Hale, Nicholas and Higham, Nicholas J. and Trefethen, Lloyd N. (2008) Computing $A^\alpha$, $\log(A)$ and Related Matrix Functions by Contour Integrals. SIAM Journal on Numerical Analysis, 46 (5). pp. 25052523. ISSN 00361429
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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:  Article 

Uncontrolled Keywords:  matrix function, contour integral, quadrature, rational approximation, trapezoid rule, Cauchy integral, conformal map 
Subjects:  MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theory MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis 
Depositing User:  Nick Higham 
Date Deposited:  18 Aug 2008 
Last Modified:  20 Oct 2017 14:12 
URI:  https://eprints.maths.manchester.ac.uk/id/eprint/1136 
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Computing $A^\alpha$, $\log(A)$ and Related Matrix Functions by Contour Integrals. (deposited 20 Aug 2007)
 Computing $A^\alpha$, $\log(A)$ and Related Matrix Functions by Contour Integrals. (deposited 18 Aug 2008) [Currently Displayed]
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