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New strategies for determining backward perturbation bound of approximate two‐sided Krylov subspaces
Author(s) -
Wang Yunjie,
Wu Gang
Publication year - 2020
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.2324
Subject(s) - linear subspace , orthonormal basis , krylov subspace , mathematics , perturbation (astronomy) , generalized minimal residual method , invariant subspace , upper and lower bounds , norm (philosophy) , invariant (physics) , linear system , pure mathematics , mathematical analysis , physics , quantum mechanics , political science , law , mathematical physics
Summary Given a nonsymmetric matrix A ∈ ℝ n × nand two unit norm vectors, the two‐sided Krylov subspace methods construct a pair of bases for two Krylov subspaces with respect to A and A T , respectively. In practical calculations, however, the two subspaces spanned by the computed bases may not be Krylov subspaces. Given two subspaces and ℒ , in [ G. Wu et al , Toward backward perturbation bounds for approximate dual Krylov subspaces , BIT, 53 (2013), pp. 225‐239], the authors considered how to determine a backward perturbation E whose norm is as small as possible, such that and ℒ are Krylov subspaces of A + E and ( A + E ) T , respectively. However, as the two bases used are biorthonormal, their results are nonoptimal in terms of unitarily invariant norms, and the perturbation bound can be greatly overestimated. In this work, we revisit this problem and use orthonormal bases instead of biorthonormal bases to derive new perturbation bounds. We propose two new strategies, the first one focuses on choosing optimal orthonormal basis matrices, and the second one resorts to solving small‐sized generalized Sylvester matrix equations. Numerical experiments show that our bounds improve the existing one substantially.