On Apr 30, 6:58 am, rjf <fate...@gmail.com> wrote: > On Apr 29, 10:58 am, Robert Bradshaw <rober...@math.washington.edu> > wrote: > > > On Apr 29, 2010, at 8:30 AM, rjf wrote: > > > > (RJF)Again, I see no definition of what you mean by accuracy in the result > > > of polynomial multiplication.The easiest position to take is that of > > > MPFR-- > > considering the inputs as exact rationals, how far off is the output > > (say, coefficient by coefficient) from the actual result. One could > > also use other measures, such as an L^2 norm over some compact region > > (like [-1,1]). What may make sense for many applications is some kind > > of a "slopped absolute" precision, as has been discussed with p-adic > > polynomials. These would have good behavior on |x| < a or |x| > a > > depending on the slope. > > There is a large literature on the meaning of the "approximate GCD" of > two polynomials with floating-point coefficients. The term > "stability" > also has a pretty good definition, and it does not seem to correspond > to how you are using it here.
Approximate GCD? That's a curious concept. What is it used for? I can't imagine defining a GCD in this context as divisibility is an exact phenomenon. I hear the term numerical stability used quite a lot. The two contexts I've encountered it are in linear algebra and in differential equations. Actually, I might have heard it used in polynomial evaluation too. I happily admit that I've no idea of a definition in the case of polynomial multiplication. However, regardless of what the correct terminology is, the problem I am studying is precisely the one stated, which I took to be the subject of the thread. I'm not interested in numerical stability in any of the other senses that I know of, at this point. I don't even claim that what I am studying is a useful notion, But feel free to suggest some references if there is something I ought to read. Oh wait, every single published article out there is nonsense. Damn, I forgot. You don't recommend any articles. Bill. > > > > > It all really depends on your application, and currently we don't > > define, let alone make guarantees, on the precision of the result (and > > nor does Maxima as far as I understand). > > Maxima was designed around exact arithmetic, and generally offers to > convert floats to their corresponding exact rationals before doing > anything > that requires arithmetic. It makes no claims about floats per se, > partly > because saying anything sensible is a challenge, compounded by > particular issues that are not mathematical but practical .. overflow, > different machine floating point formats, etc > > > We do, however, try to avoid > > algorithms that are known to be numerically unstable. > > The lack of serious understanding of numerical computation in Sage is > unfortunate. It might be cured by someone taking an interest in the > topic > and taking a course or two. > > > > > Fortunately for us, many asymptotically fast algorithms do have > > cutoffs well within the size ranges we regularly encounter, and low > > enough that it makes a significant difference in the runtime. > > There are a few, but not many compared to the huge number that you > will encounter > if you read certain theoretical journals. > > If you care about specifics of the failure of theoretical computer > scientists > to address computation, write to me privately. > > RJF > > -- > To post to this group, send an email to sage-devel@googlegroups.com > To unsubscribe from this group, send an email to > sage-devel+unsubscr...@googlegroups.com > For more options, visit this group athttp://groups.google.com/group/sage-devel > URL:http://www.sagemath.org -- To post to this group, send an email to sage-devel@googlegroups.com To unsubscribe from this group, send an email to sage-devel+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-devel URL: http://www.sagemath.org
