lambertW: Lambert W function (principal branch)

In RTMBdist: Distributions Compatible with Automatic Differentiation by 'RTMB'

Lambert W function (principal branch)

Description

Solves W(x) e^{W(x)} = x for the principal branch W_0.

Usage

lambertW(x)

Arguments

x	vector of evaluation points, x \ge -1/e.

Details

This implementation allows for automatic differentiation with RTMB.

The value is obtained by Halley iteration, which converges to machine precision in a handful of steps over the whole domain. For AD, the function is registered as an atomic operation via ADjoint with the analytic derivative

W'(x) = \frac{1}{e^{W(x)} (1 + W(x))},

expressed through the returned value rather than through x. Written this way the derivative is itself an AD-able expression, so derivatives of every order are available; in particular the third-order derivatives that the gradient of a Laplace approximation requires.

The principal branch is defined for x \ge -1/e, with W_0(-1/e) = -1 and W_0(x) \ge -1 throughout. Values below -1/e return NaN with a warning. The derivative is infinite at the branch point x = -1/e.

Value

The principal branch of the Lambert W function evaluated at x.

Examples

lambertW(exp(1)) # 1
lambertW(0) # 0
x <- c(0.5, 1, 10, 1000)
lambertW(x) * exp(lambertW(x)) - x # ~ 0

RTMBdist documentation built on Oct. 31, 2026, 5:07 p.m.