The Solution of S exp(S) = A is Not Always the Lambert W Function of A

Corless, Robert M. and Ding, Hui and Higham, Nicholas J. and Jeffrey, David J. (2007) The Solution of S exp(S) = A is Not Always the Lambert W Function of A. [MIMS Preprint]

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Abstract

We study the solutions of the matrix equation $S\exp(S) = A$. Our motivation comes from the study of systems of delay differential equations $y'(t) = A y(t-1)$, which occur in some models of practical interest, especially in mathematical biology. This paper concentrates on the distinction between \emph{evaluating a matrix function} and \emph{solving a matrix equation}. In particular, it shows that the matrix Lambert $W$ function evaluated at the matrix $A$ does not represent all possible solutions of $S\exp(S) = A$. These results can easily be extended to more general matrix equations.

Item Type: MIMS Preprint
Additional Information: © ACM, (2007). This is the author's version of the work. It is posted here by permission of ACM for your personal use. Not for redistribution. The definitive version was published in PUBLICATION, {VOL#, ISS#, (DATE)} http://doi.acm.org/10.1145/nnnnnn.nnnnnn
Uncontrolled Keywords: Matrix function; Lambert W function; nonlinear matrix equation
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: 28 May 2007
Last Modified: 08 Nov 2017 18:18
URI: http://eprints.maths.manchester.ac.uk/id/eprint/810

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