Backward perturbation analysis and residual-based error bounds for the linear response eigenvalue problem
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The numerical solution of a large scale linear response eigenvalue problem is often accomplished by computing a pair of deflating subspaces associated with the interesting part of the spectrum. This paper is concerned with the backward perturbation analysis for a given pair of approximate deflating subspaces or an approximate eigenquadruple. Various optimal backward perturbation bounds are obtained, as well as bounds for approximate eigenvalues computed through the pair of approximate deflating subspaces or approximate eigenquadruple. These results are reminiscent of many existing classical ones for the standard eigenvalue problem.
KeywordsLinear response eigenvalue problem Eigenvalue approximation Rayleigh–Ritz approximation Backward perturbation Error bound Deflating subspace
Mathematics Subject Classification65L15 65F15 81Q15 15A18 15A42
The authors are grateful to the editor and two anonymous referees for their careful reading and helpful comments and suggestions, which have improved the paper considerably. The explicit formulas of \(S_K\) and \(S_M\) and their derivations in Theorem 3.2 are in fact due to one of the referees. Zhang was supported in part by the National Natural Science Foundation of China NSFC-11101257, NSFC-11371102, and the Basic Academic Discipline Program, the 11th five year plan of 211 Project for Shanghai University of Finance and Economics. Part of this work is done while Zhang was a visiting scholar at the Department of Mathematics, University of Texas at Arlington from February 2013 to January 2014. Lin was supported in part by National Center of Theoretical Science, Taiwan and ST Yau Center, Chiao Tung University, Taiwan. Li was supported in part by NSF grants DMS-1115834 and DMS-1317330, by a Research Gift Grant from Intel Corporation, and by National Center of Theoretical Science, Taiwan while he visited in December 2013.
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