The Lambert W Function

Computes the Lambert W function for real values.

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.

Details

The Lambert \(W\) function is the root of the equation \(W(z) \exp(W(z)) = z\) for complex \(z\). If \(z\) is real and \(-1/e < z < 0\) then it has two possible real values, and currently only the upper branch (often called \(W_0\)) is computed so that a value that is \(\geq -1\) is returned.

Value

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

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.

Author

T. W. Yee

Note

If convergence does not occur then increase the value of maxit and/or tolerance.

Yet to do: add an argument lbranch = TRUE to return the lower branch (often called \(W_{-1}\)) for real \(-1/e \leq z < 0\); this would give \(W(z) \leq -1\).

See also

log, exp, bell. There is also a package called LambertW.

Examples

 if (FALSE) { # \dontrun{
curve(lambertW, -exp(-1), 3, xlim = c(-1, 3), ylim = c(-2, 1),
      las = 1, col = "orange", n = 1001)
abline(v = -exp(-1), h = -1, lwd = 2, lty = "dotted", col = "gray")
abline(h = 0, v = 0, lty = "dashed", col = "blue") } # }