## Corless, Knuth and Lambert W

Attendees at the SIAM Annual Meeting in Boston last month had the opportunity to meet Donald Knuth. He was there to give the John von Neumann lecture, about which I reported at SIAM News.

At the Sunday evening Welcome Reception I captured this photo of Don and Rob Corless (whose graduate textbook on numerical analysis I discussed here).

Don and Rob are co-authors on the classic paper

Robert M. Corless, Gaston N. Gonnet, D. E. G. Hare and David J. Jeffrey and Donald Knuth, On the Lambert $W$ Function, Adv. in Comput. Math. 5, 329-359, 1996

The Lambert W function is a multivalued function $W_k(x)$, with a countably infinite number of branches, that solves the equation $x e^x = a$. According to Google Scholar this is Don’s most-cited paper. Here is a diagram of the ranges of the branches of $W_k(x)$, together with values of $W_k(1)$ (+), $W_k(10 + 10i)$ (×), and $W_k(-0.1)$ (o).

This is to be compared with the the corresponding plot for the logarithm, which consists of horizontal strips of height $2\pi$ with boundaries at odd multiples of $\pi$.

Following the Annual Meeting, Rob ran a conference Celebrating 20 years of the Lambert W function at the University of Western Ontario.

Rob co-authored with David Jeffrey an article on the Lambert W function for the Princeton Companion to Applied Mathematics. The article summarizes the basic theory of the function and some of its many applications, which include delay differential equations. Rob and David note that

The Lambert $W$ function crept into the mathematics literature unobtrusively, and it now seems natural there.

I have worked on generalizing the Lambert W function to matrices, as discussed in

Robert M. Corless, Hui Ding, Nicholas J. Higham and David J. Jeffrey, 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, ACM Publications, pp. 116-121, 2007.

Massimiliano Fasi, Nicholas J. Higham and Bruno Iannazzo, An Algorithm for the Matrix Lambert W Function, SIAM J. Matrix Anal. Appl., 36, 669-685, 2015.

The diagram above is from the latter paper.

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