Condition number of the Krylov bases and subspaces

Abstract

The problems of numerical analysis with large sparse matrices often involve a projection of this matrix onto a Krylov subspace to obtain a smaller matrix, which is used to solve the initial problem. The subspace depends on the matrix and on an arbitrary vector. We consider a method to study the sensitivity of the Krylov subspace to a matrix perturbation. This method includes a definition of the condition numbers for the computation of the Krylov basis and the Krylov subspace. A practical method for estimating these numbers is provided. It is based on the solution of a large triangular system.