| Preface | p. xi |
| Acknowledgments | p. xiii |
| Direct Solution Methods | p. 1 |
| Introduction: Networks and Structures | p. 2 |
| Gaussian Elimination and Matrix Factorization | p. 7 |
| Range and Nullspace | p. 14 |
| Practical Considerations | p. 18 |
| Solution of Tridiagonal Systems of Equations | p. 24 |
| Exercises | p. 30 |
| References | p. 44 |
| Theory of Matrix Eigenvalues | p. 46 |
| The Minimal Polynomial | p. 47 |
| Selfadjoint and Unitary Matrices | p. 54 |
| Matrix Equivalence (Similarity Transformations) | p. 56 |
| Normal and H-Normal Matrices | p. 62 |
| Exercises | p. 66 |
| References | p. 82 |
| Positive Definite Matrices, Schur Complements, and Generalized Eigenvalue Problems | p. 84 |
| Positive Definite Matrices | p. 85 |
| Schur Complements | p. 92 |
| Condition Numbers | p. 95 |
| Estimates of Eigenvalues of Generalized Eigenvalue Problems | p. 98 |
| Congruence Transformations | p. 104 |
| Quasisymmetric Matrices | p. 106 |
| Exercises | p. 107 |
| References | p. 120 |
| Reducible and Irreducible Matrices and the Perron-Frobenius Theory for Nonnegative Matrices | p. 122 |
| Reducible and Irreducible Matrices | p. 123 |
| Gershgorin Type Eigenvalue Estimates | p. 127 |
| The Perron-Frobenius Theorem | p. 134 |
| Rayleigh Quotient and Numerical Range | p. 136 |
| Some Estimates of the Perron-Frobenius Root of Nonnegative Matrices | p. 141 |
| A Leontief Closed Input-Output Model | p. 146 |
| Exercises | p. 147 |
| References | p. 156 |
| Basic Iterative Methods and Their Rates of Convergence | p. 158 |
| Basic Iterative Methods | p. 160 |
| Stationary Iterative Methods | p. 168 |
| The Chebyshev Iterative Method | p. 179 |
| The Chebyshev Iterative Method for Matrices with Special Eigenvalue Distributions | p. 185 |
| Exercises | p. 193 |
| References | p. 198 |
| M-Matrices, Convergent Splittings, and the SOR Method | p. 200 |
| M-Matrices | p. 202 |
| Convergent Splittings | p. 213 |
| Comparison Theorems | p. 215 |
| Diagonally Compensated Reduction of Positive Matrix Entries | p. 222 |
| The SOR Method | p. 230 |
| Exercises | p. 245 |
| References | p. 249 |
| Incomplete Factorization Preconditioning Methods | p. 252 |
| Point Incomplete Factorization | p. 254 |
| Block Incomplete Factorization; Introduction | p. 260 |
| Block Incomplete Factorization of M-Matrices | p. 263 |
| Block Incomplete Factorization of Positive Definite Matrices | p. 265 |
| Incomplete Factorization Methods for Block H-Matrices | p. 269 |
| Inverse Free Form for Block Tridiagonal Matrices | p. 281 |
| Symmetrization of Preconditioners and the SSOR and ADI Methods | p. 287 |
| Exercises | p. 302 |
| References | p. 310 |
| Approximate Matrix Inverses and Corresponding Preconditioning Methods | p. 314 |
| Two Methods of Computing Approximate Inverses of Block Bandmatrices | p. 315 |
| A Class of Methods for Computing Approximate Inverses of Matrices | p. 321 |
| A Symmetric and Positive Definite Approximate Inverse | p. 335 |
| Combinations of Explicit and Implicit Methods | p. 342 |
| Methods of Matrix Action | p. 350 |
| Decay Rates of (Block-) Entries of Inverses of (Block-) Tridiagonal s.p.d. Matrices | p. 356 |
| References | p. 370 |
| Block Diagonal and Schur Complement Preconditionings | p. 372 |
| The C.B.S. Constant | p. 374 |
| Block-Diagonal Preconditioning | p. 378 |
| Schur Complement Preconditioning | p. 381 |
| Full Block-Matrix Factorization Methods | p. 383 |
| Indefinite Systems | p. 394 |
| References | p. 400 |
| Estimates of Eigenvalues and Condition Numbers for Preconditioned Matrices | p. 402 |
| Upper Eigenvalue Bounds | p. 403 |
| Perturbation Methods | p. 413 |
| Lower Eigenvalue Bounds for M-Matrices | p. 418 |
| Upper and Lower Bounds of Condition Numbers | p. 425 |
| Asymptotic Estimates of Condition Numbers for Second-Order Elliptic Problems | p. 433 |
| References | p. 447 |
| Conjugate Gradient and Lanczos-Type Methods | p. 449 |
| The Three-Term Recurrence Form of the Conjugate Gradient Method | p. 452 |
| The Standard Conjugate Gradient Method | p. 459 |
| The Lanczos Method for Generating A-Orthogonal Vectors | p. 480 |
| References | p. 501 |
| Generalized Conjugate Gradient Methods | p. 504 |
| Generalized Conjugate Gradient, Least Squares Methods | p. 506 |
| Orthogonal Error Methods | p. 530 |
| Generalized Conjugate Gradient Methods and Variable (Nonlinear) Preconditioners | p. 542 |
| References | p. 555 |
| The Rate of Convergence of the Conjugate Gradient Method | p. 558 |
| Rate of Convergence Estimates Based on Min Max Approximations | p. 559 |
| Estimates Based on the Condition Number | p. 566 |
| An Estimate Based on a Ratio Involving the Trace and the Determinant | p. 575 |
| Estimates of the Rate of Convergence Using Different Norms | p. 585 |
| Conclusions | p. 591 |
| References | p. 593 |
| Appendices | |
| Matrix Norms, Inherent Errors, and Computation of Eigenvalues | p. 595 |
| Vector and Matrix Norms | p. 596 |
| Inherent Errors in Systems of Linear Algebraic Equations | p. 605 |
| Estimation and Computation of Eigenvalues | p. 609 |
| Exercises | p. 625 |
| References | p. 637 |
| Chebyshev Polynomials | p. 639 |
| References | p. 641 |
| Some Inequalities for Functions of Matrices | p. 642 |
| Convex Functions | p. 644 |
| Matrix-Convex Functions | p. 645 |
| References | p. 648 |
| Index | p. 649 |
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