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2010.18: Computing Matrix Functions

2010.18: Nicholas J. Higham and Awad H. Al-Mohy (2010) Computing Matrix Functions. Acta Numerica, 19. 159 -208. ISSN 0962-4929

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DOI: 10.1017/S0962492910000036

Abstract

The need to evaluate a function $f(A)\in\mathbb{C}^{n \times n}$ of a matrix $A\in\mathbb{C}^{n \times n}$ arises in a wide and growing number of applications, ranging from the numerical solution of differential equations to measures of the complexity of networks. We give a survey of numerical methods for evaluating matrix functions, along with a brief treatment of the underlying theory and a description of two recent applications. The survey is organized by classes of methods, which are broadly those based on similarity transformations, those employing approximation by polynomial or rational functions, and matrix iterations. Computation of the Fr\'echet derivative, which is important for condition number estimation, is also treated, along with the problem of computing $f(A)b$ without computing $f(A)$. A summary of available software completes the survey.

Item Type:Article
Additional Information:

Uncontrolled Keywords: matrix $p$th root, primary matrix function, nonprimary matrix function, Markov chain, transition matrix, matrix exponential, Schur-Parlett method, CICADA
Subjects:MSC 2000 > 15 Linear and multilinear algebra; matrix theory
MSC 2000 > 65 Numerical analysis
MIMS number:2010.18
Deposited By:Nick Higham
Deposited On:18 May 2010

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