Applied Iterative Methods (Computer Science and Applied Mathematics)

✍ Scribed by Louis A. Hageman, David M. Young

• Publisher: Academic Press
• Year: 1981
• Tongue: English
• Leaves: 404
• Category: Library

✦ Synopsis

This graduate-level text examines the practical use of iterative methods in solving large, sparse systems of linear algebraic equations and in resolving multidimensional boundary-value problems. Topics include polynomial acceleration of basic iterative methods, Chebyshev and conjugate gradient acceleration procedures applicable to partitioning the linear system into a “red/black” block form, more. 1981 ed. Includes 48 figures and 35 tables.

✦ Table of Contents

Contents......Page 3
Preface......Page 7
Acknowledgments......Page 11
Notation......Page 13
1.1 Introduction......Page 19
1.2 Vectors and Matrices......Page 20
1.3 Eigenvalues and Eigenvectors......Page 21
1.4 Vector and Matrix Norms......Page 25
1.5 Partitioned Matrices......Page 27
1.6 The Generalized Dirichlet Problem......Page 28
1.7 The Model Problem......Page 32
2.1 Introduction......Page 36
2.2 Convergence and Other Properties......Page 37
2.3 Examples of Basic Iterative Methods......Page 40
2.4 Comparison of Basic Methods......Page 51
2.5 Other Methods......Page 54
3.2 Polynomial Acceleration of Basic Iterative Methods......Page 57
3.3 Examples of Nonpolynomial Acceleration Methods......Page 61
4.1 lntroduction......Page 63
4.2 Optimal Chebyshev Acceleration......Page 64
4.3 Chebyshev Acceleration with Estimated Eigenvalue Bounds......Page 69
4.4 Sensitivity of the Rate of Convergence to the Estimated Eigenvalues......Page 73
5.1 Introduction......Page 77
5.2 The Pseudoresidual Vector delta(n)......Page 79
5.3 Basic Assumptions......Page 80
5.4 Basic Adaptive Parameter and Stopping Relations......Page 82
5.5 An Overall Computational Algorithm......Page 90
5.6 Treatment of the W-Norm......Page 92
5.7 Numerical Results......Page 97
6.1 Introduction......Page 111
6.2 Eigenvector Convergence Theorems......Page 113
6.3 Adaptive Parameter and Stopping Procedures......Page 118
6.4 An Overall Computational Algorithm Using the 2-Norm......Page 124
6.5 The Estimation of the Smallest Eigenvalue mu(N)......Page 130
6.6 Numerical Results......Page 138
6.7 Iterative Behavior When M(E) > mu(1)......Page 149
6.8 Singular and Eigenvector Deficient Problems......Page 152
7.1 Introduction......Page 156
7.2 The Conjugate Gradient Method......Page 157
7.3 The Three-Term Form of the Conjugate Gradient Method......Page 161
7.4 Conjugate Gradient Acceleration......Page 163
7.5 Stopping Procedures......Page 166
7.6 Computational Procedures......Page 169
7.7 Numerical Results?......Page 174
8.1 Introduction......Page 180
8.2 The RS-S1 and RS-CG Methods......Page 184
8.3 The CCSI and CCG Procedures......Page 188
8.4 Numerical Results......Page 207
8.5 Arithmetic and Storage Requirements......Page 217
8.6 Combined (Hybrid) Chebyshev and Conjugate Gradient Iterations......Page 219
8.7 Proofs......Page 223
9.1 Introduction......Page 227
9.2 Consistently Ordered Matrices and Related Matrices......Page 229
9.3 The SOR Method......Page 232
9.4 Eigenvector Convergence of the SOR Difference Vector......Page 237
9.5 SOR Adaptive Parameter and Stopping Procedures......Page 241
9.6 An Overall Computational Algorithm......Page 246
9.7 The SOR Method for .Problems with Red/Black Partitionings......Page 252
9.8 Numerical Results......Page 257
9.9 On the Relative Merits of Certain Partitionings and Certain Iterative Procedures......Page 264
9.10 Proofs of Theorems and Discussion of the Strategy Condition (9-5.21)......Page 271
10.1 Introduction......Page 277
10.2 The Time-Independent Two-Dimensional Problem......Page 280
10.3 The Time-Independent Three-Dimensional Problem......Page 295
10.4 The Time-Dependent Problem......Page 301
11.1 Introduction......Page 305
11.2 The Two-Group Neutron Diffusion Problem......Page 306
11.3 The Neutron Transport Equation in X-y Geometry......Page 318
11.4 A Nonlinear Network Problem......Page 336
12.1 Introduction......Page 348
12.2 Chebyshev Acceleration......Page 350
12.3 Generalized Conjugate Gradient Acceleration Procedures......Page 357
12.4 Lanczos Acceleration......Page 366
12.5 Acceleration Procedures for the GCW Method......Page 369
12.6 An Example......Page 372
Appendix A Chebyshev Acceleration Subroutine......Page 375
Appendix B CCSI Subroutine......Page 381
Appendix C SOR Subroutine......Page 386
Bibliography......Page 391
Index......Page 399