VGAM (version 0.8-7)

lambertW: The Lambert W function

Description

Computes the Lambert W function for real values.

Usage

lambertW(x, tolerance = 1e-10, maxit = 50)

Arguments

x

A vector of reals.

tolerance

Accuracy desired.

maxit

Maximum number of iterations of third-order Halley's method.

Value

• This function returns the principal branch of the $W$ function for real $z$. It returns $W(z) \geq -1$, and NA for $z < -1/e$.

Details

The Lambert $W$ function is the root of the equation $W(z) \exp(W(z)) = z$ for complex $z$. It is multi-valued if $z$ is real and $z < -1/e$. For real $-1/e \leq z < 0$ it has two possible real values, and currently only the upper branch is computed.

References

Corless, R. M. and Gonnet, G. H. and Hare, D. E. G. and Jeffrey, D. J. and Knuth, D. E. (1996) On the Lambert $W$ function. Advances in Computational Mathematics, 5(4), 329--359.

See Also

log, exp.

Examples

curve(lambertW, -exp(-1), 3, xlim = c(-1, 3), ylim = c(-2, 1), col = "red")
abline(v = -exp(-1), h = -1, lty = "dotted")
abline(h = 0, v = 0, lty = "dashed", lwd = 2)