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

Thomas Schmelzer and Lloyd N. Trefethen

Abstract

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 $f$ 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, Minchev, 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 is based on best rational approximations to $f$ on the negative real axis computed via the Carathéodory-Fejér 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 $3.5$. The second method is an application of the trapezoid rule on a Talbot-type contour.

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Key words

matrix exponential, exponential integrators, stiff semilinear parabolic PDEs, rational uniform approximation, Hankel contour, numerical quadrature

AMS subject classifications

65L05, 41A20, 30E20

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