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MathGroup Archive 2005

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Re: Re: Re: Re: Solve Limitations

  • To: mathgroup at smc.vnet.net
  • Subject: [mg63167] Re: [mg63123] Re: [mg63107] Re: [mg62980] Re: [mg62963] Solve Limitations
  • From: Pratik Desai <pdesai1 at umbc.edu>
  • Date: Fri, 16 Dec 2005 07:22:24 -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> <43A1882D.4050205@wolfram.com>
  • Sender: owner-wri-mathgroup at wolfram.com

Daniel Lichtblau wrote:

> 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

Thanks Daniel,
As I posted earlier, FindRoot was able to solve the equation I was 
mentioned in my post. This may be due to the fact that I knew what the 
roots for my equations looked like and I could give the correct initial 
starting point. The advantage of zeal, in my opinion, is that one only 
has to give a bounding rectangle from which the program automatically 
calculates the zeros and its multiplicities. Overall, ZEAL was quite 
cumbersome to use, I think one has to be quite conversant with Complex 
Analysis and Numerical methods to be able to use ZEAL correctly (I had 
the manuscript version published in Lecture notes in Mathematics) but 
still I had trouble using the code. But as you have pointed out in your 
post, Mathematica can be an ideal candidate for a robust, user friendly 
implimentation.

Thanks again for your response

Pratik


-- 
Pratik Desai 

...Moderation, as well as Regularity of Thinking, so much to be wished for in the Heads of those who imagine they come into the World only to watch and govern it?s Motion
Gulliver's Travels
by Jonathan Swift




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