Polynomial based iteration methods for symmetric linear systems:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | German |
| Published: |
Chichester [u.a.]
Wiley Teubner
1996
|
| Series: | Wiley-Teubner series advances in numerical mathematics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | 283 S. graph. Darst. |
| ISBN: | 351902604X 0471967963 |
Staff View
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| 245 | 1 | 0 | |a Polynomial based iteration methods for symmetric linear systems |c Bernd Fischer |
| 264 | 1 | |a Chichester [u.a.] |b Wiley Teubner |c 1996 | |
| 300 | |a 283 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 650 | 7 | |a polynôme Tchebychev |2 inriac | |
| 650 | 7 | |a polynôme orthogonal |2 inriac | |
| 650 | 7 | |a problème Stokes |2 inriac | |
| 650 | 7 | |a sous-espace Krylov |2 inriac | |
| 650 | 7 | |a système linéaire symétrique |2 inriac | |
| 650 | 4 | |a Equations |x Numerical solutions | |
| 650 | 4 | |a Iterative methods (Mathematics) | |
| 650 | 4 | |a Polynomials | |
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Record in the Search Index
| _version_ | 1804125200630939648 |
|---|---|
| adam_text | Contents
1 Introduction 15
1.1 Getting Started 15
General Setting 15
The Startresidual 16
An Isomorphism 16
An Approximation Problem 17
An Inner Product 18
MATLAB/MATHEMATICA Implementations 18
2 Orthogonal Polynomials 19
2.1 General Properties 19
Basic Definitions 19
Three Term Recurrence Relation 23
Stieltjes Procedure 26
Jacobi Matrix 27
Extremal Property of Orthogonal Polynomials 28
Zeros of Orthogonal Polynomials 29
Computing Zeros of Orthogonal Polynomials 30
2.2 Some Applications 30
Gaussian Quadrature 30
Moments and Distribution Functions 34
Associated Polynomials and Continued Fractions 37
2.3 A Useful Tool 44
QR Factorization; Tridiagonal Case 44
2.4 Orthogonal Residual Polynomials 48
Recurrence Relations for Orthogonal Residual Polynomials 48
Extremal Property of Orthogonal Residual Polynomials 51
2.5 Kernel Polynomials 55
Definition and Orthogonality 55
Extremal Property of Kernel Polynomials 56
12 Contents
Recurrence Relations for Kernel Polynomials 60
Zeros of Kernel Polynomials 65
Computing Zeros of Kernel Polynomials 69
2.6 Hermite Kernel Polynomials 70
Definition and Orthogonality 71
Extremal Property and a Christoffel Darboux Formula 73
Recurrence Relations for Hermite Kernel Polynomials 79
Zeros of Hermite Kernel Polynomials 83
2.7 Orthogonal and (Hermite) Kernel Polynomials 84
The ME Connection 85
The MR Connection 87
The ME MR Connection 89
3 Chebyshev and Optimal Polynomials 90
3.1 Basic Definitions 90
Green s Function 92
Equilibrium Distribution 94
Characterization of the Best Approximation 96
3.2 Chebyshev and Optimal Polynomials; One Interval 97
3.3 Chebyshev and Optimal Polynomials; Two Intervals 102
Basic Observations 102
One More Extremal Point 103
Elliptic Functions 110
A Conformal Mapping 115
Green s Function for the Union of Two Disjoint Intervals 119
The Achieser Representation of the Chebyshev Polynomials 121
3.4 Computing an Asymptotic Convergence Factor 125
The Inverse of the Elliptic Sine 126
MATLAB Implementation of SNINV 128
Evaluation of a Theta Function 129
MATLAB Implementation of ASYMPFAC 130
4 Orthogonal Polynomials and Krylov Subspaces 132
4.1 Generating a Basis; Orthonormal Case 132
Lanczos Method 134
MATLAB Implementation of LANCZOS 134
4.2 Generating a Basis; Monic Case 135
5 Estimating the Spectrum and the Distribution function 137
Contents 13
5.1 The Model Problem 137
5.2 Estimating the Spectrum 140
5.3 Approximating the Distribution Function 144
Lanczos Method and Distribution Functions 145
Monotone Spline 149
MATLAB Implementation of MPCI 151
Computing New Orthogonal Polynomials 152
6 Parameter Free Methods 155
6.1 Overview 155
6.2 Implementations Based on Three Term Recurrences 159
Basic Algorithm 159
The CG Approach 161
MATLAB Implementation of CG 162
The CR Approach 164
MATLAB Implementation of CR 165
6.3 CG/CR Applied to Indefinite Systems 166
6.4 Implementations Based on the Monic Basis 173
The STOD Approach 174
MATLAB Implementation of STOD 175
The MCR Approach 176
MATLAB Implementation of MCR 177
178
Modifications 179
6.5 Implementations Based on the Lanczos Basis 179
The SYMMLQ Approach 180
MATLAB Implementation of SYMMLQ 183
The MINRES Approach 185
MATLAB Implementation of MINRES 187
188
6.6 Residual Smoothing
188
6.7 A Non Feasible Approach 189
MATHEMATICA Computation of the Minimal Error 190
6.8 Implementations Based on Normal Equations 191
The Golub/Kahan Bidiagonalization 192
QR Factorization; Bidiagonal Case 194
The LSQR Approach 195
MATLAB Implementation of LSQR 197
The CRAIG Approach 199
14 Contents
MATLAB Implementation of CRAIG 201
6.9 Comparison of the Various Methods 202
Symmetric Spectrum 207
7 Parameter Dependent Methods 212
7.1 The Chebyshev Iteration for Symmetric Indefinite Systems 212
7.2 Methods Based on the Eigenvalue Distribution 217
8 The Stokes Problem 224
8.1 The Continuous Problem 224
Hilbert Spaces 225
The Continuous Stokes Problem 226
Variational Formulation 227
Saddle Point Problems 229
Existence and Uniqueness of Solutions 230
8.2 The Discrete Problem 231
The Linear System 234
8.3 Some Finite Element Spaces 237
9 Approximating the A Norm 248
9.1 Energy Norm 248
9.2 Approximating the A Norm of the Error 250
CG Case 250
MATLAB Implementation of cfAerr 256
Two Examples 256
General Case 259
Lower and Upper Bounds 262
10 Bibliography 263
11 Notation 274
12 Index 278
|
| any_adam_object | 1 |
| author | Fischer, Bernd 1957-2013 |
| author_GND | (DE-588)131854240 |
| author_facet | Fischer, Bernd 1957-2013 |
| author_role | aut |
| author_sort | Fischer, Bernd 1957-2013 |
| author_variant | b f bf |
| building | Verbundindex |
| bvnumber | BV010720063 |
| callnumber-first | Q - Science |
| callnumber-label | QA297 |
| callnumber-raw | QA297.8 |
| callnumber-search | QA297.8 |
| callnumber-sort | QA 3297.8 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 910 SK 915 |
| classification_tum | MAT 412f |
| ctrlnum | (OCoLC)35207804 (DE-599)BVBBV010720063 |
| dewey-full | 512/.5 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512/.5 |
| dewey-search | 512/.5 |
| dewey-sort | 3512 15 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV010720063 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T17:57:47Z |
| institution | BVB |
| isbn | 351902604X 0471967963 |
| language | German |
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| series2 | Wiley-Teubner series advances in numerical mathematics |
| spelling | Fischer, Bernd 1957-2013 Verfasser (DE-588)131854240 aut Polynomial based iteration methods for symmetric linear systems Bernd Fischer Chichester [u.a.] Wiley Teubner 1996 283 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Wiley-Teubner series advances in numerical mathematics Itération (mathématiques) ram MATLAB inriac fonction distribution inriac méthode itérative inriac polynôme Tchebychev inriac polynôme orthogonal inriac problème Stokes inriac sous-espace Krylov inriac système linéaire symétrique inriac Equations Numerical solutions Iterative methods (Mathematics) Polynomials Schwach besetzte Matrix (DE-588)4056053-3 gnd rswk-swf Iteration (DE-588)4123457-1 gnd rswk-swf Lineares Gleichungssystem (DE-588)4035826-4 gnd rswk-swf Orthogonale Polynome (DE-588)4172863-4 gnd rswk-swf Lineares Gleichungssystem (DE-588)4035826-4 s Schwach besetzte Matrix (DE-588)4056053-3 s Iteration (DE-588)4123457-1 s Orthogonale Polynome (DE-588)4172863-4 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007158400&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Fischer, Bernd 1957-2013 Polynomial based iteration methods for symmetric linear systems Itération (mathématiques) ram MATLAB inriac fonction distribution inriac méthode itérative inriac polynôme Tchebychev inriac polynôme orthogonal inriac problème Stokes inriac sous-espace Krylov inriac système linéaire symétrique inriac Equations Numerical solutions Iterative methods (Mathematics) Polynomials Schwach besetzte Matrix (DE-588)4056053-3 gnd Iteration (DE-588)4123457-1 gnd Lineares Gleichungssystem (DE-588)4035826-4 gnd Orthogonale Polynome (DE-588)4172863-4 gnd |
| subject_GND | (DE-588)4056053-3 (DE-588)4123457-1 (DE-588)4035826-4 (DE-588)4172863-4 |
| title | Polynomial based iteration methods for symmetric linear systems |
| title_auth | Polynomial based iteration methods for symmetric linear systems |
| title_exact_search | Polynomial based iteration methods for symmetric linear systems |
| title_full | Polynomial based iteration methods for symmetric linear systems Bernd Fischer |
| title_fullStr | Polynomial based iteration methods for symmetric linear systems Bernd Fischer |
| title_full_unstemmed | Polynomial based iteration methods for symmetric linear systems Bernd Fischer |
| title_short | Polynomial based iteration methods for symmetric linear systems |
| title_sort | polynomial based iteration methods for symmetric linear systems |
| topic | Itération (mathématiques) ram MATLAB inriac fonction distribution inriac méthode itérative inriac polynôme Tchebychev inriac polynôme orthogonal inriac problème Stokes inriac sous-espace Krylov inriac système linéaire symétrique inriac Equations Numerical solutions Iterative methods (Mathematics) Polynomials Schwach besetzte Matrix (DE-588)4056053-3 gnd Iteration (DE-588)4123457-1 gnd Lineares Gleichungssystem (DE-588)4035826-4 gnd Orthogonale Polynome (DE-588)4172863-4 gnd |
| topic_facet | Itération (mathématiques) MATLAB fonction distribution méthode itérative polynôme Tchebychev polynôme orthogonal problème Stokes sous-espace Krylov système linéaire symétrique Equations Numerical solutions Iterative methods (Mathematics) Polynomials Schwach besetzte Matrix Iteration Lineares Gleichungssystem Orthogonale Polynome |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007158400&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT fischerbernd polynomialbasediterationmethodsforsymmetriclinearsystems |