Evaluating matrix functions for exponential integrators via Carathéodory-Fejér approximation and contour integrals

Abstract

Among the fastest methods for solving stiff PDE are exponential integrators, which require the evaluation of , where is a negative definite matrix and is the exponential function or one of the related `` functions'' such as . Building on previous work by Trefethen and Gutknecht, Gonchar and Rakhmanov, and Lu, we propose two methods for the fast evaluation of that are especially useful when shifted systems can be solved efficiently, e.g. by a sparse direct solver. The first method method is based on best rational approximations to on the negative real axis computed via the Carathéodory-Fejér procedure, and we conjecture that the accuracy scales as , where is the number of complex matrix solves. In particular, three matrix solves suffice to evaluate to approximately six digits of accuracy. The second method is an application of the trapezoid rule on a Talbot-type contour.