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

• To: mathgroup at smc.vnet.net
• Subject: [mg117184] Re: best approximation to the LambertW(x) or exp(LambertW(x)) for large x say x > 2500
• From: DrMajorBob <btreat1 at austin.rr.com>
• Date: Thu, 10 Mar 2011 16:03:51 -0500 (EST)

```"You are solving
for x in the equation y == x * E^x."

That would be

Solve[y == x Exp@x, x]

{{x -> ProductLog[y]}}

Bobby

On Thu, 10 Mar 2011 05:12:47 -0600, Scott Hemphill
<hemphill at hemphills.net> wrote:

> 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

--
DrMajorBob at yahoo.com

```

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