Numerical Range Error Estimates for Evaluating Functions of Matrices via the Arnoldi Method

In this talk we propose explicit a priori error bounds for approaching f(A)b by help of the Arnoldi method. Here A is a large real not necessarily symmetric matrix, and f some function analytic on the field of values or numerical range W (A) = {y∗Ay : ‖y‖ = 1}. An essential tool in our work is the inequality ‖Fn(A)‖ ≤ 2 derived in [1] where Fn is the nth Faber polynomial corresponding to W (A), and ‖·‖ denotes euclidean vector norms and the induced spectral matrix norm. We show in a first step how to improve bounds given by Knizhnerman [3] and by Hochbruck and Lubich [2]. Subsequently we give some simple bounds in terms of the numerical range for the exponential function as well as for Stieltjes functions like the pth power of A.