# Numerical Methods for General and Structured Eigenvalue Problems

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

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

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].

algorithms computational methods eigenvalue matrix product structured matrix

- 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