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Summary:
Publisher Summary 1
This text introduces the fundamentals of numerical linear algebra and matrix computations to advanced undergraduates (and beyond) studying in mathematics, computer science, engineering, and other disciplines where numerical methods are used. Watkins (mathematics, Washington State U.) includes chapters on Gaussian elimination and its variants, sensitivity of linear systems, the least squares problem, the singular value decomposition, eigenvalues and eigenvectors, and iterative methods for linear systems. The material is illustrated using examples and exercises on applications (e.g. electrical circuits, mass-spring systems, and simple partial differential equations) utilizing MATLAB for solutions. For this new edition, he has simplified the presentation of Francis's algorithm (more commonly known as the implicitly shifted QR algorithm) for computing eigenvalues and eigenvectors, among other changes. Prerequisites are a first course in linear algebra and some experience with computer programming. A first course in differential equations may also prove helpful for understanding some of the examples. Annotation 漏2010 Book News, Inc., Portland, OR (booknews.com)
Publisher Summary 2
This new, modernized edition provides a clear and thorough introduction to matrix computations,a key component of scientific computingRetaining the accessible and hands-on style of its predecessor, Fundamentals of Matrix Computations, Third Edition thoroughly details matrix computations and the accompanying theory alongside the author's useful insights. The book presents the most important algorithms of numerical linear algebra and helps readers to understand how the algorithms are developed and why they work.Along with new and updated examples, the Third Editionfeatures:A novel approach to Francis' QR algorithm that explains its properties without reference to the basic QR algorithmApplication of classical Gram-Schmidt with reorthogonalizationA revised approach to the derivation of the Golub-Reinsch SVD algorithmNew coverage on solving product eigenvalue problemsExpanded treatment of the Jacobi-Davidson methodA new discussion on stopping criteria for iterative methods for solving linear equationsThroughout the book, numerous new and updated exercises鈥攔anging from routine computations and verifications to challenging programming and proofs鈥攁re provided, allowing readers to immediately engage in applying the presented concepts. The new edition also incorporates MATLAB to solve real-world problems in electrical circuits, mass-spring systems, and simple partial differential equations, and an index of MATLAB terms assists readers with understanding the basic concepts related to the software.Fundamentals of Matrix Computations, Third Edition is an excellent book for courses on matrix computations and applied numerical linear algebra at the upper-undergraduate and graduate level. The book is also a valuable resource for researchers and practitioners working in the fields of engineering and computer science who need to know how to solve problems involving matrix computations.
目录
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庐 Terms p. 643
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庐 Terms p. 643
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