Error and Perturbation Bounds for Subspaces Associated with Certain Eigenvalue Problems PDF Download
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Author: Gilbert W. Stewart
Publisher:
ISBN:
Category : Eigenvalues
Languages : en
Pages : 35
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Book Description
The paper describes a technique for obtaining error bounds for certain characteristic subspaces associated with the algebraic eigenvalue problem, the generalized eigenvalue problem, and the singular value decomposition. The method also gives perturbation bounds for isolated eigenvalues and useful information about clusters of eigenvalues. The bounds are obtained from an iterative process for generating the subspaces in question, and one or more steps of the iteration can be used to construct perturbation estimates whose error can be bounded. (Author).
Author: Gilbert W. Stewart
Publisher:
ISBN:
Category : Eigenvalues
Languages : en
Pages : 35
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Book Description
The paper describes a technique for obtaining error bounds for certain characteristic subspaces associated with the algebraic eigenvalue problem, the generalized eigenvalue problem, and the singular value decomposition. The method also gives perturbation bounds for isolated eigenvalues and useful information about clusters of eigenvalues. The bounds are obtained from an iterative process for generating the subspaces in question, and one or more steps of the iteration can be used to construct perturbation estimates whose error can be bounded. (Author).
Author: Jesse Louis Barlow
Publisher:
ISBN:
Category : Eigenvalues
Languages : en
Pages : 27
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Book Description
Abstract: "There is now a large literature on structured perturbation bounds for eigenvalue problems of the form [formula], where H and M are Hermitian. These results give relative error bounds on the i[superscript th] eigenvalue, [lambda subscript i], of the form [formula], and bound the error in the i[superscript th] eigenvector in terms of the relative gap, [formula]. In general, this theory usually restricts H to be nonsingular and M to be positive definite. We relax this restriction by allowing H to be singular. For our results on eigenvales we allow M to be positive semi-definite and for few results we allow it to be more general. For these problems, for eigenvalues that are not zero or infinity under perturbation, it is possible to obtain local relative error bounds. Thus, a wider class of problems may be characterized by this theory. The theory is applied to the SVD and some of its generalizations. In fact, for structured perturbations, our bound on generalized Hermitian eigenproblems are based upon our bounds for generalized singular value problems. Although it is impossible to give meaningful relative error bounds on eigenvalues that are not bounded away from zero, we show that the error in the subspace associated with those eigenvalues can be characterized meaningfully."
Author: Yousef Saad
Publisher: SIAM
ISBN: 9781611970739
Category : Mathematics
Languages : en
Pages : 292
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Book Description
This revised edition discusses numerical methods for computing eigenvalues and eigenvectors of large sparse matrices. It provides an in-depth view of the numerical methods that are applicable for solving matrix eigenvalue problems that arise in various engineering and scientific applications. Each chapter was updated by shortening or deleting outdated topics, adding topics of more recent interest, and adapting the Notes and References section. Significant changes have been made to Chapters 6 through 8, which describe algorithms and their implementations and now include topics such as the implicit restart techniques, the Jacobi-Davidson method, and automatic multilevel substructuring.
Author: Gene H. Golub
Publisher: JHU Press
ISBN: 1421408597
Category : Mathematics
Languages : en
Pages : 781
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Book Description
A comprehensive treatment of numerical linear algebra from the standpoint of both theory and practice. The fourth edition of Gene H. Golub and Charles F. Van Loan's classic is an essential reference for computational scientists and engineers in addition to researchers in the numerical linear algebra community. Anyone whose work requires the solution to a matrix problem and an appreciation of its mathematical properties will find this book to be an indispensible tool. This revision is a cover-to-cover expansion and renovation of the third edition. It now includes an introduction to tensor computations and brand new sections on • fast transforms • parallel LU • discrete Poisson solvers • pseudospectra • structured linear equation problems • structured eigenvalue problems • large-scale SVD methods • polynomial eigenvalue problems Matrix Computations is packed with challenging problems, insightful derivations, and pointers to the literature—everything needed to become a matrix-savvy developer of numerical methods and software. The second most cited math book of 2012 according to MathSciNet, the book has placed in the top 10 for since 2005.
Author: David S. Watkins
Publisher: SIAM
ISBN: 9780898717808
Category : Mathematics
Languages : en
Pages : 452
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Book Description
The first in-depth, complete, and unified theoretical discussion of the two most important classes of algorithms for solving matrix eigenvalue problems: QR-like algorithms for dense problems and Krylov subspace methods for sparse problems. The author discusses the theory of the generic GR algorithm, including special cases (for example, QR, SR, HR), and the development of Krylov subspace methods. This book also addresses a generic Krylov process and the Arnoldi and various Lanczos algorithms, which are obtained as special cases. Theoretical and computational exercises guide students, step by step, to the results. Downloadable MATLAB programs, compiled by the author, are available on a supplementary Web site. Readers of this book are expected to be familiar with the basic ideas of linear algebra and to have had some experience with matrix computations. Ideal for graduate students, or as a reference book for researchers and users of eigenvalue codes.
Author: Leslie Hogben
Publisher: CRC Press
ISBN: 1466507284
Category : Mathematics
Languages : en
Pages : 1906
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Book Description
With a substantial amount of new material, the Handbook of Linear Algebra, Second Edition provides comprehensive coverage of linear algebra concepts, applications, and computational software packages in an easy-to-use format. It guides you from the very elementary aspects of the subject to the frontiers of current research. Along with revisions and updates throughout, the second edition of this bestseller includes 20 new chapters. New to the Second Edition Separate chapters on Schur complements, additional types of canonical forms, tensors, matrix polynomials, matrix equations, special types of matrices, generalized inverses, matrices over finite fields, invariant subspaces, representations of quivers, and spectral sets New chapters on combinatorial matrix theory topics, such as tournaments, the minimum rank problem, and spectral graph theory, as well as numerical linear algebra topics, including algorithms for structured matrix computations, stability of structured matrix computations, and nonlinear eigenvalue problems More chapters on applications of linear algebra, including epidemiology and quantum error correction New chapter on using the free and open source software system Sage for linear algebra Additional sections in the chapters on sign pattern matrices and applications to geometry Conjectures and open problems in most chapters on advanced topics Highly praised as a valuable resource for anyone who uses linear algebra, the first edition covered virtually all aspects of linear algebra and its applications. This edition continues to encompass the fundamentals of linear algebra, combinatorial and numerical linear algebra, and applications of linear algebra to various disciplines while also covering up-to-date software packages for linear algebra computations.
Author: E. Anderson
Publisher: SIAM
ISBN: 0898714478
Category : Mathematics
Languages : en
Pages : 422
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Book Description
LAPACK is a library of numerical linear algebra subroutines designed for high performance on workstations, vector computers, and shared memory multiprocessors. Release 3.0 of LAPACK introduces new routines and extends the functionality of existing routines. The most significant new routines and functions include: 1. a faster singular value decomposition computed by divide-and-conquer 2. faster routines for solving rank-deficient least squares problems: Using QR with column pivoting using the SVD based on divide-and-conquer 3. new routines for the generalized symmetric eigenproblem: faster routines based on divide-and-conquer routines based on bisection/inverse iteration, for computing part of the spectrum 4. faster routine for the symmetric eigenproblem using "relatively robust eigenvector algorithm" 5. new simple and expert drivers for the generalized nonsymmetric eigenproblem, including error bounds 6. solver for generalized Sylvester equation, used in 5 7.computational routines used in 5 Each Users' Guide comes with a 'Quick Reference Guide' card.
Author: R. Wong
Publisher: CRC Press
ISBN: 1000154130
Category : Mathematics
Languages : en
Pages : 782
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Book Description
Papers presented at the International Symposium on Asymptotic and Computational Analysis, held June 1989, Winnipeg, Man., sponsored by the Dept. of Applied Mathematics, University of Manitoba and the Canadian Applied Mathematics Society.
Author: Gene Howard Golub
Publisher: Oxford University Press
ISBN: 0199206813
Category : Mathematics
Languages : en
Pages : 581
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Book Description
The text presents and discusses some of the most influential papers in Matrix Computation authored by Gene H. Golub, one of the founding fathers of the field. Including commentaries by leading experts and a brief biography, this text will be of great interest to students and researchers in numerical analysis and scientific computation.
Author: Raymond Chan
Publisher: OUP Oxford
ISBN: 9780199206810
Category : Mathematics
Languages : en
Pages : 584
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Book Description
The text presents and discusses some of the most influential papers in Matrix Computation authored by Gene H. Golub, one of the founding fathers of the field. The collection of 21 papers is divided into five main areas: iterative methods for linear systems, solution of least squares problems, matrix factorizations and applications, orthogonal polynomials and quadrature, and eigenvalue problems. Commentaries for each area are provided by leading experts: Anne Greenbaum, Ake Bjorck, Nicholas Higham, Walter Gautschi, and G. W. (Pete) Stewart. Comments on each paper are also included by the original authors, providing the reader with historical information on how the paper came to be written and under what circumstances the collaboration was undertaken. Including a brief biography and facsimiles of the original papers, this text will be of great interest to students and researchers in numerical analysis and scientific computation.
