Lambert W function implementation in Java

Question

I'm working on a project and have found myself in a situation where I need a function to be able to get at least an approximation of the value of W(x), the Lambert W function, where x can be any real number. I'm working in Java. I couldn't find any implementations of W in java when I searched. I am willing to code up the implementation myself if need be, but I am unsure of how that would be done right now. Any pushes in the right direction would be much appreciated.

• Graham: Also, if it makes a difference, I think that I only need W on the domain of [0,inf).

Answer 1

Take a look at this page: http://mathworld.wolfram.com/LambertW-Function.html

It lists an approximation for z>3 as well as a series expansion for the function.

You can also use Newton's method and Halley's method to approximate the function: http://en.wikipedia.org/wiki/Lambert_W_function#Numerical_evaluation

• Graham: Looking at the approximation for z>3 on the first link you posted, I understand most of the formula except for the end bit. What does the O[(L1/L2)^6] part mean? Also when it says something like L2(-2 + L2) would that be lnlnz * (-2 + lnlnz) or lnln(-2 + lnlnz)?
• tskuzzy: The +O(stuff) at the end is just an error term. You can ignore it. And L2(-2 + L2) = lnlnz * (-2 + lnlnz)
• Graham: Thanks! That clears some things up. So I could probably just use that formula for input to the function greater than 3, and find a simpler approximation formula for input between 0 and 3?
• tskuzzy: That sounds about right. You should compare all these different methods though to see which one gives you the best approximation. I don't know the answer to that question.

Answer 2

The lambert function is the reciprocal function of g(w) = w*exp(w) it verifies:

W(z)eW(z) = z.

A good way to evaluate W(z) on a given z would be to use newton raphson method :

to solve : f(Y)= Yexp(Y) - z = 0.

you will find Y = W(z) with the method

You would have to find and implementation in java of the method yourself.

Hope it helps

below an illustration from wikipedia of the method:

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