A Chebychev method for matrix function approximation

We present a Chebyshev-based method for approximating matrix functions or products of matrix functions with vectors. Our main interest is in matrix functions that arise in exponential integrators. The approach builds upon the construction of a Faber expansion of the function on an ellipse that encloses the field of values of the matrix. We derive error bounds for the proposed approximations and provide an efficient residual-based error estimator for the product of a matrix function with a vector, that can be computed efficiently using short recurrences. Since Faber polynomials rely on a priori information on the spectrum of the field of values of the matrix, we propose a novel algorithm to determine a suitable ellipse from appropriate Ritz values. Numerical examples demonstrate the effectiveness of the proposed method.