W() R function from [LambertW]

Lambert W function, its logarithm and derivative

The Lambert W function $W(z) = u$ is defined as the inverse of (see xexp)

[REMOVE_ME] $u \exp(u) = z, [REMOVE_ME_2]$

i.e., it satisfies $W(z) \exp(W(z)) = z$ .

W evaluates the Lambert W function (W), its first derivative (deriv_W), and its logarithm (log_W). All of them have a principal (branch = 0 (default)) and non-principal branch (branch = -1) solution.

W is a wrapper for lambertW0 and lambertWm1 in the lamW package.


W(z, branch = 0)

deriv_W(z, branch = 0, W.z = W(z, branch = branch))

log_deriv_W(z, branch = 0, W.z = W(z, branch = branch))

deriv_log_W(z, branch = 0, W.z = W(z, branch = branch))

log_W(z, branch = 0, W.z = W(z, branch = branch))

Arguments

z: a numeric vector of real values; note that W(Inf, branch = 0) = Inf.
branch: either 0 or -1 for the principal or non-principal branch solution.
W.z: Lambert W function evaluated at z; see Details below for why this is useful.

Returns

numeric; same dimensions/size as z.

W returns numeric, Inf (for z = Inf), or NA if $z < -1/e$ .

Note that W handles NaN differently to lambertW0 / lambertWm1 in the lamW package; it returns NA.

Description

The Lambert W function $W(z) = u$ is defined as the inverse of (see xexp)

u \exp(u) = z,

i.e., it satisfies $W(z) \exp(W(z)) = z$ .

W is a wrapper for lambertW0 and lambertWm1 in the lamW package.

Details

Depending on the argument $z$ of $W(z)$ one can distinguish 3 cases:

$z \geq 0$ : solution is unique W(z) = W(z, branch = 0)
$-1/e \leq z < 0$ : two solutions: the principal (W(z, branch = 0)) and non-principal (W(z, branch = -1)) branch;
$z < -1/e$ : no solution exists in the reals.

log_W computes the natural logarithm of $W(z)$ . This can be done efficiently since $\log W(z) = \log z - W(z)$ . Similarly, the derivative can be expressed as a function of $W(z)$ :

W'(z) = \frac{1}{(1 + W(z)) \exp(W(z))} = \frac{W(z)}{z(1 + W(z))}.

Note that $W'(0) = 1$ and $W'(-1/e) = \infty$ .

Moreover, by taking logs on both sides we can even simplify further to

\log W'(z) = \log W(z) - \log z - \log (1 + W(z))

which, since $\log W(z) = \log z - W(z)$ , simplifies to

\log W'(z) = - W(z) - \log (1 + W(z)).

For this reason it is numerically faster to pass the value of $W(z)$ as an argument to deriv_W since W(z) often has already been evaluated in a previous step.

Examples


W(-0.25) # "reasonable" input event 
W(-0.25, branch = -1) # "extreme" input event

curve(W(x, branch = -1), -1, 2, type = "l", col = 2, lwd = 2)
curve(W(x), -1, 2, type = "l", add = TRUE, lty = 2) 
abline(v = - 1 / exp(1))

# For lower values, the principal branch gives the 'wrong' solution; 
# the non-principal must be used.
xexp(-10) 
W(xexp(-10), branch = 0) 
W(xexp(-10), branch = -1)

curve(log(x), 0.1, 5, lty = 2, col = 1, ylab = "")
curve(W(x), 0, 5, add = TRUE, col = "red")
curve(log_W(x), 0.1, 5, add = TRUE, col = "blue")
grid()
legend("bottomright", c("log(x)", "W(x)", "log(W(x))"),
       col = c("black", "red", "blue"), lty = c(2, 1, 1))

References

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

W function