Computing Matrix Functions

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

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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
Uncontrolled Keywords: matrix $p$th root, primary matrix function, nonprimary matrix function, Markov chain, transition matrix, matrix exponential, Schur-Parlett method, CICADA
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 May 2010
Last Modified: 20 Oct 2017 14:12

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