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Why W?

Brian Hayes

Evaluating W

It's all very well to express the solutions of problems in terms of W, but then how do we find the value of the resulting function? In the case of logarithms and trigonometric functions, the standard method for many years was to look up the answer in a big printed table; now we push the appropriate button on a calculator. For W, however, there are no published tables, and so far no scientific calculator has a built-in Lambert W key. Several computer algebra systems know how to evaluate the W function, but if you don't have access to such software, you're on your own.

Suppose we already know how to calculate exponentials and logarithms; can we then solve the equation WeW = x? As noted above, the forward version is easy: just evaluate eW and then multiply by W. At first glance, the inverse function looks like it might be wrestled to submission by a similar tactic. If we can solve for x by calculating an exponential and then multiplying, can't we solve for W by dividing and then taking a logarithm?

Dividing both sides of the equation by W gives eW = x/W. Then, taking the logarithm of both sides produces log(eW) = log(x/W). On the left hand side, the logarithm of eW is simply W. On the right hand side we can rewrite the logarithm of a quotient as the difference of two logarithms, and so we wind up with this equation:

W = log(x) - log(W).

We have succeeded in getting W off by itself on the left side, but unfortunately there's still a log(W) on the right. Thus we don't have a closed-form solution, a formula that would allow us to plug in an x and immediately get back the corresponding W. This failure is not merely a result of my ineptitude; no algebraic wizardry will yield a finite closed-form solution.

On the other hand, the equation above is not totally worthless. If we have a guess about the value of W, then we can plug it into the right hand side of the equation to get an even better guess, then repeat the process until we're satisfied with the accuracy of the approximation. For some values of x—well away from 0—this simple iterative scheme converges quickly on the correct result. The algorithms used in computer-algebra software are more efficient, accurate and robust, but they still rely on successive approximations.

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