# Numerical Methods for General and Structured Eigenvalue Problems

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

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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
- Buy this book on publisher's site