Optimal perturbation bounds for the Hermitian eigenvalue problem

Jesse Louis Barlow, Ivan Slapničar

Research output: Contribution to journalConference article

12 Citations (Scopus)

Abstract

There is now a large literature on structured perturbation bounds for eigenvalue problems of the formvboxHx=λMx, where H and M are Hermitian. These results give relative error bounds on the ith eigenvalue, λi, of the form|λi-λ̃i||λi|,and bound the error in the ith eigenvector in terms of the relative gap,minj≠iij||λiλj|1/2. In general, this theory usually restricts H to be nonsingular and M to be positive definite. We relax this restriction by allowing H to be singular. For our results on eigenvalues weallow M to be positive semi-definite and for a few results we allow it to be more general. For these problems, for eigenvalues that are not zero or infinity under perturbation, it is possible to obtain local relative error bounds. Thus, a wider class of problems may be characterized by this theory. Although it is impossible to give meaningful relative error bounds on eigenvalues that are not bounded away from zero, we show that the error in the subspace associated with those eigenvalues can be characterized meaningfully.

Original languageEnglish (US)
Pages (from-to)19-43
Number of pages25
JournalLinear Algebra and Its Applications
Volume309
Issue number1-3
DOIs
StatePublished - Apr 15 2000
EventThe International Workshop on Accurate Solutions of Eigenvalue Problems - University Park, PA, United States
Duration: Jul 20 1998Jul 23 1998

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Perturbation Bound
Optimal Bound
Eigenvalue Problem
Eigenvalue
Relative Error
Error Bounds
Structured Perturbations
Positive semidefinite
Zero
Eigenvalues and eigenfunctions
Positive definite
Eigenvector
Subspace
Infinity
Restriction
Perturbation

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory
  • Numerical Analysis
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics

Cite this

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abstract = "There is now a large literature on structured perturbation bounds for eigenvalue problems of the formvboxHx=λMx, where H and M are Hermitian. These results give relative error bounds on the ith eigenvalue, λi, of the form|λi-λ̃i||λi|,and bound the error in the ith eigenvector in terms of the relative gap,minj≠i|λi-λ j||λiλj|1/2. In general, this theory usually restricts H to be nonsingular and M to be positive definite. We relax this restriction by allowing H to be singular. For our results on eigenvalues weallow M to be positive semi-definite and for a few results we allow it to be more general. For these problems, for eigenvalues that are not zero or infinity under perturbation, it is possible to obtain local relative error bounds. Thus, a wider class of problems may be characterized by this theory. Although it is impossible to give meaningful relative error bounds on eigenvalues that are not bounded away from zero, we show that the error in the subspace associated with those eigenvalues can be characterized meaningfully.",
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Optimal perturbation bounds for the Hermitian eigenvalue problem. / Barlow, Jesse Louis; Slapničar, Ivan.

In: Linear Algebra and Its Applications, Vol. 309, No. 1-3, 15.04.2000, p. 19-43.

Research output: Contribution to journalConference article

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N2 - There is now a large literature on structured perturbation bounds for eigenvalue problems of the formvboxHx=λMx, where H and M are Hermitian. These results give relative error bounds on the ith eigenvalue, λi, of the form|λi-λ̃i||λi|,and bound the error in the ith eigenvector in terms of the relative gap,minj≠i|λi-λ j||λiλj|1/2. In general, this theory usually restricts H to be nonsingular and M to be positive definite. We relax this restriction by allowing H to be singular. For our results on eigenvalues weallow M to be positive semi-definite and for a few results we allow it to be more general. For these problems, for eigenvalues that are not zero or infinity under perturbation, it is possible to obtain local relative error bounds. Thus, a wider class of problems may be characterized by this theory. Although it is impossible to give meaningful relative error bounds on eigenvalues that are not bounded away from zero, we show that the error in the subspace associated with those eigenvalues can be characterized meaningfully.

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