## 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.*

There is a more recent version of this eprint available. Click here to view it.

Full text available as:

PDF - Requires a PDF viewer such as GSview, Xpdf or Adobe Acrobat Reader 460 Kb |

## 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]**

- Computing $A^\alpha$, $\log(A)$ and Related Matrix Functions by Contour Integrals (deposited 20 August 2007)

Download Statistics: last 4 weeks

Repository Staff Only: edit this item