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Re:numerical-symbolic Groebner basis

• To: mathgroup at smc.vnet.net
• Subject: [mg60177] Re:numerical-symbolic Groebner basis
• From: Paul Abbott <paul at physics.uwa.edu.au>
• Date: Tue, 6 Sep 2005 01:26:48 -0400 (EDT)
• Organization: The University of Western Australia
• References: <200508251034.GAA10208@smc.vnet.net> <demmfd$rf1$1@smc.vnet.net> <200509010613.CAA08855@smc.vnet.net> <200509030606.CAA19076@smc.vnet.net> <dfgohu$n17$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

In article <dfgohu$n17$1 at smc.vnet.net>,
 Andrzej Kozlowski <akoz at mimuw.edu.pl> wrote:

> Indeed, Daniel Lichtblau succeeded many years ago in implementing a  
> numerical-symbolic Groebner basis, which is used by NSolve and  
> perhaps some other polynomial algebra functions. It works pretty well  
> as I have had the chance to check myself while writing a review of a  
> new book on "Numerical Polynomial Algebra" by Hans Stetter. It means  
> that NSolve can deal with problems that other systems cannot without  
> expert  human intervention. And the key to Daniel's implementation is  
> the Mathematica model of big num arithmetic, based on what is known  
> as "significance arithmetic". The key role is played by the fact that  
> thanks to significance aritmetic Mathematica can perform automatic  
> tracking of Precision/Accuracy. RJF has in the past written  
> dismissively of this as of something that can be of use only to  
> "sloppy people". In a sense he is of course right. Someone skilled in  
> polynomial algebra and numerical analysis (which I suspect is still a  
> set containing only a few individuals) does not need a numerical  
> Groebner basis, as Hans Stetter impressively demonstrates in his  
> book. But not all of us have the time, the sill an the patience to  
> overcome our sloppiness at dealing with rather esoteric numerical  
> issues.

Related to this topic, the paper

  Erich Kaltofen, "Challenges of Symbolic Computation: My Favorite Open 
  Problems", J. Symbolic Computation (2000) 29, 891-919

particularly Open Problem 5, is relevant.

Cheers,
Paul

_______________________________________________________________________
Paul Abbott                                      Phone:  61 8 6488 2734
School of Physics, M013                            Fax: +61 8 6488 1014
The University of Western Australia         (CRICOS Provider No 00126G)    
AUSTRALIA                               http://physics.uwa.edu.au/~paul

• References:
  • Re: inconsistency with Inequality testing and Floor
    • From: "Richard J. Fateman" <fateman@eecs.berkeley.edu>
  • Re: Re: inconsistency with Inequality testing and Floor
    • From: Andrzej Kozlowski <andrzej@akikoz.net>