# Computing $A^\alpha$, $\log(A)$ and Related Matrix Functions by Contour Integrals

Hale, Nicholas and Higham, Nicholas J. and Trefethen, Lloyd N. (2007) Computing $A^\alpha$, $\log(A)$ and Related Matrix Functions by Contour Integrals. [MIMS Preprint]

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## Abstract

New methods are proposed for the numerical evaluation of $f(\A)$ or $f(\A) b$, where $f(\A)$ is a function such as $\sqrt \A$ or $\log (\A)$ with singularities in $(-\infty,0\kern .7pt ]$ and $\A$ is a matrix with eigenvalues on or near $(0,\infty)$. The methods are based on combining contour integrals evaluated by the periodic trapezoid rule with conformal maps involving Jacobi elliptic functions. The convergence is geometric, so that the computation of $f(\A)b$ is typically reduced to one or two dozen linear system solves.

Item Type: MIMS Preprint matrix function, contour integral, quadrature, rational approximation, trapezoid rule, Cauchy integral, conformal map MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theoryMSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis Nick Higham 20 Aug 2007 08 Nov 2017 18:18 http://eprints.maths.manchester.ac.uk/id/eprint/834

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