Iterative Solution of Large Linear Systems

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ISBN: 9780127730509 | 0127730508
Cover: Hardcover
Copyright: 6/1/1971

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Supplemental Materials

Preface	p. xiii
Acknowledgments	p. xvii
Notation	p. xix
List of Fundamental Matrix Properties	p. xxi
List of Iterative Methods	p. xxiii
Introduction	p. 1
The Model Problem	p. 2
Supplementary Discussion	p. 6
Exercises	p. 6
Matrix Preliminaries	p. 7
Review of Matrix Theory	p. 7
Hermitian Matrices and Positive Definite Matrices	p. 18
Vector Norms and Matrix Norms	p. 25
Convergence of Sequences of Vectors and Matrices	p. 34
Irreducibility and Weak Diagonal Dominance	p. 36
Property A	p. 41
L-Matrices and Related Matrices	p. 42
Illustrations	p. 48
Supplementary Discussion	p. 53
Exercises	p. 55
Linear Stationary Iterative Methods	p. 63
Introduction	p. 63
Consistency, Reciprocal Consistency, and Complete Consistency	p. 65
Basic Linear Stationary Iterative Methods	p. 70
Generation of Completely Consistent Methods	p. 75
General Convergence Theorems	p. 77
Alternative Convergence Conditions	p. 80
Rates of Convergence	p. 84
The Jordan Condition Number of a 2 X 2 Matrix	p. 89
Supplementary Discussion	p. 94
Exercises	p. 95
Convergence of the Basic Iterative Methods	p. 106
General Convergence Theorems	p. 106
Irreducible Matrices with Weak Diagonal Dominance	p. 107
Positive Definite Matrices	p. 108
The SOR Method with Varying Relaxation Factors	p. 118
L-Matrices and Related Matrices	p. 120
Rates of Convergence of the J and GS Methods for the Model Problem	p. 127
Supplementary Discussion	p. 132
Exercises	p. 133
Eigenvalues of the SOR Method for Consistently Ordered Matrices	p. 140
Introduction	p. 140
Block Tri-Diagonal Matrices	p. 141
Consistently Ordered Matrices and Ordering Vectors	p. 144
Property A	p. 148
Nonmigratory Permutations	p. 153
Consistently Ordered Matrices Arising from Difference Equations	p. 157
A Computer Program for Testing for Property A and Consistent Ordering	p. 159
Other Developments of the SOR Theory	p. 162
Supplementary Discussion	p. 163
Exercises	p. 163
Determination of the Optimum Relaxation Factor	p. 169
Virtual Spectral Radius	p. 170
Analysis of the Case Where All Eigenvalues of B Are Real	p. 171
Rates of Convergence: Comparison with the Gauss-Seidel Method	p. 188
Analysis of the Case Where Some Eigenvalues of B Are Complex	p. 191
Practical Determination of [Omega subscript b]: General Considerations	p. 200
Iterative Methods of Choosing [Omega subscript b]	p. 209
An Upper Bound for [mu]	p. 211
A Priori Determination of [mu]: Exact Methods	p. 216
A Priori Determination of [mu]: Approximate Values	p. 222
Numerical Results	p. 224
Supplementary Discussion	p. 227
Exercises	p. 228
Norms of the SOR Method	p. 233
The Jordan Canonical Form of L[subscript Omega]	p. 234
Basic Eigenvalue Relation	p. 239
Determination of [double vertical line] L[subscript Omega double vertical line subscript D superscript 1/2]	p. 245
Determination of [double vertical line] L[superscript m subscript Omega b double vertical line subscript D superscript 1/2]	p. 248
Determination of [double vertical line] L[subscript Omega double vertical line subscript A superscript 1/2]	p. 255
Determination of [double vertical line] L[superscript m subscript Omega b double vertical line subscript A superscript 1/2]	p. 258
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]	p. 264
Supplementary Discussion	p. 265
Exercises	p. 266
The Modified SOR Method: Fixed Parameters	p. 271
Introduction	p. 271
Eigenvalues of L[subscript Omega, Omega']	p. 273
Convergence and Spectral Radius	p. 277
Determination of [double vertical line] L[subscript Omega, Omega' double vertical line subscript D superscript 1/2]	p. 283
Determination of [double vertical line] L[subscript Omega, Omega' double vertical line subscript A superscript 1/2]	p. 288
Supplementary Discussion	p. 291
Exercises	p. 291
Nonstationary Linear Iterative Methods	p. 295
Consistency, Convergence, and Rates of Convergence	p. 295
Periodic Nonstationary Methods	p. 300
Chebyshev Polynomials	p. 301
Supplementary Discussion	p. 304
Exercises	p. 304
The Modified SOR Method: Variable Parameters	p. 306
Convergence of the MSOR Method	p. 307
Optimum Choice of Relaxation Factors	p. 307
Alternative Optimum Parameter Sets	p. 311
Norms of the MSOR Method: Sheldon's Method	p. 315
The Modified Sheldon Method	p. 319
Cyclic Chebyshev Semi-Iterative Method	p. 321
Comparison of Norms	p. 327
Supplementary Discussion	p. 340
Exercises	p. 341
Semi-Iterative Methods	p. 344
General Considerations	p. 345
The Case Where G Has Real Eigenvalues	p. 347
J, JOR, and RF Semi-Iterative Methods	p. 355
Richardson's Method	p. 361
Cyclic Chebyshev Semi-Iterative Method	p. 365
GS Semi-Iterative Methods	p. 367
SOR Semi-Iterative Methods	p. 374
MSOR Semi-Iterative Methods	p. 376
Comparison of Norms	p. 383
Supplementary Discussion	p. 385
Exercises	p. 386
Extensions of the SOR Theory: Stieltjes Matrices	p. 391
The Need for Some Restrictions on A	p. 391
Stieltjes Matrices	p. 395
Supplementary Discussion	p. 401
Exercises	p. 401
Generalized Consistently Ordered Matrices	p. 404
Introduction	p. 404
CO(q, r)-Matrices, Property A[subscript q,r], and Ordering Vectors	p. 405
Determination of the Optimum Relaxation Factor	p. 413
Generalized Consistently Ordered Matrices	p. 418
Relation between GCO(q, r)-Matrices and CO(q, r)-Matrices	p. 419
Computational Procedures: Canonical Forms	p. 422
Relation to Other Work	p. 428
Supplementary Discussion	p. 429
Exercises	p. 430
Group Iterative Methods	p. 434
Construction of Group Iterative Methods	p. 435
Solution of a Linear System with a Tri-Diagonal Matrix	p. 441
Convergence Analysis	p. 445
Applications	p. 452
Comparison of Point and Group Iterative Methods	p. 454
Supplementary Discussion	p. 456
Exercises	p. 457
Symmetric SOR Method and Related Methods	p. 461
Introduction	p. 461
Convergence Analysis	p. 463
Choice of Relaxation Factor	p. 464
SSOR Semi-Iterative Methods: The Discrete Dirichlet Problem	p. 471
Group SSOR Methods	p. 474
Unsymmetric SOR Method	p. 476
Symmetric and Unsymmetric MSOR Methods	p. 478
Supplementary Discussion	p. 480
Exercises	p. 481
Second-Degree Methods	p. 486
Supplementary Discussion	p. 493
Exercises	p. 493
Alternating Direction Implicit Methods	p. 495
Introduction: The Peaceman-Rachford Method	p. 495
The Stationary Case: Consistency and Convergence	p. 498
The Stationary Case: Choice of Parameters	p. 503
The Commutative Case	p. 514
Optimum Parameters	p. 518
Good Parameters	p. 525
The Helmholtz Equation in a Rectangle	p. 531
Monotonicity	p. 534
Necessary and Sufficient Conditions for the Commutative Case	p. 535
The Noncommutative Case	p. 545
Supplementary Discussion	p. 547
Exercises	p. 548
Selection of Iterative Method	p. 553
Bibliography	p. 556
Index	p. 565
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