pracma (version 1.4.5)

lambertWp: Lambert's W Function

Description

Principal branch of the Lambert W function

Usage

lambertWp(z)

Arguments

z

Numeric vector of real numbers >= -1/e.

Value

• Returns the solution w of w*exp(w) = z for real z with NA if z < 1/exp(1).

Details

The Lambert W function is the inverse of x --> x e^x, which is unique for x >= -1/e. Here only the principal branch is computed for real z.

The value is calculated using an iteration that stems from applying Newton's method. This iteration is quite fast.

The function is not really vectorized, but at least returns a vector of values when presented with a numeric vector of length >= 2.

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, Vol. 5, pp. 329-359. http://www.apmaths.uwo.ca/~djeffrey/Offprints/W-adv-cm.pdf.

See Also

newtonRaphson

Examples

##  Examples
lambertWp(0)          #=> 0
lambertWp(1)          #=> 0.5671432904...  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...

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")

# 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