Error bounds and estimates for Krylov subspace approximations of Stieltjes matrix functions

When using the Lanczos method to approximate , the action of a matrix function on a vector, there is, in contrast to the solution of linear systems, no straightforward way to measure or estimate the error of the current iterate. Therefore, to be able to decide whether the desired accuracy has been reached, several different estimates and bounds for the error have been suggested, all of them specific to certain classes of functions. In this paper, we add to these results by developing a technique to compute error bounds for Stieltjes functions, using a recently suggested integral representation of the error, and we show how these bounds can be computed essentially for free, i.e., with cost independent of the iteration number and the dimension of the matrix A.