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Re: best approximation to the LambertW(x) or exp(LambertW(x)) for large x say x > 2500
Posted:
Mar 10, 2011 4:04 PM
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Scott Hemphill <hemphill@hemphills.net> writes:
> barefoot gigantor <barefoot1980@gmail.com> writes:
>
>> what is the best available approximation ( say up to 10 digits ) for
>> LambertW(x) or exp(LambertW(x)) for x > 2000
>>
>> thank you for your help
>
> It isn't immediately obvious to me what "best" means. You are solving
> for x in the equation y == x * E^x. A simple solution would be to start
> with x = Log[y], then iterate x = Log[y] - Log[x] enough times that you
> have as many correct digits as you need. The convergence is a little
> slow, though, so you could use Newton's Method instead. You could still
> start with x = Log[y], but then iterate x = (Log[y]-Log[x]+1) * x/(x+1).
To follow up, I just want to point out that for y > 2000, you only need
three iterations to get the desired accuracy. It is asymptotic, so if x
is large enough you need fewer iterations. The worst case is for y ==
2000, where three iterations gives you a relative error of approximately
2*10^-13. To sum up:
x = Log[y];
x = (Log[y]-Log[x]+1) * x/(x+1);
x = (Log[y]-Log[x]+1) * x/(x+1);
x = (Log[y]-Log[x]+1) * x/(x+1);
gives you the value of LambertW[y] to the required accuracy for all y >
2000. (Actually, it does better than that. It gives a relative error
of better than 10^-10 down to about y==32.)
Scott
--
Scott Hemphill hemphill@alumni.caltech.edu
"This isn't flying. This is falling, with style." -- Buzz Lightyear
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