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
This is the latest version of this item.
PDF
hht08.pdf Download (442kB) 
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:  http://eprints.maths.manchester.ac.uk/id/eprint/1136 
Available Versions of this Item

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]
Actions (login required)
View Item 