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 |
Available Versions of this Item
- Computing $A^\alpha$, $\log(A)$ and Related Matrix Functions by Contour Integrals (deposited 18 August 2008)
- Computing $A^\alpha$, $\log(A)$ and Related Matrix Functions by Contour Integrals (deposited 20 August 2007) [Currently Displayed]
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