Introduction to numerical analysis:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York, NY [u.a.]
McGraw-Hill
1974
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| Edition: | 2. ed. |
| Series: | International series in pure and applied mathematics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIII, 669 S. graph. Darst. |
| ISBN: | 0070287619 |
Staff View
MARC
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| 245 | 1 | 0 | |a Introduction to numerical analysis |c F. B. Hildebrand |
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| 300 | |a XIII, 669 S. |b graph. Darst. | ||
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Record in the Search Index
| _version_ | 1804122850654683136 |
|---|---|
| adam_text | CONTENTS
Preface xi
1 Introduction 1
/./ Numerical Analysis 1
1.2 Approximation 3
1.3 Errors 5
1.4 Significant Figures 10
7.5 Determinacy of Functions. Error Control 16
1.6 Machine Errors 19
1.7 Random Errors 23
1.8 Recursive Computation 28
1.9 Mathematical Preliminaries 31
1.10 Supplementary References 38
Problems 39
2 Interpolation with Divided Differences 51
2.1 Introduction 51
2.2 Linear Interpolation 52
2.3 Divided Differences 54
2.4 Second Degree Interpolation 58
vi CONTENTS
2.5 Newton s Fundamental Formula 60
2.6 Error Formulas 62
2.7 Iterated Interpolation 66
2.8 Inverse Interpolation 68
2.9 Supplementary References 70
Problems 71
3 Lagrangian Methods 80
3.1 Introduction 80
3.2 Lagrange s Interpolation Formula 81
3.3 Numerical Differentiation and Integration 85
3.4 Uniform spacing Interpolation 89
3.5 Newton Cotes Integration Formulas 91
3.6 Composite Integration Formulas 95
3.7 Use of Integration Formulas 97
3.8 Richardson Extrapolation. Romberg Integration 99
3.9 Asymptotic Behavior of Newton Cotes Formulas 103
3.10 Weighting Functions. Filon Integration 107
3.11 Differentiation Formulas 110
3.12 Supplementary References 114
Problems 116
4 Finite Difference Interpolation 129
4.1 Introduction 129
4.2 Difference Notations 130
4.3 Newton Forward and Backward difference Formulas 133
4.4 Gaussian Formulas 136
4.5 Stirling s Formula 138
4.6 Bessel s Formula 140
4.7 Everett s Formulas 143
4.8 Use of Interpolation Formulas 144
4.9 Propagation of Inherent Errors 150
4.10 Throwback Techniques 153
4.11 Interpolation Series 154
4.12 Tables of Interpolation Coefficients 158
4.13 Supplementary References 161
Problems 162
5 Operations with Finite Differences 174
5.1 Introduction 174
CONTENTS vii
5.2 Difference Operators 175
5.3 Differentiation Formulas 181
5.4 Newtonian Integration Formulas 186
5.5 Newtonian Formulas for Repeated Integration 189
5.6 Central Difference Integration Formulas 192
5.7 Subtabulation 195
5.8 Summation and Integration. The Euler Maclaurin Sum Formula 197
5.9 Approximate Summation 203
5.10 Error Terms in Integration Formulas 208
5.11 Other Representations of Error Terms 217
5.12 Supplementary References 222
Problems 222
6 Numerical Solution of Differential Equations 240
6.1 Introduction 240
6.2 Formulas of Open Type 241
6.3 Formulas of Closed Type 244
6.4 Start of Solution 245
6.5 Methods Based on Open Type Formulas 250
6.6 Methods Based on Closed Type Formulas. Prediction Correction
Methods 252
6.7 The Special Case F = Ay 257
6.8 Propagated Error Bounds 265
6.9 Application to Equations of Higher Order. Sets of Equations 269
6.10 Special Second order Equations 275
6.11 Change of Interval 280
6.12 Use of Higher Derivatives 282
6.13 A Simple Runge Kutta Method 285
6.14 Runge Kutta Methods of Higher Order 290
6.15 Boundary Value Problems 293
6.16 Linear Characteristic value Problems 297
6.17 Selection of a Method 301
6.18 Supplementary References 303
Problems 303
7 Least Squares Polynomial Approximation 314
7.1 Introduction 314
7.2 The Principle of Least Squares 314
7.3 Least Squares Approximation over Discrete Sets of Points 315
7.4 Error Estimation 318
Vlii CONTENTS
7.5 Orthogonal Polynomials 327
7.6 Legendre Approximation 329
7.7 Laguerre Approximation 332
7.8 Hermite Approximation 334
7.9 Chebyshev Approximation 336
7.10 Properties of Orthogonal Polynomials. Recursive Computation 340
7.11 Factorial Power Functions and Summation Formulas 344
7.12 Polynomials Orthogonal over Discrete Sets of Points 348
7.13 Gram Approximation 350
7.14 Example: Five Point Least Squares Approximation 353
7.15 Smoothing Formulas 357
7.16 Recursive Computation of Orthogonal Polynomials on Discrete
Sets of Points 363
7.17 Supplementary References 365
Problems 365
8 Gaussian Quadrature and Related Topics 379
8.1 Introduction 379
8.2 Hermite Interpolation 382
8.3 Hermite Quadrature 385
8.4 Gaussian Quadrature 387
8.5 Legendre Gauss Quadrature 390
8.6 Laguerre Gauss Quadrature 392
8.7 Hermite Gauss Quadrature 395
8.8 Chebyshev Gauss Quadrature 398
8.9 Jacobi Gauss Quadrature 399
8.10 Formulas with Assigned Abscissas 402
8.11 Radau Quadrature 406
8.12 Lobatto Quadrature 409
8.13 Convergence of Gaussian quadrature Sequences 412
8.14 Chebyshev Quadrature 414
8.15 Algebraic Derivations 421
8.16 Application to Trigonometric Integrals 427
8.17 Supplementary References 432
Problems 433
9 Approximations of Various Types 446
9.1 Introduction 446
9.2 Fourier Approximation: Continuous Domain 447
9.3 Fourier Approximation: Discrete Domain 452
CONTENTS IX
9.4 Exponential Approximation 457
9.5 Determination of Constituent Periodicities 462
9.6 Optimum Polynomial Interpolation with Selected Abscissas 466
9.7 Chebyshev Interpolation 469
9.8 Economization of Polynomial Approximations 471
9.9 Uniform (Minimax) Polynomial Approximation 475
9.10 Spline Approximation 478
9.11 Splines with Uniform Spacing 482
9.12 Spline Error Estimates 485
9.13 A Special Class of Splines 488
9.14 Approximation by Continued Fractions 494
9.15 Rational Approximations and Continued Fractions 498
9.16 Determination of Convergents of Continued Fractions 502
9.17 Thiele s Continued Fraction Approximations 506
9.18 Uniformization of Rational Approximations 514
9.19 Supplementary References 518
Problems 519
10 Numerical Solution of Equations 539
10.1 Introduction 539
10.2 Sets of Linear Equations 539
10.3 The Gauss Reduction 543
10.4 The Crout Reduction 545
10.5 Intermediate Roundoff Errors 549
10.6 Determination of the Inverse Matrix 553
10.7 Inherent Errors 555
10.8 Tridiagonal Sets of Equations 559
10.9 Iterative Methods and Relaxation 561
10.10 Iterative Methods for Nonlinear Equations 567
70.7/ The Newton Raphson Method 575
70.72 Iterative Methods of Higher Order 578
70.75 Sets of Nonlinear Equations 583
10.14 Iterated Synthetic Division of Polynomials. Lin s Method 588
70.75 Determinacy of Zeros of Polynomials 595
10.16 Bernoulli s Iteration 598
70.77 Graeffe s Root squaring Technique 602
70.75 Quadratic Factors. Lin s Quadratic Method 609
70.79 Bairstow Iteration 613
10.20 Supplementary References 618
Problems 621
X CONTENTS
Appendixes 640
A Justification of the Crout Reduction 640
B Bibliography 644
C Directory of Methods 659
Index 663
|
| any_adam_object | 1 |
| author | Hildebrand, Francis Begnaud 1915-2002 |
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| dewey-ones | 515 - Analysis |
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| discipline | Mathematik |
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| indexdate | 2024-07-09T17:20:26Z |
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| isbn | 0070287619 |
| language | English |
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| physical | XIII, 669 S. graph. Darst. |
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| spelling | Hildebrand, Francis Begnaud 1915-2002 Verfasser (DE-588)138317267 aut Introduction to numerical analysis F. B. Hildebrand 2. ed. New York, NY [u.a.] McGraw-Hill 1974 XIII, 669 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier International series in pure and applied mathematics Analyse numérique ram Numerical analysis Numerische Mathematik (DE-588)4042805-9 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Numerische Mathematik (DE-588)4042805-9 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005593430&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Hildebrand, Francis Begnaud 1915-2002 Introduction to numerical analysis Analyse numérique ram Numerical analysis Numerische Mathematik (DE-588)4042805-9 gnd |
| subject_GND | (DE-588)4042805-9 (DE-588)4123623-3 |
| title | Introduction to numerical analysis |
| title_auth | Introduction to numerical analysis |
| title_exact_search | Introduction to numerical analysis |
| title_full | Introduction to numerical analysis F. B. Hildebrand |
| title_fullStr | Introduction to numerical analysis F. B. Hildebrand |
| title_full_unstemmed | Introduction to numerical analysis F. B. Hildebrand |
| title_short | Introduction to numerical analysis |
| title_sort | introduction to numerical analysis |
| topic | Analyse numérique ram Numerical analysis Numerische Mathematik (DE-588)4042805-9 gnd |
| topic_facet | Analyse numérique Numerical analysis Numerische Mathematik Lehrbuch |
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