Eigenvalue Backward Errors of Polynomial Eigenvalue Problems under Structure Preserving Perturbations
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2015
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Abstract
We develop a new framework to obtain computable formulas for s tructured eigenvalue backward errors of matrix polynomials with various structures under some prespec undertake a detailed analysis of structured when the perturbations are measured with respect to st ructured eigenvalue backward errors of matrix polyn omials with Hermitian, skew alternating, -palindromic and - antipalindromic structures in terms of the maximal eigenvalue of a parameter depending Hermitian matrix. and T- palindromic polynomials of degree at most 2, T second largest eigenvalue of a parameter depending Hermitian matrix. For higher degree T and T-alternating polynomials we estimate Under the s ame framework, we generalize eigenvalue backward errors of structured matrix polynomials with respect to most cases, the lower bound gives the exact value o f the backward error when certain assumptions are met. Numerical experiments suggest exact eigenvalue backward error. Finally, we estimate real eigenpair and eigenvalue backward errors of re al matrix polynomials perturbations. If the real matrix polynomial has ad ditional s palindromic etc., then in many cases eigenvalue backward errors are c omputed with respect to perturbations that preserve these additional struct ures also. and eigenpa ir backward errors of some special block structured pencils that arise optimal control problems with respect to special st ructure preserving perturbations. In most cases, we observe that there is a significant difference between t respect to perturbations that preserve structure and those with respect to arbitr ary perturbations. Structured Matrix polynomials, Polynomial Problem, Structured Eigenvalue a new framework to obtain computable formulas for s tructured eigenvalue backward with various structures under some prespec ifie d norms. In particular, we a detailed analysis of structured e igenvalue backward errors of structured matrix poly nomials when the perturbations are measured with respect to ǁ.ǁ w,2 norm. We obtain explicit formulas for ructured eigenvalue backward errors of matrix polyn omials with Hermitian, skew - Hermitian, antipalindromic structures in terms of the maximal eigenvalue of a parameter depending Hermitian matrix. We also derive structured eigenvalue backward errors of T palindromic polynomials of degree at most 2, T -odd and T- antipalindromic pencil second largest eigenvalue of a parameter depending Hermitian matrix. For higher degree T polynomials we estimate the structured eigenvalue backward error by tight bounds. ame framework, we generalize these ideas to obtain computable bounds for structu red structured matrix polynomials with respect to ǁ.ǁ w,∞ and most cases, the lower bound gives the exact value o f the backward error when certain assumptions are suggest that these assu mptions are satisfied in practice, thus giving the real eigenpair and eigenvalue backward errors of re al matrix polynomials perturbations. If the real matrix polynomial has ad ditional s tructure like Hermitian, - alternating, T then in many cases eigenvalue backward errors are c omputed with respect to perturbations that preserve these additional struct ures also. We also compute structured eigenvalue ir backward errors of some special block structured pencils that arise in linear optimal control problems with respect to special st ructure preserving perturbations. that there is a significant difference between t he backward errors with preserve structure and those with respect to arbitr ary perturbations. Eigenvalue Backward Errors of Polynomial Eigenvalue Perturbations ” Polynomial Eigenvalue ackward Errors a new framework to obtain computable formulas for s tructured eigenvalue backward d norms. In particular, we igenvalue backward errors of structured matrix poly nomials explicit formulas for Hermitian, - antipalindromic structures in terms of the maximal eigenvalue of a structured eigenvalue backward errors of T -even antipalindromic pencil s in terms of second largest eigenvalue of a parameter depending Hermitian matrix. For higher degree T -palindromic backward error by tight bounds. these ideas to obtain computable bounds for structu red and ǁ.ǁ w,F norms. In most cases, the lower bound gives the exact value o f the backward error when certain assumptions are mptions are satisfied in practice, thus giving the real eigenpair and eigenvalue backward errors of re al matrix polynomials under real alternating, T - then in many cases eigenvalue backward errors are c omputed with respect to structured eigenvalue in linear -quadratic he backward errors with preserve structure and those with respect to arbitr ary perturbations.
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Supervisor: Shreemayee Bora
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MATHEMATICS