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

Report

Among the fastest methods for solving stiff PDE are exponential integrators, which require the evaluation of $f(A)$, where $A$ is a negative definite matrix and $f$ is the exponential function or one of the related ``$\varphi$ functions'' such as $\varphi_1(z) = (e^z-1)/z$. Building on previous work by Trefethen and Gutknecht, Gonchar and Rakhmanov, and Lu, we propose two methods for the fast evaluation of $f(A)$ that are especially useful when shifted systems $(A+zI)x=b$ can be solved efficiently, e.g. by a sparse direct solver. The first method method is based on best rational approximations to $f$ on the negative real axis computed via the Carathéodory-Fejér procedure, and we conjecture that the accuracy scales as $(9.28903\dots)^{-2n}$, where $n$ is the number of complex matrix solves. In particular, three matrix solves suffice to evaluate $f(A)$ to approximately six digits of accuracy. The second method is an application of the trapezoid rule on a Talbot-type contour.

Authors: Schmelzer, T; Trefethen, L

• Publisher: Unspecified
• Publication date: 2006-11-01