Evaluating matrix functions for exponential integrators via Caratheodory-Fejer approximation and contour integrals

Among the fastest methods for solving stiff PDE are exponential integrators, which require the evaluation of f(A), where A is a negative semidefinite matrix and is the exponential function or one of the related "ψ functions" such as [ψ.sub.i](z) = ([e.sup.z] - l)/z. Building on previous work by Trefethen and Gutknecht, Minchev, and Lu, we propose two methods for the fast evaluation of f(A) that are especially useful when shifted systems (A + zl)x = b can be solved efficiently, e.g. by a sparse direct solver. The first method is based on best rational approximations to on the negative real axis computed via the Caratheodory-Fejer procedure. Rather than using optimal poles we approximate the functions in a set of common poles, which speeds up typical computations by a factor of 2 to 3.. The second method is an application of the trapezoid rule on a Talbot-type contour. Key words. matrix exponential, exponential integrators, stiff semilinear parabolic PDEs, rational uniform approximation, Hankel contour, numerical quadrature