lambertWp | R Documentation |
Principal real branch of the Lambert W function.
lambertWp(x)
lambertWn(x)
x |
Numeric vector of real numbers |
The Lambert W function is the inverse of x --> x e^x
, with two
real branches, W0 for x >= -1/e
and W-1 for -1/e <= x < 0
.
Here the principal branch is called lambertWp
, tho other one
lambertWp
, computed for real x
.
The value is calculated using an iteration that stems from applying Halley's method. This iteration is quite fast and accurate.
The functions is not really vectorized, but at least returns a vector of
values when presented with a numeric vector of length >= 2
.
Returns the solution w
of w*exp(w) = x
for real x
with NaN
if x < 1/exp(1)
(resp. x >= 0
for the
second branch).
See the examples how values for the second branch or the complex Lambert W function could be calculated by Newton's method.
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, Vol. 5, pp. 329-359.
halley
## Examples
lambertWp(0) #=> 0
lambertWp(1) #=> 0.5671432904097838... Omega constant
lambertWp(exp(1)) #=> 1
lambertWp(-log(2)/2) #=> -log(2)
# The solution of x * a^x = z is W(log(a)*z)/log(a)
# x * 123^(x-1) = 3
lambertWp(3*123*log(123))/log(123) #=> 1.19183018...
x <- seq(-0.35, 0.0, by=0.05)
w <- lambertWn(x)
w * exp(w) # max. error < 3e-16
# [1] -0.35 -0.30 -0.25 -0.20 -0.15 -0.10 -0.05 NaN
## Not run:
xs <- c(-1/exp(1), seq(-0.35, 6, by=0.05))
ys <- lambertWp(xs)
plot(xs, ys, type="l", col="darkred", lwd=2, ylim=c(-2,2),
main="Lambert W0 Function", xlab="", ylab="")
grid()
points(c(-1/exp(1), 0, 1, exp(1)), c(-1, 0, lambertWp(1), 1))
text(1.8, 0.5, "Omega constant")
## End(Not run)
## Analytic derivative of lambertWp (similar for lambertWn)
D_lambertWp <- function(x) {
xw <- lambertWp(x)
1 / (1+xw) / exp(xw)
}
D_lambertWp(c(-1/exp(1), 0, 1, exp(1)))
# [1] Inf 1.0000000 0.3618963 0.1839397
## Second branch resp. the complex function lambertWm()
F <- function(xy, z0) {
z <- xy[1] + xy[2]*1i
fz <- z * exp(z) - z0
return(c(Re(fz), Im(fz)))
}
newtonsys(F, c(-1, -1), z0 = -0.1) #=> -3.5771520639573
newtonsys(F, c(-1, -1), z0 = -pi/2) #=> -1.5707963267949i = -pi/2 * 1i
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