Numerical Methods for General and Structured Eigenvalue Problems

✍ Scribed by Daniel Kressner (auth.)

• Publisher: Springer-Verlag Berlin Heidelberg
• Year: 2005
• Tongue: English
• Leaves: 273
• Series: Lecture Notes in Computational Science and Engineering 46
• Edition: 1
• Category: Library

✦ Synopsis

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].

✦ Table of Contents

The QR Algorithm....Pages 1-66
The QZ Algorithm....Pages 67-111
The Krylov-Schur Algorithm....Pages 113-130
Structured Eigenvalue Problems....Pages 131-214
Background in Control Theory Structured Eigenvalue Problems....Pages 215-223
Software....Pages 225-231

✦ Subjects

Computational Mathematics and Numerical Analysis; Systems Theory, Control; Computational Science and Engineering