       Re: best approximation to the LambertW(x) or exp(LambertW(x)) for large x say x > 2500

• To: mathgroup at smc.vnet.net
• Subject: [mg117138] Re: best approximation to the LambertW(x) or exp(LambertW(x)) for large x say x > 2500
• From: Scott Hemphill <hemphill at hemphills.net>
• Date: Thu, 10 Mar 2011 06:12:47 -0500 (EST)
• References: <il7qab\$du5\$1@smc.vnet.net>

```barefoot gigantor <barefoot1980 at 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).

Scott
--
Scott Hemphill	hemphill at alumni.caltech.edu
"This isn't flying.  This is falling, with style."  -- Buzz Lightyear

```

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