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Preface | p. ix |
Acknowledgments | p. xv |
Gaussian Elimination and Its Variants | p. 1 |
Matrix Multiplication | p. 1 |
Systems of Linear Equations | p. 12 |
Triangular Systems | p. 24 |
Positive Definite Systems; Cholesky Decomposition | p. 33 |
Banded Positive Definite Systems | p. 55 |
Sparse Positive Definite Systems | p. 64 |
Gaussian Elimination and the LU Decomposition | p. 71 |
Gaussian Elimination with Pivoting | p. 94 |
Sparse Gaussian Elimination | p. 107 |
Sensitivity of Linear Systems | p. 113 |
Vector and Matrix Norms | p. 114 |
Condition Numbers | p. 122 |
Perturbing the Coefficient Matrix | p. 133 |
A Posteriori Error Analysis Using the Residual | p. 137 |
Roundoff Errors; Backward Stability | p. 139 |
Propagation of Roundoff Errors | p. 149 |
Backward Error Analysis of Gaussian Elimination | p. 157 |
Scaling | p. 171 |
Componentwise Sensitivity Analysis | p. 176 |
The Least Squares Problem | p. 183 |
The Discrete Least Squares Problem | p. 183 |
Orthogonal Matrices, Rotators, and Reflectors | p. 187 |
Solution of the Least Squares Problem | p. 215 |
The Gram-Schmidt Process | p. 223 |
Geometric Approach | p. 238 |
Updating the QR Decomposition | p. 247 |
The Singular Value Decomposition | p. 259 |
Introduction | p. 260 |
Some Basic Applications of Singular Values | p. 264 |
The SVD and the Least Squares Problem | p. 273 |
Sensitivity of the Least Squares Problem | p. 279 |
Eigenvalues and Eigenvectors I | p. 289 |
Systems of Differential Equations | p. 289 |
Basic Facts | p. 305 |
The Power Method and Some Simple Extensions | p. 314 |
Similarity Transforms | p. 334 |
Reduction to Hessenberg and Tridiagonal Forms | p. 350 |
Francis's Algorithm | p. 358 |
Use of Francis's Algorithm to Calculate Eigenvectors | p. 386 |
The SVD Revisited | p. 389 |
Eigenvalues and Eigenvectors II | p. 409 |
Eigenspaces and Invariant Subspaces | p. 410 |
Subspace Iteration and Simultaneous Iteration | p. 420 |
Krylov Subspaces and Francis's Algorithm | p. 428 |
Large Sparse Eigenvalue Problems | p. 437 |
Implicit Restarts | p. 456 |
The Jacobi-Davidson and Related Algorithms | p. 466 |
Eigenvalues and Eigenvectors III | p. 471 |
Sensitivity of Eigenvalues and Eigenvectors | p. 471 |
Methods for the Symmetric Eigenvalue Problem | p. 485 |
Product Eigenvalue Problems | p. 511 |
The Generalized Eigenvalue Problem | p. 526 |
Iterative Methods for Linear Systems | p. 545 |
A Model Problem | p. 546 |
The Classical Iterative Methods | p. 554 |
Convergence of Iterative Methods | p. 568 |
Descent Methods; Steepest Descent | p. 583 |
On Stopping Criteria | p. 594 |
Preconditioners | p. 596 |
The Conjugate-Gradient Method | p. 602 |
Derivation of the CG Algorithm | p. 607 |
Convergence of the CG Algorithm | p. 615 |
Indefinite and Nonsymmetric Problems | p. 621 |
References | p. 627 |
Index | p. 635 |
Index of MATLABŪ Terms | p. 643 |
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