Fundamentals of Matrix Computations

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  • Edition: 3rd
  • Format: Hardcover
  • Copyright: 2010-07-06
  • Publisher: Wiley

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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.

Author Biography

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.

Table of Contents

Prefacep. ix
Acknowledgmentsp. xv
Gaussian Elimination and Its Variantsp. 1
Matrix Multiplicationp. 1
Systems of Linear Equationsp. 12
Triangular Systemsp. 24
Positive Definite Systems; Cholesky Decompositionp. 33
Banded Positive Definite Systemsp. 55
Sparse Positive Definite Systemsp. 64
Gaussian Elimination and the LU Decompositionp. 71
Gaussian Elimination with Pivotingp. 94
Sparse Gaussian Eliminationp. 107
Sensitivity of Linear Systemsp. 113
Vector and Matrix Normsp. 114
Condition Numbersp. 122
Perturbing the Coefficient Matrixp. 133
A Posteriori Error Analysis Using the Residualp. 137
Roundoff Errors; Backward Stabilityp. 139
Propagation of Roundoff Errorsp. 149
Backward Error Analysis of Gaussian Eliminationp. 157
Scalingp. 171
Componentwise Sensitivity Analysisp. 176
The Least Squares Problemp. 183
The Discrete Least Squares Problemp. 183
Orthogonal Matrices, Rotators, and Reflectorsp. 187
Solution of the Least Squares Problemp. 215
The Gram-Schmidt Processp. 223
Geometric Approachp. 238
Updating the QR Decompositionp. 247
The Singular Value Decompositionp. 259
Introductionp. 260
Some Basic Applications of Singular Valuesp. 264
The SVD and the Least Squares Problemp. 273
Sensitivity of the Least Squares Problemp. 279
Eigenvalues and Eigenvectors Ip. 289
Systems of Differential Equationsp. 289
Basic Factsp. 305
The Power Method and Some Simple Extensionsp. 314
Similarity Transformsp. 334
Reduction to Hessenberg and Tridiagonal Formsp. 350
Francis's Algorithmp. 358
Use of Francis's Algorithm to Calculate Eigenvectorsp. 386
The SVD Revisitedp. 389
Eigenvalues and Eigenvectors IIp. 409
Eigenspaces and Invariant Subspacesp. 410
Subspace Iteration and Simultaneous Iterationp. 420
Krylov Subspaces and Francis's Algorithmp. 428
Large Sparse Eigenvalue Problemsp. 437
Implicit Restartsp. 456
The Jacobi-Davidson and Related Algorithmsp. 466
Eigenvalues and Eigenvectors IIIp. 471
Sensitivity of Eigenvalues and Eigenvectorsp. 471
Methods for the Symmetric Eigenvalue Problemp. 485
Product Eigenvalue Problemsp. 511
The Generalized Eigenvalue Problemp. 526
Iterative Methods for Linear Systemsp. 545
A Model Problemp. 546
The Classical Iterative Methodsp. 554
Convergence of Iterative Methodsp. 568
Descent Methods; Steepest Descentp. 583
On Stopping Criteriap. 594
Preconditionersp. 596
The Conjugate-Gradient Methodp. 602
Derivation of the CG Algorithmp. 607
Convergence of the CG Algorithmp. 615
Indefinite and Nonsymmetric Problemsp. 621
Referencesp. 627
Indexp. 635
Index of MATLABŪ Termsp. 643
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