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| Preface | |
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| Acknowledgments | |
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| Direct Solution Methods | |
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| Introduction: Networks and Structures | |
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| Gaussian Elimination and Matrix Factorization | |
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| Range and Nullspace | |
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| Practical Considerations | |
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| Solution of Tridiagonal Systems of Equations | |
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| Exercises | |
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| References | |
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| Theory of Matrix Eigenvalues | |
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| The Minimal Polynomial | |
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| Selfadjoint and Unitary Matrices | |
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| Matrix Equivalence (Similarity Transformations) | |
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| Normal and H-Normal Matrices | |
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| Exercises | |
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| References | |
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| Positive Definite Matrices, Schur Complements, and Generalized Eigenvalue Problems | |
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| Positive Definite Matrices | |
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| Schur Complements | |
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| Condition Numbers | |
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| Estimates of Eigenvalues of Generalized Eigenvalue Problems | |
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| Congruence Transformations | |
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| Quasisymmetric Matrices | |
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| Exercises | |
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| References | |
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| Reducible and Irreducible Matrices and the Perron-Frobenius Theory for Nonnegative Matrices | |
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| Reducible and Irreducible Matrices | |
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| Gershgorin Type Eigenvalue Estimates | |
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| The Perron-Frobenius Theorem | |
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| Rayleigh Quotient and Numerical Range | |
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| Some Estimates of the Perron-Frobenius Root of Nonnegative Matrices | |
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| A Leontief Closed Input-Output Model | |
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| Exercises | |
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| References | |
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| Basic Iterative Methods and Their Rates of Convergence | |
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| Basic Iterative Methods | |
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| Stationary Iterative Methods | |
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| The Chebyshev Iterative Method | |
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| The Chebyshev Iterative Method for Matrices with Special Eigenvalue Distributions | |
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| Exercises | |
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| References | |
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| M-Matrices, Convergent Splittings, and the SOR Method | |
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| M-Matrices | |
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| Convergent Splittings | |
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| Comparison Theorems | |
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| Diagonally Compensated Reduction of Positive Matrix Entries | |
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| The SOR Method | |
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| Exercises | |
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| References | |
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| Incomplete Factorization Preconditioning Methods | |
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| Point Incomplete Factorization | |
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| Block Incomplete Factorization; Introduction | |
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| Block Incomplete Factorization of M-Matrices | |
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| Block Incomplete Factorization of Positive Definite Matrices | |
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| Incomplete Factorization Methods for Block H-Matrices | |
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| Inverse Free Form for Block Tridiagonal Matrices | |
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| Symmetrization of Preconditioners and the SSOR and ADI Methods | |
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| Exercises | |
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| References | |
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| Approximate Matrix Inverses and Corresponding Preconditioning Methods | |
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| Two Methods of Computing Approximate Inverses of Block Bandmatrices | |
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| A Class of Methods for Computing Approximate Inverses of Matrices | |
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| A Symmetric and Positive Definite Approximate Inverse | |
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| Combinations of Explicit and Implicit Methods | |
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| Methods of Matrix Action | |
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| Decay Rates of (Block-) Entries of Inverses of (Block-) Tridiagonal s.p.d. Matrices | |
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| References | |
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| Block Diagonal and Schur Complement Preconditionings | |
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| The C.B.S. Constant | |
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| Block-Diagonal Preconditioning | |
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| Schur Complement Preconditioning | |
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| Full Block-Matrix Factorization Methods | |
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| Indefinite Systems | |
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| References | |
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| Estimates of Eigenvalues and Condition Numbers for Preconditioned Matrices | |
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| Upper Eigenvalue Bounds | |
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| Perturbation Methods | |
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| Lower Eigenvalue Bounds for M-Matrices | |
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| Upper and Lower Bounds of Condition Numbers | |
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| Asymptotic Estimates of Condition Numbers for Second-Order Elliptic Problems | |
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| References | |
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| Conjugate Gradient and Lanczos-Type Methods | |
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| The Three-Term Recurrence Form of the Conjugate Gradient Method | |
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| The Standard Conjugate Gradient Method | |
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| The Lanczos Method for Generating A-Orthogonal Vectors | |
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| References | |
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| Generalized Conjugate Gradient Methods | |
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| Generalized Conjugate Gradient, Least Squares Methods | |
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| Orthogonal Error Methods | |
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| Generalized Conjugate Gradient Methods and Variable (Nonlinear) Preconditioners | |
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| References | |
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| The Rate of Convergence of the Conjugate Gradient Method | |
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| Rate of Convergence Estimates Based on Min Max Approximations | |
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| Estimates Based on the Condition Number | |
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| An Estimate Based on a Ratio Involving the Trace and the Determinant | |
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| Estimates of the Rate of Convergence Using Different Norms | |
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| Conclusions | |
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| References | |
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| Appendices | |
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| Matrix Norms, Inherent Errors, and Computation of Eigenvalues | |
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| Vector and Matrix Norms | |
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| Inherent Errors in Systems of Linear Algebraic Equations | |
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| Estimation and Computation of Eigenvalues | |
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| Exercises | |
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| References | |
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| Chebyshev Polynomials | |
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| References | |
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| Some Inequalities for Functions of Matrices | |
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| Convex Functions | |
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| Matrix-Convex Functions | |
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| References | |
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| Index | |