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