to the MACSYMA user on the SHARE directory (see 12). To use the
functions in these files, the user can load them in by doing
LOADFILE(ETENSR,FASL,DSK,SHARE); --- for explicit tensor manipulation.
LOADFILE(ITENSR,FASL,DSK,SHARE); --- for indicial tensor manipulation
Both of these packages enable the user to specify a metric and compute the
basic quantities of interest---Christoffel symbols, Riemann curvature tensor,
curvature invariants---in the study of Riemannian manifolds. These routines
were written primarily for research in gravitation theory; however, they may
also be of some use in other areas of physics where Riemannian geometry is
applied.
11.1 Explicit Tensor Manipulation
TSETUP() automatically loads the ETENSR package and presents several options
which are self-explanatory. First the user chooses the kind of metric to
be used---whether it will be one of the standard metrics already stored in
some file, or a power-series approximation, or some new metric to be
specified (perhaps only a slight modification of a metric already
defined). After the metric has been specified, a number of simplification
options are provided which govern the rational simplification and
factoring of the tensor components to be computed. The particular
quantity to be computed can then be indicated; the user can say whether
the results (some of which may be quite lengthy) are to be immediately
displayed or not.
Here is a sample protocol:
(C2) TSETUP();
DO YOU WANT
1 - TO CONSIDER A METRIC IN THE SPECIAL METRIC FILE?
2 - TO APPROXIMATE A METRIC WITH A POWER SERIES?
3 - TO ENTER A NEW METRIC?
TYPE 1 OR 2 OR 3
3;
SPECIFY THE COORDINATES AS A LIST OF FOUR ELEMENTS
[R,THETA,PHI,T];
DO YOU WANT
1 - TO SPECIFY A DIAGONAL METRIC?
2 - TO CHANGE A COMPONENT IN A PREVIOUSLY DEFINED
METRIC?
3 - TO SPECIFY A GENERAL (SYMMETRIC) METRIC?
TYPE 1 2 OR 3
1;
ENTER DIAGONAL MATRIX
[1, 1]
MACSYMA 11.1 Explicit Tensor Manipulation 163
-EXP(M);
[2, 2]
-R↑2;
[3, 3]
-R↑2*SIN(THETA)↑2;
[4, 4]
EXP(N);
INDICATE THE KIND OF SIMPLIFICATION YOU WANT
1 - RATIONAL SIMPLIFICATION ONLY
2 - FACTORING AND RATIONAL SIMPLIFICATION
3 - EXPANSION
TYPE 1 2 OR 3
2;
At this point the user has the option of computing various quantities
which are described below.
CHRISTOF(arg) computes the Christoffel symbols of both kinds; the arg,
determines which results are to be immediately displayed. The Christoffel
symbols of the first kind are stored in the array LCS[I,J,K]. If the
argument to CHRISTOF were LCS, all the non-zero values of LCS[I,J,K] would
be displayed.
The Christoffel symbols of the second kind (Mixed Christoffel Symbols)
are given by the array MCS[I,J,K]; in the example below, the argument MCS
was given resulting in the immediate display of all the non-zero mixed
Christoffel symbols.
(C3) CHRISTOF(MCS);
M
R
(E3) MCS = --
1, 1, 1 2
1
(E4) MCS = -
1, 2, 2 R
1
(E5) MCS = -
1, 3, 3 R
N
R
(E6) MCS = --
1, 4, 4 2
- M
(E7) MCS = - %E R
2, 2, 1
164 11.1 Explicit Tensor Manipulation MACSYMA
COS(THETA)
(E8) MCS = ----------
2, 3, 3 SIN(THETA)
- M 2
(E9) MCS = - %E R SIN (THETA)
3, 3, 1
(E10) MCS = - COS(THETA) SIN(THETA)
3, 3, 2
N - M
%E N
R
(E11) MCS = ----------
4, 4, 1 2
(D11) DONE
MOTION(dis) gives the geodesic equations of motion corresponding to a given
metric. They are stored in the array EM[I]. If the argument dis is TRUE
then these equations are displayed.
RICCICOM(dis) This function first computes the contravariant components
LR[I,J] of the Ricci tensor (LR is a mnemonic for "lower Ricci"). Then
the mixed Ricci tensor is computed using the covariant metric tensor. If
the value of the argument to RICCICOM is TRUE, then these mixed
components, RICCI[I,J] (the index I is covariant (down) and the index J is
contravariant (up) ), will be displayed directly. Otherwise,
RICCICOM(FALSE) will simply compute the entries of the array RICCI[I,J]
without presenting the results.
(C13) RICCICOM(TRUE);
(E13) RICCI
1, 1
- M 2
%E ((2 N + (N ) - M N ) R - 4 M )
R R R R R R
= -----------------------------------------
4 R
- M M
%E ((M - N ) R + 2 %E - 2)
R R
(E14) RICCI = - -------------------------------
2, 2 2
2 R
MACSYMA 11.1 Explicit Tensor Manipulation 165
- M M
%E ((M - N ) R + 2 %E - 2)
R R
(E15) RICCI = - -------------------------------
3, 3 2
2 R
(E16) RICCI
4, 4
- M 2
` %E ((2 N + (N ) - M N ) R + 4 N )
R R R R R R
= -----------------------------------------
4 R
(D16) DONE
NTERMSRCI() returns a list of pairs, whose second elements give the number of
terms in the RICCI component specified by the first elements. In this
way, it is possible to quickly find the non-zero expressions and attempt
simplification.
LRICCICOM(dis) computes the covariant components LR[I,J] of the Ricci tensor.
If the argument dis is TRUE, then the non-zero components are displayed.
EINSTEIN(dis) computes the Einstein tensor once the Christoffel symbols and
Ricci tensor have been obtained. Again, if the argument evaluates to
TRUE, then the non-zero values of the Einstein tensor G[I,J] will be
displayed. RATEINSTEIN:TRUE will perform rational simplification on these
components; if FACRAT:TRUE then the components will also be factored.
NTERMSG() gives the user a quick picture of the "size" of the Einstein tensor.
It returns a list of pairs whose second elements give the number of terms
in the components specified by the first elements.
Of course a detailed examination of the structure of the Einstein
components can be made using the powerful simplification, factoring and
extraction functions available in MACSYMA (see 6.1.1,6.2.3).
SCURVATURE() returns the scalar curvature (obtained by contracting the Ricci
tensor) of the Riemannian manifold with the given metric.
166 11.1 Explicit Tensor Manipulation MACSYMA
RIEMANN(dis) computes the Riemann curvature tensor from the given metric (the
Christoffel symbols should be obtained first using CHRISTOF). If dis is
TRUE, the non-zero components R[I,J,K,L] will be displayed. All the
indicated indices are covariant. As with the Einstein tensor, various
switches set by the user control the simplification of the Riemann
components. If RATRIEMAN:TRUE, then rational simplification will be done;
if FACRAT:TRUE then each of the components will also be factored.
(C27) RIEMANN(TRUE);
M R
R
(E27) R = - ----
1, 2, 1, 2 2
2
M R SIN (THETA)
R
(E28) R = - ----------------
1, 3, 1, 3 2
N N 2 N
- 2 %E N - %E (N ) + M %E N
R R R R R
(E29) R = ------------------------------------
1, 4, 1, 4 4
M - M 2 2
(E30) R = - (%E - 1) %E R SIN (THETA)
2, 3, 2, 3
N - M
%E N R
R
(E31) R = - ------------
2, 4, 2, 4 2
N - M 2
%E N R SIN (THETA)
R
(E32) R = - ------------------------
3, 4, 3, 4 2
(D32) DONE
RAISERIEMANN(dis) returns the contravariant components of the Riemann
curvature tensor as array elements UR[I,J,K,L]. These are displayed if
dis is TRUE.
MACSYMA 11.1 Explicit Tensor Manipulation 167
RINVARIANT() forms the scalar invariant obtained by contracting
R[I,J,K,L]*UR[I,J,K,L].
(C34) RINVARIANT();
-2M 2 2 4
(D34) %E ((4 (N ) + (4 (N ) - 4 M N ) N + (N )
R R R R R R R R
3 2 2 4 2 2 2
- 2 M (N ) + (M ) (N ) ) R + (8 (N ) + 8 (M ) ) R
R R R R R R
2 M M 4
+ 16 %E - 32 %E + 16)/(4 R )
WEYL(dis) computes the Weyl conformal tensor. If the argument dis is TRUE, the
non-zero components W[I,J,K,L] will be displayed to the user. Otherwise,
these components will simply be computed and stored. If the switch
RATWEYL is set to TRUE, then the components will be rationally simplified;
if FACRAT is TRUE then the results will be factored as well.
DSCALAR(function) computes the d'Alembertian of the scalar function.
(C41) DEPENDS(FIELD,R);
(D41) [FIELD(R)]
(C42) DSCALAR(FIELD);
(D43)
-M
%E ((FIELD N - FIELD M + 2 FIELD ) R + 4 FIELD )
R R R R R R R
- -----------------------------------------------------
2 R
DALEM(field,i,j) computes the d'Alembertian of the i,j - component of the rank
2 tensor field.
YT(f,m,n) computes the m,n component of the Yilmaz tensor defined by
n b n a n k b a
t = -2(d f d f - (1/2)d d f d f )
m m a b m a k b
168 11.1 Explicit Tensor Manipulation MACSYMA
n n k
+ d fd f - (1/2)d d fd f,
← ← ← ←
m m k
b
where f is the trace of f and repeated indices are summed. This reduces
←
a
to the Newtonian stress-energy tensor.
MACSYMA 169
[11.2] Indicial Tensor Manipulation
In ITENSR a tensor is represented as an "indexed object" . This is a
function of 3 groups of indices which represent the covariant, contravariant
and derivative indices. The covariant indices are specified by a list as the
first argument to the indexed object; the contravariant indices are specified
by a list as the second argument. If the indexed object lacks either of these
groups of indices, then the empty list, [], is given as the corresponding
argument. For example,
G([mu,nu],[])
represents an indexed object called G which has covariant indices mu,nu and no
contravariant or derivative indices.
The derivative indices, if they are present, follow as additional
arguments to the (symbolic) function representing the tensor. They are usually
not explicitly specified by the user but are created in the process of
differentiation with respect to some coordinate variable. A derivative index
is not the coordinate variable itself, but rather the coordinate index. These
indices are appended as additional arguments to the function representing the
tensor. Since it is assumed that ordinary differentiation is independent of
the order in which it is carried out, the derivative indices are sorted
alphabetically. This canonical order makes it possible for MACSYMA to
recognize that, for example, T([mu],[nu],i,j) is the same as T([mu],[nu],j,i).
Differentiation of an indexed object with respect to some coordinate whose
index does not appear as an argument to the indexed object would normally
yield zero since MACSYMA would not know that the tensor represented by the
indexed object might depend implicitly on the corresponding coordinate. This
has been remedied by modifying the existing MACSYMA function DIFF so that in
the tensor package it assumes that all indexed objects depend on any variable
of differentiation unless otherwise stated. This makes it possible for the
summation convention to be extended to derivative indices.
To specify that an indexed object is independent of all coordinate
variables, it is specified a constant by using the DECLARE function (see 8.1).
Usually, DIFF(W([],[I,J]),K) results in W([],[I,J],K); if the command
DECLARE(W,CONSTANT) had previously been given, the result of the
differentiation would be 0.
The following functions are available in the tensor package for
manipulating indexed objects. At present it is assumed that all tensor
indices are completely symmetric.
In what follows, general indexed objects will be denoted tensor,
tensor1,tensor2,... . The letters L1,L2,... denote lists which are arguments
to indexed objects. Optional arguments are enclosed in angle brackets.
SHOWTEN(exp) will display exp with the indexed objects in it shown having
covariant indices as subscripts,contravariant indices as superscripts. The
derivative indices will be displayed as subscripts, separated from the
covariant indices by a comma.
170 11.2 Indicial Tensor Manipulation MACSYMA
COMPONENTS(tensor,exp) permits one to assign an expression exp giving the
values of the components of tensor. These are automatically substituted
for the tensor whenever it occurs with all integer indices.
The tensor must be of the form T([...],[...]), which can specify a
covariant (second list empty), contravariant (first list empty), or mixed
(neither list empty) tensor. exp can be a matrix, an array or any
expression (except a single variable) involving other tensors.
If exp is a matrix, then tensor must have exactly two indices. If it
is an array, then exp and tensor must have the same number of indices. If
exp is some other expression involving tensors, then it must have the same
free indices as tensor. This expression will be used even if the indices
are not integers.
(C1) COMPONENTS(G([I,J],[]),MATRIX)$
(C2) COMPONENTS(G([],[J,K]),ARRAY)$
(C3) COMPONENTS(E([I],[J]),G([I,K])*H([],[K,J]))$
Thus, the various covariant, contravariant, and mixed components of the
∪same tensor (for example, G above) can be specified using COMPONENTS
several times. The appropriate components will be chosen when required in
a computation.
INDEXED(tensor) must be executed before assigning components to a tensor for
which a built in value already exists as with CHR1, CHR2, RIEMANN.
REMCOMPS(tensor) unbinds all values from tensor which were assigned with the
COMPONENTS function.
INDICES(exp) returns a list of two elements. The first is a list of the free
indices in exp (those that occur only once); the second is the list of
dummy indices in exp (those that occur exactly twice).
RENAME(exp, <count>) returns an expression equivalent to exp but with the
dummy indices in each term chosen from the set [#1, #2,...], if the
optional second argument is omitted. Otherwise, the dummy indices are
indexed beginning at the value of count. Each dummy index in a product
will be different; for a sum, RENAME will try to make each dummy index in
a sum the same. In addition, the indices will be sorted alphanumerically.
DUMMY(i1,i2,...) will set each index i1,i2,... to name of the form #n where n
is a positive integer. This guarantees that dummy indices which are
needed in forming expressions will not conflict with indices already in
use.
MACSYMA 11.2 Indicial Tensor Manipulation 171
COUNTER[1] determines the numerical suffix to be used in generating the
next dummy index. The prefix is determined by the option DUMMYX[#].
DEFCON(tensor1,<tensor2,tensor3>) gives tensor1 the property that the
contraction of a product of tensor1 and tensor2 results in tensor3 with
the appropriate indices. If only one argument, tensor1, is given, then
the contraction of the product of tensor1 with any indexed object having
the appropriate indices (say tensor) will yield an indexed object with
that name,i.e.tensor, and with a new set of indices reflecting the
contractions performed.
For example, if METRIC: G, then DEFCON(G) will implement the raising
and lowering of indices through contraction with the metric tensor.
More than one DEFCON can be given for the same indexed object; the
latest one given which applies in a particular contraction will be used.
CONTRACTIONS is a list of those indexed objects which have been given
contraction properties with DEFCON.
DISPCON(tensor1,tensor2,...) displays the contraction properties of the
tensori as were given to DEFCON. DISPCON(ALL) displays all the
contraction properties which were defined.
REMCON(tensor1,tensor2,...) removes all the contraction properties from the
tensori. REMCON(ALL) removes all contraction properties from all indexed
objects.
CONTRACT(exp) carries out all possible contractions in exp, which may be any
well-formed combination of sums and products. This function uses the
information given to the DEFCON function. Since all tensors are
considered to be symmetric in all indices, the indices are sorted into
alphabetical order. Also all dummy indices are renamed using the symbols
#1,#2,... to permit the expression to be simplified as much as possible
by reducing equivalent terms to a canonical form. For best results exp
should be fully expanded.
RATEXPAND (see 6.1.1) is the fastest way to expand products and powers of
sums if there are no variables in the denominators of the terms. The
TAKEGCD switch should be FALSE if gcd cancellations are unnecessary.
CANTEN(exp) reduces exp to a canonical form and simplifies the expression as
much as possible by renaming and permuting dummy indices. The expression
exp must be fully expanded.
As an example of the kind of simplification this function achieves,
consider the following sum of tensor "monomials":
(C4) P([I,J,S,V],[M,N,Q],V)*P1([Q,T],[R,S])*P2([R,L,M,N,U],[I,J,K]) +
172 11.2 Indicial Tensor Manipulation MACSYMA
P2([L,N,U],[R,M,I,J,K])*P([M,I,J,V],[N,Q,S],V)*P1([R,S,Q,T],[])$
(C6) SHOWTEN(D4);
R M I J K N Q S M N Q R S I J K
(E6) P2 P P1 + P P1 P2
L N U M I J V,V R S Q T I J S V,V Q T R L M N U
(D6) E6
Recall that all these indexed objects are assumed to be completely
symmetric in their indices. The function RENAME renames the dummy
indices:
(C7) SHOWTEN(RENAME(D4))$
#6 #7 #8 #2 #3 K #4 #5
(E7) P P2 P1
#1 #2 #3 #5,#1 #4 #6 #7 L U #8 T
#3 #4 #8 #2 #5 #6 #7 K
+ P P1 P2
#1 #5 #6 #7,#1 #2 #3 #4 T #8 L U
but is not able to notice that by a sequence of raising and lowering
operations, the original expression can be transformed into
(C8) SHOWTEN(CANTEN(D4))$
K I J M N Q R S
(E8) 2 P2 P P1
I J L M N R U Q S V,V T
METRIC(G) specifies the metric by assigning the variable METRIC:G; in
addition, the contraction properties of the metric G are set up by
executing the commands DEFCON(G), DEFCON(G,G,KDELTA).
CHR1([i,j,k]) yields the Christoffel symbol of the first kind
(g + g - g )/2
ik,j jk,i ij,k
The variable METRIC must be assigned the name of a function (which can be
either defined or undefined); in the above example, METRIC:g.
MACSYMA 11.2 Indicial Tensor Manipulation 173
CHR2([i,j],[k]) yields the Christoffel symbol of the second kind.
ks
CHR2([i,j],[k]) = g CHR1([i,j,s])
LC(L) is the permutation (or Levi-Civita) tensor which yields 1 if the list L
consists of an even permutation of integers, -1 if it consists of an odd
permutation, and 0 if some indices in L are repeated.
KDELTA(L1,L2) is the generalized Kronecker delta function. L1 and L2 are
lists of indices of the same length.
a) If L1 and L2 have a single member, say L1=[a] and L2=[b]. Then
1) if the index a is identical to the index b and they are non-
numeric, the value of the function KDELTA is the value of the
variable DIMENSION[4] ;
2) if a and b are numeric then the value of the function is 1 if
they are equal, else 0;
3) otherwise the noun form of KDELTA.
b) If L1=[a1,a2] and L2=[b1,b2]. Then the value of the function is
KDELTA([a1],[b1])*KDELTA([a2],[b2])
- KDELTA([a1],[b2])*KDELTA([a2],[b1])
c) If L1 and L2 have more than two indices the result generalizes.
RIEMANN([i,j,k],[l]) yields the Riemann curvature tensor in terms of the
Christoffel symbols of the second kind (CHR2).
Suppose the name specified by the value of METRIC corresponds to a tensor
which has been given some structure via the COMPONENTS command above; in
order to evaluate an expression involving the Riemann tensor and
incorporate this given structure of the metric explicitly into the result,
the user can simply do expression,EVAL.
Consider the following example involving a metric, G, expressed in terms
of the rank two tensors, P and E. First, the covariant and contravariant
forms of the metric are specified:
(C9) COMPONENTS(G([M,N],[]),
E([M,N],[]) + L*(2*P([],[])*E([M,N],[])-4*P([M,N],[]))
L↑2*(2*P([],[])↑2*E([M,N],[])
- 8*P([],[])*P([M,N],[])+8*P([M],[Q])*P([Q,N],[])))$
174 11.2 Indicial Tensor Manipulation MACSYMA
(C10) COMPONENTS(G([],[M,N]),
E([],[M,N]) - L*(2*P([],[])*E([],[M,N])-4*P([],[M,N]))
+ L↑2*(2*P([],[])↑2*E([],[M,N])
- 8*P([],[])*P([],[M,N])+8*P([],[M,Q])*P([Q],[N])))$
(C11) SHOWTEN(G([A,B],[]))$
2 Q 2
(E11) L (8 P P - 8 P P + 2 P E ) + L (2 P E - 4 P )
A Q B A B A B A B A B
+ E
A B
(C12) SHOWTEN(G([],[R,S]));
2 R Q S R S 2 R S R S R S
(E12) L (8 P P - 8 P P + 2 P E ) - (2 P E - 4 P ) L
Q
R S
+ E
(C13) (RATVARS(L),RATWEIGHT(L,1),RATWTLVL:2)$
The above metric is an expansion to second order of the Yilmaz exponential
metric
2*L*(I.P - 2*P)
←
G = E.e
where L is an expansion parameter, P is a rank two tensor field viewed as
←
R,S
a matrix whose components are P and whose trace is P.
To second order, the covariant and contravariant forms of the metric
should be mutually inverse. As an example of the contraction and
simplification functions CONTRACT,CANTEN one can verify this as follows:
(C14) SHOWTEN(EX:CONTRACT(RATEXPAND(G([M,R],[])*G([],[R,N]))))$
N N 2 #1 #1 N 2
(E14) KDELTA + 16 P L P - 16 P P L
M #1 M #1 M
Because RATWTLVL:2, the product of the expansions is truncated; the second
order terms are actually equal and should cancel. Using a canonical form,
the function CANTEN makes the appropriate simplification which enables the
last two terms to be compared and cancelled:
(C16) SHOWTEN(CANTEN(EX));
MACSYMA 11.2 Indicial Tensor Manipulation 175
N
(E16) KDELTA
M
The Ricci tensor is easily expressed in terms of the Riemann tensor,
using the Einstein summation convention:
(C18) RICCI:RIEMANN([I,J,K],[K])$
(C19) %,EVAL$
(C20) EXP1:CONTRACT(RATEXPAND(RICCI))$
(C21) SHOWTEN(FACTOR(EXP1))$
#1 #1 #2
(E21) - (2 P - 2 E P
J,#1 I I J,#1 #2
#1 #2 #1
+ P E E + 2 P ) L
,#1 #2 I J I,#1 J
DIFF(exp,v1,n1,v2,n2,...) is the usual MACSYMA differentiation function; it
takes the derivative of exp wrt v1 n1 times, wrt v2 n2 times, etc. For
the tensor package, the following modifications have been incorporated:
1) the derivatives of any indexed objects in exp will have the
variables vi appended as additional arguments. Then all the derivative
indices will be sorted.
2) the vi may be integers from 1 up to the value of the variable
DIMENSION[4]. This will cause the differentiation to be carried out wrt
the vith member of the list COORDINATES which should be set to a list of
the names of the coordinates, e.g., [x,y,z,t] . If COORDINATES is bound to
an atomic variable, then that variable subscripted by vi will be used for
the variable of differentiation. This permits an array of coordinate
names or subscripted names like X[1], X[2],... to be used. If
COORDINATES has not been assigned a value, then the variables will be
treated as in 1) above.
COVDIFF(exp,v1,v2,...) yields the covariant derivative of exp with respect to
the variables vi in terms of the Christoffel symbols of the second kind
(CHR2). In order to evaluate these, one can use exp, eval or
EV(exp,CHR2).
176 11.2 Indicial Tensor Manipulation MACSYMA
UNDIFF(exp) returns an expression equivalent to exp but with all derivatives
of indexed objects replaced by the noun form of the DIFF function with
arguments which would yield that indexed object if the differentiation
were carried out. This is useful when it is desired to replace a
differentiated indexed object with some function definition and then carry
out the differentiation by saying EV(...,DIFF).
LORENTZ(exp) yields exp replacing by zero those indexed objects which have a
derivative index identical to a contravariant index. This imposes the
Lorentz condition.
MAKEBOX(exp) will display exp in the same manner as SHOWTEN; however, any
tensor d'Alembertian occurring in exp will be indicated using the symbol
[]. For example, []P([M],[N]) represents G([],[I,J])*P([M],[N],I,J).
ZERO(tensor) will give tensor a property such that the
command FLUSH(expression) will, in expression, replace by zero all
occurences of the tensor involving derivative indices.
MACSYMA 177
12 The SHARE Directory
The SHARE directory contains programs, information files, etc. which are
considered to be of interest to the MACSYMA community. Most files on SHARE;
are not part of the MACSYMA system per se and must be loaded individually by
the user, e.g. LOADFILE(ITENSR,FASL,DSK,SHARE);. Many files on SHARE; were
contributed by MACSYMA users, and all MACSYMA users are encouraged to do so.
Names for files on SHARE; should be chosen as appropriate. However, the
contributor will probably want to follow the conventions discussed here. A
contributor will probably create some but not all of the following files.
Examples may be seen on the SHARE directory.
1) NAME > is the file name of the MACSYMA BATCHable programs. The > sign
indicates a numeric second filename which is increased whenever a new version
is created.
2) NAME LISP is the file name of the LISP code for the programs contained
in the file NAME >. This file is loaded into MACSYMA using LOADFILE. It was
obtained by using the TRANSLATE command or was written directly in LISP by the
contributor.
3) NAME FASL is the file name of the FASL (fast-loadable) version of NAME
LISP, and was produced from NAME LISP by using the LISP compiler. It is
loaded into MACSYMA using the LOADFILE command.
4) NAME USAGE is the name of the documentation file for the programs in
NAME >. It describes how the programs are used, inputs, outputs, options,
warnings, error messages, etc. It may mention the algorithms behind the
programs, references, and whatever else the user should know. It should
certainly indicate who programmed the routines, especially his login name. If
the NAME USAGE file does not exist, this information should be given in NAME >
or elsewhere.
5) NAME DEMO is the name of the demonstration file which may be used in
DEMOing NAME > or NAME LISP.
6) NAME OUTPUT may be used to store sample output obtained from running
NAME > on some examples or from DEMOing the NAME DEMO file.
7) Other file names may be used for information files providing some
information on some aspect of the MACSYMA system or for describing some
MACSYMA utility or for notes on some MACSYMA issues, etc.
The SHARE > file is an index to the SHARE directory and is intended to
contain a short note on each of the programs on the SHARE directory. It
should be updated by the SHARE; contributor as appropriate.
Any comments or questions about the use of the SHARE directory should be
sent to JPG.
178 MACSYMA
12.1 Simplification for ABS and SIGNUM
The file SHARE;ABSIMP > contains MACSYMA pattern-matching rules that
extend the built-in simplification rules for the ABS and SIGNUM functions.
Among other things, use is made of global relations established with the
built-in ASSUME function or by declarations such as DECLARE(M,EVEN, N,ODD) for
even or odd integers. UNITRAMP and UNITSTEP functions are also defined in
terms of ABS and SIGNUM. These routines were written by David Stoutemyer.
[12.2] Array Manipulation
The file ARRAY FASL provides various utility functions for handling
arrays.
LISTARRAY(array) returns a list of the elements of a declared array. the
order is row-major. You will get garbage if any of the elements have not
been defined yet.
FILLARRAY(array,list-or-array) fills array from list-or-array. If array is a
floating-point (integer) array then list-or-array should be either a list
of floating-point (integer) numbers or another floating-point (integer)
array. If the dimensions of the arrays are different array is filled in
row-major order. If there are not enough elements in list-or-array the
last element is used to fill out the rest of array. If there are too many
the remaining ones are thrown away. FILLARRAY returns its first argument.
REARRAY(array,dim1, ... ,dimk) can be used to change the size or dimensions of
an array. The new array will be filled with the elements of the old one
in row-major order. If the old array was too small, FALSE, 0.0 or 0 will
be used to fill the remaining elements, depending on the type of the
array. The type of the array cannot be changed.
ARRAYAPPLY(array,[sub1, ... ,subk]) is like APPLY except the first argument is
an array.
These routines were written by Charles Karney.
[12.3] Solving Differential Equations by Laplace Transforms
DESOLN LISP contains a routine, written by Richard Bogen, for solving
differential equations or systems of them by using Laplace transforms. The
call is:
MACSYMA 12.3 Solving Differential Equations by Laplace Transforms 179
DESOLVE([eq1,...,eqn],[var1,...,varn]) where the eq's are differential
equations in the dependent variables var1,...,varn. The functional
relationships must be explicitly indicated in both the equations and the
variables. For example
(C1) 'DIFF(F,X,2)=SIN(X)+'DIFF(G,X);
(C2) 'DIFF(F,X)+X↑2-F=2*'DIFF(G,X,2);
is not the proper format. The correct way is:
(C3) 'DIFF(F(X),X,2)=SIN(X)+'DIFF(G(X),X);
(C4) 'DIFF(F(X),X)+X↑2-F(X)=2*'DIFF(G(X),X,2);
The quotes are not necessary since DIFF will return the noun forms
anyway.
The call is then DESOLVE([D3,D4],[F(X),G(X)]);
If initial conditions at 0 are known, they should be supplied before
calling DESOLVE by using ATVALUE.
(C11) 'DIFF(F(X),X)='DIFF(G(X),X)+SIN(X);
D D
(D11) -- F(X) = -- G(X) + SIN(X)
DX DX
(C12) 'DIFF(G(X),X,2)='DIFF(F(X),X)-COS(X);
2
D D
(D12) --- G(X) = -- F(X) - COS(X)
2 DX
DX
(C13) ATVALUE('DIFF(G(X),X),X=0,A);
(D13) A
(C14) ATVALUE(F(X),X=0,1);
(D14) 1
(C15) DESOLVE([D11,D12],[F(X),G(X)]);
X X
(D16) [F(X)=A %E - A+1, G(X) = COS(X) + A %E - A + G(0) - 1]
/* VERIFICATION */
(C17) [D11,D12],D16,DIFF;
X X X X
(D17) [A %E = A %E , A %E - COS(X) = A %E - COS(X)]
180 MACSYMA
12.4 Exterior Calculus of Differential Forms
The exterior calculus of differential forms is a basic tool of
differential geometry developed by Elie Cartan and has important applications
in the theory of partial differential equations. The present implementation
is due to F.B. Estabrook and H.D. Wahlquist. The program is self-explanatory
and can be accessed by doing
BATCH([CARTAN,START,DSK,SHARE],ON)
which will give a description with examples.
MACSYMA 12.4 Exterior Calculus of Differential Forms 181
The next six sections describe programs written by David Stoutemyer.
12.5 Vector Analysis
The file VECT > contains a vector analysis package, VECT DEMO contains a
corresponding demonstration, and VECT ORTH contains definitions of various
orthogonal curvilinear coordinate systems.
The vector analysis package can combine and simplify symbolic expressions
including dot products and cross products, together with the gradient,
divergence, curl, and Laplacian operators. The distribution of these
operators over sums or products is under user control, as are various other
expansions, including expansion into components in any specific orthogonal
coordinate systems. There is also a capability for deriving the scalar or
vector potential of a field.
To establish indeterminate1, indeterminate2, ... as vector entities, type
DECLARE([indeterminate1, indeterminate2, ...], NONSCALAR) $
Vectors can also be represented as lists of components.
"." is the dot-product operator, "}" is the cross-product operator, GRAD
is the gradient operator, DIV is the divergence operator, CURL is the curl or
rotation operator, and LAPLACIAN is DIV GRAD.
Most non-controversial simplifications are automatic. For additional
simplification, there is a function which can be used in the form
VECTORSIMP(vectorexpression)
This function employs additional non-controversial simplifications, together
with various optional expansions according to the settings of the following
global flags:
EXPANDALL, EXPANDDOT, EXPANDDOTPLUS
EXPANDCROSS, EXPANDCROSSPLUS, EXPANDCROSSCROSS
EXPANDGRAD, EXPANDGRADPLUS, EXPANDGRADPROD
EXPANDDIV, EXPANDDIVPLUS, EXPANDDIVPROD
EXPANDCURL, EXPANDCURLPLUS, EXPANDCURLCURL
EXPANDLAPLACIAN, EXPANDLAPLACIANPLUS, EXPANDLAPLACIANPROD
All these flags have default value FALSE. The PLUS suffix refers to employing
additivity or distributivity. The PROD suffix refers to the expansion for an
182 12.5 Vector Analysis MACSYMA
operand that is any kind of product. EXPANDCROSSCROSS refers to replacing
p}(q}r) with (p.r)*q-(p.q)*r, and EXPANDCURLCURL refers to replacing CURL CURL
p with GRAD DIV p + DIV GRAD p. EXPANDCROSS:TRUE has the same effect as
EXPANDCROSSPLUS:EXPANDCROSSCROSS:TRUE, etc. Two other flags, EXPANDPLUS and
EXPANDPROD, have the same effect as setting all similarly suffixed flags true.
When TRUE, another flag named EXPANDLAPLACIANTODIVGRAD, replaces the LAPLACIAN
operator with the composition DIV GRAD. All of these flags are initially
FALSE. For convenience, all of these flags have been declared EVFLAG.
For orthogonal curvilinear coordinates, the global variables
COORDINATES[[X,Y,Z]], DIMENSION[3], SF[[1,1,1]], and SFPROD[1] are set by the
function invocation
SCALEFACTORS(coordinatetransform)
Here coordinatetransform evaluates to the form [[expression1, expression2,
...], indeterminate1, indeterminat2, ...], where indeterminate1,
indeterminate2, etc. are the curvilinear coordinate variables and where a set
of rectangular Cartesian components is given in terms of the curvilinear
coordinates by [expression1, expression2, ...]. COORDINATES is set to the
vector [indeterminate1, indeterminate2, ...], and DIMENSION is set to the
length of this vector. SF[1], SF[2], ..., SF[DIMENSION] are set to the
coordinate scale factors, and SFPROD is set to the product of these scale
factors. Initially, COORDINATES is [X, Y, Z], DIMENSION is 3, and
SF[1]=SF[2]=SF[3]=SFPROD=1, corresponding to 3-dimensional rectangular
Cartesian coordinates.
To expand an expression into physical components in the current coordinate
system, there is a function with usage of the form
EXPRESS(expression)
The result uses the noun form of any derivatives arising from expansion of the
vector differential operators. To force evaluation of these derivatives, the
built-in EV function can be used together with the DIFF evflag, after using
the built-in DEPENDS function to establish any new implicit dependencies.
The scalar potential of a given gradient vector, in the current coordinate
system, is returned as the result of POTENTIAL(givengradient)
The calculation makes use of the global variable POTENTIALZEROLOC[0], which
must be NONLIST or of the form [indeterminatej=expressionj,
indeterminatek=expressionk, ...], the former being equivalent to the nonlist
expression for all right-hand sides in the latter. The indicated right-hand
sides are used as the lower limit of integration. The success of the
integrations may depend upon their values and order. POTENTIALZEROLOC is
initially set to 0.
The vector potential of a given curl vector, in the current coordinate
system, is returned as the result of
VECTORPOTENTIAL(givencurl)
MACSYMA 12.5 Vector Analysis 183
POTENTIALZEROLOC has a similar role as for POTENTIAL, but the order of the
left-hand sides of the equations must be a cyclic permutation of the
coordinate variables.
EXPRESS, POTENTIAL, and VECTORPOTENTIAL can have a second argument like
the argument of SCALEFACTORS, causing a corresponding invocation of
SCALEFACTORS before the other computations.
12.6 Dimensional Analysis
The file DIMEN > contains functions for automatic dimensional analysis,
and file DIMEN DEMO contains a demonstration. Usage is of the form
NONDIMENSIONALIZE(list of physical quantities)
The returned value is a sufficient list of nondimensional products of
powers of the physical quantities. A physical relation between only the given
physical quantities must be expressible as a relation between the
nondimensional quantities. There are usually fewer nondimensional than
physical quantities, which reduces the number of experiments or numerical
computations necessary to establish the physical relation to a specified
resolution, in comparison with the number if all but one dependent physical
variable were independently varied. Also, the absence of any given physical
quantity in the output reveals that either the quantity is irrelevant or
others are necessary to describe the relation.
The program already knows an extensive number of relations between
physical quantities, such as VELOCITY=LENGTH/TIME. The user may over-ride or
supplement the prespecified relations by typing
DIMENSION(equation or list of equations)
where each equation is of the form indeterminate=expression, where expression
is a product or quotient of powers of none or more of the indeterminates
CHARGE, TEMPERATURE, LENGTH, TIME, or MASS. To see if a relation is already
established type
GET(indeterminate, 'DIMENSION);
The result of NONDIMENSIONALIZE usually depends upon the value of the
global variable %PURE, which is set to a list of none or more of the
indeterminates ELECTRICPERMITTIVITYOFAVACUUM, BOLTZMANNSCONSTANT,
SPEEDOFLIGHT, PLANCKSCONSTANT, GRAVITYCONSTANT, corresponding to the relation
between charge and force, temperature and energy, length and time, length and
momentum, and the inverse-square law of gravitation respectively. Each
included relation is used to eliminate one of CHARGE, TEMPERATURE, LENGTH,
TIME, or MASS from the dimensional basis. To avoid omission of a possibly
relevant nondimensional grouping, either include the relevant constant in
%PURE or in the argument of NONDIMENSIONALIZE if the corresponding physical
effect is thought to be relevant to the problem. However, the inclusion of
184 12.6 Dimensional Analysis MACSYMA
unnecessary constants, especially the latter three, tends to produce
irrelevant or misleading dimensionless groupings, defeating the purpose of
dimensional analysis. As an extreme example, if all five constants are
included in %PURE, all physical quantities are already dimensionless. %PURE
is initially set to '[ELECTRICPERMITTIVITYOFVACUUM, BOLTZMANNSCONSTANT], which
is best for most engineering work. %PURE must not include any of the other 3
constants without also including these 2.
[12.7] Analytic Optimization
We now describe a package for finding the stationary points of a
multivariate objective function, either unconstrained or subject to equality
and/or inequality constraints.
RELEVANT FILES: OPTMIZ > is a MACSYMA batch file containing the functions and
option settings for optimization. OPTMIZ DEMO is a MACSYMA batch file
demonstrating various ways of using the optimization functions. OPTMIZ OUTPUT
is a text file listing OPTMIZ DEMO together with the output that it produces
when executed.
To use this package from a MACSYMA, first type BATCH(OPTMIZ, ">", DSK,
SHARE) Then the following command is available:
STAP(OBJECTIVE, LEZEROS, EQZEROS, DECISIONVARS)
OBJECTIVE is an expression denoting the objective function or the label of
such an expression. LEZEROS is a list of expressions which are constrained to
be less than or equal to zero. Use [] if no such constraints. EQZEROS is a
list of expressions which are constrained to equal zero, or the label of such
a list. Use [] if there are no such constraints. DECISIONVARS is a list of
the decision variables or the label of such a list. One may use [] if all
variables in objective and constraintsare decision variables. For convenience,
brackets may be omitted from one-expression lists, and trailing [] arguments
may be omitted.
ROOTSEPSILON may affect the accuracy of results computed by SOLVE and
REALROOTS within STAP. The default value of 1.0E-7 for this MACSYMA global
variable is as small as practical for pdp single-precision floating-point
arithmetic. Larger values save cpu time.
The class of functions that may be used and the practical limitations on
the number of decision variables and constraints is primarily dependent upon
the capabilities of the built-in SOLVE function, which is still under
development.
[12.8] Variational Optimization
MACSYMA 12.8 Variational Optimization 185
This section describes how to use a MACSYMA variational optimization
package to analytically solve problems from the calculus of variations and the
maximum principle, including optimal control.
To use this package in a MACSYMA, first type BATCH(OPTVAR,">", DSK, SHARE)
or LOADFILE(OPTVAR, LISP, DSK, SHARE).
To derive the Euler-Lagrange equations for a calculus-of-variations
problem, type
EL(F, YLIST, TLIST)
F is an expression or the label of an expression for the integrand of the
stationary functional, augmented by Lagrange multipliers times the integrands
of any isoperimetric constraints and/or differential expressions constrained
to equal zero. The multipliers should be written as functions of the
independent variables in the latter case.
YLIST is a list of the dependent variables, or the label thereof.
TLIST is a list of the independent variables, or the label thereof.
For convenience, square brackets may be omitted from 1-element lists. EL
displays one or more E-labeled equations, then returns a list of the E-labels.
These equations are the Euler-Lagrange equations, perhaps together with first
integrals corresponding to conservation of energy and/or conservation of
momentum. The former will contain a constant of integration K[0], whereas the
latter will contain constants of integration K[I], with positive I. The
latter will immediately follow the corresponding Euler-Lagrange equation.
OPTVAR DEMO or OPTVAR OUTPUT illustrates some ways that the resulting
differential equations may be solved analytically.
To derive the Hamiltonian and auxiliary differential equations for an
optimal control problem, type
HAM(ODES)
ODES is a list of the first-order differential equations that govern the state
variables. Each differential equation must be of the form
'D(Y,T) = EXPRESSION
where Y is one of the dependent variables, T is the independent variable, and
EXPRESSION depends upon the independent, dependent, and control variables.
HAM displays two or more E-labeled expressions, then returns a list of the
E-labels. The first expression is the Hamiltonian, and the other expressiona
are the auxiliary diferential equations, together with their general
solutions, AUX[I] = K[I], whenever the Ith differential equation is of the
trivial form 'D(AUX[I],T) = 0. The K[I] are undetermined constants of
integration.
186 12.8 Variational Optimization MACSYMA
HAM is directly suitable for the autonomous time-optimal problem. Other
problems may be converted to this form by introducing extra state variables,
as described in most optimal-control texts or in the report referenced in
OPTVAR OUTPUT and OPTVAR DEMO.
[12.9] Qualitative Analysis
QUAL > contains MACSYMA functions for qualitative analysis of an
expression, QUAL DEMO contains a demonstration, and QUAL OUT contains the
output from executing the demo.
To use the functions do ALLOC(2); LOADFILE(STOUTE,">",DSK,MRG);
BATCH(QUAL, ">", DSK, SHARE);
Top-level usage is of the form QUAL(<expression>, <variables>)
where <expression> is any given expression, <variables> is a given
indeterminate or list of indeterminates. If omitted, this argument defaults
to all of the indeterminates in the first argument.
QUAL returns a list of E-labels of displayed equations, each of the form
<property name> = <property value>
where <property name> is one of the second-level function names below, and
<property value> is the value returned by that function. These second-level
functions may also be used directly. Usage is of the form
REVELATION(<expression>, <minimum>, <maximum>);
BOUNDS(<expression>);
SLOPES(<expression>, <variables>);
CURVATURE(<expression>, <variables>);
SYMMETRY(<expression>, <variables>);
PERIODS(<expression>, <variables>);
ZEROSANDPOLES(<expression>, <variables>);
STATIONARYPOINTS(<expression>, <variables>);
LIMITS(<expression>, <variables>);
where <expression>, <variables>, and their defaults are as for QUAL.
[12.10] Units Conversion
The file UNITS > contains assignments for automatic conversion to MKS
metric units. Usage example: 5*FT + METER + CM; simplifies to 2.534*METER .
Erroneously dimensionally inhomogeneous expressions are revealed by
uncollected terms. For example, 5*FT + SECOND; does not simplify to one term.
MACSYMA 12.10 Units Conversion 187
The supplied conversions comprise a rather complete set, but it should be
clear how to supplement them or produce an analagous set for conversion to
other units.
[12.11] The Eigen Package
The BATCH file EIGEN > contains a package of functions is written in top-
level MACSYMA. Its purpose is to compute right eigenvectors, right unit
eigenvectors, eigenvalues, and similarity transforms. NOTE: This package
currently will not handle systems with multiple eigenvalues. Implementation of
this capability will await the ability to handle multiple roots from SOLVE.
EVEC1(M,mu,modes) computes right eigenvectors of the matrix M, given the
eigenvalues mu; mu is a one dimensional array of the eigenvalues of M.
modes* is the order of the system. EVEC1 returns a list of lists which
are the eigenvectors; i.e., it returns a list of sublists, each of which
contains the components of an eigenvector.
EVEC2(M,mu,modes) is exactly the same as EVEC1 except that it constructs the
list of eigenvectors differently from EVEC1. It is a toss-up as to which
is faster.
INPROD(x,y) computes the real inner product of two lists (not vectors). This
inner product is the sum of the products of the respective components of
the lists. The two lists used as arguments to INPROD must be of the same
length.
UEVEC(M,mu,modes) computes the unit eigenvectors of the matrix M. That is,
the eigenvectors are of unit length as defined by INPROD(x,x). UEVEC
merely calls evec1 to compute the eigenvectors, and then calls INPROD to
compute the length of each eigenvector in turn and divides by this length.
UEVEC returns a list of vectors (not lists) which are the unit
eigenvectors. Since UEVEC calls EVEC1, the eigenvalues mu (defined as in
EVEC1) must be distinct.
EVALS(mat,lambda) computes the eigenvalues of the matrix mat and stores them
in the one dimensional array lambda. It returns the value "done". The
array lambda must be set up as an array before the call to EVALS. Also,
this function will not handle multiple eigenvalues.
SIMTRAN(mat) is the function which computes the similarity transformation
mentioned above. it merely calls the other functions such as EVALS,
UEVEC, etc. to determine the eigenvalues and right unit eigenvectors. For
188 12.11 The Eigen Package MACSYMA
a symmetric matrix with distinct eigenvalues, the matrix formed by taking
the unit eigenvectors as columns is an orthogonal matrix, as is its
transpose. Thus if A is the original symmetric matrix (with distinct
eigenvalues) and Q is the orthogonal matrix constructed as above, and QT
is the transpose of Q, then
QT*A*Q = D
where D is the diagonal matrix with the eigenvalues of A on the diagonal.
SIMTRAN takes the matrix mat and fills the global array lambda with the
eigenvalues of mat. It also fills the globally defined matrices Q and QT
with the matrices Q and QT as described above.
The array lambda and the matrices Q and QT must exist as an array and
matrices (presumably loaded with dummy info) before the call to SIMTRAN.
The matrix mat should not have multiple eigenvalues for the reasons
mentioned in previous comments.
[12.12] Elimination by Resultants
ELIM LISP contains a program, written by Richard Bogen, for eliminating
variables from equations (or expressions assumed equal to zero) by taking
successive resultants. the call is:
ELIMINATE([eq1,eq2,...,eqn],[v1,v2,...,vk])
This returns a list of n-k expressions with the k variables v1,...,vk
eliminated. First v1 is eliminated yielding n-1 expressions, then v2 is, etc.
If k=n then a single expression in a list is returned free of the variables
v1,...,vk. In this case SOLVE is called to SOLVE the last resultant for the
last variable. If there are floating point numbers in the input expressions
then KEEPFLOAT:TRUE should be done so as not to have them rationalized in
taking the resultant.
EXAMPLE
(C5) .5*X↑4+Y*X+Z;
4
(D5) Z +X Y + 0.5 X
(C6) 3*X+5*Y-Z-1;
(D6) - Z + 5 Y + 3 X - 1
(C7) Z↑2+X↑2-Y↑2+5;
2 2 2
(D7) Z - Y + X + 5
(C8) ELIMINATE([D7,D6,D5],[Y,Z]),KEEPFLOAT:TRUE;
MACSYMA 12.12 Elimination by Resultants 189
8 6 5 4 3 2
(D8) [150.0 X - 375.0 X + 50.0 X + 275.0 X + 100 X + 550 X
+ 1400 X + 3100]
[12.13] Integration of Special Forms
INTSCE LISP contains a routine, written by Richard Bogen, for integrating
products of sines,cosines and exponentials of the form
EXP(A*X+B)*COS(C*X)↑N*SIN(C*X)↑M
The call is INTSCE(expr,var) expr may be any expression, but if it is not
in the above form then the regular integration program will be invoked if the
switch ERRINTSCE[TRUE] is TRUE. If it is FALSE then INTSCE will err out.
12.14 Integral Equations
The following package was written by Richard Bogen based on some routines
of David Stoutemeyer. It is still an experimental version, untranslated and
uncompiled at present.
To load the package, do BATCH(INTEQN,">",DSK,RAB).
CAVEAT: To free some storage, a KILL(LABELS) is included in this file.
Therefore, ∪before loading the integral equation package, the user should give
names to any expressions he wants to keep.
The usage is very simple. The main function is called IEQN. It takes five
arguments though only the first one is required. If trailing arguments are
omitted they will default to preset values which will be announced.
Two types of equations are considered. An integral equation of the second
kind is of the form:
B(X)
/
[
(D3) P(X) = Q(X, P(X), I W(X, U, P(X), P(U)) dU) .
]
/
A(X)
An integral equation of the first kind is of the form:
190 12.14 Integral Equations MACSYMA
B(X)
/
[
(D4) I W(X, U, P(U)) dU = F(X)
]
/
A(X)
The unknown function in these equations is P(X) while Q,W,A, and B are given
functions of the independent variable. Although these are the general forms,
most of the solution techniques require particular forms of Q and W.
The techniques used are:
For SECONDKIND equations:
FINITERANK: for degenerate (or separable) integrands.
DIFFEQN: reduction to differential equation.
TRANSFORM: Laplace Transform for convolution types.
FREDSERIES: Fredholm-Carleman series for linear equations.
TAILOR: Taylor series for quasi-linear variable-limit equations.
NEUMANN: Neumann series for quasi-second kind equations.
COLLOCATE: collocation using a power series form for P(X)
evaluated at equally spaced points.
For FIRSTKIND equations:
FINITERANK: for degenerate integrands.
DIFFEQN: reduction to differential equation.
ABEL: for singular integrands.
TRANSFORM: see above
COLLOCATE: see above
FIRSTKINDSERIES: iteration technique similar to Neumann series.
Also, differentiation is used in certain cases to transform a FIRSTKIND into a
SECONDKIND.
The calling sequence is
IEQN(ie,unk,tech,n,guess) where ie is the integral equation; unk is the
unknown function; tech is the technique to be tried from those given above
(tech = FIRST means: try the first technique which finds a solution; tech
= ALL means: try all applicable techniques); n is the maximum number of
terms to take for TAYLOR, NEUMANN, FIRSTKINDSERIES, or FREDSERIES (it is
also the maximum depth of recursion for the differentiation method); guess
is the initial guess for NEUMANN or FIRSTKINDSERIES.
Default values for the 2nd thru 5th parameters are:
unk: P(X), where P is the first function encountered in an
MACSYMA 12.14 Integral Equations 191
integrand which is unknown to MACSYMA and X is the variable
which occurs as an argument to the first occurrence of P
found outside of an integral in the case of SECONDKIND
equations, or is the only other variable besides the variable
of integration in FIRSTKIND equations. If the attempt to
search for X fails, the user will be asked to supply the
independent variable;
tech: FIRST;
n: 1;
guess: NONE, which will cause NEUMANN and FIRSTKINDSERIES too
use F(X) as an initial guess.
The value returned by IEQN is a list of labels of solution lists. The
solution lists are printed as they are found unless the variable
IEQNPRINT[TRUE] is set to FALSE. These lists are of the form
[SOLUTION, TECHNIQUE USED, NTERMS, FLAG]
where FLAG is absent if the solution is exact. Otherwise, it is the word
APPROXIMATE or INCOMPLETE corresponding to an inexact or non-closed form
solution, respectively. If a series method was used, NTERMS gives the
number of terms taken (which could be less than the n given to IEQN if an
error prevented generation of further terms).
For examples, do BATCH(INTEXS,">",DSK,RAB) which will load an array called
EQ with about 43 sample integral equations. Then try IEQN(EQ[1]),
IEQN(EQ[30],P(X),ALL), for instance.
12.15 Numerical Techniques
[12.15.1] Numerical Integration
A program for Romberg numerical integration can be obtained by doing:
LOADFILE(ROMBRG,FASL,DSK,SHARE)
The original version of this program was written by Richard Fateman. The
current improved version of the program was written by Charles Karney. In the
1
latest version there are two ways of calling the function ROMBERG:
1) An inefficient way that resembles a call to INTEGRATE (the definite
integral version): ROMBERG(integrand,variable,lower limit,upper limit)
For example,
(C1) ROMBERG(SIN(Y),Y,1,%PI);
←←←←←←←←←←←←←←←
1. Note the change from the original version of the program
192 12.15.1 Numerical Integration MACSYMA
time= 54 msec.
(D1) 1.5403023
(C2) F(X):=1/(X↑5+X+1)$
time= 1 msec.
(C3) ROMBERG(F(X),X,1.5,0);
time= 216 msec.
2) An efficient way that is more like the old ROMBERG function:
ROMBERG(function name,lower limit,upper limit)
The first argument must be a TRANSLATEd or compiled function. (If it is
compiled it must be declared to return a FLONUM.) If the first argument is not
already TRANSLATEd, ROMBERG will not attempt to TRANSLATE it but will give an
error. As an example consider the function f defined above,
(C4) F(X):=(MODEDECLARE([FUNCTION(F),X],FLOAT),1/(X↑5+X+1));
time= 1 msec.
1
(D4) F(X) := (MODEDECLARE([FUNCTION(F), X], FLOAT), ----------)
5
X + X + 1
(C5) TRANSLATE(F)$
time= 10 msec.
(C6) ROMBERG(F,1.5,0);
time= 13 msec.
(D6) - 0.75293843
The accuracy of the integration is governed by the global variables
ROMBERGTOL[1.E-4] and ROMBERGIT[11]. ROMBERG will return a result if the
relative difference in successive approximations is less than ROMBERGTOL. It
will try halving the stepsize ROMBERGIT times before it gives up.
ROMBERG may be called recursively and thus can do double and triple
integrals. In this case , it is even more important to TRANSLATE the
functions and to use the second way of calling ROMBERG.
(C7) INTEGRATE(INTEGRATE(X*Y/(X+Y),Y,0,X/2),X,1,3);
time= 305 msec.
2
13 (2 LOG(-) + 1)
3
(D7) -----------------
3
(C8) %,NUMER;
MACSYMA 12.15.1 Numerical Integration 193
time= 6 msec.
(D8) 0.819302335
(C9) F(Y):=(MODEDECLARE([FUNCTION(F),X,Y],FLOAT),X*Y/(X+Y))$
time= 2 msec.
(C10) G(X):=(MODEDECLARE([FUNCTION(F,G),X],FLOAT),ROMBERG(F,0,X/2))$
time= 1 msec.
(C11) TRANSLATE(F,G)$
time= 6 msec.
(C12) ROMBERG(G,1,3);
time= 32 msec.
(D12) 0.8193023
[12.15.2] Fast Fourier Transforms
The following describes some FFT routines written by Tom Knight. To load
the routines do LOADFILE(FFT,FASL,DSK,SHARE); The basic functions are:
FFT(Fast Fourier Transform), IFT (Inverse Fast Fourier Transform).
These functions perform a (complex) fast fourier transform on either 1 or
2 dimensional FLOATING-POINT arrays, obtained by: ARRAY(array,FLOAT,dim1); or
n
ARRAY(array,FLOAT,dim1,dim2); for 1d arrays dim1 must equal 2 -1, and for 2d
n
arrays dim1=dim2=2 -1 (i.e. the array is square). (Recall that MACSYMA arrays
n n 2
are indexed from a 0 origin so that there will be 2 and (2 ) arrays elements
in the above two cases.)
The calling sequence is:
FFT(real-array,imag-array)
IFT(real-array,imag-array)
The real and imaginary arrays must of course be the same size. The transforms
are done in place so that real-array and imag-array will contain the real and
imaginary parts of the transform.
Other functions included in this file are:
194 12.15.2 Fast Fourier Transforms MACSYMA
POLARTORECT(magnitude-array,phase-array) converts and magnitude phase arrays
into real/imaginary form putting the real part in the magnitude array and
the imaginary part into the phase array
RECTTOPOLAR(real-array,imag-array) undoes POLARTORECT
(the above 4 functions return a list of their arguments)
[12.15.3] Roots of Equations by Interpolation
The file INTPOL FASL, created by Charles Karney, contains the function
INTERPOLATE(func,x,a,b) which finds the zero of func as x varies. The last
two args give the range to look in. The function must have a different
sign at each endpoint. If this condition is not met, the action of the of
the function is governed by INTPOLERROR[TRUE]). If INTPOLERROR is TRUE
then an error occurs, otherwise the value of INTPOLERROR is returned (thus
for plotting INTPOLERROR might be set to 0.0). Otherwise (given that
MACSYMA can evaluate the first argument in the specified range, and that
it is continuous) INTERPOLATE is guaranteed to come up with the zero (or
one of them if there is more than one zero).
The accuracy of INTERPOLATE is governed by INTPOLABS[0.0] and
INTPOLREL[0.0] which must be non-negative floating point numbers.
INTERPOLATE will stop when the first arg evaluates to something less than
or equal to INTPOLABS or if successive approximants to the root differ by
no more than INTPOLREL * <one of the approximants>. The defaults values
of INTPOLABS and INTPOLREL are 0.0 so INTERPOLATE gets as good an answer
as is possible with the single precision arithmetic we have. The first
arg may be an equation. The order of the last two args is irrelevant.
Thus
INTERPOLATE(SIN(X)=X/2,X,%PI,.1);
is equivalent to
INTERPOLATE(SIN(X)=X/2,X,.1,%PI);
The method used is a binary search in the range specified by the last two
args. When it thinks the function is close enough to being linear, it
starts using linear interpolation.
An alternative syntax has been added to interpolate, this replaces the
first two arguments by a function name. The function MUST be TRANSLATEd
or compiled function of one argument. No checking of the result is done,
so make sure the function returns a floating point number.
F(X):=(MODEDECLARE(X,FLOAT),SIN(X)-X/2.0);
MACSYMA 12.15.3 Roots of Equations by Interpolation 195
INTERPOLATE(SIN(X)-X/2,X,0.1,%PI) time= 60 msec
INTERPOLATE(F(X),X,0.1,%PI); time= 68 msec
TRANSLATE(F);
INTERPOLATE(F(X),X,0.1,%PI); time= 26 msec
INTERPOLATE(F,0.1,%PI); time= 5 msec
[12.15.4] Special Functions
The file BESSEL FASL contains routines for computing numerical values for
various special functions. If they are given non-numeric arguments they
return themselves.
Bessel Functions
The following functions compute Bessel functions of integer order for real
arguments.
J0(X) returns the value of the zeroth order Bessel function at X.
J1(X) returns the value of the Bessel function of first order at X
JN(X,N) returns the N'th order Bessel function. In addition it sets up an
array JARRAY of N+1 elements, (numbered from 0 to ABS(N)) such that
JARRAY[I] gives the value of the I'th order Bessel function with argument
X. (If N < 0 then JARRAY[I] gives the (-I)'th Bessel function).
Modified Bessel Functions
The following functions compute the Modified Bessel Functions I of integer
orders for real arguments.
I0(X) returns the value of the modified Bessel function of zeroth order.
I1(X) returns the value of the modified Bessel function of first order.
IN(X,N) works the same way as JN(X,N), except that the array is called IARRAY.
Since the modified Bessel function blows up like EXP(ABS(X)) at infinity,
they cannot be evaluated directly for ABS(X) > 83 (due to overflow). The
following functions avoid this problem:
196 12.15.4 Special Functions MACSYMA
G0(X) returns I0(X)*EXP(-ABS(X)).
G1(X) returns I1(X)*EXP(-ABS(X)).
GN(X,N) returns IN(X,N)*EXP(-ABS(X)). The array generated by GN is called
GARRAY.
Complex Bessel Function of positive fractional order
BESSEL(Z,A) returns the Bessel function J for complex Z and real A > 0.0 .
Also an array BESSELARRAY is set up such that BESSELARRAY[I] = J[I+A-
ENTIER(A)](Z)
AIRY(X) returns the Airy function Ai of real argument X.
Plasma Dispersion Function, NZETA(Z).
This function is related to the complex error function by
NZETA(Z) = %I*SQRT(%PI)*EXP(-Z↑2)*(1-ERF(-%I*Z))
NZETA(Z) returns the complex value of the Plasma Dispersion Function for
complex Z.
NZETAR(Z) returns REALPART(NZETA(Z)).
NZETAI(Z) returns IMAGPART(NZETA(Z)).
Normal distribution function
GAUSS(MEAN,SD) returns a random floating point number from a normal
distribution with mean MEAN and standard deviation SD
[12.15.5] Polynomial Zeros
The file SHARE;PRRID FASL contains the code, written by Richard Fateman,
which implements the Collins-Loos "Polynomial Real Root Isoloation by
Differentiation" algorithm, as described in the Collins-Loos paper, Proc. of
the 1976 ACM Symp. on Symbolic and Algebraic Computation. The subsequent
improvements, due to Fateman, and described in his paper in the Proc. of the
1977 MACSYMA Users' Conference, are also included. Three versions, PRRID1,
MACSYMA 12.15.5 Polynomial Zeros 197
PRRID2, and PRRID3 are included. The first of these is the original
algorithm, transcribed from "ALDES", and the second two use combinations of
exact and floating point arithmetic to achieve the same results. PRRID3
appears to be the fastest for most cases.
Input: a univariate polynomial (preferably in rational canonical form).
Output: a list of the form [ I1, M1, I2, m2, ... In, Mn] in which each Ij
is itself a list of two items, each a rational number, [Aj, Bj]. There are
exactly Mj real zeros of the input polynomial in the half-open interval (Aj,
Bj]. These programs are potentially much faster than the Sturm sequence
calculation used by realroots (the system command), for polynomials with fewer
real zeros than the maximum possible. They are slower for some polynomials
e.g. (x-1)*(x-2)* ...*(x-20). Realroots goes a bit further in that it will
refine the isolating intervals to some desired narrowness. This is not done
by the PRRID programs, but could be easily done. A Newton iteration
guaranteed to converge, can be used to refine the zeros, starting from points
based on the zeros of the second derivative of the input.
A faster program has been implemented by Bruce Char, which, however, does
not have the same precise characteristics. Using the "allroots" command,
approximate zeros are located, and then disks in the complex plane are
computed which include the actual positions of the zeros. This is much faster
in finding isolating disks, and works in the complex plane. Unfortunately,
the disks may intersect (in which case the program says so).
198 MACSYMA
13 The MACSYMA Editor
13.1 Introduction
The major features of the editor are its concise commands (few alphabetic
characters), its varied assortment of commands, concatenation of commands as
in TECO (the PDP-10 file editor), mnemonics for command names (once you know
them, R means "move in the Reverse direction" ; B means "move to the Bottom"
), and compatibility with TECO as to command names (in the case of C, D, F, G,
I, J, K, L, R, and S).
13.2 Entering the Editor
At any time while the user is inputting a command line to MACSYMA, he may
enter the input-stream editor by typing "altmode" or "escape", henceforth
denoted by <$>. The editor is given the string of characters typed so far in
the current input line. In the case of a detected syntax error, upon typing
1
<$> the entire previous command string will be given to the editor.
Alternatively, <control>-Y also retrieves the previous command string (even if
other characters have already been typed in the current C-line). Thus, the
previous command line is readily available for editing without retyping it.
One may also request the editor to edit or modify a previously accepted
input line by using the STRING function in MACSYMA. Typing STRING(Ci) will
restore the expression labeled as Ci as the current input string. This
enables the user to modify it by then immediately typing <$>. For a simpler
method, see the MYV command below.
All the commands to the editor reference a cursor (displayed as an
underscore or back-arrow, depending on the console) which is displayed within
or at either end of the string of characters currently being edited (called
the "input string" from now on). The value of the variable CURSOR determines
what character is used (see 10.7).
The editor accepts a command string which must be terminated by <$><$>. A
command string is any concatenation of one or more legal commands which will
be processed in left-to-right order. Display of the input string occurs at
the end of the processing of each command string. <$> is used to enter the
editor, to exit from the editor (as <$><$>), and to terminate insert or search
substrings. Otherwise, spurious <$>'s are ignored. Rubouts (the rubout or
delete key on the console) may be used at any point prior to command
termination to delete the last character typed in. ?? deletes the entire
←←←←←←←←←←←←←←←
1. <$> must be the first character typed on the next command line. Any other
character (except "space") causes the edit buffer to be emptied.
MACSYMA 13.2 Entering the Editor 199
command. At any point prior to command termination, the user may type a
<control>K, and the editor will reprint the characters of the command typed so
far.
Occasionally, one gets a syntax error because of omitting characters from
the end of a command (especially right parentheses). By typing <$><$><$>
immediately, as the first 3 characters of the next input line, the last
command will be automatically reproduced on the current input line at which
point one can supply the missing characters or rubout erroneous characters.
For example:
(C1) (((X+1)*X+2)*X+3$
( ( ( X + 1 ) * X + 2 ) * X + 3 ***$***
SYNTAX ERROR
PLEASE REPHRASE OR EDIT
(C1) <$>
←(((X+1)*X+2)*X+3
<$><$>
(C1) (((X+1)*X+2)*X+3 )*X+4$
(In the above line the user typed the
characters after the 3)
13.3 A Description of the Commands
Some commands may be prefixed by an integer (represented below by "n")
which usually may be positive or negative; although it may be zero as well in
the case of K, L, and W; and it must be non-negative in case of W. The
default value of n is +1. Except in the case of R, if n is positive the
commands operate toward the right of the cursor, if n is negative they operate
toward the left. I and S are two of the few commands which may be followed by
other characters, namely the characters which constitute the insert or search
strings. An error message will be printed if an illegal command substring is
encountered or if any command substring fails. In case of such error, the
processing of the current command string will be terminated at that point,
with the offending command substring indicated.
Command Mnemonic Action
Commands which move the cursor
nC Character moves the cursor past n characters.
nR Reverse moves the cursor past n characters in the reverse
direction (nR = -nC).
J or T Jump to Top moves the cursor to the beginning of the input string.
200 13.3 A Description of the Commands MACSYMA
B or ZJ Bottom moves the cursor to the end of the input string.
nL Line moves the cursor to the right of the nth carriage
return (0L moves left); e.g., L moves to the next line.
nSstring<$> moves the cursor to the right (left if n is
negative) of the nth occurrence of "string" in the
input string.
nS Search repeats the last S command given.
) or ] Move moves the cursor right from the current position over
the next balanced pair of parentheses (or brackets).
( or [ Move similar to ) or ] but moves left.
Commands which delete characters
nD Delete deletes n characters, and saves them in the "save-
register" (see the G command below).
nK Kill deletes all the characters through the nth carriage
return (0K kills left), and saves them in the "save-
register"; e.g., K deletes the remainder of this line.
M) or M] Delete similar to ) or ] but deletes the characters moved over
and saves them in the "save-register".
M( or M[ Delete similar to M) or M] but moves left.
nFRstring<$> deletes the next n occurrences of string.
(This command is a special case of the FR command below
and can be used in this way only when it is the last
argument in the command string).
Commands which insert characters
Istring<$> Insert inserts the characters "string" at the current cursor
position. The cursor is positioned at the right of the
inserted text. If no argument is given then the string
of the last I command which had one is used.
GR Get inserts at the current cursor position the characters
deleted by the last use of D, K, or M. Thus G may be
used in combination with D or K to move characters from
one place to another in the input string; or to recover
from an accidental use of D or K. There is only one
"save-register".
MACSYMA 13.3 A Description of the Commands 201
nFRstring1<$>string2<$> replaces the next n occurrences of string1
by string2. If n is 1 it may be omitted. nFR given
with no string arguments uses those from the last FR
given which had them.
MFRstring1<$>string2<$> replaces all occurrences of string1 by
string2.
YVname<$> Yank value puts into the editing buffer, the value of the argument
whose name is given, if a label or the name of a user
variable, at the current cursor position leaving the
cursor at the end of the inserted string.
YFname<$> Yank function puts into the editing buffer
the definition of the user function whose name is given
(as with YV). If the name is followed by a list of
subscripts in brackets, then the named subscripted
function is brought into the buffer. This command
provides an alternative to DISPFUN and STRING (see
10.2).
Note: If the YV or YF commands are prefixed by the letter M then the editor
will clear the buffer before yanking, and also will leave the cursor at the
head of the edit string when done.
Commands which control display of results
P Print simply reprints the input string. This is useful in
case of console problems.
nW Window controls the window size of the display, which is the
maximum number of characters displayed on each side of
the cursor. This is useful in case of slow consoles
and large input strings. 0W will cause only the cursor
to be displayed. Once set the window size remains at
that setting until it is reset. V View restores the
display to full view, which is the normal mode
(affected only by W).
Q Quit exits the editor without reprinting the just edited
string.
<$><$> will exit from the editor and is also the command string
terminator. Two examples of legal command strings are 4C3DIFOO<$><$> and
-2SBAR<$>3R<$><$>. The first moves right over four characters, deletes the
next three characters, and inserts FOO. The second searches from the current
pointer position to the beginning of the text for the second occurrence of BAR
then moves left over three characters.
202 13.3 A Description of the Commands MACSYMA
Example
(C1) X:1$
(C2) NATRIX([A,4],[-1,A/2]);
(D2) NATRIX([A, 4], [- 1, 1/2 A])
(C3)<$>
←NATRIX([A,4],[-1,A/2])
DIM<$><$>
M←ATRIX([A,4],[-1,A/2])
]2CD<$><$>
MATRIX([A,4],[←1,A/2])
<$><$> (In the line below the user typed the ;)
(C3) MATRIX([A,4],[1,A/2]);
[ A 4 ]
(D3) [ ]
[ 1 1/2 A ]
(C4) CHARPOLY(%,X);
(D4) (A - 1) (1/2 A - 1) - 4
(C5)<$>
←CHARPOLY(%,X)
S%<$>-DID3<$>CI'<$><$>
CHARPOLY(D3,'←X)
<$><$> (In the line below the user typed the ;)
(C5) CHARPOLY(D3,'X);
(D5) (A - X) (1/2 A - X) - 4
MACSYMA 203
14 Batch Functions
14.1 Introduction
The Batch set of functions in MACSYMA, namely BATCH, DEMO, and BATCON
(mnemonic for BATch CONtinue), provide a facility for executing command lines
stored on a disk file rather than in the usual on-line mode. This facility has
several uses, namely to provide a reservoir for working command lines, for
giving error-free demonstrations, or helping in organizing one's thinking in
complex problem-solving situations where modifications may be done via the
PDP-10 TECO file editor.
A batch file consists of a set of MACSYMA command lines, each with its
terminating ; or $, which may be further separated by spaces, carriage-
returns, form-feeds, and the like. The BATCH and DEMO functions have both a
simple and more complicated format, which are described below.
14.2 The Simple Format
BATCH(filename1, filename2, DSK, directory)
(The same function format holds for DEMO as well.) The arguments to BATCH
(or DEMO) in this format specify the file which is to be batched. Here, each
file is specified by two filenames of at most six characters each, the device
the file is on ( which is normally DSK), and the user file directory. E.g.
DEMO(TAYLOR,DEMO,DSK,DEMO) calls for "demonstrating" (see below) the file
TAYLOR DEMO on the DEMO disk directory. Latter arguments to the BATCH or DEMO
functions may always be omitted if they are known from previous file-
manipulating functions.
The BATCH function calls for reading in the command lines from the file
one at a time, echoing them on the user console, and executing them in turn.
Control is returned to the user console only when serious errors occur or when
the end of the file is met. Of course, the user may quit out of the file-
processing by typing control-G at any point (see 4) .
DEMO differs from BATCH only in that it pauses after the execution of each
command line, waiting for the user to type a space which tells it to go on.
If the user types any other character, file-processing will then terminate,
giving control over to the user console. (The user may actually continue
processing from the file at any time - see the BATCON function below.)
14.3 The More Complicated Format
BATCH([fn1, fn2, DSK, directory], <delay-switch>,<index-specification>)
204 14.3 The More Complicated Format MACSYMA
The arguments to BATCH or DEMO in this mode are as follows:
The first argument is the file specification (as above), enclosed in
brackets.
The second argument, the delay-switch, may be answered by ON or OFF (the
default if it is omitted). This switch has to do with the temporary inability
of LISP, the system underlying MACSYMA, to have more than one input file open
at a time. If in the course of batching in a file of command lines, execution
of a function forces a second file to be input, this would ordinarily cause an
error. However, setting the delay-switch to ON causes the entire batch file
to be read in before execution of it begins, thus preventing the error. The
default for the delay-switch is OFF, as the circumstance described above is
not frequent, it takes some time to read in a batch file, and one may always
continue batching via the BATCON function. As soon as the inability of LISP
is removed, this switch will no longer be needed.
The index-specification is given by one or two arguments, the
possibilities being: (In the following, m and n are positive integers.)
(i) m. This indicates that processing is to begin with the mth command
line in the file. Thus, the default for the index-specification is 1.
(ii) m, n. This indicates that only the mth command line through the nth
command line are to be processed.
(iii) a variable (say FOO). FOO must be non-numeric and neither TRUE nor
FALSE. This causes file-processing to begin at FOO&& (see 14.5) and
continue until the end of the file. This makes it unnecessary to count
command lines as required by (i) above.
(iv) variable (say FOO), continue-flag. The continue-flag is either ON
(the default, and unnecessary) or OFF. If OFF, this enables one to separate a
batch file into subfiles by prefixing a command line in the file with FOO&&.
By using FOO as the index-specification, one may execute only that subfile
which begins with FOO and ends with some other variable&&, or the end of file.
If the continue-flag is ON, this causes mode (iv) to operate as (iii) above.
One can see that BATCH(TAYLOR,DEMO,DSK,DEMO) and
BATCH([TAYLOR,DEMO,DSK,DEMO], OFF, 1) are equivalent.
14.4 The BATCON Function
The BATCON function is used to continue or change the last BATCH or DEMO
function, without it being necessary to mention again BATCH or DEMO, the file
specification, or the setting of the delay-switch. Of course, if one wishes
to change any of these, a new call to BATCH or DEMO is required.
The possible arguments to BATCON are as follows:
MACSYMA 14.4 The BATCON Function 205
(i) a number
(ii) number1, number2
(iii) a variable
(iv) variable, continue-flag
They are all as in 8.3. The numeric arguments may involve the variable
BATCOUNT[0] which is set to the number of the last expression BATCHed in from
the file. Thus BATCON(BATCOUNT-1) will resume BATCHing from the expression
before the last BATCHed in from before. One other mode is possible:
(v) skip-flag. The skip-flag is useful if an error has occurred while
batching, or if the user wishes to interject command lines from the console
while in DEMO-mode and then to continue processing from the file. The skip-
flag may be either TRUE or FALSE. If FALSE, this indicates that processing is
to continue with the last command line attempted (supposedly edited, in case
of error); if TRUE, this indicates that processing is to continue with the
next (untried) command line in the file.
14.5 Miscellany
(1) Comments may be added to batch files at any point, and will, of
course, be treated as such when batching in the file. A comment is any string
beginning with /* and ending with */ as in PL/I.
(2) Any command line in a batch file may begin with variable&&. This
labels that command line so that the file can be partitioned into subfiles.
If not in a subfile mode, this prefix will be treated as a comment.
(3) When using the batch functions, it is inconvenient to keep track of
which Di label MACSYMA will assign to a computation; yet later command lines
often need to refer to an earlier computation. One way to get around this, of
course, is for the user to explicitly label some of his command lines. A
function %TH is also provided, such that %TH(i), where i is positive, refers
to the result of the ith previous command line. E.g., %TH(1) and the variable
% both refer to the same computation.
(4) When BATCHing in several files it is possible for one file to
unintentionally cause an error to occur in a subsequent one by duplication of
names or settings of options. If the variable BATCHKILL[FALSE] is TRUE
however, then the effect of all previous BATCH files is nullified because a
KILL(ALL) and a RESET() will be done automatically when the next one is read
in. If BATCHKILL is bound to any other atom then a KILL(BATCHKILL) will be
done. (The default value of BATCHKILL is FALSE meaning to do nothing.)
(5) While BATCHing in a file which takes a lot of time to process the user
may leave his terminal unattended. If an error occurs he may want some
206 14.5 Miscellany MACSYMA
special action to be taken automatically. By setting the option ERRORFUN to
the name of a function of no arguments one can have that function executed
when any error occurs. Useful functions are QUIT and LOGOUT. However in the
case of LOGOUT the user should also set the switch TTYOFF to TRUE to prevent
his job from hanging up in the case it tries to output to the terminal (see
4). In addition if a file has been opened for writing, then a command to close
it should be executed before the LOGOUT. Also, the user may wish to set
DYNAMALLOC[FALSE] to TRUE (see 16) so that his job will not hang if
additional storage space is needed
If the user is executing a function of his own and would like to signal an
error he can use the functions ERROR and ERRCATCH (see 10.6).
(6) If the user does not have a directory of his own then he can use the
one called USERS to store his files. He should identify them as his in some
fashion such as using his login name for the first file names.
(7) The DEMO file directory contains many demonstration files which may be
helpful to the user in learning to use MACSYMA.
MACSYMA 207
15 Secondary Storage Functions
15.1 Introduction
There are two different reasons for wanting to use secondary storage while
running a MACSYMA. Sometimes the user's intermediate expressions are large,
and it is impossible to complete the job if all the intermediate expressions
are kept in main memory. In this case the user would like to have his
intermediate expressions written automatically onto the disk, in order to free
main memory. On the other hand, some users would like to save some
expressions onto the disk so that they can be read back into a future MACSYMA
at a later time. In this case the user would like to specify certain
expressions to be stored and to name the disk file where they are to be
stored. MACSYMA offers the user two secondary storage schemes. The user may
ask to have his expressions automatically filed away onto the disk, or he may,
by means of the SAVE and STORE functions, exercise explicit control over the
storage of expressions. These latter functions give the user more power and
flexibility at the expense of a greater effort. It is expected that the user
whose only concern is to run a big job which would not run without using
secondary storage will use the automatic storage scheme, while the user who
wishes to save expressions for use in later MACSYMAs will use the SAVE and
STORE functions.
15.2 Automatic Storage of Expressions
A - How to use it
To activate the automatic storage scheme the user merely sets the MACSYMA
option DSKUSE[FALSE] to TRUE. From this point on labelled expressions will be
written out periodically onto the disk. (A labelled expression is one which
is referred to by a line label, e.g. D4, C7, E12.) Once an expression is
written onto the disk it will no longer reside in main memory and most of the
main memory storage taken up by it will be released. When the user attempts to
reference an expression which has been stored onto the disk, MACSYMA will
retrieve the correct value from the disk file. In this scheme expressions are
copied periodically onto the disk whenever there are enough to write out (see
FILESIZE). An alternative heuristic to use in order to free some storage is
to write out all labelled expressions, values, functions, and arrays whenever
the garbage collector finds that space is getting low. This is the purpose of
the DSKGC function (see 10).
If the user is dealing with large expressions then his storage limit may
be exceeded before FILESIZE expressions have been generated. In this case the
DSKGC method should be used. If this situation does not occur and if the user
208 15.2 Automatic Storage of Expressions MACSYMA
prefers to have some control over how many expressions are saved in each file
then the other scheme should be used.
B - Cleaning up the disk
The automatic storage scheme will in general cause several disk files to
be created, which are of no further value after the user has finished running
his current MACSYMA. There is a function called REMFILE, which will delete all
the files created by the automatic storage scheme. Thus if the user does not
want these files to stay around, he should execute REMFILE() before leaving
MACSYMA. REMFILE will only delete files created in the same MACSYMA to which
the REMFILE function is given. In order to delete files created in previous
uses of MACSYMA it is necessary to use the DELFILE function (see 10).
C - Options
The user may specify how often files are written, how large they are,
what they will be named, and what gets stored in them, or he may accept the
default values for all these. The following MACSYMA options are relevant.
FILENAME - The value of this variable is the first name of the files which
are generated by the automatic disk storage scheme. The default value is the
first three characters of the user's login name concatenated with a three-
digit random number (e.g. ECR864)
STORENUM - The value of this variable, an integer, is the second name of
the last file written. Each time a file is written, this value is first
increased by 1, so it must always be an integer. It is initially set to 0.
FILESIZE - The value of this variable is the number of expressions written
into each file. The default value is 16.
DEVICE - The value of this variable is the default device. It is
initialized to DSK.
UNAME - The value of this variable is the default sname. It is
initialized to the user's login name, if he has a disk directory, and to USERS
otherwise. UNAME determines to what directory disk files will be written.
DIREC - may be used as an alias for UNAME.
DSKALL - If TRUE will cause values, functions, arrays, and rules to be
written periodically onto the disk in addition to labelled expressions. TRUE
is the default value whereas if DSKALL is FALSE then only labelled expresions
will be written.
MACSYMA 209
15.3 Explicit Storage of Expressions
15.3.1 Use of the storage functions
The functions SAVE, STORE, and FASSAVE allow the user to explicitly state
that certain expressions should be written onto the disk. These functions
also allow him to specify the file into which these expressions should be
written. They allow the user to store away arrays, function definitions,
rules, and any other type of information. The main purpose of these functions
is to allow the user to save expressions onto the disk so that they can be
read into future MACSYMAs.
SAVE and STORE are identical in all respects but one. When an expression
is STOREd it is both written onto the disk and removed from main memory. (When
the expression is referenced, of course, the correct value is retrieved from
the disk.) When an expression is SAVEd, it is written onto the disk but not
removed from main memory. The only difference between these two functions is
their effect on main memory storage.
FASSAVE is similar to SAVE but produces a FASL file in which the sharing
of subexpressions which are shared in core is preserved in the file created.
Hence, expressions which have common subexpressions will consume less space
when loaded back from a file created by FASSAVE rather than by SAVE. The user
should note that FASSAVE files are not as flexible as SAVE files since the
RESTORE function (see below) cannot be applied to them. Also if the user's
MACSYMA is already near maximal allocation, FASSAVE , which uses a
considerable amount of space and time in doing its job, may not work, whereas
SAVE still might.
SAVE, STORE, FASSAVE take any number of arguments. If the first argument
is a list it is assumed to be the file specification (e.g. [fn1, fn2, DSK,
directory]). In accordance with the standard options for file specifications,
the latter arguments may be omitted from the list and the defaults will be
assumed. If the first argument is not a list, the expressions will be written
into a file with the default filename. The value of the MACSYMA variable
FILENAME is the default first filename, and the value of the MACSYMA variable
FILENUM is the default second filename. The value of FILENUM is increased by
1 each time a file is written, so its value must always be an integer. FILENUM
is initially 0. The value of DEV is the default device, and the value of
UNAME is the default username.
All arguments to SAVE or STORE, except possibly the first, must be one of
the following:
(1) The name of an "information list" (see 8.1). SAVE(VALUES) will not
cause MACSYMA options (e.g. SHOWTIME, RATPRINT, etc.) to be saved. Also
ALIASES will be automatically saved with every use of the SAVE function if
they exist.
210 15.3.1 Use of the storage functions MACSYMA
(2) ALL When this atom is an argument every quantity associated with any
information list is written.
(3) [m,n] when this list is given as an argument, every label whose line
number lies between m and n inclusive gets written.
(4) When any other atom is an argument, it must be either an array, a
function, or have a value. It gets written onto the disk.
(5) A=B The effect is similar to the case where the argument is just B,
i.e. B gets written onto the disk. The only difference shows up if the file
is read into some future MACSYMA. In that case, the expression which is
referred to as "B" in the present MACSYMA will be referred to as "A" in the
future MACSYMA. For example, suppose the user wishes to save some expression,
say D7, for use in a future MACSYMA. He can execute STORE([FOO, BAR],
YESTERDAYSD7 = D7). D7 is now stored onto the disk. When he comes back the
following day and load in a fresh MACSYMA he merely executes LOADFILE(FOO,
BAR, DSK, ECR) and the variable YESTERDAYSD7 will take on the value which D7
had yesterday. This renaming however has no effect on the present MACSYMA,
where D7 must still be referred to as "D7". Note that if a SAVEd or STOREd
file contains labelled expressions they may conflict with expressions having
the same label in the MACSYMA into which the file is loaded. For example if
D9 is in a file which is loaded into a MACSYMA then it would replace the D9
which was already in the MACSYMA (if there was a D9 generated), or it would
itself be replaced by D9 when the new D9 was generated. To avoid this
difficulty the user should give labelled expressions a name as described in
(4) above. He could also set LINENUM in the new MACSYMA or save it from the
old one so that line numbers wouldn't conflict.
The user should note that each use of the SAVE or STORE function will
cause exactly one file to be written, regardless of the number of arguments
the function is given.
REMFILE(TRUE) will perform REMFILE() (see 15.2 - B) and in addition will
delete files created by SAVE or STORE which haven't been assigned names
explicitly by the user.
Certain MACSYMA variables (i.e. LINENUM, FILESIZE, etc.) are used to
communicate to the MACSYMA system that certain options are in effect, or to
tell the system to use certain values. These variables should not be STOREd
(though they may be SAVEd), since the system programs will not be able to
correctly retrieve their values from the disk. In general, one should not
attempt to STORE variables whose purpose is to provide information to the
system (i.e. MACSYMA options).
15.3.2 Retrieval of expressions stored on disk
1 - In the MACSYMA you are using
MACSYMA 15.3.2 Retrieval of expressions stored on disk 211
Expressions which are written onto the disk using the SAVE function also
reside in main memory, so the notion of retrieving them from the disk in the
present MACSYMA is not applicable. Expressions written onto the disk using
STORE, however, no longer reside in main memory. When such expressions are
referenced the system will always retrieve the correct value from the disk.
When a STOREd array is referenced, the array will be brought back to main
memory. Functions and values will be read from the disk correctly, but will
not be returned to main memory. If the user wants to bring an expression back
to main memory he may use the function UNSTORE. This function takes any
number of arguments. Each argument must be an atomic variable, and if this
atomic variable refers to an expression which is stored onto the disk, the
expression is returned to main memory. Of course, when an expression is
UNSTOREd, either by the user or by the system (as happens when STOREd arrays
are accessed), a copy of the expression still remains on the disk in the
assigned file.
2 - In future MACSYMAs
Files created by SAVE and STORE can be loaded into future MACSYMAs using
the LOADFILE function. This will set up in main memory all those expressions
which were written into the file. Some of the expressions will have different
names than they had in the MACSYMA where they were created, if the renaming
option (i.e. arguments of the form A=B) of the STORE or SAVE function was
used. Also, unless the FASSAVE scheme was used, expressions will generally
take up more space than they did in the MACSYMA where they were created, as
sharing among common subexpressions will be lost.
15.4 Saving a MACSYMA Overnight
Often a user in the middle of his work would like to save everything onto
the disk so he can go home and resume work tomorrow. When the user decides to
save the state of his MACSYMA, he should execute for example:
SAVE([fn1,fn2,DSK,directory],ALL)
This will write all his lines, arrays, functions, values, rules, and
aliases (if he has created any), and the current value of LINENUM into a
single disk file named fn1 fn2 (where these may be any names given by the
user). Of course, the user should choose names for his files which are
unique. If he does not have his own directory then he should use the USERS
directory and his login name for the first file name. If the automatic
storage scheme was in effect he should now execute REMFILE(); to delete
useless files from the disk. When the user comes back the next day he should
load a fresh MACSYMA and execute one of the following two functions:
LOADFILE(fn1, fn2, DSK, directory);
or RESTORE(fn1, fn2, DSK, directory);
212 15.4 Saving a MACSYMA Overnight MACSYMA
The first command will cause all expressions to be loaded into the present
MACSYMA. Whereas all the expressions may have fit into the MACSYMA in which
they were generated, they may not fit into a new MACSYMA, because common
subexpressions originally shared will not be shared in a new MACSYMA. However
if the OPTIMIZE function is used (see 6.2.3), then some sharing of common
subexpressions may be obtained. The RESTORE function does not cause the
expressions to be loaded into main memory but does permit them to be accessed
when needed. (This is as though STORE had been used on the information.) Thus
it should be used if it is not desired to bring all the expressions into main
memory at the same time.
MACSYMA 213
16 Storage Management
In the LISP system in which MACSYMA resides, the space requirements of the
user's programs and data may be increased during the execution of the
programs. This is in contrast to static storage allocation systems in which
the storage is completely allocated before the programs are executed and
consequently the storage requirements must be completely known before
execution time and cannot be changed during program execution. If they exceed
the capacity of the memory space that has been allocated for them, the
programs will not be allowed to run.
With our LISP system, a certain amount of space is initially allocated,
the programs are started running, and the amount of space utilized changes
during execution. If at some point the limit of available space is exceeded,
program execution will be terminated.
This LISP divides up the available memory spaces into several portions on
the basis of what kind of data they will contain.
BPS - (binary program space) for compiled functions and arrays.
FIXNUM - for integers which fit into one machine word.
FLONUM - for floating point numbers.
BIGNUM - the first word of numbers bigger than one machine word.
SYMBOL - for atomic symbols.
ARRAY - for array indicators.
LIST - for anything else not in the other spaces, e.g. uncompiled functions,
symbolic expressions, etc.
PDL - for several kinds of pushdown lists.
When a MACSYMA is started up, each space is initially allocated some fixed
amount. These spaces will grow as the user interacts with MACSYMA, each
particular space growing as the user causes more objects to be created which
reside in that space. For example, executing a command line which causes an
out-of-core file to be loaded mainly increases BPS and LIST space. Also new
labelled expressions are created every time a command line is executed and
these occupy LIST space. Push down lists are used to store variables, return
addresses, and other information related to the function calling mechanism.
When a space (except for BPS and the PDLs) is used up, a process is
initiated called "garbage collection" which attempts to free up storage so it
can be reused rather than trying to increase the size of the spaces. In very
simple terms, it marks every word in a particular storage space which is still
214 16 Storage Management MACSYMA
being utilized and then links up the unmarked words (termed "garbage") on a
chain to be used to store subsequently created data. If this chain is not of
a certain assigned minimal size, a special allocation routine is invoked. At
this point several possibilities can occur, among which is the possiblity of
increasing the size of the spaces. Before these are described however, there
is something to be mentioned which the user should take note of. The initial
allocations (later to be referred to as "level 0 allocations") are quite
reasonable. Many problems run quite well using these allocations. If the
user's problem does not run due to running out of storage, most often this is
due to one of the following circumstances and not to the insufficiency of the
allocations:
(1) The user has organized his problem poorly, thereby either not
solving the problem he intended to solve, biting off too much in too short
a time, or creating intermediate expression swell of perhaps incredible
proportions. He should get a feel for the size of all of his expressions
and the behavior of MACSYMA's functions on them.
(2) The user is retaining expressions in core that are useless to him.
Since MACSYMA maintains a complete history of the user's session, it does
not release the storage occupied by the user's data unless explicitly
instructed to do so. This can be accomplished in several ways. One way is
by using the functions KILL, REMVALUE, REMFUNCTION, and REMARRAY (see
10.3) which unbind an item from the expression it represents thus freeing
up the storage occupied by the expression to be reclaimed on the next
garbage collection. It is recommended that the user give a name to all
labelled expressions which he wishes to keep around for a time, and then
periodically do a KILL(LABELS). He should also KILL functions and arrays
which he no longer needs. Killing a name will not accomplish much unless
the labelled expression (D line) at which the assignment was done is also
KILLed because the two symbols are holding onto the same expresson.
If the above approach is unacceptable because the (intermediate or
final) expressions which the user needs occupy a lot of storage, he can
store them on the disk (kill the corresponding label if any) and have them
retrieved when needed. This can be accomplished in the following two ways
(see Chapter 15).
The STORE function can be used to explicitly transfer expressions from
main memory to disk. These will be brought back into main memory each
time they are referenced. Since the process of retrieving an expression
from a large file may be rather time consuming, the user can use the
UNSTORE function to bring an expression back to main memory and keep it
there. However, in general an UNSTOREd expression will occupy more space
then it originally did because several copies of common subexpressions
will be created. These were originally represented by pointers to a
single copy.
Another way to transfer expressions to the disk is by using the
automatic storage mechanism. This will cause expressions to be STOREd
automatically in an effort to conserve space. As with the explicit use of
MACSYMA 16 Storage Management 215
the STORE function, the expressions will be retrieved by MACSYMA when they
are referenced. There are two heuristics which MACSYMA uses to decide when
to store expressions. One is to STORE all values, functions, arrays, and
labelled expressions whenever available space becomes low. This is the
purpose of the DSKGC function. The other heuristic is to periodically
STORE a fixed number of labelled expressions whenever that many get
created above a minimum number which are to be kept around. The user can
utilize this option by setting the switch DSKUSE to TRUE. In addition if
DSKALL is TRUE, then all values, functions, and arrays will be written at
this time as well.
(3) A recursive infinite loop has occurred because of a bug in either
the user's code or in MACSYMA's code. Such a loop would cause storage
spaces to grow indefinitely if possible. It may be difficult for the user
to recognize that this situation has occurred, as opposed to a real need
for more space, yet such a situation can cause havoc to any dynamic
allocation scheme. By typing a control-D the user will be informed
whenever a garbage collection occurs and a printout such as the following
will occur:
;GC DUE TO ... SPACE
;2729[33%] LIST, 1935[94%] FIXNUM, 511[99%] FLONUM,
; 509[99%] BIGNUM, 629[15%] SYMBOL,
; 480[93%] ARRAY WORDS FREE
The numbers before the name of each space give the number of words of
that space which are available. The percentages refer to the ratio of the
available amount of space compared to the total amount (used plus unused).
This may be of some help in determining whether the user's computation is
doing what he expected. Typing a control-C will turn off this g.c.
printout.
If the user feels that he is not in one of the above situations and that
there is a real need for more space, he can avail himself of the following
scheme:
When a space, e.g. LIST is exhausted, MACSYMA will print:
You have run out of LIST space
Do you want more?
Type ALL; NONE; a level-no.; or the name of a space;
At this point the user can type a control-A and enter a (MACSYMA-BREAK).
He can then follow any of the procedures mentioned in (2) above. After
getting out of the break by typing EXIT; he can then type OK; which indicates
that the user believes he has freed up enough space thus making reallocation
unnecessary. If he is wrong, he will get the above message again. He can also
reply with the name of a space, i.e. one of FIXNUM, FLONUM, BIGNUM, SYMBOL, or
216 16 Storage Management MACSYMA
ARRAY which will cause the size of that space to be expanded. If he replies
with LIST or ALL then not only will list space be increased, but so will every
other space. This is because if he needs more list space, then he probably
needs more of the other spaces as well. Replying LIST also increments the
"allocation level" by one. There are 5 equi-spaced levels of allocation,
ranging from the initial level 0 to level 4. The user can also boost his
MACSYMA up to that particular level of allocation immediately by replying with
that level number.
At level 4 the maximal allocation possible on the computer is just about
exhausted so that there is no higher level. Also, the amount of core space
devoted to the allocatible storage spaces is obviously inversely proportional
to the number and size of out-of-core files which are loaded in. If many out-
of-core files are loaded in, allocation level 4 will not even be attainable.
The user may wish for this reason alone to continue now and then with a fresh
MACSYMA. If the user's computation exceeds this level of storage it will
error out with the message "...STORAGE CAPACITY EXCEEDED". Since the size of
the spaces can't be decreased, it is important not to increase them unless it
is necessary.
If the user knows initially that his problem will require much space, the
function ALLOC is provided. ALLOC takes any number of arguments which are the
same as the replies to the "run out of space" question above. It increases
allocations accordingly.
Lastly, if he is running a MACSYMA disowned, or for other reasons wishes
storage space to be increased automatically as needed without having any
questions asked, the user may set the switch DYNAMALLOC [FALSE] to TRUE which
will allocate additional space whenever necessary.
Like the other spaces BPS and the PDLs cannot be decreased. BPS will
continue to grow until the MACSYMA runs out of address space so caution should
be exercised in causing out-of-core files to be loaded. For example, once the
integration file is loaded into the user's MACSYMA it is there to stay even if
he no longer uses it. The allocation of the PDLs is sufficiently large, so
that if they are ever caused to overflow it is probably due to a recursive
infinite loop in the user's programs.
MACSYMA 217
17 Simple Plotting Functions
The MACSYMA functions PLOT,PARAMPLOT,GRAPH, and MULTIGRAPH produce
character plots of specified functions and sets of data points. (They can
also be used to produce output files for plotting on the Calcomp plotter or
XGP attached to the PDP-10 used by the Artificial Intelligence Group). The
format of these functions and the variables used by the corresponding routines
are described below:
Variables
LINEL - width of graphing area in terms of number of characters.
PLOTHEIGHT - height of graph in terms of number of characters.
XAXIS[FALSE] - if set to TRUE will cause the Y=0 axis to be displayed.
YAXIS[FALSE] - if set to TRUE will cause the X=0 axis to be displayed.
Formats for PLOT and PARAMPLOT
PLOT(F(x), x, low, high) - plots the expression F(x) over the domain low < x <
high.
PLOT(F(x), x, low, high, INTEGER) - as above, but plots F(x) only for integer
values of x.
PLOT(F(x), x, [x1, x2, x3, ..., xn]) - plots the function F(x) for the values
x1,x2,x3,...,xn.
The first argument to PLOT may also be a list of functions rather than
just a single function. This permits several functions to be plotted on the
same set of axes. Three optional final arguments may also be given. They are:
(1) an X axis label (quoted string or name), (2) a Y axis label, and (3) a
list of plotting characters used for the given function(s) enclosed in "s.
(Note also that if a special symbol such as ; , $ , etc. is used it must be
preceded by a \.) An * will be used to plot any functions which are not given
a particular plotting character.
PARAMPLOT(f1(t),f2(t),t,low,high) plots the plane curve f(t) = ( f1(t) , f2(t)
) with parameter t. The syntax is basically like that of plot. For
example,
PARAMPLOT(COS(T),SIN(T),T,0,2*%PI)
218 17 Simple Plotting Functions MACSYMA
plots a circle. Also several curves may be presented at the same time by
using the following syntax
PARAMPLOT([f1(t), g1(t), ... h1(t)], [f2(t), g2(t), ..., h2(t)], t, low, high,
[list of plotting characters]) - plots the plane curves f(t) = (f1,f2),
g(t) = (g1,g2),...,h(t) = (h1,h2) using the specified plotting characters
or the default "*". For example,
PARAMPLOT([COS(T),COS(T)+7],[SIN(T),SIN(T)],
T,0,2*%PI,["@"])
plots two circles.
The user may wish to TRANSLATE or COMPFILE (see 10.8) the functions to be
plotted as they might be evaluated many times.
Formats for GRAPH and MULTIGRAPH:
GRAPH([x1, x2, x3, ..., xn], [y1, y2, y3, ..., yn]) - Graphs the two sets of
data points.
GRAPH([[x1, y1], [x2, y2], ..., [xn, yn]]) - Graphs the points specified by
the list of coordinate pairs.
GRAPH(xset, [yset1, yset2,..., ysetn],optional-args) - allows graphing of one
x-domain with several y-ranges; e.g. GRAPH([0,1],[[0, 1],[1,2]],["&"]).
MULTIGRAPH([ [xset1, yset1], ..., [xsetn, ysetn] ], optional-args) - allows
the user to produce a scatter-graph involving several x-domains each with
a single y-range; e.g. MULTIGRAPH([ [[0,1],[0,1]], [[3,4],[1,2]] ],
["&"]).
The three optional final arguments mentioned above under PLOT may also be
used with GRAPH and MULTIGRAPH.
The plot produced by the above functions is a character plot on a
coordinate system defined by axes along the minimum x and y values of the
plot. The x and y coordinates are independently scaled to optimally use the
specified graphing area. Note that this may distort the shape of the graph
e.g., a circle could become an ellipse. The origin of the graph (left-hand
corner) is given on the graph by the values of XORG and YORG; the computed
increments (= one character) are given by the values of XDELTA and YDELTA and
the maximum X and Y values are given by XMAX and YMAX. The axes are labeled
with the number sequence 0,2,4,6,8,0,2,4,... as an aid in counting the number
of increments from the origin.
MACSYMA 17 Simple Plotting Functions 219
When a graph is completed, the user must type a single character (on non-
printing consoles), such as space or carriage return, to return control to
MACSYMA.
Examples
(C1) XAXIS:YAXIS:TRUE$
(C2) PLOT([1/(X↑2+1),X↑2-1],X,-2,2,[@]);
4[ * .
[ . *
2[ .
[ * .
0[ . *
[ * .
8[ . *
[ * .
6[ . *
[ * .
4[ . *
[ * .
2[ @@@@@@@ *
[ * @@@ . @@@
0[ @@ . @@ *
[ * @@ . @@@
8[ @@@ . @@ *
[ * @@ . @@@
6[ @@@@ . @@@*
[ @@@@ . *@@@@
4[ @@@@@@ * . @@@@@@
[@@@ * . * @@@
2[ . *
[................*.................................................
0[ * . *
[ * . *
8[ . *
[ * . *
6[ * . *
[ * . *
4[ ** . *
[ * . **
2[ ** . *
[ ** . ***
0[ *******
-------------------------------------------------------------------
0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 0 2 4
XORG=-2.0 YORG=-0.999744 XDELTA=0.062 YDELTA=0.091
XMAX=1.96800017 YMAX=3.0
220 17 Simple Plotting Functions MACSYMA
(C2) POLARPLOT(RHO,NUMBREV):=BLOCK([THETA, LIMIT, X, Y, P, NUMER,
RATPRINT], NUMER:TRUE, RATPRINT:FALSE, THETA:0.0, X:Y:[],
LIMIT:72*NUMBREV, FOR I:1 THRU LIMIT DO
( P:RHO(THETA), X:CONS(P*COS(THETA),X),
Y:CONS(P*SIN(THETA),Y), THETA:THETA+%PI/36.0),
GRAPH(X,Y,X,Y))$
(C3) XAXIS:YAXIS:FALSE$
(C4) F(T):=4-COS(8.0/3.0*T)$
MACSYMA 17 Simple Plotting Functions 221
(C5) POLARPLOT(F,3)$
6[ * * * * * * * *
[ * * * * * * * * * * *
4[ * * * * * *
[ * * * * * * ** * * * * * * * * * *** * * * ** * *
Y 2[ * * ** ** * *
[ ** * * * ** * ** * * * *
0[* * *** *** * *
[** * * * * * * **
8[ * * * *
[** * * * * * * **
6[* * *** *** * *
[ * ** * * * ** * ** *
4[ * * * *** *** * **
[ * * * ** * * * * * * * * * *** ** * *
2[ ** * * * * * * * * * *
[ * * * * * * * *
0[ * * * * * * * * * * *
---------------------------------------------------------------------------
0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 0 2
XORG=-4.77137667 YORG=-4.77137756 XDELTA=0.14 YDELTA=0.6
XMAX=4.7713773 YMAX=4.77137697
X
222 MACSYMA
18 Complex Plotting and Graphics---PLOT2
18.1 Introduction
This is a description of the functions contained in the following 6 files:
APLOT2 > ARC CFFK, TEKPLT > ARC CFFK, PLOT3D > ARC CFFK
WORLD > ARC CFFK, PRINT > ARC CFFK, IFFUN > ARC CFFK.
They will be loaded up as needed when most of the functions below are used.
However to get the "complete" PLOT2 environment, LOADFILE(PLOT2,LISP,
DSK,SHARE) is recommended. This loads in complete AUTOLOAD properties for the
functions described below. The effect of loading this file will be to cause
the PLOT2 package to be automatically loaded when you need it. If you use
PLOT2 regularly, you should consider including PLOT2 LISP DSK SHARE in your
MACSYM (INIT) file; it will take up neglegible room in your MACSYM.
The capabilities of the routines described here include plotting of
several curves on a single graph, plotting several graphs in different
positions on the screen, saving plots, replotting plots with different scales
without having to recompute any points, plotting of 3 dimensional surfaces,
plotting of user defined dashed lines and symbols.
The devices supported are: the Tektronix 4010 and 4013, the Imlac PDS 1
and PDS 4 (using ARDS graphics conventions), the XGP and the Gould line-
printer (at MIT) and printing and display consoles in a "preview" mode.
18.2 Two-Dimensional Plotting
PLOT2(y-exprs,variable,var-range,optionals-args) plots y-exprs on the y axis
as variable (the x axis) takes on values specified by var-range.
y-exprs can take one of two forms:
i) expr plots a curve of expr against variable
ii) [expr1,expr2, .. ,expri, .. ,exprn] plots n curves of
expri against variable. expri gets evaluated in the context
FLOAT(EV(expri,variable=value gotten from var-range, NUMER)).
It is an error if this doesn't result in a floating point
number.
var-range can have the following forms:
i) low,high, where low and high evaluate to numbers. low may
be either greater or less than high. variable will take on
CALCOMPNUM values equally spaced between low and high.
MACSYMA 18.2 Two-Dimensional Plotting 223
Note that the first argument will be evaluated at low
first e.g. PLOT2(1/X,X,-1,-3); calculates 1/(-1.0)
before 1/(-3.0). This will only make a difference if the
computation of the first arg changes a variable which
changes the value returned by subsequent computation.
Whether or not low < high, min(low,high) will be on the
left of the plot.
ii) low,high,INTEGER. As in i), except variable will take on
all integer values between low and high inclusive.
iii) [val1,val2, ... , valn] . variable takes the values
specified by the list
iv) arrayname where arrayname is the name of a declared
floating-point one-dimensional array (i.e. declared by
ARRAY(arrayname,FLOAT, max-index);). variable takes the
values from arrayname[0] thru arrayname[max-index] (max-index
is the maximum index of arrayname
optional-args can be any of the following:
i) X-Label, Y-Label or Title descriptor
ii) Line type descriptor
iii) FIRST, SAME and LAST
iv) POLAR, LOG, LINLOG, LOGLIN
The optional arguments may appear in any order. The rule for evaluation
of the optional args is as follows. If the argument is atomic it gets
evaluated. The resulting arguments are the ones that get used.
If you want to plot more than 3-4 curves on the same plot investigate
using the NOT3D option to PLOT3D (see 18.4)
Examples
PLOT2(SIN(X),X,-%PI,%PI); plots sin(X) against X as X takes on
CALCOMPNUM values between -%PI to %PI
PLOT2(X!,X,0,6,INTEGER); plots X! as X takes integral values
between 0 and 6
F(X):=SQRT(X);
PLOT2(F(X),X,[-2,3,100.12]); plots F(X) as X takes the values in the
values in the list
PLOT2([X+1,X↑2+1],X,-1,1); plots 2 curves on top of each other
PLOT2(y-funs,var-range,optionals-args) is the alternative form for PLOT2. y-
funs must be a function of one argument or a list of functions of one
argument. The functions must be either translated or compiled functions
which return a floating point number when it is given floating point arg
(or integer arg if the INTEGER arg to PLOT2 is given). This form of PLOT2
acts as though you had given a argument to the y-funs, and also specified
224 18.2 Two-Dimensional Plotting MACSYMA
that argument as the variable in the form above. E.g. PLOT2(F,-2,2); acts
like PLOT2(F(X),X,-2,2); This is supposed to provide a quicker evaluation
of the first arg and for that reason NO checking is done on the result.
If the wrong kind of number is returned, the resulting plot will not be
meaningful.
TRANSLATE:TRUE;
F(X):=(MODEDECLARE(X,FLOAT),EXP(-X*X));
PLOT2(F,-2,2);
PLOT2(F,[-2,-1,0,1,2]);
ARRAY(V,FLOAT,10);
FOR I FROM 0 THRU 10 DO V[I]:FLOAT(I*I);
PLOT2(F,V);
GRAPH2(x-lists,y-lists,optional-args) plots points specified by the first x-
lists and y-lists. The format for x-lists can be one of
i) [x-pt1,x-pt2, .. ,x-pti, .. ,x-ptn] where x-pti evaluates
to a number
ii) arrayname where arrayname is the name of a declared one-
dimensional array of floating point numbers
iii) 2d-arrayname where 2d-arrayname is the name of a
declared two-dimensional array of floating point numbers
(i.e. by ARRAY(2d-arrayname, FLOAT, max-row-index,max-col-
index);)
iv) [x-list1,x-list2, .. ,x-listi, .. ,x-listk] where x-listi
can have the form of either i) or ii).
The format of y-lists is similar. The format of optional-args is the same
as for PLOT2.
Note that GRAPH2 performs the same job as the MACSYMA function
MULTIGRAPH and that GRAPH2 is thus slightly incompatible with GRAPH FASL.
PARAMPLOT2(x-exprs,y-exprs,variable,var-range,optional-args) plots x-exprs as
the x coordinate against y-exprs as the y coordinate.
The format for the first two arguments is the same as that for the first
argument to PLOT2. Thus if x-exprs is [x-expr1,x-expr2,...,x-expri,...,x-
exprn] and y-exprs is [y-expr1,y-expr2, .. ,y-expri, .. ,y-exprk], then
max(n,k) curves will be plotted. They will be (assuming n > k):
x-expr1 vs. y-expr1, .. ,x-exprk vs. y-exprk,
x-expr(k+1) vs. y-exprk, .. ,x-exprn vs. y-exprk
The format for the remaining arguments is the same as for PLOT2.
MACSYMA 18.2 Two-Dimensional Plotting 225
PARAMPLOT2(x-funs.y-funs,var-range,optional-args) efficiently evaluates its
first 2 arguments in the same way that the alternative form of PLOT2
works.
Examples
TRANSLATE:TRUE; causes automatic translation
F(X):=(MODEDECLARE(X,FLOAT),COS(X));
G(X):=(MODEDECLARE(X,FLOAT),SIN(X));
PARAMPLOT2(F,G,0,2*%PI); plots F(x) vs G(x) as x goes from 0 to 2*%PI
PARAMPLOT2(SIN(T),COS(T),T,0,2*%PI);
plots sin(T) for the x-axis and cos(T) for the y-axis as T takes on CALCOMPNUM
(see 18.2) values between 0 and 2*%PI. (If EQUALSCALE is TRUE this draws a
circle.)
CALCOMPNUM[20] is the number of points PLOT2 and PARAMPLOT2 plot when given
the low,high type of variable range. The default if you load up
PLOT2,LISP,DSK,SHARE is 20, which is sufficient for trying things out. 100 is
a suitable value for a final hard copy.
18.3 What to Type When PLOT2 has Finished Plotting
When a plot is finished the bell on your terminal will be dinged. (In
fact, this happens only if WAIT is TRUE and if you're plotting on the
terminal.) The plotting function is now waiting for you to type something
before it exits and prints the next C-Label on your nice plot. It does
nothing with most characters; they are left to be part of the next C-Line.
However, the following characters are read and interpreted specially.
space exits the plotting function.
return clears the screen and then exits.
tab causes the previous plot to be replotted.
This is useful if the line is noisy, or, in conjunction with
control-A if various plotting parameters need to be changed.
linefeed sends out a hardcopy signal.
At present this can only be used to generate hardcopy on the
Tektronix hardcopy unit or on the Gould line-printer.
rubout names the plot.
PLOT2 types out "Enter name of plot" you reply plotname; this is
identical to typing a space instead of a rubout, followed by
NAMEPLOT(plotname);.
control-A
This is not really read by the plotting function, but enables you
226 18.3 What to Type When PLOT2 has Finished Plotting MACSYMA
to enter a control-A break and change various options, before
typing tab to have the plot replotted. Make sure the plot has
finished before you type this.
PLOTBELL[TRUE] when FALSE inhibits the dinging of the bell.
While your plot is coming out your terminal is in a rather strange state (e.g.
not echoing characters). Thus it is OK to type ahead to MACSYMA, but the ONLY
way you should interrupt the plot is with control-↑. E.g. do not use control-
A until the plot has finished.
18.4 Three-Dimensional Plotting
PLOT3D(z-exprs,x-var,var-range,y-var,var1-range,optional-args) makes a 3-
dimensional plot of z-exprs against x-var and y-var. The plot consists of
curves of y-exprs against x-var (the x coordinate) with y-var (the y
coordinate) held fixed. Perspective is used and curves further away from
the viewer have those parts of them which are hidden by the closer curves
removed.
The format of y-exprs is the same as for PLOT2. The context of evaluation
is FLOAT(EV(expri,x-var=value gotten from var-range,y-var=value gotten
from var1-range,NUMER)).
The format for var-range and var1-range is the same as for PLOT2
except that if var1-range is of the form low,high then y-var will take on
CALCOMPNUM1 values.
The format of optional-args is the same as for PLOT2 except that
additional options NOT3D, \3D, CONTOUR are available.
PLOT3D(z-funs,var-range,var1-range,optional-args) is analogous to the
alternative form for PLOT2. z-funs must be a function or list of
functions of 2 arguments, which must return a floating point argument when
given floating point (integer, if the INTEGER argument is used for either
var-range or var1-range) arguments. The functions must be translated or
compiled. If you expect to make several 3D plots this form is
recommended.
A simple example is
TRANSLATE:TRUE; causes automatic translation
G(X,Y):=(MODECLARE(X,FLOAT),EXP(-X*X-Y*Y)); defines a funtion G
PLOT3D(G,-2,2,-2,2); plots it
GRAPH3D(x-lists,y-lists,z-listsoptional-args) takes 3 arguments (GRAPH2 takes
2) and interprets them as lists of x, y, and z points which it uses to
draw lines using the 3d transformations. It can be used to add lines
(e.g. axes) to your 3D plot. The hidden line routines are not used.
MACSYMA 18.4 Three-Dimensional Plotting 227
NOT3D
In this section we describe the option NOT3D. As an example, consider
PLOT3D(SIN(X)+A,X,-%PI,%PI,A,[-2,3,4,6],NOT3D)
which plots sin(X)+A for X from -%PI to %PI (CALCOMPNUM[20] points) and A
taking the values in the list. This is equivalent to:
PLOT2([SIN(X)-2,SIN(X)+3,SIN(X)+4,SIN(X)+6],X,-%PI,%PI)
but requires less typing.
The additional NOT3D argument to PLOT3D, causes exactly the same points as
in the bare PLOT3D to be calculated. Instead of plotting a 3-dimensional
representation of the data, the data is plotted in a 2D one. Specifically 1
2D curve of z vs. x for each y value, and so is a convenient way to plot
several curves on the same plot.
PERSPECTIVE, REVERSE, VIEWPT, and CENTERPLOT
The following options govern the type of perspective view given.
PERSPECTIVE[TRUE], if FALSE causes a non-perspective view to be taken.
This is equivalent to extending the viewing position out to infinity along a
line connecting the origin and VIEWPT.
REVERSE[FALSE], if TRUE cause a left-handed coordinate system to be
assumed.
VIEWPT and CENTERPLOT determine the perspective view taken. They are
defaulted to be unbound. VIEWPT may be set to a list of 3 numbers and gives
the point from which the projection should be made. CENTERPLOT may likewise
be set to a list of 3 numbers and gives a point on the line of sight. The
projection will be made onto a plane perpendicular to a line joining VIEWPT
and CENTERPLOT.
If VIEWPT and CENTERPLOT are unbound (the default) then they will be
chosen as follows: the extreme values of the coordinates are determined. This
gives the two points min: [xmin,ymin,zmin], max:[xmax,ymax,zmax]. CENTERPLOT
is chosen as (min+max)/2, and VIEWPT is chosen as max+3*(max-min). The view
is then one in which the z axis is vertical, the x axis is increasing towards
you to the left and the y axis is increasing towards you to the right.
If CENTERPLOT is FALSE then the old type of perspective view will be given
(like setting the x and z components of CENTERPLOT to the corresponding
components of VIEWPT).
18.5 Using the XGP from PLOT2
228 18.5 Using the XGP from PLOT2 MACSYMA
To get plots out on the XGP, simply do PLOTMODE(XGP, ...) where ... is the
correct plot mode for your terminal (i.e. GR for Grinnell TV's, T for
Tektronix, or D for character display terminals like VT52's). Then you can
use linefeed at the end of the display of a plot, or HARDCOPY(); to cause a
hardcopy to be submitted to the XGP (it is processed by the Gould spooler
1
first). If MIT-AI is up and the queuing for the XGP is successful, you will
receive a message from the XGP spooler when your plot is printed. If MIT-AI
is down, your plot will be processed into an XGP scan file, .GLPT.; > SCN, and
the Gould spooler will send you a warning message telling you that MIT-AI is
down. You must copy the SCN file to MIT-AI yourself when MIT-AI comes up and
queue it by doing
:XGP ;SCAN AI:dir;* SCN
where dir is the directory on MIT-AI to which you copied the files (use
MACSYM; if you don't know of any other directory to use).
2
If you use the NAMEFILE command instead of HARDCOPY or linefeed, you can
print the plot file later by doing
:GTPL dir;fn1 fn2/a/o[x]
where dir, fn1, fn2 are the directory, first file name, and second file name
of the plot file stored with NAMEFILE. PLOTLFTMAR[128] and PLOTBOTMAR[320]
adjust the bottom margin and left margin for the XGP plots. These default to
3
values such that the plots will fit comfortably on an 8 1/2 x 11 page.
There are many other features of PLOT2 (such as three-dimensional and
contour plotting) which the user can learn about by reading SHARE;PLOT2 USAGE
and SHARE;PLOT2 RECENT. As an example, consider
(PLOTMODE(XGP,D),WINDOW:[100,900,0,950])$
(CALCOMPNUM:CALCOMPNUM1:40,VIEWPT:[-30,-20,5])$
PLOT3D((X↑3+Y↑4-0.2*X)*EXP(-X↑2-Y↑2)+0.3*EXP(-(X-1.225)↑2-Y↑2),
X,-3,3,Y,-3,3);
NAMEFILE(PLOT,TEST,DSK,CFFK);
The resulting plot is shown on the next page
←←←←←←←←←←←←←←←
1. The computer at the MIT AI Lab
2. NAMEFILE(filespec) simply copies the PLOT2 scratch file to filespec
3. Whose units are given in increments of 1/200 th of an inch
230 MACSYMA
19 Debugging in MACSYMA
When the user's command lines, especially functions and BLOCK programs, do
not do what is expected or generate errors, MACSYMA offers several debugging
alternatives:
(1) The user may trace function calls by typing TRACE(fun1,fun2,...),
where the funi are either MACSYMA or user-defined functions. This will cause
a printout of the function name and its arguments each time it is entered, and
of the function name and the value it returns each time it is exited. A count
which is the level of recursion is also printed. Usually, this is all the
tracing power the user will need, although MACSYMA offers the full
capabilities of the LISP tracing package including conditional and breakpoint
tracing. This will not be described here - for information see [Mn1].
MACSYMA uses trace-syntax very similar to that of LISP.
To check which functions are currently under trace, the user may type
TRACE(). To remove tracing of functions use UNTRACE(fun1,fun2,...). To
untrace all previously traced functions type UNTRACE(). Since the TRACE
package takes up some of the user's workspace in core, when finished with it
the user should type REMTRACE(). It can always be reloaded again if
necessary.
(2) The assignment of variables can be traced by setting the variable
SETCHECK to a list of variables (which can be subscripted). When a variable
on the list is bound (either with : or :: or function argument binding) then a
message -- variable "SET TO" value -- will be printed. If the variable
SETCHECKBREAK is set to TRUE then a (MACSYMA-BREAK) will be caused each time a
variable on the SETCHECK list is bound.
(3) By setting the variable REFCHECK[FALSE] to TRUE, the user will be
informed when each of his variables which has a value comes up for evaluation
for the first time during the course of a computation. This has a dual
purpose. The user will be informed of evaluations he may not have been aware
of which are the result of assignments he made long ago. It also gives him a
sort of chronological trace of his computations which may be helpful in
finding out where an error has occurred.
(4) By setting the variable PREDERROR to TRUE, the user will be informed
of predicates of IF-THEN-ELSE statements which failed to evaluate to either
TRUE or FALSE.
(5) The user may have variables which he intends not to use purely
symbolically, i.e. they are to have values all the time. By typing
DECLARE([var1,var2,...],BINDTEST) MACSYMA will give the user an error whenever
any of the vari appear in a computation unbound. To remove a BINDTEST
declaration, the user may use the function REMOVE. (see 8.1)
(6) When an error occurs in the course of a computation, MACSYMA prints
out an error message and terminates the computation. The error message
MACSYMA 19 Debugging in MACSYMA 231
sometimes displays the offending expression to help the user pinpoint the
source of error. This is inconvenient if the expression is large, especially
when the user has a slow terminal. Gradually this scheme is being replaced by
a scheme that does not display the offending expression, but rather sets
ERREXP to it, and prints out the error message followed by the message "ERREXP
contains the offending expression".
The user may find it helpful to investigate the environment at the place
of the error. To do so, type DEBUGMODE:TRUE or DEBUGMODE:ALL and repeat the
computation. This enters a special debugging mode which will "break" or pause
when an error occurs. This mode may be terminated by typing DEBUGMODE:FALSE.
When an error occurs in debugging mode, (ERROR-BREAK) is printed. MACSYMA is
then waiting for the user to type something. He may type any command line
just as if he were at "top level". The command lines will be evaluated in the
environment of the error. If the user had done DEBUGMODE:ALL, he may now type
BACKTRACE;, and MACSYMA will print out a backtrace, which is a list of the
function calls the user is currently in together with the arguments they were
called with, ordered from most recent to earliest i.e., when reversed, this
list shows a trace beginning from the initial function and ending at the last
call entered including only those function calls from which the user still has
not exited. To exit from the MACSYMA error-break and return to "top-level",
type EXIT;.
The user may also enter the error-break at any point, by typing control-A
or by executing the function BREAK (see 10.6). This will simply cause his
computation to pause, while he investigates at will. %% refers to the last
computed result while in the MACSYMA break. % still refers to the last result
computed at top-level. Upon typing EXIT;, the computation will resume. If he
wants to quit a computation begun in a control-A break without quitting the
top-level suspended computation, the user can type control-X.
During a break one may type TOPLEVEL;. This will cause top-level MACSYMA
to be entered recursively. Labels will now be bound as usual. Everything
will be identical to the previous top-level state except that the computation
which was interrupted is saved. The function TOBREAK() will cause the break
which was left by typing TOPLEVEL; to be re-entered. If TOBREAK is given any
argument whatsoever, then the break will be exited, which is equivalent to
typing TOBREAK() immediately followed by EXIT;.
In the following example, an attempt is made to define a function ROOT
which finds an approximate root to an expression using Newton-Raphson
iteration.
(C1) ROOT(F,V):=BLOCK([VAL,FUN,DER],DER:DIFF(F,V),VAL:0.0,
TEST,FUN:SUBST(VAL,V,F),IF ABS(FUN)<5.0E-7 THEN
RETURN(VAL),DER:SUBST(VAL,V,DER),VAL:VAL-FUN/DER,
GO(TEST))$
(C2) NUMER:TRUE$
(C3) F:SIN(%PI*X)-%PI*(X-1)$
232 19 Debugging in MACSYMA MACSYMA
(C4) ROOT(F,X);
3.3116898E+8 ARG TOO BIG FOR ACCURACY - SIN
(C5) DEBUG:TRUE$
(C6) DEBUGMODE:TRUE$
(C7) ''C4;
F has value
V has value
VAL has value
FUN has value
DER has value
3.3116898E+8 ARG TOO BIG FOR ACCURACY - SIN
(ERROR-BREAK)
←VAL;
1.0541436E+8
←DER;
- 2.98023224E-8
←TRACE(SUBST)$
←''C4;
(The numerical value of %PI is present below due to NUMER being set to TRUE
above.)
1 ENTER SUBST [0.0, X, SIN(3.1415927 X) - 3.1415927 (X - 1)]
1 EXIT SUBST: 3.1415927
1 ENTER SUBST [0.0, X, 3.1415927 COS(3.1415927 X) - 3.1415927]
1 EXIT SUBST: - 2.98023224E-8
1 ENTER SUBST [1.0541436E+8, X, SIN(3.1415927 X) - 3.1415927 (X - 1)]
3.3116898E+8 ARG TOO BIG FOR ACCURACY - SIN
try again
(This message is due to an error-break occurring within another error break.)
←EXIT;
exited from the break
(C8) <$>
←''C4
MYFROOT<$><$>
MACSYMA 19 Debugging in MACSYMA 233
(The user uses the MYF command of the MACSYMA editor to insert the definition
of ROOT into the edit buffer. The editor is then used to insert an IF
statement to test for DER being close to 0. The actual editing work is not
shown.)
(C8) ROOT(F,V):=BLOCK([VAL,FUN,DER],DER:DIFF(F,V,1),VAL:0.0,
TEST,FUN:SUBST(VAL,V,F),IF ABS(FUN)<5.0E-7 THEN
RETURN(VAL),DER:SUBST(VAL,V,DER),IF ABS(DER)<5.E-8
THEN ERROR("DERIVATIVE IS ZERO"),VAL:VAL-FUN/DER,
GO(TEST))$
(C9) UNTRACE();
(D9) [SUBST]
(C10) DEBUG:FALSE$
(C11) ''C4;
DERIVATIVE IS ZERO
QUIT (This is due to the ERROR function being executed.)
234 MACSYMA
Appendix I - Multics MACSYMA
MACSYMA is also implemented on the Multics operating system which runs on
Honeywell Series 60 machines (the 68/80 in particular). It is the same MACSYMA
as on the ITS system; the only differences that exist are due to the
interaction of the LISP in which the MACSYMA is imbedded with a different
operating system. Since this manual is mainly written for using a MACSYMA on
ITS, one has to be aware of these differences if one is to use the manual
while using a MACSYMA on Multics.
Procedures for logging in and out will not be given here. It is assumed
that if one has access to a Multics one will either know the procedures for
logging in or will be able to obtain adequate documentation in order to learn
how to do so.
1) To use a MACSYMA on a Multics one simply invokes the "macsyma" command.
However since this command may not be in one's default search rules, it may be
necessary to link to the command or to refer to it by its full pathname. For
example, on the Multics at MIT, the full pathname of the command is
">udd>ap>library>macsyma".
When the command is invoked, the MACSYMA system is loaded and prompts the
user in the standard manner.
2) All input must end with a newline (return) after the semi-colon or
dollar-sign is typed.
3) The terminator character for the editor is "&" instead of altmode.
4) Control characters are not entered in the manner described in the rest
of this manual but are instead entered by the conventions of the Multics
implementation of LISP. This is done by using the "attention" or "break"
button on the console followed by the letter of the alphabet for the control
character you want, followed by a newline character. i.e. when you hit the
attention key, the system will type "CTRL/", and you should respond by typing
the appropriate character. If you want a control-Z, for example, you would
type the letter "Z" followed by a newline. The various control characters have
the same meaning as on ITS MACSYMA. The only one that is different is
control-Z which rather than "returning" to a monitor level instead calls a new
invocation of the monitor at a higher level i.e. it is the same thing as a
normal Multics quit signal. A "start" command will start the macsyma moving
again.
5) There are two different ways of referring to files in the Multics
hierarchy. MACSYMA commands that take file names as arguments can be given the
Multics file name enclosed in double quotes as a single argument. An
alternative form of reference is one that maps the ITS way of referring to
files with four arguments into a reasonable Multics filename. Thus:
(c1) batch("random.demo");
(c1) batch(">udd>Project>JRNurd>random.demo");
MACSYMA I Multics MACSYMA 235
(c1) batch(random,demo);
(c1) batch(random,demo,dsk,">udd>Project>JRNurd");
would all refer to the same file if the current working directory were
">udd>Project>JRNurd".
(6) To use the Multics plotting package, set the variable MULTGRAPH to
TRUE.
236 MACSYMA
Appendix II - A MACSYMA Grammar Primer
The concepts MACSYMA deals with are primarily mathematical, and its
grammar has been designed to reflect this emphasis by making the
representation of expressions as natural as possible. All the usual
mathematical operators are predefined, and MACSYMA commands are expressed
exclusively in functional notation. In addition, MACSYMA provides a flexible
syntax extension capability for users who require additional operators.
For the purposes of this appendix, the internal representation of a
MACSYMA expression can best be described in terms of function calls. All
expressions are represented as appropriately nested function calls; all
actions are the result of function evaluations. This primer is intended to
introduce the reader to MACSYMA's syntax and and syntax extension capabilities
and to help him utilize these features most effectively in syntactically
expressing the semantics of the job he wants done.
It is assumed here that the reader is already familiar with the rules of
formation for lexemes, i.e. integers, real numbers, atoms, and strings. At
present these rules are somewhat confusing and should be mastered before
proceeding (see 2.3) .
The Parser
When a user types a string of characters to MACSYMA, it is first broken up
into lexemes by a lexical-scanning program. For example the input "IF X>0
THEN X ELSE -X" becomes (IF X > 0 THEN X ELSE - X). This sequence of lexemes
is then passed to an "extended operator precedence parser with types" and
converted into MACSYMA's internal representation, i.e. suitably nested
function calls. In this case, the result would be as follows:
"IF"(">"(X,0),X,"-"(X))
Such functional notation is always legal MACSYMA syntax, and it will be
used throughout this appendix, as above, to represent the meaning of various
syntactic constructions.
Every lexeme in MACSYMA is either an "operator", a "delimiter", or an
"operand". The operators and delimiters taken together are sometimes referred
to as the "keywords" of MACSYMA. With each operator is associated a specific
parsing function which prescribes how its arguments are to be selected from
the input. Thus, for example, the arguments of an infix operator like ">" are
to be found, one to the left and one to the right of the operator. The
resulting internal representation is a function call of the operator on its
arguments. For example, in the sentence above, ">" has as arguments "X" and
"0", and the internal representation of "X>0" is ">"(X,0).
MACSYMA II A MACSYMA Grammar Primer 237
A delimiter is a reserved lexeme used by certain operators to mark their
arguments. A delimiter may not be used as an operand but has no special
parsing function associated with it. In the sentence above, "THEN" and "ELSE"
are delimiters, used by the operator "IF" to mark its second and third
arguments.
An operand is a lexeme with no special parsing properties. All lexemes,
unless otherwise specified, are operands. Operands serve as the arguments of
operators to form function calls which may then in term serve as arguments to
other operators. In the example sentence, "X" and "0" are operands; ">"(X,0),
X, and "-"(X) also appear as operands to the operator "IF". Note that any
operator may be used as an operand by enclosing it in double quotes, e.g.
INFIX("&").
The process of parsing is one of recognizing the operators, delimiters,
and operands in a sequence of lexemes and correctly identifying the arguments
to the operators in order to construct the function nesting implicit in the
sequence. There are only seven types of operators in MACSYMA, most of which
should be familiar to mathematicians. These seven types are described below.
PREFIX OPERATORS
A PREFIX operator is one which signifies a function of one argument, which
argument immediately follows an occurrence of the operator. Some examples
are:
NOT TRUE means "NOT" (TRUE)
- A means "-" (A)
This resembles the usual functional notation except that the parentheses
surrounding the argument are here unnecessary. Of course any expression may
be embedded to arbitrary depth within another as in the following.
NOT NOT TRUE means "NOT" ("NOT" (TRUE)) = TRUE
POSTFIX OPERATORS
POSTFIX operators like the PREFIX variety denote functions of a single
argument, but in this case the argument immediately precedes an occurrence of
the operator in the input string.
3 ! means "!" (3)
A !! means "!!" (A)
INFIX OPERATORS
238 II A MACSYMA Grammar Primer MACSYMA
INFIX operators are used to denote functions of two arguments, one given
before the operator and one after. Again some examples:
A↑2 means "↑" (A,2)
3↑3 > 10 means ">" ("↑" (3,3), 10)
A variation of the INFIX operator is the NARY.
NARY OPERATORS
An NARY operator is used to denote a function of any number of arguments,
each of which is separated by an occurrence of the operator.
A*B*C means "*" (A,B,C)
A>B AND TRUE AND C<D means "AND"(">"(A,B),TRUE,"<"(C,D))
are all examples of NARY operators.
SPECIAL OPERATORS
NARY operators are useful for functions whose arguments are in one way or
other homogeneous. For other functions of many arguments, special forms are
required. A familiar example is the conditional statement.
IF A>2 THEN A-1 ELSE A means "IF"(">"(A,2),"-"(A,1),A)
Here the operator "IF" denotes a function of three arguments; the first is
found immediately after the "IF"; the others are introduced arguments
signalled by the occurrence of the delimiters associated with "IF", namely
"THEN" and "ELSE". Once again, delimiters are not in themselves operators but
are merely used by operators to mark introduced arguments. Using a delimiter
out of the context of the operators for which it was defined will result in a
syntax error.
Another example of an operator with introduced arguments is the iteration
statement. Here the delimiters precede their defining operator.
FOR I FROM 2 STEP 3 THRU 10 UNLESS A>10 DO PRINT(A)
In this example each of the indicated segments contributes an argument to
the "DO" function. It happens that in MACSYMA any of these arguments may be
omitted, or if given they can be given in any order. Thus the following are
all legal, though not necessarily equivalent, sentences.
THRU 10 DO S:S↑2
FOR I FROM 2 THRU 5 DO PRINT (A[I])
MACSYMA II A MACSYMA Grammar Primer 239
THRU 5 UNLESS A>1000 DO A:A!
When arguments are omitted as above or like the "ELSE" argument of the
"IF" operator, the corresponding "holes" are filled with predetermined default
values. These are listed in the dictionary below. Also the "DO" statement
has some additional flexibility. The "STEP" argument can be replaced by a
"NEXT" expression which denotes what the iteration variable is to be set to on
each pass thru the loop rather than the value by which it is to be
incremented; and there are permitted arbitrarily many "WHILE" or "UNLESS"
clauses as termination conditions. Some examples with answers:
THRU 3 DO PRINT (A)
A
A
A
DONE
FOR I STEP 2 THRU 3 DO PRINT (I)
1
3
DONE
FOR I NEXT I+2 UNLESS I>3 DO PRINT (I)
1
3
DONE
NOFIX OPERATORS
NOFIX operators are used to denote functions of no arguments. The mere
presence of such an operator in a sentence will cause the corresponding
function to be evaluated.
QUIT means "QUIT" ()
LOGOUT means "LOGOUT" ()
Care should be taken in using these operators, however, since they tend to
look much like variables but semantically are very different.
MATCHFIX OPERATORS
MATCHFIX operators are used to denote functions of any number of arguments
which are passed to the function as a list. The arguments occur between the
main operator and its "matching" delimiter. For example:
[) A, B, C (] means [)" (A,B,C)
240 II A MACSYMA Grammar Primer MACSYMA
A legal sentence in MACSYMA is a correct sequence of operators from these
seven categories and their operands. By "correct" here we mean that due
respect has been shown the type of the operator, e.g. not giving two arguments
to a PREFIX operator, and that the two sole grammatical rules in the language
have not been violated. These rules concern the "binding powers" of MACSYMA's
keywords and the "parts of speech" legal in each argument slot.
BINDING POWERS
The binding powers of keywords are used to resolve ambiguities of argument
association such as that in the following example.
- 233 !
Is this "-"("!"(233)) or "!"("-"(233)); or, in other words, which operator
gets the operand "233" and which, the resulting function call? It is a
convention in MACSYMA that the keyword with the higher binding power gets the
disputed argument and the other is then applied to the result. In this case,
the "left binding power" of "!" (160) is greater than the "right binding
power" of "-" (100); and so "233" is associated with "!" and the resulting
function call becomes the argument for "-".
Each keyword must possess a left and a right binding power to resolve such
conflicts. Some of these numbers are superfluous, such as the left binding
power of a prefix operator; and in such cases the binding power is arbitrarily
taken to be 200. Currently the range of binding powers is 0 to 200.
PARTS OF SPEECH
From natural language the notion of "part of speech" should be familiar.
MACSYMA also has parts of speech and constraints on which parts of speech are
legal in various contexts. Whereas binding powers are necessary to resolve
ambiguities af argument assignment, parts of speech exist solely to detect
unintentional syntax errors.
Every operator possesses a part of speech constraint on each of
its argument slots. Any operand filling a slot must satisfy the
associated constraint, or a syntax error will result.
There are only three parts of speech predefined in MACSYMA, namely EXPR,
CLAUSE, and ANY. An EXPR is essentially a mathematical expression; a CLAUSE,
a mathematical predicate or a command. Thus "A+B" is an EXPR but not a
CLAUSE; and "A+B>2" is a CLAUSE but not an EXPR. The part of speech ANY is
used to signify objects which may be either CLAUSEs or EXPRs, such as "F(X)".
The parts of speech required by MACSYMA's predefined operators are listed in
the dictionary below.
MACSYMA II A MACSYMA Grammar Primer 241
SYNTAX EXTENSION
While MACSYMA's syntax should be adequate for most ordinary applications,
it is possible to define new operators or eliminate predefined ones that get
in the user's way. The extension mechanism is rather straightforward and
should be evident from the examples below.
(C1) PREFIX("DDX")$
(C2) DDX Y$ means "DDX"(Y)
(C3) INFIX("<-")$
(C4) A<-DDX Y$ means "<-"(A,"DDX"(Y))
An appreciation of the concepts and rules introduced in this primer should
be all that is necessary to use the syntax extension capabilities
successfully. The only form of syntax extension available is the definition
of new operators. For each of the types of operator except SPECIAL, there is
a corresponding creation function that will give the lexeme specified the
corresponding parsing properties. Thus "PREFIX("DDX")" will make "DDX" a
prefix operator just like "-" or "NOT". Of course, certain extension
functions require additional information such as the matching keyword for a
matchfix operator. In addition, binding powers and parts of speech must be
specified for all keywords defined. This is done by passing additional
arguments to the extension functions. If a user does not specify these
additional parameters, MACSYMA will assign default values. The six extension
functions with binding powers and parts of speech defaults (enclosed in
brackets) are summarized below.
PREFIX(operator, rbp[180], rpos[ANY], pos[ANY])
POSTFIX(operator, lbp[180], lpos[ANY], pos[ANY])
INFIX(operator, lbp[180], rbp[180], lpos[ANY], rpos[ANY],pos[ANY])
NARY(operator, bp[180], argpos[ANY], pos[ANY])
NOFIX|operator(pos[ANY]|)
MATCHFIX(operator, match, argpos[ANY], pos[ANY])
The defaults have been provided so that a user who does not wish to
concern himself with parts of speech or binding powers may simply omit those
arguments to the extension functions. Thus the following are all equivalent.
PREFIX("DDX",180,ANY,ANY)$
PREFIX("DDX",180)$
PREFIX("DDX")$
242 II A MACSYMA Grammar Primer MACSYMA
It is also possible to remove the syntax properties of an operator by
using the functions REMOVE or KILL. Specifically, "REMOVE("DDX",OP)" or
"KILL("DDX")" will return "DDX" to operand status; but in the second case all
the other properties of "DDX" will also be removed.
The following is an example of syntax extension to permit the use of set
notation.
(C1) MATCHFIX("{","}")$
(C2) INFIX("|")$
(C3) {X|X>0};
(D3) {X|X>0}
(C4) {X|X<2};
(D4) {X|X<2}
(C5) INFIX(".U.")$
(C6) INFIX(".I.")$
Now assuming the functions ".U." and ".I." have been appropriately
defined as union and intersection, the following interaction can occur.
(C7) D3.U.D4;
(D7) UNIVERSE
(C8) D3.I.D4;
(D8) {X|X>0 AND X<2}
(C9) {1,2,3}$
(C10) {3,4,5}$
(C11) D9.U.D9.I.D10;
(D11) {3}
Line C11 was parsed as ((D9.U.D9).I.D10) whereas the usual convention
would call for the alternate parsing (D9.U.(D9.I.D10)), which would have
resulted in {1,2,3} as value. The problem here is that the default binding
powers for ".U." and ".I." are identical; so the parser associates them in
left to right order. To obtain the usual parsing, the syntax definitions in
lines C5 and C6 must give ".I." a higher left binding power than ".U."'s right
binding power as in the following.
(C12) INFIX(".U.",100,100)$
(C13) INFIX(".I.",120,120)$
(C14) D9.U.D9.I.D10;
(D14) {1,2,3}
(C15) REMOVE(".U.",OPERATOR)$
MACSYMA II A MACSYMA Grammar Primer 243
(C16) D9.U.D10;
Syntax error
D9 .U. ***$*** D10
Please rephrase or edit
A DICTIONARY OF MACSYMA'S KEYWORDS
The following is a list of all the keywords in MACSYMA, categorized with
respect to type. With each keyword is given the information necessary to
recreate its syntactic behavior. The abbreviations "lbp", "rbp", "lpos",
"rpos", "bp", and "pos" stand for "left binding power", "right binding power",
left part of speech", "right part of speech", "binding power", and "part of
speech". The reader should consult the text of this primer to understand the
significance of these parameters. It should also be noted that some lexemes,
like "-", have two syntactic types.
PREFIX OPERATORS
rbp rpos pos
' 190 ANY ANY
'' 190 ANY ---
+ 100 EXPR EXPR
- 100 EXPR EXPR
NOT 70 CLAUSE CLAUSE
POSTFIX OPERATORS
lbp lpos pos
! 160 EXPR EXPR
!! 160 EXPR ANY
INFIX OPERATORS
lbp lpos rbp rpos pos
# 80 ANY 80 ANY CLAUSE
** 140 EXPR 139 EXPR EXPR
. 130 ANY 129 ANY ANY
: 180 ANY 20 ANY ANY
:: 180 ANY 20 ANY ANY
:= 180 ANY 20 ANY ANY
< 80 EXPR 80 EXPR CLAUSE
<= 80 EXPR 80 EXPR CLAUSE
= 80 EXPR 80 EXPR CLAUSE
> 80 EXPR 80 EXPR CLAUSE
>= 80 EXPR 80 EXPR CLAUSE
↑ 140 EXPR 139 EXPR EXPR
↑↑ 135 ANY 134 ANY ANY
244 II A MACSYMA Grammar Primer MACSYMA
NARY OPERATORS
bp argpos pos
* 120 EXPR EXPR
+ 100 EXPR EXPR
, 10 ANY ANY
- 100 EXPR EXPR
/ 120 EXPR EXPR
AND 60 CLAUSE CLAUSE
OR 50 CLAUSE CLAUSE
SPECIAL OPERATORS
lbp lpos rbp rpos pos
( 200 ANY MATCHFIX for right arg ANY
[ 200 ANY MATCHFIX for right arg ANY
DO 200 25 ANY ANY
FOR optional
FROM optional
IN optional
STEP or NEXT optional
THRU optional
WHILE any number of occurrences
UNLESS any number of occurrences
IF 200 45 CLAUSE ANY
THEN
ELSE optional
MATCHFIX OPERATORS
match argpos pos
( ) ANY ANY
[ ] ANY ANY
DELIMITERS
op lbp rbp rpos
ELSE IF 5 25 ANY
FOR DO 30 200 ANY
FROM DO 30 95 EXPR
IN DO 30 95 ANY
NEXT DO 30 45 ANY
STEP DO 30 95 EXPR
THEN IF 5 25 ANY
THRU DO 30 95 EXPR
UNLESS DO 30 45 CLAUSE
WHILE DO 30 45 CLAUSE
MACSYMA 245
Appendix III - Illustrative Examples
This appendix shows a complete interaction with MACSYMA. Three examples
are given. First an ordinary second-order differential equation is solved by
two methods. (1) by using pattern matching and solving the characteristic
equation and (2) by using Laplace transforms. The second example shows the
conversion of an expression from one coordinate system to another and the
third example shows a truncated power series solution of a differential
equation.
Example 1
(C1) BATCH(SOLDER,DEMO,DSK,DEMO);
(C2) /* THE FOLLOWING ROUTINE RETURNS THE HOMOG.-PART SOLN.
TO 2ND ORDER LINEAR DIFF'L EQNS. WITH CONST. COEFFS. */
MATCHDECLARE([B,C],RATNUMP)$
MATCOM FASL DSK MACSYM BEING LOADED
LOADING DONE
(C3) MATCHDECLARE(F,FREEOF(U))$
(C4) ALIAS(D,DIFF)$
(C5) DEFMATCH(SOLDE,'D(U,X,2) + B*'D(U,X) + C*U = F,U,X)$
(C6) SOLDER(EQN,U,X) :=
BLOCK([B,C,F,DISC,R1,R2,ALPHA,BETA],
IF SOLDE(EQN,U,X) = FALSE THEN RETURN(FALSE),
DISC: B↑2 - 4*C, ALPHA: -B/2,
IF DISC=0 THEN RETURN(%E↑(ALPHA*X)*(A1+A2*X)),
BETA: SQRT(DISC)/2,
IF DISC > 0
THEN (R1: ALPHA + BETA, R2: ALPHA - BETA,
RETURN(A1*%E↑(R1*X) + A2*%E↑(R2*X)))
ELSE (BETA: SQRT(-1)*BETA,
RETURN(%E↑(ALPHA*X) * (A1*COS(BETA*X)
+ A2*SIN(BETA*X)))))$
(C7) /* AN EXAMPLE - THE METHOD OF UNDETERMINED COEFFS. FOR
OBTAINING THE PARTICULAR SOLN. AS WELL */
246 III Illustrative Examples MACSYMA
DE: 'D(Y,X,2) - 'D(Y,X) - 6*Y = SIN(X);
2
D Y DY
(D7) --- - -- - 6 Y = SIN(X)
2 DX
DX
(C8) YH(X) := ''(SOLDER(%,Y,X));
- 2 X 3 X
(D8) YH(X) := A2 %E + A1 %E
(C9) YP(X) := B1*SIN(X) + B2*COS(X)$
(C10) YG(X) := YH(X) + YP(X)$
(C11) PLUGIN: EV(DE,DIFF,EXPAND,Y=YP(X));
(D11) B2 SIN(X) - 7 B1 SIN(X) - 7 B2 COS(X)
- B1 COS(X) = SIN(X)
(C12) EQN1: COEFF(PLUGIN,SIN(X));
(D12) B2 - 7 B1 = 1
(C13) EQN2: COEFF(PLUGIN,COS(X));
(D13) - 7 B2 - B1 = 0
(C14) GLOBALSOLVE: TRUE$
(C15) SOLN: LINSOLVE([EQN1,EQN2],[B1,B2]);
SOLUTION
7
(E15) B1 : - --
50
1
(E16) B2 : --
50
(D16) [E15, E16]
(C17) YG(X);
7 SIN(X) COS(X) 3 X - 2 X
(D17) - -------- + ------ + A1 %E + A2 %E
50 50
MACSYMA III Illustrative Examples 247
(C18) /* PLUGGING IN INITIAL CONDITIONS OF Y(0)=1
AND Y'(0)=0 */
EQN1: YG(0) = 1;
1
(D18) A2 + A1 + -- = 1
50
(C19) DIFF(YG(X),X);
SIN(X) 7 COS(X) 3 X - 2 X
(D19) - ------ - -------- + 3 A1 %E - 2 A2 %E
50 50
(C20) EQN2: EV(%,X=0) = 0;
7
(D20) - 2 A2 + 3 A1 - -- = 0
50
(C21) SOLN: LINSOLVE([EQN1,EQN2],[A1,A2]);
SOLUTION
21
(E21) A1 : --
50
14
(E22) A2 : --
25
(D22) [E21, E22]
(C23) YG(X);
3 X - 2 X
7 SIN(X) COS(X) 21 %E 14 %E
(D23) - -------- + ------ + -------- + -----------
50 50 50 25
(C24) /* RESETTING OF OPTIONS */
GLOBALSOLVE: FALSE$
(D25) BATCH DONE
(C26) "SOLUTION BY LAPLACE TRANSFORMS"$
(C27) SUBST(Y(X),Y,DE);
248 III Illustrative Examples MACSYMA
2
D D
(D27) --- Y(X) - -- Y(X) - 6 Y(X) = SIN(X)
2 DX
DX
(C28) [ATVALUE(Y(X),X=0,1), ATVALUE('DIFF(Y(X),X),X=0,0)];
(D28) [1, 0]
(C29) LAPLACE(D29,X,S);
LAPLAC FASL DSK MACSYM being loaded
loading done
2
(D29) S LAPLACE(Y(X), X, S) - S LAPLACE(Y(X), X, S)
1
- 6 LAPLACE(Y(X), X, S) - S + 1 = ------
2
S + 1
(C30) LINSOLVE([%],['LAPLACE(Y(X),X,S)]);
Solution
3 2
S - S + S
(E30) LAPLACE(Y(X), X, S) = ----------------------
4 3 2
S - S - 5 S - S - 6
(D30) [E30]
(C31) ILT(E30,S,X);
3 X -2X
7 SIN(X) COS(X) 21 %E 14 %E
(D31) Y(X) = - -------- + ------ + -------- + --------
50 50 50 25
MACSYMA III Illustrative Examples 249
Example 2
(C1) BATCH(C2CYL,DEMO,DSK,DEMO);
(C2) /* CONVERSION OF THE LAPLACIAN FROM CARTESIAN COORDS. TO
CYLINDRICAL COORDS. */
/* CAUSE DERIVATIVES TO DISPLAY WITH SUBSCRIPTS */
DERIVABBREV:TRUE$
(C3) /* ORDER X,Y, AND Z SO THEY WILL BE GROUPED NICELY */
ORDERLESS(Z,Y,X)$
(C4) /* U(X,Y,Z) BECOMES U(R,T,Z) IN CYLINDRICAL COORDINATES
R STANDS FOR RHO AND T FOR THETA */
DEPENDS(U,[R,T,Z])$
(C5) /* INPUT THE TRANSFORMATION RULES FROM THE
CARTESIAN SYSTEM TO THE CYLINDRICAL SYSTEM */
GRADEF(R,X,X/R)$
(C6) GRADEF(R,Y,Y/R)$
(C7) GRADEF(T,X,-Y/R↑2)$
(C8) GRADEF(T,Y,X/R↑2)$
(C9) /* SET EXPOP TO CAUSE PARENTHESIZED EXPRESSIONS
TO BE EXPANDED AUTOMATICALLY */
EXPOP:1$
(C10) /* NOW JUST INPUT THE LAPLACIAN IN CART. COORDS.,
AND LET THE CHAIN RULE DO ITS THING */
DIFF(U,X,2)+DIFF(U,Y,2)+DIFF(U,Z,2);
2 2 2 2 2 2
X U Y U X U Y U 2 U X U Y U
T T T T R R R R R R R
(D10) ------- + ------- + ------- + ------- + ---- - ----- - ----- + U
4 4 2 2 R 3 3 Z Z
R R R R R R
250 III Illustrative Examples MACSYMA
(C11) SUBST(R↑2-X↑2,Y↑2,%);
U U
T T R
(D11) ---- + U + -- + U
2 R R R Z Z
R
(D12) BATCH DONE
MACSYMA III Illustrative Examples 251
Example 3
The following differential equation:
T↑4*B(T)↑3*DIFF(B(T),T,2)+(1-K*T↑2)*B(T)↑4-T↑4=0
is known to have a solution of the form:
B(T)=T+A3*T↑3+A5*T↑5+...+A11*T↑11+...
valid for small T. The problem is to find the coefficients A3 through A11 as
functions of K. We use RATWEIGHT and RATWTLVL to truncate on powers of T above
14. (This problem originated in "Bessel Functions for Large Arguments" by
Goldstein and Thaler, in Math. Tables and Other Aids to Computation, (now
called Mathematics of Computation) XII, no. 61, p.18, January 1958.)
(C3) EQ:T↑4*B(T)↑3*DIFF(B(T),T,2)+(1-K*T↑2)*B(T)↑4-T↑4;
2
4 3 D 2 4 4
(D3) T B (T) (--- B(T)) + (1 - K T ) B (T) - T
2
DT
C4) TRIAL:T+SUM(A[2*I+1]*T↑(2*I+1),I,1,5);
11 9 7 5 3
(D4) A T + A T + A T + A T + A T + T
11 9 7 5 3
(C5) POWERDISP:TRUE$
(C6) RATWEIGHT(T,1)$
(C7) RATWTLVL:14$
(C8) EQ,B(T)=TRIAL,DIFF;
4 4 3 5 7 9
(D8) - T + T (6 A T + 20 A T + 42 A T + 72 A T + 110 A T )
3 5 7 9 11
3 5 7 9 11 3
(T + A T + A T + A T + A T + A T )
3 5 7 9 11
2 3 5 7 9 11 4
+ (1 - K T ) (T + A T + A T + A T + A T + A T )
3 5 7 9 11
252 III Illustrative Examples MACSYMA
a
(C9) EXPANDEDEQ:RAT(%,T);
6 2 8
(D9)/R/ (4 A - K) T + (6 A + 6 A + 4 A - 4 A K) T
3 3 3 5 3
2 3 2 10
+ (18 A + 4 A + (20 + 12 A ) A + 4 A + ( - 6 A - 4 A ) K) T
3 3 3 5 7 3 5
3 4 2 2
+ (18 A + A + (78 A + 12 A ) A + 6 A + (42 + 12 A ) A + 4 A
3 3 3 3 5 5 3 7 9
3 12
+ ( - 4 A - 12 A A - 4 A ) K) T
3 3 5 7
4 2 3 2
+ (6 A + (96 A + 4 A ) A + (60 + 12 A ) A
3 3 3 5 3 5
2
+ (144 A + 12 A + 12 A ) A + (72 + 12 A ) A + 4 A
3 3 5 7 3 9 11
4 2 2 14
+ ( - A - 12 A A - 6 A - 12 A A - 4 A ) K) T
3 3 5 5 3 7 9
(C10) COEFF(EXPANDEDEQ,T,6);
(D10)/R/ 4 A - K
3
(C11) ANS3:SOLVE(%,A[3]);
Solution
K
(E11) A = -
3 4
(D11) [E11]
(C12) COEFF(EXPANDEDEQ,T,8);
2
(D12)/R/ 6 A + 6 A + 4 A - 4 A K
3 3 5 3
(C13) %,ANS3;
2
- 32 A - 12 K + 5 K
5
(D13)/R/ - ----------------------
8
MACSYMA III Illustrative Examples 253
(C14) SOLVE(%,A[5]);
Solution
2
- 12 K + 5 K
(E14) A = --------------
5 32
(D14) [E14]
(C15) /* ETC*/
FOR I:3 THRU 11 STEP 2 DO
COEFFICIENT[I]:COEFF(EXPANDEDEQ,T,I+3)$
(C16) FOR I:3 THRU 11 STEP 2 DO
(SOL[I]:ANS:SOLVE(COEFFICIENT[I],A[I]),
FOR J:I+2 STEP 2 THRU 11 DO
COEFFICIENT[J]:EV(COEFFICIENT[J],ANS))$
(C21) RATEXPAND:TRUE$
(C22) FOR I:3 THRU 11 STEP 2 DO PRINT(RATSIMP(EV(SOL[I])))$
K
[A = -]
3 4
2
3 K 5 K
[A = - --- + ----]
5 8 32
2 3
15 K 37 K 15 K
[A = ---- - ----- + -----]
7 8 32 128
2 3 4
315 K 1821 K 611 K 195 K
[A = - ----- + ------- - ------ + ------]
9 16 128 256 2048
2 3 4 5
2835 K 2223 K 29811 K 4199 K 663 K
[A = ------ - ------- + -------- - ------- + ------]
11 8 8 512 1024 8192
254 MACSYMA
Appendix IV - Glossary For The Programming Novice
algorithm - a method, specified with sufficient precision to be programmed for
a computer, to resolve any one of a well-defined class of problems.
arguments - the expressions which are the values of the formal parameters when
a function is called.
assignment - the process of associating a value with a variable.
atomic - (in the sense of high level programming languages) cannot be broken
down into smaller parts, e.g. a number, a string, or a name.
binding - the process of assigning values to the formal parameters in a
function definition or to the local parameters in a block in such a
way that, upon exit from the function or block, the previous values of
the parameters are restored.
bound variable - a variable which has been assigned a value (see binding).
break point - a point at which a computation is temporarily suspended and
control returned to the console, permitting the user to explore the
state of the computation.
bug - an error in a program caused by improper coding which may be due to
unanticipated types of arguments being given to the program, faulty
logic, etc.
command line - the input line typed to MACSYMA, terminated by ; or $.
constant - any number or atomic symbol whose value does not vary, or an
expression made up only of such quantities.
CRE form - canonical rational expression form. This is one of the several
← ← ←
internal representations of MACSYMA expressions (see 4.1). It is
especially suitable for rational expressions (polynomials or ratios of
polynomials). It is also contagious in that whenever any expression
is added to or multiplied by a CRE form the result will be in CRE
form.
DDT - originally a program used for debugging of other programs but modified
in ITS to include the functions of a monitor.
default value - the initial value of a variable before any assignments to it.
MACSYMA IV Glossary For The Programming Novice 255
evaluation - the process of replacing variables and function calls in an
expression by their values.
expansion - the transformation of a product of sums into a sum of products by
applying the distributive laws.
expression - a syntactically legal sequence of characters composed of
constants, variables, functions, and operators.
flag - a variable whose value is usually either TRUE or FALSE, e.g. NUMER.
formal parameters - the atomic variables appearing in the function header (the
left-hand side of a function definition).
hashing - a method of storing sparse vectors and arrays through the use of a
function, which for given arguments produces a number in a range of
possible values.
indicator - the name of a property, e.g. GRADEF.
internal representation - the representation of MACSYMA expressions in LISP.
ITS - the time sharing system used on the PDP-10 at Project MAC.
LISP - a list processing programming language used extensively in non-
←←← ←
numerical applications. The LISP in which MACSYMA resides is called
MACLISP.
local parameters - atomic variables which are bound within a function or
block.
property - a piece of information known about or associated with a variable.
Some specific properties are used in MACSYMA for simplification and
evaluation, e.g. the GRADEF property for SIN.
property list - for a given variable, the set of all properties associated
with the variable.
quoted string - a sequence of characters enclosed in quotation marks, used as
a comment, message, etc.
rational expression - a polynomial or the ratio of two polynomials. Sometimes
used as a synonym for CRE form.
scalar - an expression which is not (or is assumed not to be) a list or
matrix.
semantics - rules for determining the meaning of any legal (syntactically
correct) sentence in a language.
simplification - the process of reducing the "complexity" of an expression
256 IV Glossary For The Programming Novice MACSYMA
(relative to some criterion or measure) by applying known (or assumed)
relations in the form of rules which transform the original expression
into an "equivalent" one.
string - a sequence of characters consisting of digits, letters, special
characters ($,%,#) or break characters (space,tab).
subscripted function - a type of array each of whose elements is a function
expression.
subscripted variable - a variable, e.g. A[0], in subscripted form.
switch - a variable which can take on only a small number of values (usually
just two). It is used to determine which branch of a condition to
follow.
syntax - rules for determining whether a sequence of characters is a legal
sentence in that language. If not, then a parsing error results. In
MACSYMA, the rules are implemented as a parsing procedure which
converts an input string into MACSYMA's internal representation.
terminator - a character which signals the end of a sequence of characters.
In a MACSYMA command line the terminators are semi-colon and dollar
sign.
variable - an atomic symbol. A variable that evaluates to a value is an
assigned variable. If no value has been assigned to a variable, the
variable itself is returned as the result of evaluation.
MACSYMA 257
Appendix V - Bibliography and References
[A1] A.C.M. Proceedings of the Second Symposium on Symbolic and Algebraic
Manipulation, Los Angeles, Calif., March, 1971. +
[A2] A.C.M. Communications - August 1971 Vol. 14 No. 8 +
[A3] A.C.M. Journal - October 1971 Vol. 18 No. 4 +
[Av] Avgoustis, Yannis. "Symbolic Laplace Transforms of Special Functions."
Proceedings of the MACSYMA Users' Conference, NASA, Berkeley, CA, July
1977.
[Ba1] Barton,D.A. and R. Zippel "A Polynomial Decomposition Algorithm"
Proceedings of the 1976 ACM Symposium on Symbolic and Algebraic
Computation, August 10-12, 1976. +
[Be1] Berlekamp, E. R. "Factoring Polynomials Over Large Finite Fields" -
Mathematics of Computation Vol. 24 No. 111 July 1970
[Br1] Brown, W. S. "On Euclid's Algorithm and the Computation of Polynomial
Greatest Common Divisors" in [A3]
[Ca1] Caviness,B.F. and R. Fateman, "Simplification of Radical Expressions"
Proceedings of the 1976 ACM Symposium on Symbolic and Algebraic
Computation, August 10-12, 1976. +
[Co1] Collins, G. E. "The Calculation of Multivariate Polynomial Resultants"
in [A3]
[Do] Doohovskoy, Alexander. "Varieties of Operator Manipulation." Proceedings
of the MACSYMA Users' Conference, NASA, Berkeley, CA, July 1977.
[Ea1] Eastlake, D., et. al. "ITS 1.5 Reference Manual" A.I. Memo 161A July
1969 *
[Fa1] Fateman, R. J. "The User-Level Semantic Matching Capability in MACSYMA"
- in [A1]
[Fa2] --- "Essays in Algebraic Simplification" - Ph.D. Thesis MAC TR-95 April
1972 **
[Fa3] --- "Rationally Simplifying Non-Rational Expressions" SIGSAM Bulletin
July 1972 No. 23 *
258 V Bibliography and References MACSYMA
[Fa4] --- "On the Implementation of Modular Algorithms" SIAM National Meeting.
June 1973
[Fa5] --- "On the Computation of Powers of Sparse Polynomials" MIT Studies in
Applied Mathematics Vol. 53 No. 2 June 1974 *
[Fa6] --- "Polynomial Multiplication, Powers, and Asymptotic Analysis, Some
Comments" - SIAM Journal of Computing Vol. 3 Sept. 1974, pp.196-213.
[Fa7] --- "On the Multiplication of Poisson Series" , Celestial Mechanics, Vol
10, No. 2, Oct. 1974, pp. 243-247.
[Fa8] --- "A Case History in Interactive Problem Solving" SIGSAM Bulletin No.
28, Dec. 1973 +
[Fa9] --- "The MACSYMA Big-Floating-Point Arithmetic System" Proceedings of
the 1976 ACM Symposium on Symbolic and Algebraic Computation, August
10-12, 1976. +
[Fa10] --- "An Approach to Automatic Asymptotic Expansions" Proceedings of the
1976 ACM Symposium on Symbolic and Algebraic Computation, August 10-
12, 1976. +
[Gene1] Genesereth, Michael R. "The Difficulties of Using MACSYMA and the
Function of User Aids." Proceedings of the MACSYMA Users' Conference,
NASA, Berkeley, CA, July 1977.
[Gene2] --- "An Automated Consultant for MACSYMA." Proceedings of the MACSYMA
Users' Conference, NASA, Berkeley, CA, July 1977.
[Gent1] Gentleman, W.M. and S.C. Johnson, "The Evaluation of Determinants by
Expansion by Minors and the General Problem of Substitution"
Mathematics of Computation,Volume 28,Number 126,April 1974,pp. 543-
548.
[Go1] Golden, Jeffrey P. "MACSYMA's Symbolic Ordinary Differential Equation
Solver." Proceedings of the MACSYMA Users' Conference, NASA, Berkeley,
CA, July 1977.
[Go2] --- "The Evaluation of Atomic Variables in MACSYMA." Proceedings of the
MACSYMA Users' Conference, NASA, Berkeley, CA, July 1977.
[Kn1] Knuth, D. "Seminumerical Algorithms" in The Art of Computer Programming,
←←← ←←← ←← ←←←←←←←← ←←←←←←←←←←←
Vol. 2 Addison-Wesley Publishers 1969
[Lew1] Lewis, Ellen, "An Introduction to ITS for the MACSYMA User", September,
1977. *
[Lew2] --- "User Aids for MACSYMA." Proceedings of the MACSYMA Users'
Conference, NASA, Berkeley, CA, July 1977.
MACSYMA V Bibliography and References 259
[Ma1] Martin, W. A. "Computer Input/Output of Mathematical Expressions" in
[A1]
[Ma2] --- "Determining the Equivalence of Algebraic Expressions by Hash
Coding" in [A1, A3]
[Ma3] --- "Symbolic Mathematical Laboratory" - Ph.D. Thesis MAC TR-36 January
1967 AD-657-283 ***
[Ma4] --- and Fateman, R. J. - "The MACSYMA System" in [A1]
[Mn1] Moon, D., et. al "LISP Reference Manual" *
[Mo1] Moses, J. "Algebraic Simplification - A Guide for the Perplexed" - in
[A1, A2]
[Mo2] --- "Symbolic Integration - The Stormy Decade" - in [A1, A2]
[Mo3] --- "Towards a General Theory of Special Functions" Communications of
the Assocation for Computing Machinery July 1972 Vol. 15 No. 7 +
[Mo4] --- "Symbolic Integration" - Ph.D. Thesis - MAC TR-47 December 1967 AD-
662-666 ***
[Mo5] --- "MACSYMA Primer", October 1977 *
[Mo6] --- and Yun, D. Y. "The EZ GCD Algorithm" - Procedings of the ACM
convention August 1973. +
[Mo7] --- "The Evolution of Algebraic Manipulation Algorithms" Proceedings of
IFIP conference August 1974, Stockholm.
[Mo8] --- "MACSYMA - The Fifth Year" SIGSAM Bulletin - August 1974. *
[Mo9] --- "An Introduction to the Risch Integration Algorithm." Proceedings of
the ACM Annual Conference, ACM, Houston, Tx., Oct. 1976.
[Mo10] --- "The Variety of Variables in Mathematical Expressions." Proceedings
of the MACSYMA Users' Conference, NASA, Berkeley, CA, July 1977.
[Mo11] --- and Cohen, Jacques. "Summation of Rational Exponential Expressions
in Closed Form." Proceedings of the MACSYMA Users' Conference, NASA,
[Os1] Osman, E. "DDT Reference Manual" - A.I. Memo 147A Sept 1971 *
[St1] Steele, Guy L., Jr. "Data Representations in PDP-10 MACLISP."
Proceedings of the MACSYMA Users' Conference, NASA, Berkeley, CA, July
1977.
[St2] --- "Fast Arithmetic in MACLISP." Proceedings of the MACSYMA Users'
Conference, NASA, Berkeley, CA, July 1977.
260 V Bibliography and References MACSYMA
[Tr1] Trager,B. "Algebraic Factoring and Rational Function Integration"
Proceedings of the 1976 ACM Symposium on Symbolic and Algebraic
Computation, August 10-12, 1976. +
[Tr2] --- and D.Y.Y.Yun "Completing nth Powers of Polynomials" Proceedings of
the 1976 ACM Symposium on Symbolic and Algebraic Computation, August
10-12, 1976. +
[Wa1] Wang, P. "Automatic Computation of Limits" - in [A1]
[Wa2] --- "Application of MACSYMA to an Asymptotic Expansion Problem" -
Proceedings of the ACM Annual Conference August 1972 Vol. 2 +
[Wa3] --- "Evaluation of Definite Integrals by Symbolic Manipulation" - Ph.D.
Thesis - MAC TR-92 October 1971 **
[Wa4] --- and Rothschild, L. "Factoring Multivariate Polynomials over the
Integers" - Mathematics of Computation, vol. 29 (131) , pp. 935-950.
*
[Wa5] Wang,P. "Multivariate Polynomials over Algebraic Number Fields" -
Mathematics of Computation, Vol. 30, (134),pp. 324-336.
[Wa6] --- "On the Expansion of Sparse Symbolic Determinants" Proceedings of
the International Confference on System Sciences, Jan. 4-6,1977
Honolulu, Hawaii
[Wa7] --- "An Improved Multivariate Polynomial Factoring Algorithm"
Mathematics of Computation (to appear)
[Wa8] --- and T. Minamikawa "Taking Advantage of Zero Entries in the Exact
Inverse of Sparse Matrices, Proceedings of the 1976 ACM Symposium on
Symbolic and Algebraic Computation, August 10-12, 1976. +
[Wa9] --- "Preserving Sparseness in Multivariate Polynomial Factorization."
Proceedings of the MACSYMA Users' Conference, NASA, Berkeley, CA, July
1977.
[Wa10] --- "Matrix Computations in MACSYMA." Proceedings of the MACSYMA Users'
Conference, NASA, Berkeley, CA, July 1977.
[Wh1] White, Jon L. "Lisp: Program is Data - A Historical Perspective on
MACLISP." Proceedings of the MACSYMA Users' Conference, NASA,
Berkeley, CA, July 1977.
[Wh2] --- "Lisp: Data is Program - A Tutorial in LISP." Proceedings of the
MACSYMA Users' Conference, NASA, Berkeley, CA, July 1977.
[Yu1] Yun, D. Y. "The Hensel Lemma in Symbolic Manipulation" Ph.D. Thesis -
MAC TR-138, November 1974 *
MACSYMA V Bibliography and References 261
[Yu2] --- "On Algorithms for Solving Systems of Polynomial Equations" SIGSAM
Bulletin - Sept. 1973 +
[Zi1] Zippel, R. "Power Series Expansions in MACSYMA" Proceedings of the
Conference on Mathematical Software II Purdue University, May 29, 1974
[Zi2] --- "Univariate Power Series Expansions in Algebraic Manipulation"
Proceedings of the 1976 ACM Symposium on Symbolic and Algebraic
Computation, August 10-12, 1976. +
[Zi3] --- "Radical Simplification Made Easy." →Proceedings of the MACSYMA
Users' Conference_, NASA, Berkeley, CA, July 1977.
262 V Bibliography and References MACSYMA
Where to Obtain Publications
* Available from:
Mathlab Group
MIT Laboratory for Computer Science NE43-828
545 Technology Square
Cambridge, Mass. 02139
** Available for a charge from:
MIT Laboratory for Computer Science
Publications - NE43-417A
545 Technology Square
Cambridge, Mass. 02139
*** Available for a charge from:
National Technical Information Service (NTIS)
Operations Division
Springfield, Va. 22151
(Include AD number given in Bibliography with order)
+ Available from:
The Association for Computing Machinery (ACM)
1133 Avenue of the Americas
New York, N.Y. 10036
MACSYMA 263
Appendix VI - MACSYMA Functions and Argument Evaluation
1
Most MACSYMA functions, including all user-defined functions, are
processed by MACSYMA's evaluator in a straightforward manner: the arguments to
the function are evaluated (left-to-right), the function is applied to the
evaluated arguments, and then the result is returned.
However, there are two classes of functions which are not subsumed under
this simple scheme. In the first class, some or all of the arguments are NOT
evaluated. This class includes
ALIAS ALLOC APPENDFILE ARRAY BATCH BATCON CLOSEFILE
COMPFILE DECLARE DEFINE DELFILE DEMO DISPFUN DISPLAY
FASSAVE GRADEF KILL LABELS LOADFILE LOCAL MATCHDECLARE
MODEDECLARE PLAYBACK QPUT REMARRAY REMFILE REMFUNCTION
REMOVE REMRULE REMVALUE RESTORE SAVE STORE STRING STRINGOUT
TIME TRACE TRANSLATE UNSTORE UNTRACE WRITEFILE
There is another class of functions which control the evaluation of their
arguments; "control" involves the order of evaluation of the arguments as well
as their form. For example, order of evaluation is important in SUM, PROD,
and the plotting functions; on the other hand, the EV command takes advantage
of the structure of the arguments as well. This class includes
CATCH ERRCATCH EV FORTRAN FORTMX FULLMAP FULLMAPL
GRIND PROD SUBSTINPART SUBSTPART SUM
and the plotting functions.
In addition, certain MACSYMA "operators" (see Appendix II) do not evaluate
some of their arguments, such as ":", ":=", "'". On the other hand, MACSYMA
constructs such as BLOCK, the various forms of DO, and logical operators such
as AND,OR, IF...THEN...ELSE control the evaluation of their arguments
←←←←←←←←←←←←←←←
1. More flexibility for user-defined functions will soon be available
MACSYMA INDEX i
%%: [] 144 , 231
%: [] 34 , 231
%E: [] 33 , 37
%EDISPFLAG: [FALSE] 156
%EMODE: [TRUE] 39
%ENUMER: [FALSE] 39
%I: [] 33 , 37
%PI: [] 33 , 37
%TH (i) 154*
ABCONVTEST: [FALSE] 63
ABS (X) 38* , 43, 178
ABSBOXCHAR: [!] 38
ACOS 41
ACOSH 41
ACOT 41
ACOTH 41
ACSC 41
ACSCH 41
ACTIVATE (cont1, cont2, ...) 125*
ADDROW (M,l) 90*
AIRY (X) 196*
ALARMCLOCK (arg1, arg2, arg3) 155*
ALGEBRAIC: [FALSE] 102 , 132
ALGEPSILON: [10 ] 84
ALGSYS ([exp1, exp2, ...], [var1, var2, ...]) 84*
ALIAS 131 , 155*
ALIASES: [] 131
ALLOC 216
ALLROOTS (poly) 82*
ALPHABETIC 132
APPEND (list1, list2, ...) 129*
APPENDFILE (filename1, filename2, DSK, directory) 150*
APPLY (function, list) 127*
APPLY1 (exp, rule1, ..., rulen) 139*
APPLY2 (exp, rule1, ..., rulen) 139*
APPLYB1 (exp, rule1, ..., rulen) 140*
ARRAY (name, dim1, dim2, ..., dimk) 12*
ARRAYAPPLY (array,[sub1, ... ,subk]) 178*
ARRAYINFO 13 , 15, 132*
ARRAYMAKE (name,[i1,i2,...]) 127*
ARRAYS: [] 13 , 15, 131, 145
ASEC 41
ASECH 41
ASIN 41
ASINH 41
ASKEXP: [] 124
ASKSIGN (exp) 124*
ASSUME (pred1, pred2, ...) 122*
AT (exp, list) 73*
ATAN 41
ATAN2 (Y,X) 41*
ATANH 41
ATOM (exp) 135*
ATVALUE (form, list, value) 73* , 115
AUGCOEFMATRIX ([eq1, ...], [var1, ...]) 91*
BACKSUBST: [TRUE] 83
BACKTRACE: [] 145
BASE: [10] 157
BATCH 35 , 147, 149*, 203, 204, 205
BATCHKILL: [FALSE] 205
BATCON (argument) 149* , 203, 204
BATCOUNT: [0] 205
BERLEFACT: [TRUE] 54
BERN (X) 40*
BERNPOLY (v, n) 120*
BESSEL (Z,A) 196*
BETA (X, Y) 40*
BFLOAT (X) 38* , 132
BFLOATP (exp) 135*
BFTORAT: [FALSE] 97
BFTRUNC: [TRUE] 156
BINDTEST 132
BINOMIAL (X, Y) 39*
BLOCK ([v1, ... vk], statement1,..., statementj) 22* , 24, 27, 230
BOTHCASES: [FALSE] 156
BOTHCOEF (exp, var) 75*
BOX (exp) 68*
BOXCHAR: ["] 68
BREAK (arg1, ...) 144* , 154*, 231
BREAKUP: [TRUE] 86
BUG (message) 4*
CABS (exp) 79*
CALCOMPNUM: [20] 225 , 227
CANTEN (exp) 171*
CARG 43 , 79*
CATCH (exp1,...,expn) 153*
CAUCHYSUM[FALSE] 57
CENTERPLOT: [] 227
CF (exp) 120*
CFDISREP (list) 120*
CFEXPAND (x) 120*
CFLENGTH: [1] 120
CHANGEVAR (exp,f(x,y),y,x) 64*
CHARPOLY (M, var) 92*
CHR1 ([i,j,k]) 172*
CHR2 ([i,j],[k]) 173*
CHRISTOF (arg) 163*
CLOSEFILE (filename1, filename2) 150*
COEFF (exp, v, n) 74*
COEFMATRIX ([eq1, ...], [var1, ...]) 90*
COL (M,i) 91*
COMBINE (exp) 51*
COMPFILE ([filespec],f1,f2,...) 160* , 218
COMPGRIND: [FALSE] 160
COMPONENTS (tensor,exp) 170*
CONCAT (arg1, arg2, ...) 154*
CONS (exp, list) 129*
CONSTANT 132
CONSTANTP (exp) 135*
CONTENT (p1, var1, ..., varn) 100*
CONTEXT: [GLOBAL] 125
CONTEXTS: [[]] 125
CONTINUE 5 , 35
CONTOUR: [] 226
CONTRACT (exp) 171*
COPYLIST (L) 90*
COPYMATRIX (M) 90*
COS 41
COSH 41
COT 41
COTH 41
COUNTER: [1] 171
COVDIFF (exp,v1,v2,...) 175*
CSC 41
CSCH 41
CURL 181
CURSOR: [←] 157 , 198
DALEM (field,i,j) 167*
DEACTIVATE (cont1, cont2, ...) 125*
DEBUGMODE: [FALSE] 144
DECLARE 30 , 94, 123*, 132*, 144*, 169
DEFCON (tensor1,<tensor2,tensor3>) 171*
DEFINE (function(arguments),body) 32* , 153*
DEFMATCH 131 , 138*, 145, 148
DEFRULE 131 , 139*, 145, 148
DEFTAYLOR (function, exp) 111*
DELETE (exp1, exp2) 80* , 130
DELFILE (file-specification) 149* , 208
DEMO 147 , 149*, 203, 204
DEMOIVRE: [FALSE] 39 , 132
DENOM (exp) 80* , 98*
DEPENDENCIES: [] 131
DEPENDS (funlist1,varlist1,funlist2,varlist2,...) 60* , 131
DERIVABBREV: [FALSE] 59
DERIVDEGREE (exp, dv, iv) 78*
DERIVLIST 45
DESCRIBE (command) 4* , 147*
DESOLVE ([eq1,...,eqn],[var1,...,varn]) 179*
DETERMINANT (M) 92*
DETOUT 44
DETOUT: [FALSE] 94 , 132
DEVICE: [] 208
DIAGMATRIX (n, x) 90*
DIFF 44 , 59*, 175*
DIMENSION (equation or list of equations) 183*
DIMENSION: [4] 173
DIREC: [] 208
DISP (exp1,exp2, ...) 146*
DISPCON (tensor1,tensor2,...) 171*
DISPFLAG: [TRUE] 87 , 156
DISPFORM (exp) 68*
DISPFUN 11 , 13, 145*
DISPLAY (exp1, exp2, ...) 145*
DISPRULE (rulename) 145*
DISPTERMS (exp) 146*
DIV 181
DIVIDE (p1, p2, var1, ..., varn) 99*
DIVSUM (n,k) 121*
DO 23 , 26, 27
DOALLMXOPS: [TRUE] 94
DOMXMXOPS: [FALSE] 94
DONTFACTOR: [] 53
DOSCMXOPS: [FALSE] 94
DOSCMXPLUS: [FALSE] 94
DOTASSOC: [TRUE] 94
DOTCONSTRULES: [TRUE] 95
DOTDISTRIB: [FALSE] 95
DOTEXPTSIMP: [TRUE] 95
DOTSCRULES: [FALSE] 94
DPART (exp, n1, ..., nk) 68*
DSCALAR (function) 167*
DSKALL: [] 208
DSKGC: [FALSE] 157 , 207
DSKUSE: [FALSE] 207
DUMMY (i1,i2,...) 170*
DUMMYX: [#] 171
DYNAMALLOC: [FALSE] 206
ECHELON (M) 91*
EINSTEIN (dis) 165*
EL (F, YLIST, TLIST) 185*
ELIMINATE ([eq1,eq2,...,eqn],[v1,v2,...,vk]) 188*
EMATRIX (m, n, x, i, j) 90*
ENDCONS (exp, list) 129*
ENTERMATRIX 18 , 89*
ENTIER (X) 38*
ERF (X) 40*
ERFFLAG: [TRUE] 63
ERRCATCH (exp1, exp2, ...) 153*
ERREXP: [] 231
ERRINTSCE: [TRUE] 189
ERROR (arg1, arg2, ...) 153*
ERRORFUN: [FALSE] 153 , 206
EULER (X) 40*
EV (exp, arg1, ..., argn) 44* , 70, 132
EVAL 44
EVALS (mat,lambda) 187*
EVEC1 (M,mu,modes) 187*
EVEC2 (M,mu,modes) 187*
EVENP (exp) 135*
EVFLAG 132
EVFLAG: [] 45
EVFUN 132
EXAMPLE (command) 4* , 147*
EXP (X) 39*
EXPAND 44 , 47*
EXPON: [0] 47
EXPONENTIALIZE: [FALSE] 41 , 132
EXPOP: [0] 47
EXPRESS (expression) 182*
EXPT 10
EXPTDISPFLAG: [TRUE] 156
EXPTISOLATE: [FALSE] 75
EXPTSUBST: [FALSE] 69
EZGCD (p1, p2, ...) 100*
FACRAT: [] 165 , 166, 167
FACTCOMB (exp) 119*
FACTLIM: [-1] 10
FACTOR (exp) 53* , 132
FACTORFLAG: [FALSE] 53 , 132
FACTOROUT (exp,var1,var2,...) 55*
FACTORSUM (exp) 55*
FACTS (context) 125*
FALSE: [] 37
FASSAVE (args) 150* , 209
FASTTIMES (p1, p2) 99*
FEATUREP (a,f ) 123*
FFT (real-array,imag-array) 193*
FIB (X) 40*
FILENAME: [] 208
FILESIZE: [] 208 , 210
FILLARRAY (array,list-or-array) 178*
FIRST (exp) 80* , 130
FLOAT (exp) 38* , 45
FLOAT2BF: [FALSE] 38
FLOAT: [] 132
FLOATNUMP (exp) 135*
FLUSH (expression) 176*
FOR 24 , 26
FORGET (pred1, pred2, ...) 123*
FORTMX (name,matrix) 147*
FORTRAN (exp) 147* , 150
FORTRAN: [FALSE] 150
FORTSPACES: [FALSE] 147
FPPREC: [16] 7 , 38
FREEOF (x1, x2, ..., exp) 136*
FULLMAP (fn, exp1, ...) 128*
FULLMAPL (fn, list1, ...) 128*
FUNCTIONS: [] 11 , 131, 145
FUNMAKE (name,[arg1,...,argn]) 127*
G0 (X) 196*
G1 (X) 196*
GAMMA (X) 40*
GAMMALIM: [1000000] 40
GAUSS (MEAN,SD) 196*
GCD (p1, p2, var1, ...) 100*
GCD: [EZ] 48 , 100
GCFACTOR (n) 54*
GENFACT (X, Y, Z) 40*
GENINDEX: [I] 157
GENINDEX[I] 57
GENMATRIX 18 , 89*
GENSUMNUM[0] 57
GET (a, i) 133*
GETCHAR (a, i) 154*
GFACTOR (exp) 56*
GFACTORSUM (exp) 56*
GLOBALSOLVE: [FALSE] 83
GLOG (X) 39*
GN (X,N) 196*
GO 22 , 24
GRAD 181
GRADEF (f(x1, ..., xn), g1, ..., gn) 61* , 62*, 131
GRADEFS: [] 131
GRAPH ([x1, x2, x3, ..., xn], [y1, y2, y3, ..., yn]) 218*
GRAPH2 (x-lists,y-lists,optional-args) 224*
GRAPH3D (x-lists,y-lists,z-listsoptional-args) 226*
GRIND (arg) 147*
GRIND: [FALSE] 147 , 155
HALFANGLES: [FALSE] 113
HAM (ODES) 185*
HIPOW (exp, v) 78*
HORNER (exp, var) 99*
I0 (X) 195*
I1 (X) 195*
IBASE: [10] 157
IDENT (n) 90*
IEQN (ie,unk,tech,n,guess) 190*
IEQNPRINT: [TRUE] 191
IF 20
IFT (real-array,imag-array) 193*
ILT (exp, lvar, ovar) 116*
IMAGPART 43 , 78*
IN (X,N) 195*
INCHAR: [C] 157
INDEXED (tensor) 170*
INDICES (exp) 170*
INF: [] 37 , 62
INFEVAL 44
INFINITY: [] 37
INFIX (operator, lbp[180], rbp[180], lpos[ANY], rpos[ANY],pos[ANY]) 241*
INFLAG: [FALSE] 80
INFOLISTS: [] 131 , 148, 157
INPART (exp,n1,...,nk) 67*
INPROD (x,y) 187*
INTEGERP (exp) 135*
INTEGRATE (exp, var) 62* , 63
INTERPOLATE (func,x,a,b) 194*
INTFACLIM: [1000] 54
INTOPOIS (A) 105*
INTPOLABS: [0.0] 194
INTPOLERROR: [TRUE] 194
INTPOLREL: [0.0] 194
INTSCE (expr,var) 189*
IRREDUCIBLE (exp) 56*
IS (pred) 123*
ISOLATE (exp, var) 75*
ISQRT (X) 39*
J0 (X) 195*
J1 (X) 195*
JACOBI (p,q) 121*
JN (X,N) 195*
KDELTA (L1,L2) 173*
KEEPFLOAT: [FALSE] 48 , 97, 132
KILL 13 , 15, 148*
LABELS (char) 155*
LABELS: [] 131 , 157
LAMBDA 14
LAPINT (exp,ovar) 117*
LAPLACE (exp, ovar, lvar) 115*
LAPLACIAN 181
LAST (exp) 80* , 130
LASTTIME: [] 156
LC (L) 173*
LDEFINT 63 , 65*
LDISP (exp1,exp2,...) 146*
LDISPLAY (exp1,exp2,...) 145*
LENGTH (exp) 80* , 130
LET (prod, repl, predname, arg1, arg2, ..., argn) 140*
LETRAT: [FALSE] 140
LETRULES (name) 141*
LETSIMP (exp) 140*
LHOSPITALLIM: [4] 64
LHS (eqn) 79*
LIMIT 63 , 64*
LINECHAR: [E] 157
LINEL: [] 35 , 157, 217
LINENUM: [] 157 , 210
LINSOLVE ([exp1, exp2, ...], [var1, var2, ...]) 83*
LISTARITH: [TRUE] 18 , 93, 132
LISTARRAY (array) 178*
LISTCONSTVARS: [FALSE] 74
LISTOFVARS (exp) 74*
LISTP (exp) 136*
LMXCHAR: [] 93
LOADFILE 35 , 149*, 160, 210
LOADPRINT: [TRUE] 156
LOCAL 22 , 126*, 153*
LOG (X) 39*
LOGARC: [FALSE] 41
LOGCONTRACT 39 , 56*, 132
LOGEXPAND: [FALSE] 39 , 132
LOGNUMER: [FALSE] 39
LOGOUT () 152* , 206
LOGSIMP: [TRUE] 39
LOPOW (exp, v) 78*
LORENTZ (exp) 176*
LPART (label, exp, n1, ..., nk) 69*
LRICCICOM (dis) 165*
MAIL (message) 4*
MAKEBOX (exp) 176*
MAKEFACT (exp) 119*
MAKEGAMMA (exp) 120*
MAP (fn, exp1, exp2, ...) 127*
MAPATOM (expr) 128*
MAPERROR: [TRUE] 127
MAPLIST (fn, exp1, exp2, ...) 128*
MATCHDECLARE (patternvar, predicate, ...) 136*
MATCHFIX (operator, match, argpos[ANY], pos[ANY]) 241*
MATRIX 18 , 89*
MATRIXMAP (fn, M) 90*
MATRIXP (exp) 136*
MAX (X1, X2, ...) 38*
MAXAPPLYDEPTH: [10000] 139
MAXAPPLYHEIGHT: [10000] 140
MAXNEGEX: [1000] 47
MAXPOSEX: [1000] 47
MAXPRIME: [489318] 121
MAXTAYORDER: [TRUE] 108
MEMBER (exp, list) 130*
METRIC (G) 172*
METRIC: [] 172
MIN (X1, X2, ...) 38*
MINF: [] 37 , 62
MINFACTORIAL (exp) 119*
MINOR (M, i, j) 91*
MOD (p) 100*
MODEDECLARE (y1, mode1, y2, mode2, ...) 158*
MODRESULT: [FALSE] 101
MODULUS: [FALSE] 100
MOTION (dis) 164*
MULTIGRAPH ([ [xset1, yset1], ..., [xsetn, ysetn] ], optional-args) 218*
MULTIPLICITIES: [NOT%SET%YET] 86
MULTTHRU (exp) 51*
MYOPTIONS: [[]] 8 , 131
NARY (operator, bp[180], argpos[ANY], pos[ANY]) 241*
NEWDET (M,n) 92*
NOEVAL 44
NOFIX|operator (pos[ANY]|) 241*
NOLABELS: [FALSE] 156
NONDIMENSIONALIZE (list of physical quantities) 183*
NONSCALAR 132
NONSCALAR: [] 18
NONSCALARP (exp) 135*
NOSTRING 147
NOT3D: [] 226 , 227
NOUN 132
NOUNDISP: [FALSE] 156
NOUNIFY (f) 68*
NOUNS 45
NROOTS (poly, low, high) 82*
NTERMS (exp) 81*
NTERMSG () 165*
NTERMSRCI () 165*
NUM (exp) 80* , 98*
NUMBERP (exp) 135*
NUMER 45
NUMER: [] 132 , 134
NUMERVAL (var1, exp1, var2, exp2, ...) 134*
NUMFACTOR (exp) 77*
NUSUM (exp,var,low,high) 58*
NZETA (Z) 196*
NZETAI (Z) 196*
NZETAR (Z) 196*
ODDP (exp) 135*
ODE2 (diffeq,depvar,indvar) 66*
OFF: [] 204
ON: [] 204
OPTIMIZE (exp) 158* , 212
OPTIONSET: [FALSE] 156
ORDERGREAT 131 , 151*
ORDERGREATP (exp1,exp2) 152*
ORDERLESS 131 , 151*
ORDERLESSP (exp1,exp2) 152*
OUTCHAR: [D] 157
OUTOFPOIS (A) 105*
PARAMPLOT (f1(t),f2(t),t,low,high) 217* , 218*
PARAMPLOT2 (x-exprs,y-exprs,variable,var-range,optional-args) 224* , 225*
PART (exp, n1, ..., nk) 67*
PARTFRAC (exp, var) 53*
PARTITION (exp, var) 56*
PARTSWITCH: [FALSE] 72
PERSPECTIVE: [TRUE] 227
PFEFORMAT: [FALSE] 156
PICKAPART (exp,depth) 76*
PIECE: [] 72
PLAYBACK (arg) 146* , 147, 155
PLOG (X) 39*
PLOT (F(x), x, low, high) 217*
PLOT2 (y-exprs,variable,var-range,optionals-args) 222* , 223*
PLOT3D (z-exprs,x-var,var-range,y-var,var1-range,optional-args) 226*
PLOTBELL: [TRUE] 226
PLOTBOTMAR: [320] 228
PLOTHEIGHT: [] 157 , 217
PLOTLFTMAR: [128] 228
POISDIFF (A, B) 105*
POISEXPT (A, B) 105*
POISINT (A, B) 105*
POISLIM[5] 107
POISMAP (series, sinfn, cosfn) 106*
POISPLUS (A, B) 105*
POISSIMP (A) 105*
POISSUBST (A, B, C) 105* , 106*
POISTIMES (A, B) 105*
POISTRIM () 105*
POLARFORM 43 , 79*
POLARTORECT (magnitude-array,phase-array) 194*
POLYDECOMP (poly,var) 88*
POLYFACTOR: [FALSE] 83
POLYSIGN (X) 38*
POSTFIX (operator, lbp[180], lpos[ANY], pos[ANY]) 241*
POTENTIAL (givengradient) 182*
POWERDISP: [FALSE] 156
POWERSERIES (exp, var, pt) 111*
PRED 45
PRED: [] 132
PREDERROR: [TRUE] 21 , 123, 144, 230
PREFIX (operator, rbp[180], rpos[ANY], pos[ANY]) 241*
PRIME (n) 121*
PRIMER () 4*
PRINT (exp1, exp2, ...) 146*
PRINTPOIS (A) 105*
PRINTPROPS (a, i) 133*
PRODUCT (exp, ind, lo, hi) 58*
PROGRAMMODE: [FALSE] 87
PROPERTIES (a) 133*
PROPS: [] 131
PROPVARS (prop) 132*
PSEXPAND: [FALSE] 103
PSI (X) 40*
PUT (a, p, i) 133*
QPUT (a, p, i) 133*
QUAL (<expression>, <variables>) 186*
QUIT () 153* , 206
QUNIT (n) 121*
QUOTIENT (p1, p2, var1, ...) 100*
RADCAN (exp) 50* , 132
RADPRODEXPAND: [TRUE] 38 , 50
RADSUBSTFLAG: [FALSE] 70
RAISERIEMANN (dis) 166*
RANDOM (X) 40*
RANK (M) 92*
RAT (exp, v1, ..., vn) 96*
RATALGDENOM: [TRUE] 102 , 132
RATCOEF (exp, v, n) 74*
RATDENOM (exp) 98*
RATDENOMDIVIDE: [TRUE] 48
RATDIFF (exp, var) 101*
RATDISREP (exp) 97*
RATEINSTEIN: [] 165
RATEPSILON: [2.0E-8] 96
RATEXPAND (exp) 48* , 97, 132
RATEXPAND: [FALSE] 48
RATFAC: [FALSE] 29 , 96
RATMX: [FALSE] 93
RATNUMER (exp) 98*
RATNUMP (exp) 135*
RATP (exp) 136*
RATPRINT: [TRUE] 97
RATRIEMAN: [] 166
RATSIMP (exp) 49* , 52, 97, 132
RATSIMPEXPONS: [FALSE] 49
RATSUBST (a, b, c) 70*
RATVARS (var1, var2, ..., varn) 96*
RATVARS: [] 96
RATWEIGHT (v1, w1, ..., vn, wn) 98*
RATWEIGHTS: [[]] 98
RATWEYL: [] 167
RATWTLVL : [FALSE] 98
READ (string1, ...) 153*
READONLY (string1,...) 153*
REALONLY: [TRUE] 85
REALPART 43 , 78*
REALROOTS (poly, bound) 82*
REALROOTS: [TRUE] 84
REARRAY (array,dim1, ... ,dimk) 178*
RECTFORM 43 , 79*
RECTTOPOLAR (real-array,imag-array) 194*
REFCHECK: [FALSE] 144 , 230
REM (a, i) 134*
REMAINDER (p1, p2, var1, ...) 100*
REMARRAY 13 , 15, 148*, 149
REMBOX (exp, arg) 69*
REMCOMPS (tensor) 170*
REMCON (tensor1,tensor2,...) 171*
REMFILE () 151* , 208, 211
REMFUNCTION (f1, f2, ...) 148* , 149
REMLET (prod, name) 141*
REMOVE (a1, f1, a2, f2, ...) 123* , 133*, 148*, 230
REMRULE (function, rulename) 148* , 149
REMTRACE () 144* , 230
REMVALUE 13 , 148*, 149
RENAME (exp, <count>) 170*
RESET () 154*
RESIDUE (exp, var, val) 65*
REST (exp, n) 80* , 130
RESTORE (file-specification) 151* , 212
RESULTANT (p1, p2, var) 100*
RETURN 22 , 24, 26
REVEAL (exp,depth) 77* , 146*
REVERSE (list) 130*
REVERSE: [FALSE] 227
RHS (eqn) 80*
RICCICOM (dis) 164*
RIEMANN (dis) 166* , 173*
RINVARIANT () 167*
RISCH (exp, var) 63*
RMXCHAR: [] 93
ROMBERG (integrand,variable,lower limit,upper limit) 191* , 192*
ROMBERGIT: [11] 192
ROMBERGTOL: [1.E-4] 192
ROOTSCONTRACT 38 , 57*
ROOTSEPSILON: [1.0E-7] 82
ROW (M, i) 91*
RULES: [] 131 , 137
SAVE 35 , 147, 150*, 207, 209, 211
SAVEDEF: [TRUE] 159
SAVEFACTORS: [FALSE] 54
SCALARMATRIXP: [TRUE] 94
SCALEFACTORS (coordinatetransform) 182*
SCANMAP (function,exp) 129*
SCURVATURE () 165*
SEC 41
SECH 41
SEND (message) 4*
SETCHECK: [FALSE] 144
SETCHECKBREAK: [FALSE] 145
SETELMX (x, i, j, M) 90*
SHOWTEN (exp) 169*
SHOWTIME: [FALSE] 35 , 155
SIGN (exp) 123*
SIGNUM (X) 38* , 178
SIMP 44
SIMP: [] 132
SIMPSUM: [] 132
SIMTRAN (mat) 187*
SIN 41
SINH 41
SOLVE 45 , 86*, 87*
SOLVEDECOMPOSES: [TRUE] 87
SOLVEEXPLICIT: [FALSE] 87
SOLVEFACTORS: [TRUE] 86
SOLVERADCAN: [FALSE] 86
SOLVETRIGWARN: [TRUE] 87
SORT (list,optional-predicate) 152*
SPARSE: [FALSE] 94
SQFR (exp) 55*
SQRT (X) 38* , 121
SQRTDISPFLAG: [TRUE] 156
SRRAT (exp) 103*
STAP (OBJECTIVE, LEZEROS, EQZEROS, DECISIONVARS) 184*
STARDISP: [FALSE] 157
STATUS (arg) 154*
STORE (args) 150* , 207, 209
STORENUM: [] 208
STRING (exp) 147* , 155, 198
STRINGOUT 99 , 147, 150*, 155
SUBMATRIX (m1, ..., M, n1, ...) 91*
SUBST (a, b, c) 69*
SUBSTINPART (x, exp, n1, ...) 71*
SUBSTPART (x, exp, n1, ..., nk) 71*
SUBVARP (exp) 135*
SUM (exp, ind, lo, hi) 57*
SUMSPLITFACT: [TRUE] 119
TAN 41
TANH 41
TAYLOR (exp,[var1,pt1,ord1],[var2,pt2,ord2],...) 103* , 107*, 109*, 110*
TAYLORDEPTH: [3] 107
TAYLORINFO (exp) 111*
TELLRAT (poly1,...,polyn) 102*
TELLSIMP 131 , 137*, 145, 148
TELLSIMPAFTER 131 , 137*, 145, 148
THROW (exp) 154*
TIME (Di1, Di2, ...) 152*
TLDEFINT (exp,var,low,high) 65*
TLIMIT (exp,var,val,dir) 65*
TLIMSWITCH: [FALSE] 65
TOPLEVEL: [] 231
TOTALDISREP (exp) 98*
TOTIENT (n) 121*
TRACE (name1, name2, ...) 144*
TRANSBIND: [TRUE] 159
TRANSLATE (f1, f2, ...) 159* , 218
TRANSLATE: [FALSE] 159
TRANSPOSE (M) 91*
TRANSRUN: [TRUE] 159
TRIANGULARIZE (M) 92*
TRIGEXPAND (exp) 113* , 132
TRIGEXPAND: [FALSE] 113 , 132
TRIGREDUCE (exp, var) 114* , 132
TRIGSIGN: [TRUE] 41
TRUE: [] 37
TSETUP () 162*
TTYOFF: [FALSE] 34
UEVEC (M,mu,modes) 187*
UNAME: [] 208
UNDIFF (exp) 176*
UNITRAMP 178
UNITSTEP 178
UNKNOWN (exp) 47*
UNLABELED: [] 69
UNORDER () 152*
UNSTORE (name1, ...) 151*
UNSUM (fun,n) 58*
UNTRACE (name1, ...) 144*
VALUES: [] 9 , 131
VECTORPOTENTIAL (givencurl) 182*
VECTORSIMP (vectorexpression) 181*
VERBIFY (f) 68*
VERBOSE: [FALSE] 111
VERSION: [267] 157
VIEWPT: [] 227
WEYL (dis) 167*
WRITEFILE (DSK, directory) 150*
XAXIS: [FALSE] 217
XTHRU (exp) 52*
YAXIS: [FALSE] 217
YT (f,m,n) 167*
ZERO (tensor) 176*
ZEROBERN: [TRUE] 40
ZEROEQUIV (exp,var) 124*
ZETA (X) 40*
ZUNDERFLOW: [TRUE] 11
3D: [] 226