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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 ..."
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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.
Parallel Krylov solvers for the polynomial eigenvalue problem
, 2015
"... Abstract. Polynomial eigenvalue problems are often found in scientific computing applications. When the coefficient matrices of the polynomial are large and sparse, usually only a few eigenpairs are required and projection methods are the best choice. We focus on Krylov methods that operate on the c ..."
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Abstract. Polynomial eigenvalue problems are often found in scientific computing applications. When the coefficient matrices of the polynomial are large and sparse, usually only a few eigenpairs are required and projection methods are the best choice. We focus on Krylov methods that operate on the companion linearization of the polynomial, but exploit the block structure with the aim of being memoryefficient in the representation of the Krylov subspace basis. The problem may appear in the form of a lowdegree polynomial (quartic or quintic, say) expressed in the monomial basis, or a highdegree polynomial (coming from interpolation of a nonlinear eigenproblem) expressed in a nonmonomial basis. We have implemented a parallel solver in SLEPc covering both cases, that is able to compute exterior as well as interior eigenvalues via spectral transformation. We discuss important issues such as scaling and restart, and illustrate the robustness and performance of the solver with some numerical experiments.
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
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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
Scaling, sensitivity and stability in the numerical solution
"... of quadratic eigenvalue problems ..."
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The MIMS Secretary
, 2007
"... to this paper the theory of determinants was well developed, and Cauchy had shown that the eigenvalues of a real matrix are real (in the context of quadratic forms). Yet the idea that an array of numbers had algebraic properties that merited study in their own right was first put forward by Cayley. ..."
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to this paper the theory of determinants was well developed, and Cauchy had shown that the eigenvalues of a real matrix are real (in the context of quadratic forms). Yet the idea that an array of numbers had algebraic properties that merited study in their own right was first put forward by Cayley. The term “matrix ” had already been coined in 1850 by James Joseph Sylvester (1814– 1897) [20]. The names of Cayley and Sylvester are of course well known to any student of linear algebra and matrix analysis, through eponymous objects such as the Cayley–Hamilton theorem, the Cayley transformation, Sylvester’s inertia theorem, and the Sylvester equation. The lives of these two mathematicians have been well documented during the century or so since their deaths, for example in [1, Chap, 21], [13], [14], [15], [18]. So what is the significance of these two new biographies, both published in 2006 by Johns Hopkins University Press? There are two answers. First, both authors are the leading experts on the respective mathematicians, having spent much of their lives studying their work, their voluminous
Contents
, 2000
"... Introduction and methodology 3 Referral to obstetric care 6 Birth environment 7 Prelabour rupture of membranes at term 10 ..."
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Introduction and methodology 3 Referral to obstetric care 6 Birth environment 7 Prelabour rupture of membranes at term 10