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Perturbation, Extraction and Refinement of Invariant Pairs for Matrix Polynomials
, 2010
"... Generalizing the notion of an eigenvector, invariant subspaces are frequently used in the context of linear eigenvalue problems, leading to conceptually elegant and numerically stable formulations in applications that require the computation of several eigenvalues and/or eigenvectors. Similar benefi ..."
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Cited by 8 (2 self)
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Generalizing the notion of an eigenvector, invariant subspaces are frequently used in the context of linear eigenvalue problems, leading to conceptually elegant and numerically stable formulations in applications that require the computation of several eigenvalues and/or eigenvectors. Similar benefits can be expected for polynomial eigenvalue problems, for which the concept of an invariant subspace needs to be replaced by the concept of an invariant pair. Little has been known so far about numerical aspects of such invariant pairs. The aim of this paper is to fill this gap. The behavior of invariant pairs under perturbations of the matrix polynomial is studied and a firstorder perturbation expansion is given. From a computational point of view, we investigate how to best extract invariant pairs from a linearization of the matrix polynomial. Moreover, we describe efficient refinement procedures directly based on the polynomial formulation. Numerical experiments with matrix polynomials from a number of applications demonstrate the effectiveness of our extraction and refinement procedures. 1
Perturbation, Computation and Refinement of Invariant Pairs for Matrix Polynomials
, 2009
"... Generalizing the notion of an eigenvector, invariant subspaces are frequently used in the context of linear eigenvalue problems, leading to conceptually elegant and numerically stable formulations in applications that require the computation of several eigenvalues and/or eigenvectors. Similar benefi ..."
Abstract

Cited by 3 (2 self)
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Generalizing the notion of an eigenvector, invariant subspaces are frequently used in the context of linear eigenvalue problems, leading to conceptually elegant and numerically stable formulations in applications that require the computation of several eigenvalues and/or eigenvectors. Similar benefits can be expected for polynomial eigenvalue problems, for which the concept of invariant subspaces needs to be replaced by the concept of invariant pair. Little is known so far about numerical aspects of such invariant pairs. The aim of this paper is to fill this gap. The behavior of invariant pairs under perturbations of the matrix polynomial is studied and a firstorder perturbation expansion is given. From a computational point of view, we investigate how to best extract invariant pairs from a linearization of the matrix polynomial. Moreover, we describe efficient refinement procedures directly based on the polynomial formulation. Numerical experiments with matrix polynomials from a number of applications demonstrate the effectiveness of our extraction and refinement procedures.
ELA ON WILKINSON’S PROBLEM FOR MATRIX PENCILS∗
"... Abstract. Suppose that an nbyn regular matrix pencil A − λB has n distinct eigenvalues. Then determining a defective pencil E−λF which is nearest to A−λB is widely known as Wilkinson’s problem. It is shown that the pencil E − λF can be constructed from eigenvalues and eigenvectors of A − λB when A ..."
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Abstract. Suppose that an nbyn regular matrix pencil A − λB has n distinct eigenvalues. Then determining a defective pencil E−λF which is nearest to A−λB is widely known as Wilkinson’s problem. It is shown that the pencil E − λF can be constructed from eigenvalues and eigenvectors of A − λB when A − λB is unitarily equivalent to a diagonal pencil. Further, in such a case, it is proved that the distance from A − λB to E − λF is the minimum “gap ” between the eigenvalues of A − λB. As a consequence, lower and upper bounds for the “Wilkinson distance ” d(L) from a regular pencil L(λ) with distinct eigenvalues to the nearest nondiagonalizable pencil are derived. Furthermore, it is shown that d(L) is almost inversely proportional to the condition number of the most illconditioned eigenvalue of L(λ).
ISSN 17499097Perturbation, Computation and Refinement of Invariant Pairs for Matrix Polynomials
, 2009
"... Generalizing the notion of an eigenvector, invariant subspaces are frequently used in the context of linear eigenvalue problems, leading to conceptually elegant and numerically stable formulations in applications that require the computation of several eigenvalues and/or eigenvectors. Similar benefi ..."
Abstract
 Add to MetaCart
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Generalizing the notion of an eigenvector, invariant subspaces are frequently used in the context of linear eigenvalue problems, leading to conceptually elegant and numerically stable formulations in applications that require the computation of several eigenvalues and/or eigenvectors. Similar benefits can be expected for polynomial eigenvalue problems, for which the concept of invariant subspaces needs to be replaced by the concept of invariant pair. Little is known so far about numerical aspects of such invariant pairs. The aim of this paper is to fill this gap. The behavior of invariant pairs under perturbations of the matrix polynomial is studied and a firstorder perturbation expansion is given. From a computational point of view, we investigate how to best extract invariant pairs from a linearization of the matrix polynomial. Moreover, we describe efficient refinement procedures directly based on the polynomial formulation. Numerical experiments with matrix polynomials from a number of applications demonstrate the effectiveness of our extraction and refinement procedures. 1
; E&5CAC4ADCADEE Perturbation, Extraction and Refinement of Invariant Pairs for Matrix Polynomials
, 2010
"... pairs for matrix polynomials ..."
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