The Mathematical Institute, University of Oxford, Eprints Archive

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

Schmelzer, Thomas and Trefethen, Lloyd N. (2006) Evaluating matrix functions for exponential integrators via Carathéodory-Fejér approximation and contour integrals. Technical Report. Unspecified. (Submitted)

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Abstract

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.

Item Type:Technical Report (Technical Report)
Subjects:A - C > Approximations and expansions
D - G > Functions of a complex variable
H - N > Numerical analysis
Research Groups:Numerical Analysis Group
ID Code:1103
Deposited By:Lotti Ekert
Deposited On:11 May 2011 10:59
Last Modified:11 May 2011 10:59

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