Numerical Methods for General and Structured Eigenvalue Problems

  • Daniel Kressner

Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 46)

Table of contents

  1. Front Matter
    Pages I-XIV
  2. Pages 1-66
  3. Pages 67-111
  4. Pages 225-231
  5. Back Matter
    Pages 233-264

About this book

Introduction

The purpose of this book is to describe recent developments in solving eig- value problems, in particular with respect to the QR and QZ algorithms as well as structured matrices. Outline Mathematically speaking, the eigenvalues of a square matrix A are the roots of its characteristic polynomial det(A??I). An invariant subspace is a linear subspace that stays invariant under the action of A. In realistic applications, it usually takes a long process of simpli?cations, linearizations and discreti- tions before one comes up with the problem of computing the eigenvalues of a matrix. In some cases, the eigenvalues have an intrinsic meaning, e.g., for the expected long-time behavior of a dynamical system; in others they are just meaningless intermediate values of a computational method. The same applies to invariant subspaces, which for example can describe sets of initial states for which a dynamical system produces exponentially decaying states. Computing eigenvalues has a long history, dating back to at least 1846 when Jacobi [172] wrote his famous paper on solving symmetric eigenvalue problems. Detailed historical accounts of this subject can be found in two papers by Golub and van der Vorst [140, 327].

Keywords

algorithms computational methods eigenvalue matrix product structured matrix

Authors and affiliations

  • Daniel Kressner
    • 1
  1. 1.Institut für MathematikTechnische Universität BerlinBerlinGermany

Bibliographic information

  • DOI https://doi.org/10.1007/3-540-28502-4
  • Copyright Information Springer-Verlag Berlin/Heidelberg 2005
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-540-24546-9
  • Online ISBN 978-3-540-28502-1
  • Series Print ISSN 1439-7358
  • About this book