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. In: ISSAC '07: Proceedings of the 2007 International Symposium on Symbolic and Algebraic Computation, July 29 - August 01, 2007, Waterloo, Ontario, Canada.

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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: Conference or Workshop Item (Paper)
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 Proceedings of the 2007 international symposium on Symbolic and algebraic computation 2007, Waterloo, Ontario, Canada July 29 - August 01, 2007. http://doi.acm.org/10.1145/1277548.1277565
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: 17 Aug 2007
Last Modified: 20 Oct 2017 14:12
URI: https://eprints.maths.manchester.ac.uk/id/eprint/833

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