Getting that X with the Glog function and Lambert’s W

Facing a simple, yet frustrating formula like this

xe^{ax}=b

and the task to solve it for x left me googling around for hours until I found salvation in Wolfram Alpha, Wikipedia, and a nice blog post with R-syntax to solve a similar equation.

Using the results from Wolfram Alpha I was able to find the solution with the ‘gsl’ library

# install.packages("gsl")
library(gsl)

# create some example data
dat <- data.frame(a = 0.109861, x = 10)
# a is set so that b is roughly 30. 
# Lazy as I am I used Excel and its solver ability to find numbers

# to check if b is close to 30. Using the initial formula
dat$b <- dat$x * exp(dat$a * dat$x)
dat

# solve for x2 and see if x and x2 are similar and close to 10
dat$x2 <- lambert_W0(dat$a * dat$b)/dat$a
dat

#  a        x  b2       x2
#1 0.109861 10 29.99993 10.00001

# Hurray!

Sometimes life can be so easy (after a long time searching for the right results….).

Appendix: Improvements

After revisiting this article some time later, I wondered what the speed is compared to Dan Kelley’s (see comment below) alternative. After firing up some repetitions using microbenchmark I got the following:

library(gsl)
library(rootSolve)
library(microbenchmark)
library(ggplot2)

dat <- data.frame(a = 0.109861, x = 10)
dat$b <- dat$x * exp(dat$a * dat$x)

f <- function(x, a, b) x*exp(a*x) - b

autoplot(microbenchmark(
               lambertW = dat$x2 <- lambert_W0(dat$a * dat$b)/dat$a,
               uniroot = dat$x3 <- uniroot.all(f, interval = c(0, 100), a = dat$a, b = dat$b),
               times = 10000))

speed-comparison

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3 thoughts on “Getting that X with the Glog function and Lambert’s W

  1. […] Getting that X with the Glog function and Lambert’s W Facing a simple, yet frustrating formula like this xe^{ax}=b and the task to solve it for x left me googling around for hours until I found salvation in Wolfram Alpha, Wikipedia, and a nice blog post with R-syntax to solve a similar equation. […]

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