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Computational Science, Engineering & Technology Series
ISSN 17593158 CSETS: 18
COMPUTATIONAL METHODS FOR ACOUSTICS PROBLEMS Edited by: F. Magoulès
Chapter 7
Theory and Numerical Methods for Eigenvalue Problems K. Meerbergen
K.U. Leuven, Department of Computer Science, Leuven, Belgium K. Meerbergen, "Theory and Numerical Methods for Eigenvalue Problems", in F. Magoulès, (Editor), "Computational Methods for Acoustics Problems", SaxeCoburg Publications, Stirlingshire, UK, Chapter 7, pp 181206, 2008. doi:10.4203/csets.18.7
Keywords: algebraic eigenvalue problems, generalised eigenvalue problem, quadratic
eigenvalue problem, Lanczos method, Arnoldi method, spectral transformation.
Abstract
This chapter reviews the theory on the algebraic eigenvalue problem, and in particular,
the theory on the linear definite generalised eigenvalue problem (stiffnessmass), and
the quadratic eigenvalue problem (stiffnessdampingmass). Numerical methods
are presented for solving large scale problems, where the focus is on Krylov methods:
Lanczos and Arnoldi, and the spectral transformation. Important notions such
as inertia and sparse LDL^{T} factorisation are also touched on. Examples are included
to illustrate the theory, as well as a bibliographical note with references to techniques
other than those discussed in this chapter.
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