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- The solution to W exp W = A is not always LambertW(A) Robert M. Corless, Hui Ding, and David Jeffrey, Paper submitted to ISSAC 2007.
**Abstract:**The solutions of the matrix equation $S\exp S = A$ are studied. This is motivated by 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 \textsl{evaluating a matrix function} and \textsl{solving a matrix equation}. As is well-known in the theory of computing matrix functions, there can be difficult, exceptional cases. This paper examines some of these; specifically, it shows that the matrix Lambert $W$ function evaluated at the matrix $A$ does not represent all possible solutions of $S\exp S = A$.

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