Extended and rational Hessenberg methods for the evaluation of matrix functions

Some Krylov subspace methods for approximating the action of matrix functions are presented in this paper. The main idea of these techniques is to project the approximation problem onto a subspace of much smaller dimension. Then the matrix function operation is performed with a much smaller matrix. These methods are projection methods that use the Hessenberg process to generate bases of the approximation spaces. We also use the introduced methods to solve shifted linear systems. Some numerical experiments are presented in order to show the efficiency of the proposed methods.