Towards backward perturbation bounds for approximate dual Krylov subspaces

Given a matrix A , n by n , and two subspaces K and L of dimension m , we consider how to determine a backward perturbation E whose norm is as small as possible, such that k and L are Krylov subspaces of A + E and its adjoint, respectively. We first focus on determining a perturbation matrix for a g...

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Bibliographic Details
Published inBIT Numerical Mathematics Vol. 53; no. 1; pp. 225 - 239
Main Authors Wu, Gang, Wei, Yimin, Jia, Zhi-gang, Ling, Si-tao, Zhang, Lu
Format Journal Article
LanguageEnglish
Published Dordrecht Springer Netherlands 01.03.2013
Subjects
Computational Mathematics and Numerical Analysis
Mathematics
Mathematics and Statistics
Numeric Computing
Krylov subspace
65F10
65F15
Perturbation matrix
Dual Krylov subspaces
Dual Krylov decomposition
Two-sided Lanczos decomposition
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Summary:Given a matrix A , n by n , and two subspaces K and L of dimension m , we consider how to determine a backward perturbation E whose norm is as small as possible, such that k and L are Krylov subspaces of A + E and its adjoint, respectively. We first focus on determining a perturbation matrix for a given pair of biorthonormal bases, and then take into account how to choose an appropriate biorthonormal pair and express the Krylov residuals as a perturbation of the matrix  A . Specifically, the perturbation matrix is globally optimal when A is Hermitian and K = L . The results show that the norm of the perturbation matrix can be assessed by using the norms of the Krylov residuals and those of the biorthonormal bases. Numerical experiments illustrate the efficiency of our strategy.
ISSN:0006-3835
1572-9125
DOI:10.1007/s10543-012-0402-4