Error Estimates and Evaluation of Matrix Functions via the Faber Transform

The need to evaluate expressions of the form f(A) or f(A)b, where f is a nonlinear function, A is a large sparse n × n matrix, and b is an n-vector, arises in many applications. This paper describes how the Faber transform applied to the field of values of A can be used to determine improved error bounds for popular polynomial approximation methods based on the Arnoldi process. Applications of the Faber transform to rational approximation methods and, in particular, to the rational Arnoldi process also are discussed.