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. > > 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 at http://groups.google.com/group/sage-devel URL: http://www.sagemath.org
