Iterative Solution of Large Linear Systems describes the systematic development of a substantial portion of the theory of iterative methods for solving large linear systems, with emphasis on practical techniques. The focal point of the book is an analysis of the convergence properties of the successive overrelaxation (SOR) method as applied to a linear system where the matrix is "consistently ordered". Comprised of 18 chapters, this volume begins by showing how the solution of a certain partial differential equation by finite difference methods leads to a large linear system with a sparse matrix. The next chapter reviews matrix theory and the properties of matrices, as well as several theorems of matrix theory without proof. A number of iterative methods, including the SOR method, are then considered. Convergence theorems are also given for various iterative methods under certain assumptions on the matrix A of the system. Subsequent chapters deal with the eigenvalues of the SOR method for consistently ordered matrices; the optimum relaxation factor; nonstationary linear iterative methods; and semi-iterative methods. This book will be of interest to students and practitioners in the fields of computer science and applied mathematics.

Table of Contents:
Preface Acknowledgments Notation List of Fundamental Matrix Properties List of Iterative Methods 1. Introduction 1.1. The Model Problem Supplementary Discussion Exercises 2. Matrix Preliminaries 2.1. Review of Matrix Theory 2.2. Hermitian Matrices and Positive Definite Matrices 2.3. Vector Norms and Matrix Norms 2.4. Convergence of Sequences of Vectors and Matrices 2.5. Irreducibility and Weak Diagonal Dominance 2.6. Property A 2.7. L-Matrices and Related Matrices 2.8. Illustrations Supplementary Discussion Exercises 3. Linear Stationary Iterative Methods 3.1. Introduction 3.2. Consistency, Reciprocal Consistency, and Complete Consistency 3.3. Basic Linear Stationary Iterative Methods 3.4. Generation of Completely Consistent Methods 3.5. General Convergence Theorems 3.6. Alternative Convergence Conditions 3.7. Rates of Convergence 3.8. The Jordan Condition Number of a 2 x 2 Matrix Supplementary Discussion Exercises 4. Convergence of the Basic Iterative Methods 4.1. General Convergence Theorems 4.2. Irreducible Matrices with Weak Diagonal Dominance 4.3. Positive Definite Matrices 4.4. The SOR Method with Varying Relaxation Factors 4.5. L-Matrices and Related Matrices 4.6. Rates of Convergence of the J and GS Methods for the Model Problem Supplementary Discussion Exercises 5. Eigenvalues of the SOR Method for Consistently Ordered Matrices 5.1. Introduction 5.2. Block Tri-Diagonal Matrices 5.3. Consistently Ordered Matrices and Ordering Vectors 5.4. Property A 5.5. Nonmigratory Permutations 5.6. Consistently Ordered Matrices Arising from Difference Equations 5.7. A Computer Program for Testing for Property A and Consistent Ordering 5.8. Other Developments of the SOR Theory Supplementary Discussion Exercises 6. Determination of the Optimum Relaxation Factor 6.1. Virtual Spectral Radius 6.2. Analysis of the Case Where All Eigenvalues of B Are Real 6.3. Rates of Convergence: Comparison with the Gauss-Seidel Method 6.4. Analysis of the Case Where Some Eigenvalues of B Are Complex 6.5. Practical Determination of ?b: General Considerations 6.6. Iterative Methods of Choosing ?b 6.7. An Upper Bound for µ 6.8. A Priori Determination of µ: Exact Methods 6.9. A Priori Determination of µ: Approximate Values 6.10. Numerical Results Supplementary Discussion Exercises 7. Norms of the SOR Method 7.1. The Jordan Canonical Form of L ? 7.2. Basic Eigenvalue Relation 7.3. Determination of ?L ??D1/2 7.4. Determination of ?Lm ?b?D1/2 7.5. Determination of ?L ??A1/2 7.6. Determination of ?Lm ?b?A1/2 7.7. Comparison of ?Lm ?b?D1/2 and ?Lm ?b?A1/2 Supplementary Discussion Exercises 8. The Modified SOR Method: Fixed Parameters 8.1. Introduction 8.2. Eigenvalues of L?, ?1 8.3. Convergence and Spectral Radius 8.4. Determination of ?L?, ?1?D1/2 8.5. Determination of ?L?, ?1?A1/2 Supplementary Discussion Exercises 9. Nonstationary Linear Iterative Methods 9.1. Consistency, Convergence, and Rates of Convergence 9.2. Periodic Nonstationary Methods 9.3. Chebyshev Polynomials Supplementary Discussion Exercises 10. The Modified SOR Method: Variable Parameters 10.1. Convergence of the MSOR Method 10.2. Optimum Choice of Relaxation Factors 10.3. Alternative Optimum Parameter Sets 10.4. Norms of the MSOR Method: Sheldon's Method 10.5. The Modified Sheldon Method 10.6. Cyclic Chebyshev Semi-Iterative Method 10.7. Comparison of Norms Supplementary Discussion Exercises 11. Semi-Iterative Methods 11.1. General Considerations 11.2. The Case Where G Has Real Eigenvalues 11.3. J, JOR, and RF Semi-Iterative Methods 11.4. Richardson's Method 11.5. Cyclic Chebyshev Semi-Iterative Method 11.6. GS Semi-Iterative Methods 11.7. SOR Semi-Iterative Methods 11.8. MSOR Semi-Iterative Methods 11.9. Comparison of Norms Supplementary Discussion Exercises 12. Extensions of the SOR Theory: Stieltjes Matrices 12.1. The Need for Some Restrictions on A 12.2. Stieltjes Matrices Supplementary Discussion Exercises 13. Generalized Consistently Ordered Matrices 13.1. Introduction 13.2. CO(q, r)-Matrices, Property Aq,r, and Ordering Vectors 13.3. Determination of the Optimum Relaxation Factor 13.4. Generalized Consistently Ordered Matrices 13.5. Relation Between GCO(q, r)-Matrices and CO(q, r)-Matrices 13.6. Computational Procedures: Canonical Forms 13.7. Relation to Other Work Supplementary Discussion Exercises 14. Group Iterative Methods 14.1. Construction of Group Iterative Methods 14.2. Solution of a Linear System with a Tri-Diagonal Matrix 14.3. Convergence Analysis 14.4. Applications 14.5. Comparison of Point and Group Iterative Methods Supplementary Discussion Exercises 15. Symmetric SOR Method and Related Methods 15.1. Introduction 15.2. Convergence Analysis 15.3. Choice of Relaxation Factor 15.4. SSOR Semi-Iterative Methods: The Discrete Dirichlet Problem 15.5. Group SSOR Methods 15.6. Unsymmetric SOR Method 15.7. Symmetric and Unsymmetric MSOR Methods Supplementary Discussion Exercises 16. Second-Degree Methods Supplementary Discussion Exercises 17. Alternating Direction Implicit Methods 17.1. Introduction: The Peaceman-Rachford Method 17.2. The Stationary Case: Consistency and Convergence 17.3. The Stationary Case: Choice of Parameters 17.4. The Commutative Case 17.5. Optimum Parameters 17.6. Good Parameters 17.7. The Helmholtz Equation in a Rectangle 17.8. Monotonicity 17.9. Necessary and Sufficient Conditions for the Commutative Case 17.10. The Noncommutative Case Supplementary Discussion Exercises 18. Selection of Iterative Method Bibliography Index