Can someone show me how to solve Lambert functions, such as the one here?

Question

I would like to understand what process (steps) are required to arrive at the answer of 43.559... as shown in the following equation. I have looked at Wikipedia and I have also looked at the MathWorld website, however I don't see any examples on how to move from the beginning of the problem to end of the problem in such as way that allows me to actually use the formula. What are the steps I need to take to solve the following problem (as is solved below)?

n=−8ln2W−1(−8ln2)≈43.559260

P/S If any steps involve using a calculator can you even go as far as telling me exactly what you typed in at each step? I have not done math in a while so baby-steps would help me the most. Thank you.

Answer 1

If I properly remember your first post, you wanted to solve $$x-8\log_2(x)=0$$ for which the solutions are $$x_1=-\frac{8 W_{-1}\left(-\frac{\log (2)}{8}\right)}{\log (2)}\approx 43.55926044$$ $$x_2=-\frac{8 W\left(-\frac{\log (2)}{8}\right)}{\log (2)} \approx 1.099997030$$ As gammatester answered, Lambert function is one of the many special functions (and this one is very fascinating to me). If you have access to it, you are done.

But, let us suppose that this is not the case. In such a case, say that your problem is just to solve for zero $$f(x)=x-\log_2(x)$$ A simple approach is to use Newton method which, starting from a "reasonable" guess $x_0$ will update it according to $$x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$$ which, in your case, write $$x_{n+1}=\frac{8 x_n (\log (x_n)-1)}{x_n \log (2)-8}$$ Let us try using $x_0=20$; the following iterates are the obtained : $54.4636$, $43.8990$, $43.5597$, $43.5593$ which is the solution for six significant figures.

Since you looked at the Wikipedia page, you probably noticed that

• there are faster methods, such as Halley's (but the philosophy is the same)
• good estimates can be generated (I did not use them here)

If there is any item you would like me to eleborate, just post.

Answer 2

The Lambert W function is just a special function like the Gamma or Error function. If you want to compute it with a programming language, you may have a look D. Veberic, Having Fun with Lambert W(x) Function, which contains basic information and C++ code for doubles. You can find more links at Darko's page.