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Re: Re: InverseFunction[]

  • To: mathgroup at smc.vnet.net
  • Subject: [mg42187] Re: [mg42160] Re: InverseFunction[]
  • From: Murray Eisenberg <murray at math.umass.edu>
  • Date: Sat, 21 Jun 2003 20:57:05 -0400 (EDT)
  • Organization: Mathematics & Statistics, Univ. of Mass./Amherst
  • References: <bbt23g$nj3$1@smc.vnet.net> <bc1jh1$bjp$1@smc.vnet.net> <200306110749.DAA02507@smc.vnet.net> <bc7pet$6f0$1@smc.vnet.net> <paul-000944.15584413062003@news.uwa.edu.au> <bcjv91$hv3$1@smc.vnet.net> <200306210649.CAA13213@smc.vnet.net>
  • Reply-to: murray at math.umass.edu
  • Sender: owner-wri-mathgroup at wolfram.com

Yes, you're correct: I did a copy-and-paste from the wrong line in the 
Help Browser window.  Indeed:

   ProductLog uses high­order Newton's method starting from rational 
approximations and asymptotic expansions.

Paul Abbott wrote:
> In article <bcjv91$hv3$1 at smc.vnet.net>,
>  Murray Eisenberg <murray at math.umass.edu> wrote:
> 
> 
>>Well, to take the example of the function you pose for comparison, Sin: 
>>  The section "Some Notes on Internal Implementation" in the Mathematica 
>>Book's "Mathematica Reference Guide", says:
>>
>>    Exponential and trigonometric functions use Taylor series,
>>    stable recursion by argument doubling, and functional relations.
>>
>>But I'm guilty of not scrolling down far enough on that same page to 
>>have discovered:
>>
>>   PolyLog uses Euler-Maclaurin summation, expansions in terms
>>   of incomplete gamma functions and numerical quadrature.
> 
> 
> PolyLog not ProductLog?
> 
> ProductLog uses high-order Newton's method starting from rational 
> approximations and asymptotic expansions. 
> 
> 
>>That's the sort of thing I was looking for and somehow had previously 
>>overlooked!
> 
> 
> So you were only interested in how Mathematica computes a function 
> numerically? This was not clear from your original question:
> 
> 
>>But what is the definition of the function as Mathematica knows it?
> 
> 
> To me the definition of a function is not how it is computed numerically.
> 
> Cheers,
> Paul
> 

-- 
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Murray Eisenberg                     murray at math.umass.edu
Mathematics & Statistics Dept.
Lederle Graduate Research Tower      phone 413 549-1020 (H)
University of Massachusetts                413 545-2859 (W)
710 North Pleasant Street            fax   413 545-1801
Amherst, MA 01003-9305


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