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 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.