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>