Lambert’s W function and the generalised logarithm

November 16, 2011

(This article was first published on dahtah » R, and kindly contributed to R-bloggers)

Yesterday I ran into an equation that was a sum of an exponential and a linear term:

$\displaystyle \alpha x+\beta=\delta e^{x}$

It doesn’t take long to figure out that there is no analytical solution, and so I set out to write some crappy numerical code. After wasting some time with a fixed point iteration that did not really work, it occured to me that I most probably wasn’t the first person out there trying to solve such a simple equation. Indeed not.

The equation above has a solution in terms of a special function called Lambert’s W, and an nicer-looking one in terms of its cousin the generalised log (introduced by D. Kalman here).

Just like ${\log x}$ is the inverse of ${\exp x}$ , ${\mbox{glog}x}$ is the inverse of ${x^{-1}\exp x}$ , and Lambert’s ${W}$ is the inverse of ${x\exp x}$ . Neither glog nor W can be computed analytically, but fast implementations for ${W}$ are available (for R, it’s in the GSL package), and:

$\displaystyle \mbox{glog}\left(x\right)=-W\left(-1/x\right)$

In terms of the generalised log function the solution to the equation is:

$\displaystyle \mbox{glog}\left(\frac{\alpha}{\delta}\exp\left(\frac{\beta}{\alpha}\right)\right)-\frac{\beta}{\alpha}$

The (easy) proof is on page 5 of Kalman’s article. Here’s some R code:

require(gsl)
solve.lexpeq <- function(alpha,beta,delta)
{
v <- beta/alpha
-lambert_W0(-(delta/alpha)*exp(-v)) -v
}

So where does this turn up in statistics? Well, one example is finding the Maximum A Posterior estimate of a Poisson mean, if you put a Gaussian prior on the log of the mean.