Concise numerical mathematics:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English German |
| Veröffentlicht: |
Providence, RI
American Mathematical Soc.
2003
|
| Schriftenreihe: | Graduate studies in mathematics
57 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Includes bibliographical references (p. 443-447) and index |
| Beschreibung: | XIV, 453 S. graph. Darst. |
| ISBN: | 082182953X 0821834142 |
Internformat
MARC
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| 240 | 1 | 0 | |a Numerische Mathematik kompakt |
| 245 | 1 | 0 | |a Concise numerical mathematics |c Robert Plato |
| 264 | 1 | |a Providence, RI |b American Mathematical Soc. |c 2003 | |
| 300 | |a XIV, 453 S. |b graph. Darst. | ||
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| 490 | 1 | |a Graduate studies in mathematics |v 57 | |
| 500 | |a Includes bibliographical references (p. 443-447) and index | ||
| 650 | 4 | |a Analyse numérique | |
| 650 | 7 | |a Análise numérica |2 larpcal | |
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Datensatz im Suchindex
| _version_ | 1804129752891523072 |
|---|---|
| adam_text | Contents
Preface to the English Edition xi
Preface to the German Edition xiii
Chapter 1. Interpolation by Polynomials 1
§1.1. General prerequisites and Landau symbols 1
§1.2. Existence and uniqueness of an interpolating polynomial 3
§1.3. Neville s algorithm 6
§1.4. Newton s interpolation formula, divided differences 8
§1.5. The interpolation error 11
§1.6. Chebyshev polynomials 14
Additional topics and literature 18
Exercises 19
Chapter 2. Spline Functions 23
§2.1. Introductory remarks 23
§2.2. Interpolating linear spline functions 24
§2.3. Minimality properties of cubic spline functions 25
§2.4. The calculation of interpolating cubic spline functions 27
§2.5. Error estimates for interpolating cubic splines 33
Additional topics and literature 38
Exercises 38
Chapter 3. The Discrete Fourier Transform and Its Applications 41
§3.1. Discrete Fourier transform 41
v
vi Contents
§3.2. Applications of the discrete Fourier transform 43
§3.3. Fast Fourier transform (FFT) 49
Additional topics and literature 56
Exercises 57
Chapter 4. Solution of Linear Systems of Equations 59
§4.1. Triangular systems 59
§4.2. Gaussian elimination 61
§4.3. The factorization PA = LR 66
§4.4. LR factorization 74
§4.5. Cholesky factorization for positive definite matrices 76
§4.6. Banded matrices 79
§4.7. Norms and error estimates 81
§4.8. The factorization A = QS 91
Additional topics and literature 100
Exercises 100
Chapter 5. Nonlinear Systems of Equations 105
§5.1. Preliminary remarks 105
§5.2. The one dimensional case (N = 1) 107
§5.3. Banach s fixed point theorem 109
§5.4. Newton s method 112
Additional topics and literature 121
Exercises 121
Chapter 6. The Numerical Integration of Functions 123
§6.1. Quadrature by interpolation formulas 124
§6.2. Special quadrature by interpolation formulas 125
§6.3. The error due to quadrature by interpolation 129
§6.4. Degree of exactness for the closed Newton Cotes formulas,
n even 132
§6.5. Composite Newton Cotes formulas 137
§6.6. Asymptotic form of the composite trapezoidal rule 141
§6.7. Extrapolation methods 142
§6.8. Gaussian quadrature 146
§6.9. Appendix: Proof of the asymptotic form for the composite
trapezoidal rule 155
Contents vii
Additional topics and literature 159
Exercises 159
Chapter 7. Explicit One Step Methods for Initial Value Problems
in Ordinary Differential Equations 161
§7.1. An existence and uniqueness theorem 162
§7.2. Theory of one step methods 163
§7.3. One Step methods 166
§7.4. Analysis of round off error 170
§7.5. Asymptotic expansion of the approximations 172
§7.6. Extrapolation methods for one step methods 178
§7.7. Step size control 182
Additional topics and literature 186
Exercises 186
Chapter 8. Multistep Methods for Initial Value Problems
of Ordinary Differential Equations 189
§8.1. Fundamental terms 189
§8.2. The global discretization error for multistep methods 192
§8.3. Specific linear multistep methods preparations 201
§8.4. Adams method 204
§8.5. Nystrom and Milne Simpson methods 210
§8.6. BDF method 214
§8.7. Predictor corrector methods 216
§8.8. Linear homogeneous difference equations 222
§8.9. Stiff differential equations 232
Additional topics and literature 241
Exercises 242
Chapter 9. Boundary Value Problems for Ordinary Differential
Equations 247
§9.1. Problem setting, existence, uniqueness 247
§9.2. Difference methods 250
§9.3. Galerkin methods 260
§9.4. Simple shooting methods 274
Additional topics and literature 276
Exercises 277
viii Contents
Chapter 10. Jacobi, Gauss Seidel and Relaxation Methods for the
Solution of Linear Systems of Equations 281
§10.1. Iteration methods for the solution of linear systems of
equations 281
§10.2. Linear fixed point iteration 282
§10.3. Some special classes of matrices and their properties 287
§10.4. The Jacobi method 289
§10.5. The Gauss Seidel method 292
§10.6. The relaxation method and first convergence results 295
§10.7. The relaxation method for consistently ordered matrices 300
Additional topics and literature 305
Exercises 305
Chapter 11. The Conjugate Gradient and GMRES Methods 311
§11.1. Prerequisites 311
§11.2. The orthogonal residual approach (11.2) for positive
definite matrices 313
§11.3. The CG method for positive definite matrices 316
§11.4. The convergence rate of the CG method 319
§11.5. The CG method for the normal equations 323
§11.6. Arnoldi process 324
§11.7. Realization of GMRES on the basis of the Arnoldi process 328
§11.8. Convergence rate of the GMRES method 333
§11.9. Appendix 1: Krylov subspaces 334
§11.10. Appendix 2: Interactive program systems with
multifunctionality 335
Additional topics and literature 336
Exercises 337
Chapter 12. Eigenvalue Problems 339
§12.1. Introduction 339
§12.2. Perturbation theory for eigenvalue problems 339
§12.3. Localization of eigenvalues 343
§12.4. Variational formulation for eigenvalues of symmetric
matrices 346
§12.5. Perturbation results for the eigenvalues of symmetric
matrices 349
Contents ix
§12.6. Appendix: Factorization of matrices 350
Additional topics and literature 351
Exercises 351
Chapter 13. Numerical Methods for Eigenvalue Problems 355
§13.1. Introductory remarks 355
§13.2. Transformation to Hessenberg form 357
§13.3. Newton s method for the calculation of the eigenvalues
of Hessenberg matrices 362
§13.4. The Jacobi method for the off diagonal element
reduction for symmetric matrices 366
§13.5. The QR algorithm 373
§13.6. The LR algorithm 386
§13.7. The vector iteration 387
Additional topics and literature 389
Exercises 390
Chapter 14. Peano s Error Representation 393
§14.1. Introductory remarks 393
§14.2. Peano kernels 394
§14.3. Applications 397
Additional topics and literature 398
Exercises 398
Chapter 15. Approximation Theory 401
§15.1. Introductory remarks 401
§15.2. Existence of a best approximation 402
§15.3. Uniqueness of a best approximation 404
§15.4. Approximation theory in spaces with a scalar product 408
§15.5. Uniform approximation of continuous functions by
polynomials of maximum degree n — 1 411
§15.6. Applications of the alternation theorem 415
§15.7. Haar spaces, Chebyshev systems 417
Additional topics and literature 420
Exercises 420
Chapter 16. Computer Arithmetic 423
§16.1. Number representations 423
x Contents
§16.2. General floating point number systems 424
§16.3. Floating point number systems in practical applications 429
§16.4. Rounding, truncating 432
§16.5. Arithmetic in floating point number systems 436
Additional topics and literature 441
Bibliography 443
Index 449
|
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| spelling | Plato, Robert Verfasser (DE-588)112333826 aut Numerische Mathematik kompakt Concise numerical mathematics Robert Plato Providence, RI American Mathematical Soc. 2003 XIV, 453 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate studies in mathematics 57 Includes bibliographical references (p. 443-447) and index Analyse numérique Análise numérica larpcal Numerical analysis Numerische Mathematik (DE-588)4042805-9 gnd rswk-swf 1\p (DE-588)4123623-3 Lehrbuch gnd-content Numerische Mathematik (DE-588)4042805-9 s b DE-604 Graduate studies in mathematics 57 (DE-604)BV009739289 57 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010149044&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Plato, Robert Concise numerical mathematics Graduate studies in mathematics Analyse numérique Análise numérica larpcal Numerical analysis Numerische Mathematik (DE-588)4042805-9 gnd |
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| title | Concise numerical mathematics |
| title_alt | Numerische Mathematik kompakt |
| title_auth | Concise numerical mathematics |
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| title_full | Concise numerical mathematics Robert Plato |
| title_fullStr | Concise numerical mathematics Robert Plato |
| title_full_unstemmed | Concise numerical mathematics Robert Plato |
| title_short | Concise numerical mathematics |
| title_sort | concise numerical mathematics |
| topic | Analyse numérique Análise numérica larpcal Numerical analysis Numerische Mathematik (DE-588)4042805-9 gnd |
| topic_facet | Analyse numérique Análise numérica Numerical analysis Numerische Mathematik Lehrbuch |
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