Error bounds for Lanczos approximations of rational functions of matrices

Having good estimates or even bounds for the error in computing approximations to expressions of the form $f(A)v$ is very important in practical applications. In this paper we consider the case that $A$ is Hermitian and that $f$ is a rational function. We assume that the Lanczos method is used to compute approximations for $f(A)v$ and we show how to obtain a posteriori upper and lower bounds on the $ell_2$-norm of the approximation error. These bounds are computed by minimizing and maximizing a rational function whose coefficients depend on the iteration step. We use global optimization based on interval arithmetic to obtain these bounds and include a number of experimental results illustrating the quality of the error estimates.