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MathGroup Archive 2005

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Re: Re: Re: Re: Solve Limitations

  • To: mathgroup at smc.vnet.net
  • Subject: [mg63162] Re: [mg63123] Re: [mg63107] Re: [mg62980] Re: [mg62963] Solve Limitations
  • From: Daniel Lichtblau <danl at wolfram.com>
  • Date: Fri, 16 Dec 2005 07:22:17 -0500 (EST)
  • References: <IRGQVT$2C607F9DAA7468FE284C86E7560B5F2C@bol.com.br> <A67108E9-A365-40E5-856F-610C5E0BAEF1@mimuw.edu.pl> <200512140936.EAA02453@smc.vnet.net> <200512150806.DAA19469@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Pratik Desai wrote:
> Andrzej Kozlowski wrote:
> [...]
>>
>>As it is, there are two functions, Solve and Reduce, which are  
>>optimised for different purposes and which use different (though  
>>intersecting) sets of algorithms. It is up to the user to judiciously  
>>choose the function that is the best suited to the problem at hand.
>>
>>Andrzej Kozlowski
>>
>>
> 
> To state the obvious, in general roots of analytic functions are hard to 
> find. I had the misfoutune to come across a nasty complex trancendental 
> equation. I found this Fortran Code ZEAL (Zeros of Analytic Functions) 
> quite invaluable. Needless to say, Solve, Reduce did not help much.
> http://cpc.cs.qub.ac.uk/summaries/ADKW_v1_0.html
> 
> A Mathematica implimentation of this software would come a long way in 
> helping us poor engineers deal with such trancendental equations. The 
> system that I was dealing with has obvious practical significance, the 
> only hinderance being the lack of tools such as root solvers such as 
> ZEAL. Any takers??
> 
> Pratik
> 
> 
> PS: Zeal not only can find the zeros of f(z) but also gives one the 
> values for f(z) with high degre of precision

Solve and Reduce use symbolic methods appropriate for polynomial 
systems. Some extend to generalizations of polynomials but these methods 
will not apply to arbitrary analytic functions.

For the case of one analytic function in one variable, a simple variant 
of the method used in ZEAL might be found at the URLs below. Also in 
those threads are other approaches to handling analytic functions, 
including multiple start Newton's method root finding, series expansion 
for polynomial approximation, and homotopy continuation.

http://forums.wolfram.com/mathgroup/archive/2001/Jun/msg00444.html

http://groups.google.com/group/sci.math.symbolic/msg/c88809cbbc7fcb32?hl=en&;

With regard to the methods in ZEAL and at the URLs I should note that 
there are various issues to consider with regard to detecting and 
handling multiplicity, numeric control of the quadrature, choice of a 
bounding curve over which to integrate, and more. I do not pretend to 
have a "production quality" code that will do all this in sensible ways. 
   It is an interesting question as to what work would go into such a 
project.


Daniel Lichtblau
Wolfram Research


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