Mathematical Experiments on the Computer (eBook)

Front Cover 1
Mathematical Experiments on the Computer 4
Copyright Page 5
Contents 8
Preface 14
Acknowledgments 18
Part I: EXPERIMENTS ON THE COMPUTER 20
Chapter 1. Definition of the Subject 22
Chapter 2. History of the Project 26
Chapter 3. Case Studies 29
Experiment 1: From Statistics 29
Experiment 2: From Linear Algebra 37
Experiment 3: An Energy Minimization Problem 43
Experiment 4: Neural Networks (Static Problem) 49
Experiment 5: Limit Theorems on Groups 55
Experiment 6: Pattern Restoration 59
Experiment 7: Modeling Language Acquisition 63
Experiment 8: Study of Invariant Curves 67
Experiment 9: Neural Networks (Dynamic Problem) 72
Experiment 10: High-Dimensional Geometry 75
Experiment 11: Strategy of Proofs 77
Chapter 4. Other Areas 79
Chapter 5. Software Requirements 81
Chapter 6. Hardware Requirements 84
Chapter 7. Practical Advice 87
Part II: A MATHEMATICIAN'S GUIDE TO APL 90
Chapter 8. Language for Mathematical Experiments 92
1. Mathematical Languages 92
2. Requirements for the Language 93
3. The Choice of APL 94
4. How to Learn APL 95
Chapter 9. MathematicaI Objects—Data Structures 97
1. Hierarchies of Objects 97
2. Scalars 98
3. Vectors 106
4. Sets 111
5. Matrices 112
6. Other Arrays 116
Chapter 10. Functions—APL Statements 119
1. Praxis of Writing APL Code 119
2. Numerical Mixed Functions 121
3. Structural Mixed Functions 123
4. Mixed Functions for Selection 127
5. Mixed Functions for Selection Information 132
6. Transformations of Data 137
Chapter 11. APL Operators 141
1. The Notion of an Operator 141
2. Reduction 141
3. Scan 146
4. Axis Operator 148
5. Inner Product 149
6. Outer Product 152
Chapter 12. APL Programs—Defined Functions 159
1. Definition Mode 159
2. Flow of Control 166
3. Managing Memory 170
4. Writing Code Praxis 172
Part III: DESIGNING MATHEMATICAL EXPERIMENTS 190
Chapter 13. Designing the Experiment 192
1. Choice of Data Structures 192
2. Looking for the Unexpected 194
3. Prepare for Continued Experiments 195
4. Interpreting the Results 197
Chapter 14. An Experiment in Heuristic Asymptotics 200
1. Guessing About Asymptotics 200
2. Synthesis of the Patterns 201
3. Recognition of Monotonic Patterns 206
4. Some Experiments 214
5. Oscillatory Patterns 222
6. Experiments with Oscillating Patterns 235
7 . Conclusions 240
Part IV: MATHEMATICAL SOFTWARE FOR EXPERIMENTS 242
Chapter 15. Organizing a Program Library 244
1. Why a Library? 244
2. External Documentation 245
3. Internal Documentation 246
4. Comments in the Code 247
5. Program Style 247
Chapter 16. Algebra 248
ADD 248
BALANCE 249
CHOLESKYC 252
CHOLSKYR 254
CONVERT 256
CUBIC 257
DIVBY 259
ELMHES 260
EVAL and EVAL2 262
HSHLDR 264
INVERSE 267
JACOBI 268
MATMULT 272
MAGNITUDE 273
MULLERM 274
MULT 277
MINUS 278
PDIVBY 279
POLY 281
POLYC 282
POLFACT 284
POWER 288
QUARTIC 289
REVERSE1 291
REVERSE2 292
SYNDIV AND SYNDIV1 293
TQL 295
Chapter 17. Analysis 300
BSPL 300
CONFORMAL 303
COSFILON and SINFILON 307
DETERMINANT 311
GRAMS 312
INTEGRAL 315
INTEQ 317
KUTTA 320
MRFFT and MRFFTl 324
NEWTON 327
ROOT 328
SIMPSON 330
SPLINE 332
SPLINT 336
SPLDER 338
TRAPEZOIDAL 339
Chapter 18. Arithmetic 342
CONFRAC and RATIONAL 342
FACTOR 344
GCD and SCM 347
PRIMGEN and PRIMGN 349
Chapter 19. Asymptotics 352
ASYMPTOTICS 353
CDIVDIFF 355
CGAUTOR 356
CONCAVEMAJ and CONVEXMIN 358
CORDER1 and CORDER2 359
FIT 361
FITO 362
FWEIGHT 363
GDIFFEQ 363
G10, G11, G12, G13, G21, G22, G23, G24, G25, G26, G27 365
LSQ 366
MON 367
MONMAJ and MONMIN 369
NORM 369
OSC 370
PRINTOUT 371
RERROR 373
Chapter 20. Geometry 374
ADDC and ADDC1 375
AREA 377
ATAN 378
AXF and CIRCUM 379
DCNP, DECOMP, DEQUAD 380
DIST, DISTANCE, DISTPT 384
GEODESIC, CHRISTOFFEL, GEOAUX 386
HULL1 389
LINSERIES 391
MATRANS 392
PERIM 393
POLY, SFACE, STFORM 394
SUPPORT 396
ROTATE, SCALE, TRANS 397
RMVFVT 399
VADD 400
Chapter 21. Graphs 401
CONNECT 401
CONVERT, MATMAK 403
REACH, COMREACH 405
Chapter 22. Probability 410
BERN 410
BETA 411
BIN 412
BINDF and BINFR 414
CHI2 416
CHISQ 417
CONVOL 419
DISC 421
DISCRETE 423
EQDIST 426
EXPON 427
FISHER 429
GAMMA 430
GAUSSDF 435
GAUSS and SGAUSS 437
GEOM 439
MARKOV 441
MULTINOM 443
NEGBIN 444
PARETO 446
POIS 447
POISSONFR, POISSONFR2, POISSONDF, POISSONDF2 449
SAMPLE 454
STUDENT 456
UNIF 457
Chapter 23. Special Functions 459
CHEB1 and CHEB2 459
ERF 462
HAAR 463
HERMITE 464
LAGUERRE 466
LEGENDRE 467
Chapter 24. Statistics 470
ANALYZE 470
FREQTAB1 472
FREQTAB2 473
HISTOGRAM1 and HISTOGRAM2 475
HISTOSPLINE 478
KOLMOG 482
MODE 484
QUANT 485
RUNS1 486
RUNS2 489
SPECTR1 492
STEMLEAF 496
SIGN 498
TRENDS 501
Chapter 25. Utilities 503
COMPOSE, EDIT 503
LISTALL 505
LISTDV 506
PRINTAB 509
SORT1 511
SPLICE 512
TIMER 514
TITLEC 515
References 517
Solutions to Exercises 521
Index 538