A significantly revised and improved introduction to a critical aspect of scientific computation Matrix computations lie at the heart of most scientific computational tasks. For any scientist or engineer doing large-scale simulations, an understanding of the topic is essential. Fundamentals of Matrix Computations, Second Edition explains matrix computations and the accompanying theory clearly and in detail, along with useful insights. This Second Edition of a popular text has now been revised and improved to appeal to the needs of practicing scientists and graduate and advanced undergraduate students. New to this edition is the use of MATLAB for many of the exercises and examples, although the Fortran exercises in the First Edition have been kept for those who want to use them. This new edition includes: * Numerous examples and exercises on applications including electrical circuits, elasticity (mass-spring systems), and simple partial differential equations * Early introduction of the singular value decomposition * A new chapter on iterative methods, including the powerful preconditioned conjugate-gradient method for solving symmetric, positive definite systems * An introduction to new methods for solving large, sparse eigenvalue problems including the popular implicitly-restarted Arnoldi and Jacobi-Davidson methods With in-depth discussions of such other topics as modern componentwise error analysis, reorthogonalization, and rank-one updates of the QR decomposition, Fundamentals of Matrix Computations, Second Edition will prove to be a versatile companion to novice and practicing mathematicians who seek mastery of matrix computation.
Author(s): David S. Watkins
Edition: 2
Publisher: Wiley-Interscience
Year: 2002
Language: English
Pages: 640
Contents......Page 6
Preface......Page 10
Acknowledgments......Page 14
1.1 Matrix Multiplication......Page 16
1.2 Systems of Linear Equations......Page 26
1.3 Triangular Systems......Page 38
1.4 Positive Definite Systems; Cholesky Decomposition......Page 47
1.5 Banded Positive Definite Systems......Page 69
1.6 Sparse Positive Definite Systems......Page 78
1.7 Gaussian Elimination and the LU Decomposition......Page 85
1.8 Gaussian Elimination with Pivoting......Page 108
1.9 Sparse Gaussian Elimination......Page 121
2 Sensitivity of Linear Systems......Page 126
2.1 Vector and Matrix Norms......Page 127
2.2 Condition Numbers......Page 135
2.3 Perturbing the Coefficient Matrix......Page 148
2.4 A Posteriori Error Analysis Using the Residual......Page 152
2.5 Roundoff Errors; Backward Stability......Page 154
2.6 Propagation of Roundoff Errors......Page 163
2.7 Backward Error Analysis of Gaussian Elimination......Page 172
2.8 Scaling......Page 186
2.9 Componentwise Sensitivity Analysis......Page 190
3.1 The Discrete Least Squares Problem......Page 196
3.2 Orthogonal Matrices, Rotators, and Reflectors......Page 200
3.3 Solution of the Least Squares Problem......Page 227
3.4 The Gram-Schmidt Process......Page 235
3.5 Geometric Approach......Page 254
3.6 Updating the QR Decomposition......Page 264
4 The Singular Value Decomposition......Page 276
4.1 Introduction......Page 277
4.2 Some Basic Applications of Singular Values......Page 281
4.3 The SVD and the Least Squares Problem......Page 290
4.4 Sensitivity of the Least Squares Problem......Page 296
5.1 Systems of Differential Equations......Page 304
5.2 Basic Facts......Page 320
5.3 The Power Method and Some Simple Extensions......Page 329
5.4 Similarity Transforms......Page 349
5.5 Reduction to Hessenberg and Tridiagonal Forms......Page 364
5.6 The QR Algorithm......Page 371
5.7 Implementation of the QR algorithm......Page 387
5.8 Use of the QR Algorithm to Calculate Eigenvectors......Page 407
5.9 The SVD Revisited......Page 411
6.1 Eigenspaces and Invariant Subspaces......Page 428
6.2 Subspace Iteration, Simultaneous Iteration, and the QR Algorithm......Page 435
6.3 Eigenvalues of Large, Sparse Matrices, I......Page 448
6.4 Eigenvalues of Large, Sparse Matrices, II......Page 466
6.5 Sensitivity of Eigenvalues and Eigenvectors......Page 477
6.6 Methods for the Symmetric Eigenvalue Problem......Page 491
6.7 The Generalized Eigenvalue Problem......Page 517
7.1 A Model Problem......Page 536
7.2 The Classical Iterative Methods......Page 545
7.3 Convergence of Iterative Methods......Page 559
7.4 Descent Methods; Steepest Descent......Page 574
7.5 Preconditioners......Page 586
7.6 The Conjugate-Gradient Method......Page 591
7.7 Derivation of the CG Algorithm......Page 596
7.8 Convergence of the CG Algorithm......Page 605
7.9 Indefinite and Nonsymmetric Problems......Page 611
Appendix: Some Sources of Software for Matrix Computations......Page 618
References......Page 620
C......Page 626
E......Page 627
J......Page 628
P......Page 629
S......Page 630
W......Page 631
P......Page 632
Y......Page 633
