> > If I have understood this correctly, and the suggestion is to use > > Buchberger's algorithm to compute Grobner bases in Sage's symbolic > > ring, which includes limited precision floating point numbers > > This is not what William meant (well, I think :). You can compute in the > symbolic ring with exact coefficients: > I believe the issue is that the symbolic ring includes limited- precision numbers, so you have the option of using limited-precision coefficients, and the question then becomes what to do if the user has done this. If limited-precision coefficients are introduced into a Buchberger algorithm just like that, the output is not always going to be something sensible.
A straightforward way to deal with this is to convert floating point coefficients to fractions before running Buchberger's algorithm, though that has issues of its own, e.g. if the coefficients are approximations to sin(x) and cos(x) for some x, then it should be true that the sum of their squares minus one is zero, but this will only be approximately true (i.e. it is false in a precise setting) and therefore Buchberger's algorithm on the (precise) fractions may not compute what one might expect. If I recall correctly, I heard about this example in a presentation by a person from the apCoCoA team, and they were presenting it to illustrate that extending precision adaptively in Buchberger's algorithm has the issue that you have to know where the input coefficients came from in order to regenerate them with the increased precision. It may be sensible to refuse to compute Grobner bases in the presence of limited-precision coefficients, unless some dangerous-sounding option is turned on by the user. Cheers Bjarke H. Roune www.broune.com --~--~---------~--~----~------------~-------~--~----~ To post to this group, send an email to [email protected] To unsubscribe from this group, send an email to [email protected] For more options, visit this group at http://groups.google.com/group/sage-devel URLs: http://www.sagemath.org -~----------~----~----~----~------~----~------~--~---
