Spectral Computations for Bounded Operators: 1st Edition (Hardback) book cover

Spectral Computations for Bounded Operators

1st Edition

By Mario Ahues, Alain Largillier, Balmohan Limaye

Chapman and Hall/CRC

400 pages | 150 B/W Illus.

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Hardback: 9781584881964
pub: 2001-02-26
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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.

Reviews

"This book gives a careful account of the theory underlying methods for numerical computation of approximations of eigenvalues, eigenvectors and generalized eigenvectors of bounded linear operators in infinite-dimensional space. The authors have been substantial contributors to the field, and the book gives some emphasis to topics on which they have worked. . . As well as being a valuable reference for mathematicians working on the development of analysis of numerical methods, the book is also suitable as a graduate text for students who have done a first course on functional analysis."

-Alan L. Andrews

Table of Contents

SPECTRAL DECOMPOSITION

Genera Notions

Decompositions

Spectral Sets of Finite Type

Adjoint and Product Spaces

SPECTRAL APPROXIMATION

Convergence of operators

Property U

Property L

Error Estimates

IMPROVEMENT OF ACCURACY

Iterative Refinement

Acceleration

FINITE RANK APPROXIMATIONS

Approximations Based on Projection

Approximations of Integral Operators

A Posteriori Error Estimates

MATRIX FORMULATIONS

Finite Rank Operators

Iterative Refinement

Acceleration

Numerical Examples

MATRIX COMPUTATIONS

QR Factorization

Convergence of a Sequence of Subspaces

QR Methods and Inverse Iteration

Error Analysis

REFERENCES

INDEX

Each chapter also includes exercises

About the Series

Applied Mathematics

Learn more…

Subject Categories

BISAC Subject Codes/Headings:
MAT003000
MATHEMATICS / Applied
SCI040000
SCIENCE / Mathematical Physics