Fast reliable algorithms for matrices with structure:
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| Format: | Book |
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| Language: | English |
| Published: |
Philadelphia
SIAM
1999
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVI, 342 S. |
| ISBN: | 0898714311 |
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| 245 | 1 | 0 | |a Fast reliable algorithms for matrices with structure |c ed. by T. Kailath ... |
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| adam_text | CONTENTS
PREFACE xiii
NOTATION xv
1 DISPLACEMENT STRUCTURE AND ARRAY ALGORITHMS 1
Thomas Kailath
1.1 Introduction 1
1.2 Toeplitz Matrices 2
1.3 Versions of Displacement Structure 6
1.4 Application to the Fast Evaluation of Matrix Vector Products 12
1.5 Two Fundamental Properties 13
1.6 Fast Factorization of Structured Matrices 15
1.6.1 Schur Reduction (Deflation) 15
1.6.2 Close Relation to Gaussian Elimination 16
1.6.3 A Generalized Schur Algorithm 17
1.6.4 Array Derivation of the Algorithm 18
1.6.5 An Elementary Section 23
1.6.6 A Simple Example 24
1.7 Proper Form of the Fast Algorithm 25
1.7.1 Positive Lengths 25
1.7.2 Negative Lengths 26
1.7.3 Statement of the Algorithm in Proper Form 26
1.7.4 An Associated Transmission Line 27
1.7.5 Shift Structured (F = Z) Positive Definite Matrices 27
1.7.6 The Classical Schur Algorithm 28
1.7.7 The Special Case of Toeplitz Matrices 30
1.8 Some Applications in Matrix Computation 30
1.8.1 Going Beyond F = Z 31
1.8.2 Simultaneous Factorization of T and T~l 31
1.8.3 QR Factorization of Structured Matrices 32
1.8.4 Avoiding Back Substitution in Linear Equations 33
1.9 Look Ahead (Block) Schur Algorithm 34
1.10 Fast Inversion of Structured Matrices 37
1.10.1 Schur Construction (Inflation) 37
1.10.2 Statement of the Fast Inversion Algorithm 39
1.10.3 Algorithm Development 40
1.10.4 An Example 41
viii Contents
1.10.5 Proper Form of the Algorithm 42
1.10.6 Application to Pick Matrices 44
1.10.7 The Degenerate Case 44
1.11 Non Hermitian Toeplitz Like Matrices 45
1.12 Schur Algorithm for Hankel Like Matrices 47
1.13 Structured Matrices and Pivoting 48
1.13.1 Incorporating Pivoting into Generalized Schur Algorithms 49
1.13.2 Transformations to Cauchy Like Structures 50
1.13.3 Numerical Issues 51
1.14 Some Further Issues 52
1.14.1 Incorporating State Space Structure 52
1.14.2 Iterative Methods 53
1.14.3 Interpolation Problems 54
1.14.4 Inverse Scattering 55
2 STABILIZED SCHUR ALGORITHMS 57
Shivkumar Chandrasekaran and AH H. Sayed
2.1 Introduction 57
2.2 Contributions and Problems 58
2.3 Related Works in the Literature 58
2.3.1 Notation 60
2.3.2 Brief Review of Displacement Structure 60
2.4 The Generalized Schur Algorithm 62
2.5 A Limit to Numerical Accuracy 64
2.6 Implementations of the Hyperbolic Rotation 65
2.6.1 Direct Implementation 66
2.6.2 Mixed Downdating 66
2.6.3 The Orthogonal Diagonal Procedure 67
2.6.4 The H Procedure 68
2.7 Implementation of the Blaschke Matrix 69
2.7.1 The Case of Diagonal F 69
2.7.2 The Case of Strictly Lower Triangular F 70
2.8 Enforcing Positive Definiteness 71
2.9 Accuracy of the Factorization 71
2.10 Pivoting with Diagonal F 73
2.10.1 A Numerical Example 74
2.10.2 The Case of Positive F 74
2.10.3 The Nonpositive Case 74
2.11 Controlling the Generator Growth 75
2.12 Solution of Linear Systems of Equations 75
2.12.1 Computing the Matrix R 75
2.12.2 Iterative Refinement 76
2.13 Enhancing the Robustness of the Algorithm 77
2.13.1 Enhancing the OD Method 77
2.13.2 Enhancing the H Procedure 78
2.13.3 A Numerical Example 78
2.14 Results for the Positive Definite Case 79
2.A Pseudocode for the Stable Schur Algorithm 81
Contents ix
3 FAST STABLE SOLVERS FOR STRUCTURED LINEAR SYSTEMS 85
All H. Sayed and Shivkumar Chandrasekaran
3.1 Introduction 85
3.2 Overview of the Proposed Solution 86
3.3 The Generalized Schur Algorithm for Indefinite Matrices 87
3.4 Fast QR Factorization of Shift Structured Matrices 90
3.4.1 The Toeplitz Case 91
3.4.2 Other Augmentations 92
3.5 Well Conditioned Coefficient Matrices 93
3.5.1 Implementation of the Hyperbolic Rotation 94
3.5.2 Avoiding Breakdown 95
3.5.3 Error Analysis of the Last Steps 96
3.5.4 Summary 97
3.5.5 Solving the Linear System of Equations 97
3.6 Ill Conditioned Coefficient Matrices 97
3.6.1 Solving the Linear System of Equations 98
3.6.2 Conditions on the Coefficient Matrix 99
3.7 Summary of the Algorithm 99
4 STABILITY OF FAST ALGORITHMS FOR STRUCTURED
LINEAR SYSTEMS 103
Richard P. Brent
4.1 Introduction 103
4.1.1 Outline 104
4.1.2 Notation 105
4.2 Stability and Weak Stability 105
4.2.1 Gaussian Elimination with Pivoting 106
4.2.2 Weak Stability 107
4.2.3 Example: Orthogonal Factorization 108
4.3 Classes of Structured Matrices 108
4.3.1 Cauchy and Cauchy Like Matrices 109
4.3.2 Toeplitz Matrices 109
4.4 Structured Gaussian Elimination 109
4.4.1 The GKO Toeplitz Algorithm 110
4.4.2 Error Analysis 111
4.4.3 A General Strategy 112
4.5 Positive Definite Structured Matrices 112
4.5.1 The Bareiss Algorithm for Positive Definite Matrices 113
4.5.2 Generalized Schur Algorithms 113
4.6 Fast Orthogonal Factorization 114
4.6.1 Use of the Seminormal Equations 115
4.6.2 Computing Q Stably 115
4.6.3 Solution of Indefinite or Unsymmetric Structured Systems 115
4.7 Concluding Remarks 116
5 ITERATIVE METHODS FOR LINEAR SYSTEMS WITH MATRIX
STRUCTURE 117
Raymond H. Chan and Michael K. Ng
5.1 Introduction 117
5.2 The CG Method 118
x Contents
5.3 Iterative Methods for Solving Toeplitz Systems 121
5.3.1 Preconditioning 122
5.3.2 Circulant Matrices 123
5.3.3 Toeplitz Matrix Vector Multiplication 124
5.3.4 Circulant Preconditioners 125
5.4 Band Toeplitz Preconditioners 130
5.5 Toeplitz Circulant Preconditioners 132
5.6 Preconditioners for Structured Linear Systems 133
5.6.1 Toeplitz Like Systems 133
5.6.2 Toeplitz Plus Hankel Systems 137
5.7 Toeplitz Plus Band Systems 139
5.8 Applications 140
5.8.1 Linear Phase Filtering 140
5.8.2 Numerical Solutions of Biharmonic Equations 142
5.8.3 Queueing Networks with Batch Arrivals 144
5.8.4 Image Restorations 147
5.9 Concluding Remarks 149
5.A Proof of Theorem 5.3.4 150
5.B Proof of Theorem 5.6.2 151
6 ASYMPTOTIC SPECTRAL DISTRIBUTION OF TOEPLITZ
RELATED MATRICES 153
Paolo Tilli
6.1 Introduction 153
6.2 What Is Spectral Distribution? 153
6.3 Toeplitz Matrices and Shift Invariance 157
6.3.1 Spectral Distribution of Toeplitz Matrices 158
6.3.2 Unbounded Generating Function 162
6.3.3 Eigenvalues in the Non Hermitian Case 163
6.3.4 The Szego Formula for Singular Values 164
6.4 Multilevel Toeplitz Matrices 166
6.5 Block Toeplitz Matrices 170
6.6 Combining Block and Multilevel Structure 174
6.7 Locally Toeplitz Matrices 175
6.7.1 A Closer Look at Locally Toeplitz Matrices 178
6.7.2 Spectral Distribution of Locally Toeplitz Sequences 182
6.8 Concluding Remarks 186
7 NEWTON S ITERATION FOR STRUCTURED MATRICES 189
Victor Y. Pan, Sheryl Branham, Rhys E. Rosholt, and Ai Long Zheng
7.1 Introduction 189
7.2 Newton s Iteration for Matrix Inversion 190
7.3 Some Basic Results on Toeplitz Like Matrices 192
7.4 The Newton Toeplitz Iteration 194
7.4.1 Bounding the Displacement Rank 195
7.4.2 Convergence Rate and Computational Complexity 196
7.4.3 An Approach Using / Circulant Matrices 198
7.5 Residual Correction Method 200
7.5.1 Application to Matrix Inversion 200
7.5.2 Application to a Linear System of Equations 201
Contents xi
7.5.3 Application to a Toeplitz Linear System of Equations 201
7.5.4 Estimates for the Convergence Rate 203
7.6 Numerical Experiments 204
7.7 Concluding Remarks 207
7.A Correctness of Algorithm 7.4.2 208
7.B Correctness of Algorithm 7.5.1 209
7.C Correctness of Algorithm 7.5.2 209
8 FAST ALGORITHMS WITH APPLICATIONS TO MARKOV
CHAINS AND QUEUEING MODELS 211
Dario A. Bini and Beatrice Meini
8.1 Introduction 211
8.2 Toeplitz Matrices and Markov Chains 212
8.2.1 Modeling of Switches and Network Traffic Control 214
8.2.2 Conditions for Positive Recurrence 215
8.2.3 Computation of the Probability Invariant Vector 216
8.3 Exploitation of Structure and Computational Tools 217
8.3.1 Block Toeplitz Matrices and Block Vector Product 218
8.3.2 Inversion of Block Triangular Block Toeplitz Matrices 221
8.3.3 Power Series Arithmetic 223
8.4 Displacement Structure 224
8.5 Fast Algorithms 226
8.5.1 The Fast Ramaswami Formula 227
8.5.2 A Doubling Algorithm 227
8.5.3 Cyclic Reduction 230
8.5.4 Cyclic Reduction for Infinite Systems 234
8.5.5 Cyclic Reduction for Generalized Hessenberg Systems 239
8.6 Numerical Experiments 241
9 TENSOR DISPLACEMENT STRUCTURES AND POLYSPECTRAL
MATCHING 245
Victor S. Grigorascu and Phillip A. Regalia
9.1 Introduction 245
9.2 Motivation for Higher Order Cumulants 245
9.3 Second Order Displacement Structure 249
9.4 Tucker Product and Cumulant Tensors 251
9.5 Examples of Cumulants and Tensors 254
9.6 Displacement Structure for Tensors 257
9.6.1 Relation to the Polyspectrum 258
9.6.2 The Linear Case 261
9.7 Polyspectral Interpolation 264
9.8 A Schur Type Algorithm for Tensors 268
9.8.1 Review of the Second Order Case 268
9.8.2 A Tensor Outer Product 269
9.8.3 Displacement Generators 272
9.9 Concluding Remarks 275
xii Contents
10 MINIMAL COMPLEXITY REALIZATION OF STRUCTURED
MATRICES 277
Patrick Dewilde
10.1 Introduction 277
10.2 Motivation of Minimal Complexity Representations 278
10.3 Displacement Structure 279
10.4 Realization Theory for Matrices 280
10.4.1 Nerode Equivalence and Natural State Spaces 283
10.4.2 Algorithm for Finding a Realization 283
10.5 Realization of Low Displacement Rank Matrices 286
10.6 A Realization for the Cholesky Factor 289
10.7 Discussion 293
A USEFUL MATRIX RESULTS 297
Thomas Kailath and AH H. Sayed
A.I Some Matrix Identities 298
A.2 The Gram Schmidt Procedure and the QR Decomposition 303
A.3 Matrix Norms 304
A.4 Unitary and ./ Unitary Transformations 305
A.5 Two Additional Results 306
B ELEMENTARY TRANSFORMATIONS 309
Thomas Kailath and Alt H. Sayed
B.I Elementary Householder Transformations 310
B.2 Elementary Circular or Givens Rotations 312
B.3 Hyperbolic Transformations 314
BIBLIOGRAPHY 321
INDEX 339
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| spelling | Fast reliable algorithms for matrices with structure ed. by T. Kailath ... Philadelphia SIAM 1999 XVI, 342 S. txt rdacontent n rdamedia nc rdacarrier Datenverarbeitung Algorithms Matrices Data processing Algorithmus (DE-588)4001183-5 gnd rswk-swf Matrix Mathematik (DE-588)4037968-1 gnd rswk-swf Matrix Mathematik (DE-588)4037968-1 s Algorithmus (DE-588)4001183-5 s DE-604 Kailath, Thomas Sonstige oth HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009176067&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Fast reliable algorithms for matrices with structure Datenverarbeitung Algorithms Matrices Data processing Algorithmus (DE-588)4001183-5 gnd Matrix Mathematik (DE-588)4037968-1 gnd |
| subject_GND | (DE-588)4001183-5 (DE-588)4037968-1 |
| title | Fast reliable algorithms for matrices with structure |
| title_auth | Fast reliable algorithms for matrices with structure |
| title_exact_search | Fast reliable algorithms for matrices with structure |
| title_full | Fast reliable algorithms for matrices with structure ed. by T. Kailath ... |
| title_fullStr | Fast reliable algorithms for matrices with structure ed. by T. Kailath ... |
| title_full_unstemmed | Fast reliable algorithms for matrices with structure ed. by T. Kailath ... |
| title_short | Fast reliable algorithms for matrices with structure |
| title_sort | fast reliable algorithms for matrices with structure |
| topic | Datenverarbeitung Algorithms Matrices Data processing Algorithmus (DE-588)4001183-5 gnd Matrix Mathematik (DE-588)4037968-1 gnd |
| topic_facet | Datenverarbeitung Algorithms Matrices Data processing Algorithmus Matrix Mathematik |
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