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2007.103: Computing $A^\alpha$, $\log(A)$ and Related Matrix Functions by Contour Integrals

2007.103: Nicholas Hale, Nicholas J. Higham and Lloyd N. Trefethen (2007) Computing $A^\alpha$, $\log(A)$ and Related Matrix Functions by Contour Integrals.

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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
Uncontrolled Keywords:matrix function, contour integral, quadrature, rational approximation, trapezoid rule, Cauchy integral, conformal map
Subjects:MSC 2000 > 15 Linear and multilinear algebra; matrix theory
MSC 2000 > 65 Numerical analysis
MIMS number:2007.103
Deposited By:Nick Higham
Deposited On:20 August 2007

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