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Computing matrix functions

Published online by Cambridge University Press:  10 May 2010

Nicholas J. Higham  and
Awad H. Al-Mohy
Nicholas J. Higham
Affiliation:
School of Mathematics, University of Manchester, Manchester, M13 9PL, UK E-mail: higham@maths.man.ac.uk, almohy@maths.man.ac.uk
Awad H. Al-Mohy
Affiliation:
School of Mathematics, University of Manchester, Manchester, M13 9PL, UK E-mail: higham@maths.man.ac.uk, almohy@maths.man.ac.uk
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The need to evaluate a function f(A) ∈ ℂn×n of a matrix A ∈ ℂn×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.

Type
Research Article
Information
Acta Numerica , Volume 19 , May 2010 , pp. 159 - 208
Copyright
Copyright © Cambridge University Press 2010

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References

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