There is an approximation offered by Winitzki (Winitzki, 2003 - http://dl.acm.org/citation.cfm?id=1756839), which provides the complex value of the principal branch with a relative error of less than 0.01 in the entire complex plane:

W(z) ~ {2ln(1 + By) − ln[1 + Cln(1 + Dy)] + E} / {1 + 1 / [2ln(1 + By) + 2A]},

where A ~ 2.344, B ~ 0.8842, C ~ 0.9294, D ~ 0.5106, E ~ -1.213, and y2 = 2ez + 2.

If you need greater accuracy, I guess a simple numerical algorithm can be used, seeded with the above value, to refine the estimate of W(z) to arbitrary precision.

Dear Abdelhay,

I think Victor's response is very clear but you should view the following discussion:

http://mathoverflow.net/questions/57819/best-approximation-to-the-lambertwx-or-explambertwx

called “best approximation to the LambertW(x) or exp(LambertW(x))” which is conducted on the MathOverflow forum. From this valuable text you can get to know for example that:

“The Lambert W(x) function is implemented in various software packages, as ProductLog[x] in Mathematica, for example. And Mathematica can compute numerical values for specific x out to as many digits as you like”

Thank you very much for your cooperation. it is very help for me.

If you look at wiki http://en.wikipedia.org/wiki/Lambert_W_function, there are both taylor and asymptotic expansions that can be very helpful

.

Th 2.8 of the reference below gives explicit O(loglog(x)/log(x)) upper and lower bounds for x >= e

https://rgmia.org/papers/v10n2/lambert-v2.pdf

.

Hello

How can we appoint positive branch of lambert function?

Dear Mohammad,

You may find my note at Arxiv 1504.01964 helpful. It provides a fast way to compute v = W(u) if u > 0 and all you want is the positive branch. That is, the branch which is called W_0 in the Corless, et al, paper and which is called W_p in the NIST Handbook of Mathematical Functions (page 111). Here is some detail:

Suppose v = W(u) is to be computed, for u > 0 and v > 0. Change to log-log coordinates, by setting v = exp(y) and u = exp(x). That is, y = ln(v) and x=ln(u). Use the method described in the Arxiv note to compute the function called g in that note, which satisfies y = g(x) = ln(W(exp(x))). The name of that function is the logWright function, since W(exp(x)) is called the Wright function.

The computation of y = g(x) should be quite accurate for only a couple of iterations. See details in the note.

Now, given y, compute v = exp(y).

Hence you do this...

Given u > 0 That is, you must verify the input argument is valid.

Compute v = W(u) = exp(g(ln(u))) using your implementation of g function.

Hope that helps. Best wishes, Ken Roberts.