Numerical Methods for Scientists and Engineers

CONTENTS 6
PREFACE 10
PART I Fundamentals and Algorithms 12
1 AN ESSAY ON NUMERICAL METHODS 14
1.1 THE FIVE MAIN IDEAS 14
1.2 SECOND-LEVEL IDEAS 16
1.3 THE FINITE DIFFERENCE CALCULUS 19
1.4 ON FINDING FORMULAS 20
1.5 CLASSICAL NUMERICAL ANALYSIS 23
1.6 MODERN NUMERICAL METHODS- FOURIER APPROXIMATION 24
1.7 OTHER CLASSES OF FUNCTIONS USED IN APPROXIMATIONS 28
1.8 MISCELLANEOUS 28
1.9 REFERENCES 29
2 NUMBERS 30
2.1 INTRODUCTION 30
2.2 THE THREE SYSTEMS OF NUMBERS 31
2.3 FLOATING-POINT NUMBERS 32
2.4 HOW NUMBERS COMBINE 35
2.5 THE RELATIONSHIP TO MATHEMATICS AND STATISTICS 38
2.6 THE STATISTICS OF ROUNDOFF 38
2.7 THE BINARY REPRESENTATION OF NUMBERS 40
2.8 THE FREQUENCY DISTRIBUTION OF MANTISSAS 44
2.9 THE IMPORTANCE OF THE RECIPROCAL DISTRIBUTION 49
2.10 HAND CALCULATION 50
3 FUNCTION EVALUATION 52
3.1 INTRODUCTION 52
3.2 THE EXAMPLE OF THE QUADRATIC EQUATION 52
3.3 REARRANGEMENT OF FORMULAS 54
3.4 SERIES EXPANSIONS 57
3.5 USE OF MACHINE TO DECIDE 59
3.6 THE MEAN VALUE THEOREM 60
3.7 SYNTHETIC DIVISION 62
3.8 ROUNDOFF EFFECTS 64
3.9 COMPLEX NUMBERS—QUADRATIC FACTORS 66
3.10 REPEATED EVALUATIONS 68
3.11 OVERFLOW AND UNDERFLOW 68
4 REAL ZEROS 70
4.1 INTRODUCTION 70
4.2 GRAPHICAL SOLUTION 71
4.3 THE BISECTION METHOD 73
4.4 THE METHOD OF FALSE POSITION 75
4.5 MODIFIED FALSE POSITION 76
4.6 NEWTON'S METHOD 79
4.7 THE CONVERGENCE OF NEWTON'S METHOD 81
4.8 INVARIANT ALGORITHMS 83
4.9 REMARKS ON COMPARING ALGORITHMS 85
4.10 TRACKING ZEROS 87
5 COMPLEX ZEROS 89
5.1 INTRODUCTION 89
5.2 THE CRUDE METHOD 91
5.3 AN EXAMPLE USING THE CRUDE METHOD 92
5.4 THE CURVES u = 0 AND v = 0 AT A ZERO 93
5.5 A PAIR OF EXAMPLES OF u = 0 AND 0 = 0 CURVES 97
5.6 GENERAL RULES FOR THE u = 0 AND v = 0 CURVES 100
5.7 THE PLAN FOR AN IMPROVED SEARCH METHOD 101
5.8 TRACKING A« = 0 CURVE 103
5.9 THE REFINEMENT PROCESS 104
5.10 MULTIPLE ZEROS IN TRACKING 106
5.11 FUNCTIONS OF TWO VARIABLES 108
6 *ZEROS OF POLYNOMIALS 109
6.1 WHY STUDY THIS SPECIAL CASE? 106
6.2 INVARIANCE PRINCIPLE 112
6.3 THE PLAN 113
6.4 PREPROCESSING THE POLYNOMIAL 113
6.5 THE REAL ZEROS 115
6.6 PLAN FOR FINDING COMPLEX ZEROS 117
6.7 BAIRSTOW'S METHOD 119
6.8 CONVERGENCE OF BAIRSTOW'S METHOD 121
6.9 MULTIPLE ZEROS 122
7 LINEAR EQUATIONS AND MATRIX INVERSION 123
7.1 INTRODUCTION 123
7.2 GAUSSIAN1 ELIMINATION—SIMPLIFIED VERSION 124
7.3 PIVOTING 126
7.4 GAUSS-JORDAN ELIMINATION 127
7.5 SCALING 127
7.6 INVARIANT SCALING—ANALYSIS OF VARIANCE 129
7.7 RANK 130
7.8 ILL-CONDITIONED SYSTEMS 132
7.9 THE RIGHT-HAND SIDES CAN CAUSE ILL-CONDITIONING 136
7.10 A DISCUSSION OF GAUSSIAN ELIMINATION 137
7.11 MATRIX INVERSION 1 139
7.12 MATRIX INVERSION 2 140
8 *RANDOM NUMBERS 143
8.1 WHY RANDOM NUMBERS? 143
8.2 SOME USES OF RANDOM NUMBERS 144
8.3 SOURCES OF RANDOM NUMBERS 147
8.4 THE RANDOM-NUMBER GENERATOR 149
8.5 TESTING A RANDOM-NUMBER GENERATOR 152
8.6 OTHER DISTRIBUTIONS 153
8.7 RANDOM MANTISSAS 154
8.8 SWINDLES 156
8.9 NOISE SIMULATION 156
9 THE DIFFERENCE CALCULUS 157
9.1 INTRODUCTION 157
9.2 THE DIFFERENCE OPERATOR 160
9.3 REPEATED DIFFERENCES 162
9.4 THE DIFFERENCE TABLE 164
9.5 TABULATING A POLYNOMIAL AT A REGULAR SPACING 166
9.6 THE FACTORIAL NOTATION 168
*9.7 STIRLING NUMBERS OF THE FIRST KIND 171
*9.8 STIRLING NUMBERS OF THE SECOND KIND 172
9.9 ALTERNATE NOTATIONS 173
9.10 AN EXAMPLE OF AN INTEGRAL EQUATION 175
10 ROUNDOFF ESTIMATION 177
10.1 WHY ROUNDOFF AGAIN? 177
10.2 RANGE ARITHMETIC (INTERVAL ARITHMETIC1) 178
10.3 ERROR PROPAGATION IN A DIFFERENCE TABLE 180
10.4 THE STATISTICS OF ROUNDOFF 181
10.5 CORRELATION IN THE kTH DIFFERENCES 184
10.6 ESTIMATION OF ROUNDOFF IN A TABLE 185
10.7 ISOLATED ERRORS 189
10.8 SYSTEMATIC ERRORS 191
11 THE SUMMATION CALCULUS 192
11.1 INTRODUCTION 192
11.2 SUMMATION BY PARTS 194
11.3 SUMMATION OF POWERS OF x 195
11.4 GENERATING FUNCTIONS 197
11.5 SUMS OF POWERS OF n AGAIN 198
11.6 THE BERNOULLI NUMBERS 200
12 INFINITE SERIES 203
12.1 INTRODUCTION 203
12.2 KUMMER'Si COMPARISON METHOD 206
12.3 SOME STANDARD SERIES 207
12.4 THE RIEMANN ZETA FUNCTION 209
12.5 ANOTHER INTEGRAL EQUATION 210
12.6 EULER'S METHOD 212
12.71 IMPROVING THE CONVERGENCE OF SEQUENCES 216
12.8 INTEGRALS AS APPROXIMATIONS TO SUMS 218
12.9 THE DIGAMMA FUNCTION 219
13 DIFFERENCE EQUATIONS 222
13.1 INTRODUCTION 222
13.2 FIRST-ORDER DIFFERENCE EQUATIONS WITH CONSTANT COEFFICIENTS 233
13.3 THE GENERAL FIRST-ORDER LINEAR DIFFERENCE EQUATION 225
13.4 THE FIBONACCI EQUATION 226
13.5 ANOTHER EXAMPLE OF A SECOND-ORDER LINEAR EQUATION 228
13.6 AN EXAMPLE OF A SYSTEM OF EQUATIONS 229
13.7 A SYSTEM OF EQUATIONS WITH VARIABLE COEFFICIENTS 230
13.8 SECOND-ORDER RECURRENCE RELATIONS 232
PART II Polynomial Approximation-Classical Theory 236
14 POLYNOMIAL INTERPOLATION 238
14.1 ORIENTATION 238
14.2 INTERPOLATION 239
14.3 INTERPOLATION USING ONLY FUNCTION VALUES 241
14.4 THE VANDERMONDE DETERMINANT 244
14.5 LAGRANGE* INTERPOLATION 246
14.6 ERROR OF POLYNOMIAL APPROXIMATIONS 247
14.7 DIFFICULTY OF POLYNOMIAL APPROXIMATION 249
14.8 ON SELECTING SAMPLE POINTS 252
14.9 SUBTABULATION 252
15 FORMULAS USING FUNCTION VALUES 254
15.1 INTRODUCTION 254
15.2 FORMULAS USING INTERPOLATION 255
15.3 THE TAYLOR-SERIES METHOD OF FINDING FORMULAS 257
15.4 THE DIRECT METHOD OF FINDING FORMULAS 259
15.5 THE INVERSE VANDERMONDE 262
15.6 UNIVERSAL MATRICES 264
15.7 SUMMARY OF THE DIRECT METHOD 267
15.8 APPENDIX 268
16 ERROR TERMS 269
16.1 THE NEED OF AN ERROR ESTIMATE 269
16.2 THREE BACKGROUND IDEAS 270
16.3 THE BASIC METHOD APPROACH 272
16.4 THE INFLUENCE FUNCTION 273
16.5 WHEN G(s) HAS A CONSTANT SIGN 275
16.6 THE PRACTICAL EVALUATION OF G(s) 279
16.7 WHEN G(s) IS NOT OF CONSTANT SIGN 280
16.8 THE FLAW IN THE TAYLOR-SERIES APPROACH 283
16.9 A CASE STUDY 284
17 FORMULAS USING DERIVATIVES 288
17.1 INTRODUCTION 288
17.2 HERMITEi INTERPOLATION 289
17.3 THE DIRECT METHOD 291
17.4 THE HERMITE UNIVERSAL MATRICES 292
17.5 SOME EXAMPLES 294
17.6 BIRKHOFF INTERPOLATION AND FORMULAS 298
17.7 AN EXAMPLE OF A NONINTERPOLATORY FORMULA 300
17.8 AN EXPERIMENT IN COMPARING THE VALUE OF DERIVATIVES 303
18 FORMULAS USING DIFFERENCES 307
18.1 USE OF DIFFERENCES 307
18.2 NEWTON'S INTERPOLATION FORMULA 308
18.3 AN ALTERNATIVE FORM FOR THE DIVIDED DIFFERENCE TABLE 311
18.4 NEWTON'S FORMULA AT EQUAL SPACES 313
18.5 INTERPOLATION IN TABLES 314
18.6 THE LOZENGE DIAGRAM 315
18.7 REMARKS ON THESE FORMULAS 319
18.8 MISCELLANEOUS INTERPOLATION FORMULAS 319
18.9 THE HAMMING-PINKHAM INTEGRATION FORMULA 321
18.10 THE DERIVATION OF THE FORMULAS 323
19 *FORMULAS USING THE SAMPLE POINTS AS PARAMETERS 328
19.1 INTRODUCTION 328
19.2 SOME EXAMPLES 329
19.3 GAUSS' QUADRATURE (INTEGRATION)—FORMAL 333
19.4 GAUSS' QUADRATURE—ANALYSIS 334
19.5 THE ERROR TERM 336
19.6 THREE SPECIAL CASES 338
19.7 GIVEN SOME SAMPLE POINTS 339
19.8 CHEBYSHEV INTEGRATION 341
19.9 RALSTON INTEGRATION 343
19.10 GAUSSIAN INTEGRATION USING DERIVATIVES 345
19.11 AN ALGORITHMIC APPROACH TO FINDING FORMULAS 346
20 COMPOSITE FORMULAS 350
20.1 INTRODUCTION 350
20.2 POLYNOMIAL APPROXIMATION AGAIN 351
20.3 THE NEWTON-COTES FORMULAS 353
20.4 REMARKS ON SOME FORMULAS 355
20.5 COMPOSITE FORMULAS 356
20.6 COMPOSITE OR HIGH-ACCURACY FORMULA? 358
20.7 GREGORY-TYPE FORMULAS 358
20.8 COMPOSITE INTERPOLATION 360
20.9 THE CUBIC SPLINE EQUATIONS 360
20.10 COMPARISON WITH POLYNOMIAL INTERPOLATION 363
21 INDEFINITE INTEGRALS—FEEDBACK 368
21.1 INTRODUCTION 368
21.2 SOME SIMPLE FORMULAS FOR INDEFINITE INTEGRALS 370
21.3 A GENERAL APPROACH 372
21.4 TRUNCATION ERROR 373
21.5 STABILITY 377
21.6 CORRELATED ROUNDOFF NOISE 380
21.7 SUMMARY 382
21.8 SOME GENERAL REMARKS 384
21.9 EXPERIMENTAL VERIFICATION OF STABILITY 386
*21.10 AN EXAMPLE OF A CONVOLUTION INTEGRAL WHICH ILLUSTRATES THE CONCEPT OF STABILITY 386
21.11 INSTABILITY IN ALGORITHMS 389
22 INTRODUCTION TO DIFFERENTIAL EQUATIONS 390
22.1 THE SOURCE AND MEANING OF DIFFERENTIAL EQUATIONS 390
22.2 THE DIRECTION FIELD 391
22.3 THE NUMERICAL SOLUTION 393
22.4 AN EXAMPLE 396
22.5 STABILITY OF THE PREDICTOR ALONE 398
22.6 STABILITY OF THE CORRECTOR 399
22.7 SOME GENERAL REMARKS 401
22.8 SYSTEMS OF EQUATIONS 402
23 A GENERAL THEORY OF PREDICTOR-CORRECTOR METHODS 404
23.1 INTRODUCTION 404
23.2 TRUNCATION ERROR 406
23.3 STABILITY 407
23.4 ROUNDOFF NOISE 411
23.5 THE THREE-POINT PREDICTOR 412
23.6 MILNE-TYPE PREDICTORS 413
23.7 ADAMS-BASHFORTH-TYPE PREDICTORS 415
23.8 GENERAL REMARKS ON THE CHOICE OF A METHOD 416
23.9 CHOICE OF PREDICTOR 417
23.10 SELECTED FORMULAS 418
23.11 DESIGNING A SYSTEM 419
23.12 NUMERICAL VERIFICATION 421
24 SPECIAL METHODS OF INTEGRATING ORDINARY DIFFERENTIAL EQUATIONS 423
24.1 INTRODUCTION AND OUTLINE 423
24.2 RUNGE-KUTTA METHODS 424
24.3 SECOND-ORDER-EQUATION METHODS WHEN y' IS MISSING 425
24.4 LINEAR EQUATIONS 427
24.5 A METHOD WHICH USES y', y", AND y"' VALUES 428
24.6 WHEN THE SOLUTION IS NOT EASILY APPROXIMATED BY A POLYNOMIAL 430
24.7 CONSERVATION LAWS 431
24.8 STIFF EQUATIONS 432
24.9 PROBLEMS WITH WIDELY DIFFERENT TIME CONSTANTS 433
24.10 TWO-POINT PROBLEMS 424
25 LEAST SQUARES: THEORY 438
25.1 INTRODUCTION 438
25.2 THE PRINCIPLE OF LEAST SQUARES 440
25.3 OTHER CHOICES BESIDES LEAST SQUARES 442
25.4 THE NORMAL LAW OF ERRORS 443
25.5 THE LEAST-SQUARES STRAIGHT LINE 446
25.6 POLYNOMIAL CURVE FITTING 448
25.7 NONPOLYNOMIAL LEAST SQUARES AND OTHER GENERALIZATIONS 452
25.8 A COMPARISON OF LEAST SQUARES AND POWER-SERIES EXPANSION 453
25.9 CONCLUDING REMARKS ON LEAST SQUARES 454
26 ORTHOGONAL FUNCTIONS 455
26.1 INTRODUCTION 455
26.2 SOME EXAMPLES OF ORTHOGONAL SYSTEMS OF FUNCTIONS 456
26.3 LINEAR INDEPENDENCE AND ORTHOGONALITY 459
26.4 LEAST-SQUARES FITS AND THE FOURIER COEFFICIENTS 461
26.5 BESSEL'S1 INEQUALITY AND COMPLETENESS 462
26.6 ORTHOGONAL POLYNOMIALS 463
26.7 THE LEGENDREi POLYNOMIALS 466
26.8 ORTHOGONAL POLYNOMIALS AND GAUSSIAN QUADRATURE 468
27 LEAST SQUARES: PRACTICE 470
27.1 GENERAL REMARKS ON THE POLYNOMIAL SITUATION 470
27.2 USE OF THE THREE-TERM RECURRENCE RELATION 471
27.3 THE CONSTRUCTION OF QUASI-ORTHOGONAL POLYNOMIALS 473
27.4 ON THE DEGREE OF THE POLYNOMIAL TO USE 475
27.5 NONLINEAR PARAMETERS 477
27.6 LEAST SQUARES WITH RESTRAINTS: CONTINUATION OF THE EXAMPLE IN SEC. 9.10 478
27.7 SMOOTHING BY LEAST-SQUARES FITTING 479
27.8 ANOTHER FAULT OF LEAST-SQUARES FITTING 480
28 CHEBYSHEV APPROXIMATION: THEORY 481
28.1 THE DEFINITION OF CHEBYSHEV POLYNOMIALS 481
28.2 CHEBYSHEV POLYNOMIALS OVER A DISCRETE SET OF POINTS 483
28.3 FIRST PROPERTIES OF THE CHEBYSHEV POLYNOMIALS 484
28.4 FURTHER PROPERTIES OF THE CHEBYSHEV POLYNOMIALS 486
28.5 THE CHEBYSHEV CRITERION 488
28.6 FURTHER IDENTITIES 490
28.7 THE SHIFTED CHEBYSHEV POLYNOMIALS 492
29 CHEBYSHEV APPROXIMATION: PRACTICE 494
29.1 ECONOMIZATION 494
29.2 ON FINDING A CHEBYSHEV EXPANSION (ECONOMIZATION) 496
29.3 THE DIRECT EVALUATION OF THE COEFFICIENTS 497
29.4 A DIRECT METHOD 499
29.5 THE CHEBYSHEV EXPANSION OF AN INTEGRAL 499
29.6 LANCZOS' t PROCESS 501
29.7 THE DIRECT METHOD FOR DIFFERENTIAL EQUATIONS 503
29.8 THE EVALUATION OF CHEBYSHEV EXPANSIONS 504
29.9 THROWBACK 504
29.10 LEVELING THE ERROR CURVE 505
30 *RATIONAL FUNCTION APPROXIMATION 506
30.1 INTRODUCTION 506
30.2 THE DIRECT APPROACH 507
30.3 LEAST-SQUARES FITTING BY RATIONAL FUNCTIONS 508
30.4 CHEBSYHEV APPROXIMATION BY RATIONAL FUNCTIONS 509
30.5 RECIPROCAL DIFFERENCES 509
PART III Fourier Approximation— Modern Theory 512
31 FOURIER SERIES: PERIODIC FUNCTIONS 514
31.1 ORIENTATION 514
31.2 THE EFFECT OF SAMPLING—ALIASING 516
31.3 THE CONTINUOUS FOURIER EXPANSION 518
31.4 THE COMPLEX FORM OF THE FOURIER SERIES 520
31.5 THE FINITE FOURIER SERIES 521
31.6 RELATION OF THE DISCRETE AND CONTINUOUS EXPANSIONS 524
31.7 THE POWER SPECTRUM 526
31.8 INTERPOLATION OF PERIODIC FUNCTIONS 527
31.9 INTEGRATION 531
31.10 THE GENERAL-OPERATOR APPROACH 533
31.11 SOME REMARKS ON THE GENERAL METHOD 536
32 CONVERGENCE OF FOURIER SERIES 538
32.1 THE IMPORTANCE OF CONVERGENCE 538
32.2 STRAIGHT-LINE APPROXIMATION 539
32.3 FUNCTIONS HAVING CONTINUOUS HIGHER DERIVATIVES 540
32.4 IMPROVING THE CONVERGENCE 541
32.5 THE GIBBS PHENOMENON* 543
32.6 LANCZOS' o FACTORS 545
32.7 THE a FACTORS IN THE GENERAL CASE 546
32.8 A COMPARISON OF CONVERGENCE METHODS 547
32.9 LANCZOS' DIFFERENTIATION TECHNIQUE 549
32.10 SUMMARY 549
33 THE FAST FOURIER TRANSFORM 550
33.1 THE DIRECT CALCULATION 550
33.2 INTRODUCTION TO THE FAST FOURIER TRANSFORM (FFT) 551
33.3 THE CENTRAL IDEA OF THE FAST FOURIER TRANSFORM 552
33.4 THE FAST FOURIER TRANSFORM IN PRACTICE 553
33.5 DANGERS OF THE FOURIER TRANSFORM 554
33.6 FOURIER ANALYSIS USING 12 POINTS 554
33.7 COSINE EXPANSIONS 557
33.8 LOCAL FOURIER SERIES 557
34 THE FOURIER INTEGRAL: NONPERIODIC FUNCTIONS 559
34.1 OUTLINE AND PURPOSE OF CHAPTER 559
34.2 NOTATION 560
34.3 SUMMARY OF RESULTS 561
34.4 THE FOURIER INTEGRAL 565
34.5 SOME TRANSFORM PAIRS 566
34.6 BAND-LIMITED FUNCTIONS AND THE SAMPLING THEOREM 568
34.7 THE CONVOLUTION THEOREM 570
34.8 THE EFFECT OF A FINITE SAMPLE SIZE 572
35 A SECOND LOOK AT POLYNOMIAL APPROXIMATION—FILTERS 573
35.1 PURPOSE OF CHAPTER 573
35.2 ROUNDOFF NOISE 574
35.3 DERIVATIVES 576
35.4 INTEGRATION—A FIRST LOOK 577
35.5 SMOOTHING, AN EXAMPLE OF DESIGN 578
35.6 LEAST-SQUARES SMOOTHING 581
35.7 CHEBYSHEV SMOOTHING 582
35.8 THE FOURIER INTEGRAL 584
35.9 SUMMARY 585
36 "INTEGRALS AND DIFFERENTIAL EQUATIONS 586
36.1 INTRODUCTION 586
36.2 SIMPLE RECURSIVE INTEGRATION FORMULAS 587
36.3 THE TRANSFER-FUNCTION APPROACH TO INTEGRATION FORMULAS 588
36.4 GENERAL INTEGRATION FORMULAS 592
36.5 DIFFERENTIAL EQUATIONS 594
36.6 CHEBYSHEV DESIGN OF INTEGRATION FORMULAS: THEORY 596
36.7 SOME DETAILS OF CHEBYSHEV DESIGN 598
36.8 SUMMARY 602
37 'DESIGN OF DIGITAL FILTERS 603
37.1 BACKGROUND 603
37.2 A NONRECURSIVE CLASS OF DIGITAL SMOOTHING FILTERS 604
37.3 AN ESSAY ON SMOOTHING 608
37.4 DIFFERENTIATION FILTERS 610
37.5 RECURSIVE FILTERS 612
38 'QUANTIZATION OF SIGNALS 614
38.1 INTRODUCTION 614
38.2 THE GRAY CODE 615
38.3 THE STATISTICAL DISTRIBUTION OF VALUES 618
38.4 NOISE DUE TO QUANTIZATION 620
38.5 THE QUANTIZATION THEOREM 622
38.6 THE POOR MAN'S FOURIER SERIES 623
38.7 SOME GENERAL REMARKS ON QUANTIZATION EFFECTS 624
PART IV Exponential Approximation 626
39 SUMS OF EXPONENTIALS 628
39.1 INTRODUCTION 628
39.2 LINEAR INDEPENDENCE 629
39.3 KNOWN EXPONENTS 630
39.4 UNKNOWN EXPONENTS 631
39.5 LEAST-SQUARES FITTING 634
39.6 PRONY'S METHOD WITH CONSTRAINTS 634
39.7 WARNINGS 635
39.8 EXPONENTIALS AND POLYNOMIALS 637
39.9 ERROR TERMS 638
40 *THE LAPLACE TRANSFORM 639
40.1 WHAT IS THE LAPLACE TRANSFORM? 639
40.2 SOME EXAMPLES OF LAPLACE TRANSFORMS 640
40.3 SOME GENERAL PROPERTIES OF LAPLACE TRANSFORMS 641
40.4 PERIODIC FUNCTIONS 643
40.5 APPROXIMATION OF LAPLACE TRANSFORMS 643
40.6 COMPLEX FREQUENCIES 645
40.7 A FORMULA FOR NUMERICAL INTEGRATION1 646
40.8 MIDPOINT FORMULAS 649
40.9 EXPERIMENTAL RESULTS 650
40.10 FOURIER TRANSFORMS 650
41 *SIMULATION AND THE METHOD OF ZEROS AND POLES 651
41.1 INTRODUCTION 651
41.2 SIMULATION LANGUAGES 652
41.3 SPECIAL METHODS3 653
41.4 THE FREQUENCY APPROACH AGAIN 654
41.5 THE z TRANSFORM 655
PART V Miscellaneous 658
42 APPROXIMATIONS TO SINGULARITIES 660
42.1 INTRODUCTION 660
42.2 SOME EXAMPLES OF INTEGRALS WITH SINGULARITIES 661
42.3 A SINGULARITY IN A LINEAR DIFFERENTIAL EQUATION 663
42.4 GENERAL REMARKS 666
43 OPTIMIZATION 668
43.1 INTRODUCTION 668
43.2 REVIEW OF CALCULUS RESULTS 670
43.3 LAGRANGE MULTIPLIERS 673
43.4 THE CURSE OF DIMENSION 675
43.5 THE GRADIENT 676
43.6 FOLLOWING THE GRADIENT 678
43.7 ESTIMATING THE GRADIENT 680
43.8 SOME PRACTICAL OBSERVATIONS 681
43.9 THE FLETCHER-POWELL METHOD 682
43.10 OPTIMIZATION SUBJECT TO LINEAR CONSTRAINTS 684
43.11 OTHER METHODS 686
44 LINEAR INDEPENDENCE 688
44.1 INTRODUCTION 688
44.2 LINEAR EQUATIONS 689
44.3 SAMPLING AND LINEAR INDEPENDENCE 691
44.4 POWERS OF x 693
44.5 ORTHOGONAL POLYNOMIALS AND LEAST SQUARES 693
44.6 WHAT SAMPLES? 695
44.7 WHICH BASIS OF FUNCTIONS? 695
45 EIGENVALUES AND EIGENVECTORS OF HERMITIAN MATRICES1 697
45.1 WHAT ARE EIGENVALUES AND EIGENVECTORS? 697
45.2 NOTATION AND HERMITIAN MATRICES 699
45.3 SIMILARITY REDUCTIONS 701
45.4 ORTHOGONAL TRANSFORMATIONS 703
45.5 HOUSEHOLDER TRANSFORMATIONS 704
45.6 TRIDIAGONALIZATION 705
45.7 THE QR ALGORITHM 710
45.8 OVERDETERMINED SYSTEMS OF LINEAR EQUATIONS 711
N+l THE ART OF COMPUTING FOR SCIENTISTS AND ENGINEERS 713
N + l.l IMPORTANCE OF THE TOPIC 713
N + 1.2 WHAT ARE WE GOING TO DO WITH THE ANSWER? 714
N + 1.3 WHAT DO WE KNOW? 716
N + 1.4 DESIGNING THE COMPUTATION ROUTINE 714
N + 1.5 ITERATION OF THE ABOVE STEPS 717
N + 1.6 A CODE OF ETHICS 718
N + 1.7 ESTIMATION OF THE EFFORT NEEDED TO SOLVE THE PROBLEM 718
N + 1.8 LEARNING FROM CHANGES IN THE PLAN 719
N + 1.9 THE OPEN SHOP PHILOSOPHY 720
N + 1.10 CLOSING REMARKS 721
BIBLIOGRAPHY 722
INDEX 726