A numerical method for polynomial eigenvalue problems using contour integral
TL;DR: In this article, a numerical method using contour integral to solve polynomial eigenvalue problems (PEPs) is proposed, which finds eigenvalues contained in a certain domain which is defined by a surrounding integral path.
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Abstract: We propose a numerical method using contour integral to solve polynomial eigenvalue problems (PEPs). The method finds eigenvalues contained in a certain domain which is defined by a surrounding integral path. By evaluating the contour integral numerically along the path, the method reduces the original PEP into a small generalized eigenvalue problem, which has the identical eigenvalues in the domain. When the contour integral is approximated numerically, eigenvalues on the periphery of the path are also obtained. Error analysis shows that, even though condition numbers of those exterior eigenvalues can be huge, the interior eigenvalues are calculated less erroneously. Four numerical examples are presented, which confirm the theoretical predictions.
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James Demmel,Jack Dongarra,Axel Ruhe,Henk A. van der Vorst,Zhaojun Bai +4 more
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TL;DR: This book discusses iterative projection methods for solving Eigenproblems, and some of the techniques used to solve these problems came from the literature on Hermitian Eigenvalue.
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TL;DR: This work surveys the quadratic eigenvalue problem, treating its many applications, its mathematical properties, and a variety of numerical solution techniques.
A projection method for generalized eigenvalue problems using numerical integration
Tetsuya Sakurai,Hiroshi Sugiura +1 more
TL;DR: In this article, a method for finding certain eigenvalues of a generalized eigenvalue problem that lie in a given domain of the complex plane is proposed, which projects the matrix pencil onto a subspace associated with the eigen values that are located in the domain via numerical integration.
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