New strategies for determining backward perturbation bound of approximate two‐sided Krylov subspaces

Abstract

Summary Given a nonsymmetric matrix A∈ℝn×n and two unit norm vectors, the two‐sided Krylov subspace methods construct a pair of bases for two Krylov subspaces with respect to A and AT, respectively. In practical calculations, however, the two subspaces spanned by the computed bases may not be Krylov subspaces. Given two subspaces 𝒦 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 𝒦 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.