# Computing Matrix Functions

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

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 matrix $p$th root, primary matrix function, nonprimary matrix function, Markov chain, transition matrix, matrix exponential, Schur-Parlett method, CICADA MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theoryMSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis Nick Higham 18 May 2010 20 Oct 2017 14:12 http://eprints.maths.manchester.ac.uk/id/eprint/1451