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Re: numeric Groebner bases et al [Was Re: Numerical accuracy/precision - this is a bug or a feature?]

• To: mathgroup at smc.vnet.net
• Subject: [mg120302] Re: numeric Groebner bases et al [Was Re: Numerical accuracy/precision - this is a bug or a feature?]
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Mon, 18 Jul 2011 06:12:49 -0400 (EDT)
• References: <201107171003.GAA17256@smc.vnet.net>

On 17 Jul 2011, at 12:03, Daniel Lichtblau wrote:

> Brief history; Around 1994 Kiyoshi Shirayanaga wrote about an 
implementation of such inside Maple. As I'd been working on a new GB 
implementation for Mathematica 3, and as his article discussed various 
tactics that amounted to an emulation of significance tracking, I 
thought I would try to put this into the new release.

This reminds me of one just one example of how Richard tries to twist 
things that others write in a way that is convenient to him (or perhaps 
he really does not understand things that I assume must to be obvious to 
him...).

I wrote:

> As for things like numerical Groebner basis etc, you keep arguing that 
it probably can be done in some other way (without significance 
arithmetic) and in a certain, rather trivial, sense you undoubtedly 
right. But the fact is, as Daniel has pointed out, that, in practice, 
nobody seems to have done it in another way, and not for want of trying. 
So again here we have the distinction between "theory" and "practical 
application" but except that this time the shoe is on the other foot.

Note the "rather trivial sense". Richard then "interpreted" it in his 
inimitable way:

> Even you seem to agree
> that Mathematica's particular kind of
> bigfloat arithmetic is not essential to this. So I don't see your 
point.

My point was obvious: the "rather trivial" way (in the case of software) 
of avoiding the use of something is by essentially "emulating it". The 
mathematical analogue of this is this: theorem B is usually deduced from 
theorem A. However, it is almost always easy to produce a a proof of B 
"without using A" by including in the proof of B all the essential 
things in the proof of A that are needed to deduce B. It is rare that 
one will get away with a claim of having obtained an "independent" proof 
by this method. A really independent proof is something else altogether.

Clearly one can emulate "significance tracking" in fixed precision 
arithmetic and thus implement Daniel's version of Groebner basis in, for 
example, Maple. It will almost certainly require clumsier and longer 
code. The answer to the question: is there a really independent way of 
achieving the same final effect? is not known to me and I certainly did 
not imply that it was. And I think Richard should have known "what I 
mean" perfectly well, but it was not a convenient thing to know for him 
a that moment.

Andrzej Kozlowski

• References:
  • numeric Groebner bases et al [Was Re: Numerical accuracy/precision - this is a bug or a feature?]
    • From: Daniel Lichtblau <danl@wolfram.com>