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Re: Re: Solving the system with inexact coefficients

  • To: mathgroup at smc.vnet.net
  • Subject: [mg99504] Re: [mg99484] Re: [mg99335] Solving the system with inexact coefficients
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Thu, 7 May 2009 06:31:38 -0400 (EDT)
  • References: <200905060928.FAA02082@smc.vnet.net>

Semenov's method, which is implemented in my demonstrations

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

and

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

can find (with provable certainty) all the solutions of a system of n- 
equations in n-variables that line in an n-dimensional arbitrary  
rectangular box. It is only required that the functions defining these  
equations have continuous second derivatives - they need not be  
analytic. My implementation only deals with the case n=2 or the  
complex one dimensional case, but this restriction is only due to the  
need of graphical interpretation in a demonstration. The method is  
thus considerably more general that Mathematica's Root function. Like  
the Root function it can't definitely determine the presence of  
multiple roots. It is of course much slower than the built in Root  
function, but that's normal for user defined functions.

Andrzej Kozlowski



On 6 May 2009, at 18:28, Ted Ersek wrote:

> A few people replied to a problem on how to find trancedental roots  
> of a
> function in one dimension, and recommended my RootSearch package.
>
> However, based on
> http://blog.wolfram.com/2008/12/18/mathematica-7-johannes-kepler-and-transcendental-roots/#more-880
>
> The built-in Root function is the prefeered approach in Mathematica 7
> (provided Root can handle the problem).
> What trancedental roots can the Root function handle?
> Well the above blog seems to indicate Root can find roots of a
> holomorphic function of a single variable.
> The blog says a holomorphic function is essentially polynomials of
> infinite degree.
>
> I say the built-in Root function is prefeered over RootSearch because
> the Root objects are tightly integrates into Mathematica. Also I  
> believe
> the algorithms used by the Root function are gaurenteed to work
> (neglecting the effect of bugs). On the other hand my RootSearch  
> package
> relies on taking a finite number of samples. RootSeach has been  
> shown to
> be relaible, but it can be tricked by a pathological problem.
>
> Notice the built-in Root function has been able to find the roots of a
> polynomial of any finite degree for a long time (those roots are
> algebraic numbers). More about the built-in Root function can be found
> at
> http://reference.wolfram.com/mathematica/ref/Root.html
>
> My RootSeach function is still the method of choice for finding  
> roots in
> one dimension when Root in Mathematica 7 can't be used.
>
> Regards,
>
>       Ted Ersek
>
> --------------------------------------------------------------
>
>



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