Iterative Solution of Large Linear Systems [NOOK Book]

Overview


This self-contained treatment offers a systematic development of the theory of iterative methods. Its focal point resides in an analysis of the convergence properties of the successive overrelaxation (SOR) method, as applied to a linear system with a consistently ordered matrix. The text explores the convergence properties of the SOR method and related techniques in terms of the spectral radii of the associated matrices as well as in terms of certain matrix norms. Contents include a review of matrix theory and ...
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Iterative Solution of Large Linear Systems

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Overview


This self-contained treatment offers a systematic development of the theory of iterative methods. Its focal point resides in an analysis of the convergence properties of the successive overrelaxation (SOR) method, as applied to a linear system with a consistently ordered matrix. The text explores the convergence properties of the SOR method and related techniques in terms of the spectral radii of the associated matrices as well as in terms of certain matrix norms. Contents include a review of matrix theory and general properties of iterative methods; SOR method and stationary modified SOR method for consistently ordered matrices; nonstationary methods; generalizations of SOR theory and variants of method; second-degree methods, alternating direction-implicit methods, and a comparison of methods. 1971 edition.
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Product Details

  • ISBN-13: 9780486153339
  • Publisher: Dover Publications
  • Publication date: 6/26/2013
  • Series: Dover Books on Mathematics
  • Sold by: Barnes & Noble
  • Format: eBook
  • Pages: 608
  • File size: 45 MB
  • Note: This product may take a few minutes to download.

Table of Contents

Preface xiii
Acknowledgments xvii
Notation xix
List of Fundamental Matrix Properties xxi
List of Iterative Methods xxiii
1. Introduction 1
1.1. The Model Problem 2
Supplementary Discussion 6
Exercises 6
2. Matrix Preliminaries 7
2.1. Review of Matrix Theory 7
2.2. Hermitian Matrices and Positive Definite Matrices 18
2.3. Vector Norms and Matrix Norms 25
2.4. Convergence of Sequences of Vectors and Matrices 34
2.5. Irreducibility and Weak Diagonal Dominance 36
2.6. Property A 41
2.7. L-Matrices and Related Matrices 42
2.8. Illustrations 48
Supplementary Discussion 53
Exercises 55
3. Linear Stationary Iterative Methods 63
3.1. Introduction 63
3.2. Consistency, Reciprocal Consistency, and Complete Consistency 65
3.3. Basic Linear Stationary Iterative Methods 70
3.4. Generation of Completely Consistent Methods 75
3.5. General Convergence Theorems 77
3.6. Alternative Convergence Conditions 80
3.7. Rates of Convergence 84
3.8. The Jordan Condition Number of a 2 X 2 Matrix 89
Supplementary Discussion 94
Exercises 95
4. Convergence of the Basic Iterative Methods 106
4.1. General Convergence Theorems 106
4.2. Irreducible Matrices with Weak Diagonal Dominance 107
4.3. Positive Definite Matrices 108
4.4. The SOR Method with Varying Relaxation Factors 118
4.5. L-Matrices and Related Matrices 120
4.6. Rates of Convergence of the J and GS Methods for the Model Problem 127
Supplementary Discussion 132
Exercises 133
5. Eigenvalues of the SOR Method for Consistently Ordered Matrices 140
5.1. Introduction 140
5.2. Block Tri-Diagonal Matrices 141
5.3. Consistently Ordered Matrices and Ordering Vectors 144
5.4. Property A 148
5.5. Nonmigratory Permutations 153
5.6. Consistently Ordered Matrices Arising from Difference Equations 157
5.7. A Computer Program for Testing for Property A and Consistent Ordering 159
5.8. Other Developments of the SOR Theory 162
Supplementary Discussion 163
Exercises 163
6. Determination of the Optimum Relaxation Factor 169
6.1. Virtual Spectral Radius 170
6.2. Analysis of the Case Where All Eigenvalues of B Are Real 171
6.3. Rates of Convergence: Comparison with the Gauss-Seidel Method 188
6.4. Analysis of the Case Where Some Eigenvalues of B Are Complex 191
6.5. Practical Determination of [Omega subscript b]: General Considerations 200
6.6. Iterative Methods of Choosing [Omega subscript b] 209
6.7. An Upper Bound for [mu] 211
6.8. A Priori Determination of [mu]: Exact Methods 216
6.9. A Priori Determination of [mu]: Approximate Values 222
6.10. Numerical Results 224
Supplementary Discussion 227
Exercises 228
7. Norms of the SOR Method 233
7.1. The Jordan Canonical Form of L[subscript Omega] 234
7.2. Basic Eigenvalue Relation 239
7.3. Determination of [double vertical line] L[subscript Omega double vertical line subscript D superscript 1/2] 245
7.4. Determination of [double vertical line] L[superscript m subscript Omega b double vertical line subscript D superscript 1/2] 248
7.5. Determination of [double vertical line] L[subscript Omega double vertical line subscript A superscript 1/2] 255
7.6. Determination of [double vertical line] L[superscript m subscript Omega b double vertical line subscript A superscript 1/2] 258
7.7. Comparison of [double vertical line] L[superscript m subscript Omega b double vertical line subscript D superscript 1/2] and [double vertical line] L[superscript m subscript Omega b double vertical line subscript A superscript 1/2] 264
Supplementary Discussion 265
Exercises 266
8. The Modified SOR Method: Fixed Parameters 271
8.1. Introduction 271
8.2. Eigenvalues of L[subscript Omega, Omega'] 273
8.3. Convergence and Spectral Radius 277
8.4. Determination of [double vertical line] L[subscript Omega, Omega' double vertical line subscript D superscript 1/2] 283
8.5. Determination of [double vertical line] L[subscript Omega, Omega' double vertical line subscript A superscript 1/2] 288
Supplementary Discussion 291
Exercises 291
9. Nonstationary Linear Iterative Methods 295
9.1. Consistency, Convergence, and Rates of Convergence 295
9.2. Periodic Nonstationary Methods 300
9.3. Chebyshev Polynomials 301
Supplementary Discussion 304
Exercises 304
10. The Modified SOR Method: Variable Parameters 306
10.1. Convergence of the MSOR Method 307
10.2. Optimum Choice of Relaxation Factors 307
10.3. Alternative Optimum Parameter Sets 311
10.4. Norms of the MSOR Method: Sheldon's Method 315
10.5. The Modified Sheldon Method 319
10.6. Cyclic Chebyshev Semi-Iterative Method 321
10.7. Comparison of Norms 327
Supplementary Discussion 340
Exercises 341
11. Semi-Iterative Methods 344
11.1. General Considerations 345
11.2. The Case Where G Has Real Eigenvalues 347
11.3. J, JOR, and RF Semi-Iterative Methods 355
11.4. Richardson's Method 361
11.5. Cyclic Chebyshev Semi-Iterative Method 365
11.6. GS Semi-Iterative Methods 367
11.7. SOR Semi-Iterative Methods 374
11.8. MSOR Semi-Iterative Methods 376
11.9. Comparison of Norms 383
Supplementary Discussion 385
Exercises 386
12. Extensions of the SOR Theory: Stieltjes Matrices 391
12.1. The Need for Some Restrictions on A 391
12.2. Stieltjes Matrices 395
Supplementary Discussion 401
Exercises 401
13. Generalized Consistently Ordered Matrices 404
13.1. Introduction 404
13.2. CO(q, r)-Matrices, Property A[subscript q,r], and Ordering Vectors 405
13.3. Determination of the Optimum Relaxation Factor 413
13.4. Generalized Consistently Ordered Matrices 418
13.5. Relation between GCO(q, r)-Matrices and CO(q, r)-Matrices 419
13.6. Computational Procedures: Canonical Forms 422
13.7. Relation to Other Work 428
Supplementary Discussion 429
Exercises 430
14. Group Iterative Methods 434
14.1. Construction of Group Iterative Methods 435
14.2. Solution of a Linear System with a Tri-Diagonal Matrix 441
14.3. Convergence Analysis 445
14.4. Applications 452
14.5. Comparison of Point and Group Iterative Methods 454
Supplementary Discussion 456
Exercises 457
15. Symmetric SOR Method and Related Methods 461
15.1. Introduction 461
15.2. Convergence Analysis 463
15.3. Choice of Relaxation Factor 464
15.4. SSOR Semi-Iterative Methods: The Discrete Dirichlet Problem 471
15.5. Group SSOR Methods 474
15.6. Unsymmetric SOR Method 476
15.7. Symmetric and Unsymmetric MSOR Methods 478
Supplementary Discussion 480
Exercises 481
16. Second-Degree Methods 486
Supplementary Discussion 493
Exercises 493
17. Alternating Direction Implicit Methods 495
17.1. Introduction: The Peaceman-Rachford Method 495
17.2. The Stationary Case: Consistency and Convergence 498
17.3. The Stationary Case: Choice of Parameters 503
17.4. The Commutative Case 514
17.5. Optimum Parameters 518
17.6. Good Parameters 525
17.7. The Helmholtz Equation in a Rectangle 531
17.8. Monotonicity 534
17.9. Necessary and Sufficient Conditions for the Commutative Case 535
17.10. The Noncommutative Case 545
Supplementary Discussion 547
Exercises 548
18. Selection of Iterative Method 553
Bibliography 556
Index 565
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