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2007.227: Perturbation theory for homogeneous polynomial eigenvalue problems

2007.227: Françoise Tisseur and Jean-Pierre Dedieu (2003) Perturbation theory for homogeneous polynomial eigenvalue problems. Linear Algebra and its Applications, 385 (1). pp. 71-74. ISSN 0024-3795

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DOI: 10.1016/S0024-3795(01)00423-2

Abstract

We consider polynomial eigenvalue problems P(A,alpha,beta)x=0 in which the matrix polynomial is homogeneous in the eigenvalue (alpha,beta)&unknown;C2. In this framework infinite eigenvalues are on the same footing as finite eigenvalues. We view the problem in projective spaces to avoid normalization of the eigenpairs. We show that a polynomial eigenvalue problem is well-posed when its eigenvalues are simple. We define the condition numbers of a simple eigenvalue (alpha,beta) and a corresponding eigenvector x and show that the distance to the nearest ill-posed problem is equal to the reciprocal of the condition number of the eigenvector x. We describe a bihomogeneous Newton method for the solution of the homogeneous polynomial eigenvalue problem (homogeneous PEP)

Item Type:Article
Uncontrolled Keywords:Mathematical Subject Codes] 65F15; [Mathematical Subject Codes] 15A18; Polynomial eigenvalue problem; Matrix polynomial; Quadratic eigenvalue problem; Condition number
Subjects:MSC 2000 > 15 Linear and multilinear algebra; matrix theory
MSC 2000 > 65 Numerical analysis
MIMS number:2007.227
Deposited By:Ms Helen Kirkbright
Deposited On:05 December 2007

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