Numerical Methods for General and Structured Eigenvalue Problems

✍ Scribed by Daniel Kressner (auth.)

• Book ID: 127422417
• Publisher: Springer
• Year: 2005
• Tongue: English
• Weight: 2 MB
• Edition: 1
• Category: Library
• City: Berlin; New York
• ISBN: 3540285024
• DOI: 10.1007/3-540-28502-4

✦ 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].

✦ Subjects

Computational Science and Engineering