Towards backward perturbation bounds for approximate dual Krylov subspaces (Q1944009)

The authors recall Arnoldi, Lanczos and Krylov decompositions. Under the assumption that for two given subspaces \(\mathcal{K}\) and \(\mathcal{L}\) a biorthonormal pair of bases \(V_m\) and \(W_m\) always exists, they define the dual Krylov subspaces with respect to the given matrix \(A\) and the dual Krylov decomposition with respect to \(A\) as a generalization of the Krylov decomposition. If \(A\) is an extended tridiagonal matrix, the dual Krylov decomposition is called ``the two-sided Lanczos decomposition'' -- a generalization of the Lanczos decomposition. The authors consider a pair of biorthonormal bases \(V_m\) and \(W_m\) and determine a backward perturbation matrix \(E\) whose norm is as small as possible, such that \(\mathcal{K}\) and \(\mathcal{L}\) are Krylov subspaces of \(A+E\) and its adjoint, respectively. A framework for determining the perturbation matrix \(E\) is given in Algorithm 2.1. The numerical experiments demonstrate the efficiency of the presented strategy for determining the backward perturbation.