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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