ABSTRACT
In this paper we study methods to obtain bounds or approximations of elements of a matrix f(A) where A is a symmetric positive definite matrix and f is a smooth function. These methods are based on the use of quadrature rules and the Lanczos algorithm for diagonal elements and the block Lanczos or the non-symmetric Lanczos algorithms for the non diagonal elements. We give some theoretical results on the behavior of these methods based on results for orthogonal polynomials as well as analytical bounds and numerical experiments on a set of matrices for several functions f.
