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Re: FindRoots?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg112299] Re: FindRoots?
  • From: Andrzej Kozlowski <akozlowski at gmail.com>
  • Date: Wed, 8 Sep 2010 00:59:15 -0400 (EDT)

On 7 Sep 2010, at 18:40, Andrzej Kozlowski wrote:

>
> On 7 Sep 2010, at 10:02, Ingolf Dahl wrote:
>
>> But anyway, I consider it as snobbery to say that Reduce is superior to
>> RootSearch. They do not play in the same division, and perform different
>> tasks.
>
> Why should it matter to me what you call it? Is that an argument?
>
> If you read what I have written on this subject you may noticed that there are ways to do essentially the same things that RootSearch attempts to do and obtain the complete set of roots (provably complete) (I am not referring to using Reduce).  That, I do call "superior" and really I don't care if anyone thinks it's "snobbery" or not.
>
> Andrzej Kozlowski


In case my meaning is still unclear, I will once again repeat something that I have written already on this forum several times on various occasions.

If Wolfram really wanted a method that solves numerical transcendental equations that could implement one of several existing sophisticated modern algorithms, including the one due to Semenev (2007) that is the subject of my demonstrations:

http://demonstrations.wolfram.com/SolvingSystemsOfTranscendentalEquations/

and

http://demonstrations.wolfram.com/SemenovsAlgorithmForSolvingSystemsOfNonlinearEquations/

Unlike the SearchRoot package this algorithm finds the complete set of solutions of any system of n equations in n unknowns satisfying certain mild conditions. I have no problem at all calling it "superior" to what is implemented in the SearchRoot package. Moreover, it can solve systems of equations (yes, systems) which Mathematica is at present is unable even to attempt (one such example is given in the first demonstration above) and neither of course can SearchRoot. The set of solutions is provably complete and the user does not need to set any options.

My implementation of Semenev's algorithm is probably inefficient as it was only meant to demonstrate the method and not to be used in practice. Semenev's paper gives accurate complexity estimates for the method which show that it is quite practical if the number of equations (and unknowns) is not too large.

It would be trivial for Wolfram to implement this algorithm and it would not require leasing anything from anyone, with all the problems that this usual entails.

Andrzej Kozlowski


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