Catalogue


Spectral computations for bounded operators /
Mario Ahues, Alain Largillier, Balmohan V. Limaye.
imprint
Boca Raton : Chapman & Hall/CRC, c2001.
description
xvii, 382 p. : ill. ; 25 cm.
ISBN
1584881968 (alk. paper)
format(s)
Book
Holdings
More Details
imprint
Boca Raton : Chapman & Hall/CRC, c2001.
isbn
1584881968 (alk. paper)
catalogue key
4352270
 
Includes bibliographical references and index.
A Look Inside
About the Author
Author Affiliation
Mario Ahues is Professor in the Department of Mathematics at the Universite de Saint-Etienne in France Alain Largillier is Professor in the Department of Mathematics at the Universite de Saint-Etienne in France Balmohan Limaye is a Professor in the Department of Mathematics at the Indian Institute of Technology Bombay in India
Reviews
This item was reviewed in:
SciTech Book News, June 2001
To find out how to look for other reviews, please see our guides to finding book reviews in the Sciences or Social Sciences and Humanities.
Summaries
Main Description
Exact eigenvalues, eigenvectors, and principal vectors of operators with infinite dimensional ranges can rarely be found. Therefore, one must approximate such operators by finite rank operators, then solve the original eigenvalue problem approximately. This book addresses this issue of solving eigenvalue problems for operators on infinite dimensional spaces. From a review of classical spectral theory, through approximation techniques, to ideas for further research that would extend the results described, this volume serves as both a text for graduate students and as a source of state-of-the-art results for research scientists.
Main Description
Exact eigenvalues, eigenvectors, and principal vectors of operators with infinite dimensional ranges can rarely be found. Therefore, one must approximate such operators by finite rank operators, then solve the original eigenvalue problem approximately. Serving as both an outstanding text for graduate students and as a source of current results for research scientists, Spectral Computations for Bounded Operators addresses the issue of solving eigenvalue problems for operators on infinite dimensional spaces. From a review of classical spectral theory through concrete approximation techniques to finite dimensional situations that can be implemented on a computer, this volume illustrates the marriage of pure and applied mathematics. It contains a variety of recent developments, including a new type of approximation that encompasses a variety of approximation methods but is simple to verify in practice. It also suggests a new stopping criterion for the QR Method and outlines advances in both the iterative refinement and acceleration techniques for improving the accuracy of approximations. The authors illustrate all definitions and results with elementary examples and include numerous exercises. Spectral Computations for Bounded Operators thus serves as both an outstanding text for second-year graduate students and as a source of current results for research scientists.
Table of Contents
Prefacep. ix
Notationp. xiii
Spectral Decompositionp. 1
General Notionsp. 1
Decompositionsp. 21
Spectral Sets of Finite Typep. 33
Adjoint and Product Spacesp. 41
Adjoint Spacep. 41
Product Spacep. 48
Exercisesp. 64
Spectral Approximationp. 69
Convergence of Operatorsp. 69
Property Up. 76
Property Lp. 81
Error Estimatesp. 90
Exercisesp. 106
Improvement of Accuracyp. 113
Iterative Refinementp. 114
General Remarksp. 114
Refinement Schemes for a Simple Eigenvaluep. 119
Refinement Schemes for a Cluster of Eigenvaluesp. 134
Accelerationp. 148
Motivationp. 148
Higher Order Spectral Approximationp. 156
Simple Eigenvalue and Cluster of Eigenvaluesp. 162
Dependence on the Order of the Spectral Analysisp. 176
Exercisesp. 181
Finite Rank Approximationsp. 185
Approximations Based on Projectionsp. 185
Truncation of a Schauder Expansionp. 187
Interpolatory Projectionsp. 190
Orthogonal Projections on Subspaces of Piecewise Constant Functionsp. 195
Finite Element Approximationp. 197
Approximations of Integral Operatorsp. 199
Degenerate Kernel Approximationp. 200
Approximations Based on Numerical Integrationp. 203
Weakly Singular Integral Operatorsp. 222
A Posteriori Error Estimatesp. 230
Exercisesp. 240
Matrix Formulationsp. 247
Finite Rank Operatorsp. 248
Singularity Subtractionp. 257
Uniformly Well-Conditioned Basesp. 264
Iterative Refinementp. 275
Accelerationp. 282
Numerical Examplesp. 300
Exercisesp. 309
Matrix Computationsp. 315
QR factorizationp. 316
Householder symmetriesp. 319
Hessenberg Matricesp. 324
Convergence of a Sequence of Subspacesp. 327
Basic Definitionsp. 327
Krylov Sequencesp. 331
QR Methods and Inverse Iterationp. 336
The Francis-Kublanovskaya QR Methodp. 336
Simultaneous Inverse Iteration Methodp. 342
Error Analysisp. 344
Condition Numbers or Forward Error Analysisp. 344
Stability or Backward Error Analysisp. 347
Relative Error in Spectral Computations and Stopping Criteria for the Basic QR Methodp. 362
Relative Error in Solving a Sylvester Equationp. 369
Referencesp. 373
Indexp. 379
Table of Contents provided by Syndetics. All Rights Reserved.

This information is provided by a service that aggregates data from review sources and other sources that are often consulted by libraries, and readers. The University does not edit this information and merely includes it as a convenience for users. It does not warrant that reviews are accurate. As with any review users should approach reviews critically and where deemed necessary should consult multiple review sources. Any concerns or questions about particular reviews should be directed to the reviewer and/or publisher.

  link to old catalogue

Report a problem