Acta Numerica

Research Article

Computing matrix functions

Nicholas J. Highama1 and Awad H. Al-Mohya1

a1 School of Mathematics, University of Manchester, Manchester, M13 9PL, UK E-mail: higham@maths.man.ac.uk, almohy@maths.man.ac.uk

The need to evaluate a function f(A) xs2208 xs2102n×n of a matrix A xs2208 xs2102n×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 Fréchet 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.