Numerical linear algebra is far too broad a subject to treat in a single introductory volume. Stewart has chosen to treat algorithms for solving linear systems, linear least squares problems, and eigenvalue problems involving matrices whose elements can all be contained in the high-speed storage of a computer. By way of theory, the author has chosen to discuss the theory of norms and perturbation theory for linear systems and for the algebraic eigenvalue problem. These choices exclude, among other things, the solution of large sparse linear systems by direct and iterative methods, linear programming, and the useful Perron-Frobenious theory and its extensions. However, a person who has fully mastered the material in this book should be well prepared for independent study in other areas of numerical linear algebra.
Author(s): Stewart G. W.
Series: Computer Science and Applied Mathematics
Publisher: Academic Press
Year: 1973
Language: English
Pages: 456
Tags: Математика;Вычислительная математика;Вычислительные методы линейной алгебры;
Stewart G. W.Introduction to Matrix Computations (Computer Science and Applied Mathematics)(AP,1973)(ISBN 0126703507)(600dpi)(456p) ......Page 4
Copyright ......Page 5
Contents vii ......Page 8
Preface, ix ......Page 10
Acknowledgments, xiii ......Page 14
1. PRELIMINARIES 1 ......Page 15
1. The Space R^n, 2 ......Page 16
2. Linear Independence, Subspaces, and Bases, 9 ......Page 23
3. Matrices, 20 ......Page 34
4. Operations with Matrices, 29 ......Page 43
5. Linear Transformations and Matrices, 46 ......Page 60
6. Linear Equations and Inverses, 54 ......Page 68
7. A Matrix Reduction and Some Consequences, 63 ......Page 77
2. PRACTICALITIES 68 ......Page 82
1. Errors, Arithmetic, and Stability, 69 ......Page 83
2. An Informal Language, 83 ......Page 97
3. Coding Matrix Operations, 93 ......Page 107
3. THE DIRECT SOLUTION OF LINEAR SYSTEMS 105 ......Page 119
1. Triangular Matrices and Systems, 106 ......Page 120
2. Gaussian Elimination, 113 ......Page 127
3. Triangular Decomposition, 131 ......Page 145
4. The Solution of Linear Systems, 144 ......Page 158
5. The Effects of Rounding Error, 148 ......Page 162
4. NORMS, LIMITS, AND CONDITION NUMBERS 160 ......Page 174
1. Norms and Limits, 161 ......Page 175
2. Matrix Norms, 173 ......Page 187
3. Inverses of Perturbed Matrices, 184 ......Page 198
4. The Accuracy of Solutions of Linear Systems, 192 ......Page 206
5. Iterative Refinement of Approximate Solutions of Linear Systems, 200 214......Page
5. THE LINEAR LEAST SQUARES PROBLEM 208 ......Page 222
1. Orthogonality, 209 ......Page 223
2. The Linear Least Squares Problem, 217 ......Page 231
3. Orthogonal Triangularization, 230 ......Page 244
4. The Iterative Refinement of Least Squares Solutions, 245 ......Page 259
6. EIGENVALUES AND EIGENVECTORS 250 ......Page 264
1. The Space Cn, 251 ......Page 265
2. Eigenvalues and Eigenvectors, 262 ......Page 276
3. Reduction of Matrices by Similarity Transformations, 275 ......Page 289
4. The Sensitivity of Eigenvalues and Eigenvectors, 289 ......Page 303
5. Hermitian Matrices, 307 ......Page 321
6. The Singular Value Decomposition, 317 ......Page 331
7. THE QR ALGORITHM 327 ......Page 341
1. Reduction to Hessenberg and Tridiagonal Forms, 328 ......Page 342
2. The Power and Inverse Power Methods, 340 ......Page 354
3. The Explicitly Shifted QR Algorithm, 351 ......Page 365
4. The Implicitly Shifted QR Algorithm, 368 ......Page 382
5. Computing Singular Values and Vectors, 381 ......Page 395
6. The Generalized Eigenvalue Problem A — ?A, 387......Page 401
Appendix 1. The Greek Alphabet and Latin Notational Correspondents, 395 ......Page 409
Appendix 2. Determinants, 396 ......Page 410
Appendix 3. Rounding-Error Analysis of Solution of Triangular Systems and of Gaussian Elimination, 405 ......Page 419
Appendix 4. Of Things Not Treated, 413 ......Page 427
Bibliography, 417 ......Page 431
Index of Notation, 425 ......Page 439
Index of Algorithms, 427 ......Page 441
Index, 429 ......Page 443
cover......Page 1
