The Symmetric Eigenvalue Problem

✍ Scribed by Beresford N. Parlett

• Publisher: Society for Industrial Mathematics
• Year: 1987
• Tongue: English
• Leaves: 426
• Series: Classics in Applied Mathematics
• Edition: Reprint
• Category: Library

✦ Synopsis

According to Parlett, ‘Vibrations are everywhere, and so too are the eigenvalues associated with them. As mathematical models invade more and more disciplines, we can anticipate a demand for eigenvalue calculations in an ever richer variety of contexts’. Anyone who performs these calculations will welcome the reprinting of Parlett's book (originally published in 1980). In this unabridged, amended version, Parlett covers aspects of the problem that are not easily found elsewhere. The chapter titles convey the scope of the material succinctly. The aim of the book is to present mathematical knowledge that is needed in order to understand the art of computing eigenvalues of real symmetric matrices, either all of them or only a few. The author explains why the selected information really matters and he is not shy about making judgments. The commentary is lively but the proofs are terse.

✦ Table of Contents

The Symmetric Eigenvalue Problem......Page 1
Notation......Page 2
ISBN 0-89871-402-8......Page 8
Contents......Page 11
Preface to the First Edition......Page 19
Preface to the Classics Edition......Page 23
Introduction......Page 25
1 Basic Facts About Self-Adjoint Matrices......Page 29
2 Tasks, Obstacles, and Aids......Page 49
3 Counting Eigenvalues......Page 71
4 Simple Vector Iterations......Page 89
5 Deflation......Page 115
6 Useful Orthogonal Matrices (Tools of the Trade)......Page 121
7 Tridiagonal Form......Page 147
8 The QL and QR Algorithms......Page 179
9 Jacobi Methods......Page 217
10 Eigenvalue Bounds......Page 229
11 Approximations from a Subspace......Page 257
12 Krylov Subspaces......Page 289
13 Lanczos Algorithms......Page 315
14 Subspace Iteration......Page 351
15 The General Linear Eigenvalue Problem......Page 367
Appendix A Rank-One and Elementary Matrices......Page 397
Appendix B Chebyshev Polynomials......Page 399
Annotated bibliography......Page 403
References......Page 407
Index......Page 421