Krylov subspace methods: principles and analysis
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| Main Authors: | , |
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| Format: | Book |
| Language: | English |
| Published: |
Oxford
Oxford Uni. Press
2013
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| Edition: | 1. ed. |
| Series: | Numerical mathematics and scientific computation
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XV, 391 S. graph. Darst. 24 cm |
| ISBN: | 9780199655410 |
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MARC
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| adam_text | Titel: Krylov subspace methods
Autor: Liesen, Jörg
Jahr: 2013
Krylov Subspace Methods
Principles andAnalysis
JOrg Liesen
Technical University of Berlin
Zden£kStrako$
Charles University in Prague
OXFORD
UNIVERSITY PRESS
CONTENTS
1. Introduction 1
1.1. Solving Real-world Problems 2
1.2. AShort Recollection ofIdeas 7
1.3. Specification ofthe Subject and Basic Notation 10
2. KrylovSubspace Methods 12
2.1. ProjectionProcesses 12
2.2. How Krylov Subspaces Come into Play 19
2.3. Mathematical Characterisation of Some Krylov Subspace Methods 22
2.4. The Arnoldi and Lanczos Algorithms 26
2.4.1. Arnoldi andHermitian Lanczos 26
2.4.2. Non-HermitianLanczos 31
2.4.3. Historical Note: The Gram-Schmidt Method 33
2.5. Derivation ofSome Krylov Subspace Methods 36
2.5.1. Derivation of CG Using the Hermitian Lanczos Algorithm 36
2.5.2. CG and the Galerkin Finite Element Method 42
2.5.3. CG and the Minimisation of a Quadratic Functional 47
2.5.4. Hermitian Indefinite Matrices andthe SYMMLQ, Method 54
2.5.5. Minimising the Residual Norm: MINRES and GMRES 57
2.5.6. Further Krylov Subspace Methods 61
2.5.7. Historical Note: A N. Krylovand the Early History ofKrylov
Subspace Methods 64
2.6. Summary and Outlook: Linear Projections and Nonlinear Behaviour 69
3. Matching Moments and Model Reduction View 71
3.1. Stieltjes Moment Problem 73
3.2. Model Reduction via Orthogonal Polynomials 76
3.2.1. Derivation ofthe Gauss-Christoffel Quadrature 77
3.2.2. Moment Matching and Convergence Properties
ofthe Gauss-Christoffel Quadrature 84
3.2.3. Historical Note: Gauss Fundamental Idea, an EarlyApplication,
and Later Developments 87
xiv Contents
3.3. Orthogonal Polynomials and ContinuedFractions 89
3.3.1. Three-term Recurrence and the Interlacing Property 89
3.3.2. Continued Fractions 96
3.3.3. The Gauss-Christoflfel Quadrature for Analytic Functions 101
3.3.4. Summary ofthe Previous Mathematical Development 103
3.3.5. Historical Note: Chebyshev, Markov, Stieltjes, and the Moment
Problem 104
3.3.6. Historical Note: Orthogonal Polynomials and Three-term
Recurrences 107
3.4. Jacobi Matrices 108
3.4.1. Algebraic Properties ofJacobi Matrices 109
3.4.2. The Persistence Theorem, Stabilisation ofNodes and Weights
in the Gauss-Christoffel Quadrature 121
3.4.3. Historical Note: The Origin and EarlyApplications
ofJacobi Matrices 130
3.5. Model Reduction via Matrices: Hermitian Lanczos and CG 136
3.6. Factorisation ofthe MatrixofMoments 142
3.7. Vorobyev s Moment Problem 145
3.7.1. Application to the Hermitian Lanczos Algorithm
and the CG Method 146
3.7.2. Application to the Non-Hermitian Lanczos Algorithm 148
3.7.3. Application to the ArnoldiAlgorithm 151
3.7.4. Matching Moments and Generalisations ofthe Gauss-Christoffel
Quadrature 153
3.8. Matching Moments and Projection Processes 156
3.9. Model Reduction ofLarge-scale Dynamical Systems 160
3.9.1. Approximation ofthe Transfer Function 161
3.9.2. Estimates in Quadratic Forms 165
4. Short Recurrences for Generating Orthogonal Krylov Subspace Bases 168
4.1. The Existence of Conjugate Gradient Like Descent Methods 169
4.2. Cyclic Subspaces and theJordan Canonical Form 172
4.2.1. Invariant Subspaces and the CyclicDecomposition 173
4.2.2. TheJordan Canonical Form and the Length ofthe Krylov
Sequences 185
4.2.3. Historical Note: Classical Results ofLinearAlgebra 188
4.3. Optimal Short Recurrences 189
4.4. Sufficient Conditions 193
4.5. Necessary Conditions 196
4.6. Matrix Formulation and Equivalent Characterisations 204
4.7. Short Recurrences and the Number ofDistinct Eigenvalues 209
4.8. Other Types ofRecurrences 211
4.8.1. The Isometric ArnoldiAlgorithm 211
4.8.2. (s +2, t)-term Recurrences 215
4.8.3. Generalised Krylov Subspaces 219
4.9. Remarks on Functional Analytic Representations 222
Contents xv
5. Cost of Computations Using Krylov Subspace Methods 227
5.1. Seeing the Context is Essential 228
5.2. Direct and Iterative Algebraic Computations 238
5.2.1. Historical Note: Are the Lanczos Method and the CG Method Direct
or Iterative Methods? 241
5.3. Computational Cost ofIndividual Iterations 245
5.4. Particular Computations and Complexity 248
5.5. CloserLook at the Concept of Convergence 250
5.5.1. Linear StationaryIterative Methods 250
5.5.2. Richardson Iteration and Chebyshev Semi-iteration 252
5.5.3. Historical Note: Semi-iterative andKrylov Subspace Methods 254
5.5.4. Krylov Subspace Methods: Nonlinear Methods for Linear
Problems 257
5.6. CG inExact Arithmetic 258
5.6.1. Expressions for the CG Errors and Their Norms 260
5.6.2. Eigenvalue-based Convergence Results for CG 265
5.6.3. Illustrations ofthe Convergence Behaviour ofCG 271
5.6.4. Outlying Eigenvalues and Superlinear Convergence 275
5.6.5. Clustered Eigenvalues and the Sensitivity ofthe Gauss-Christoffel
Quadrature 280
5.7. GMRES in Exact Arithmetic 285
5.7.1. Ritz Values and HarmonicRitzValues 286
5.7.2. Convergence Descriptions Based on Spectral Information 289
5.7.3. More GeneralApproaches 294
5.7.4. Any Nonincreasing Convergence Curve is Possible
with Any Eigenvalues 297
5.7.5. Convection-diffusionExample 303
5.7.6. Asymptotic Estimates for the GMRES Convergence 306
5.8. RoundingErrors and Backward Stability 308
5.8.1. Rounding Errors in Direct Computations 311
5.8.2. Historical Note: Wilkinson and the Backward Error Concept 315
5.8.3. The Backward Error Concept in Iterative Computations 316
5.9. RoundingErrors in the CG Method 320
5.9.1. Delay of Convergence 320
5.9.2. Delay of Convergence can Invalidate Composite
Convergence Bounds 326
5.9.3. MaximalAttainable Accuracy 328
5.9.4. Back to the Poisson Model Problem 331
5.10. Rounding Errors in the GMRES Method 338
5.10.1. The Choice ofthe Basis Affects Numerical Stability 338
5.10.2. Does the OrthogonalisationAlgorithm Matter? 340
5.10.3. MGS GMRES is Backward Stable 343
5.11. Omitted Issues and an Outlook 345
References 349
Index ofHistorical Personalities 385
Index ofTechnical Terms 388
|
| any_adam_object | 1 |
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| isbn | 9780199655410 |
| language | English |
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| owner_facet | DE-83 DE-19 DE-BY-UBM DE-706 DE-11 DE-384 |
| physical | XV, 391 S. graph. Darst. 24 cm |
| publishDate | 2013 |
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| spelling | Liesen, Jörg Verfasser (DE-588)1022275577 aut Krylov subspace methods principles and analysis Jörg Liesen ; Zdeněk Strakoš 1. ed. Oxford Oxford Uni. Press 2013 XV, 391 S. graph. Darst. 24 cm txt rdacontent n rdamedia nc rdacarrier Numerical mathematics and scientific computation Krylov-Verfahren (DE-588)4425226-2 gnd rswk-swf Krylov-Verfahren (DE-588)4425226-2 s DE-604 Strakoš, Zdenek 1959- Verfasser (DE-588)1029110697 aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025383444&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Liesen, Jörg Strakoš, Zdenek 1959- Krylov subspace methods principles and analysis Krylov-Verfahren (DE-588)4425226-2 gnd |
| subject_GND | (DE-588)4425226-2 |
| title | Krylov subspace methods principles and analysis |
| title_auth | Krylov subspace methods principles and analysis |
| title_exact_search | Krylov subspace methods principles and analysis |
| title_full | Krylov subspace methods principles and analysis Jörg Liesen ; Zdeněk Strakoš |
| title_fullStr | Krylov subspace methods principles and analysis Jörg Liesen ; Zdeněk Strakoš |
| title_full_unstemmed | Krylov subspace methods principles and analysis Jörg Liesen ; Zdeněk Strakoš |
| title_short | Krylov subspace methods |
| title_sort | krylov subspace methods principles and analysis |
| title_sub | principles and analysis |
| topic | Krylov-Verfahren (DE-588)4425226-2 gnd |
| topic_facet | Krylov-Verfahren |
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