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## 2007.90: The Solution of S exp(S) = A is Not Always the Lambert W Function of A

2007.90: Robert M. Corless, Hui Ding, Nicholas J. Higham and David J. Jeffrey (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) © 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 Matrix function; Lambert W function; nonlinear matrix equation MSC 2000 > 15 Linear and multilinear algebra; matrix theoryMSC 2000 > 65 Numerical analysis 2007.90 Nick Higham 17 August 2007