Fundamentals of Matrix Computations

Fundamentals of Matrix Computations, Third Edition thoroughly details matrix computations and the accompanying theory alongside the author's useful insights. Featuring many new and updated examples and exercises that use the MATLABr language, this revision presents the most important algorithms of numerical linear algebra and helps readers to understand how the algorithms are developed and why they work. It also includes modern coverage of Singular Value Decomposition, a streamlined discussion of the Gram-Schmidt process, and a discussion on balancing the eigenvalue problem. Practicing scientists and graduate and advanced undergraduate students will find this popular book more than meets their needs.

David S. Watkins, PhD, is Professor in the Department of Mathematics at Washington State University. He has published more than 100 articles in his areas of research interest, which include numerical linear algebra, numerical analysis, and scientific computing.

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 MATLAB® Terms	p. 643
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Fundamentals of Matrix Computations

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