Well I got the polynomial convolution working with the Fast Hartley Transform. It seems to pass my primitive test code.
As an example of a first timing, 1000 iterations of multiplying polynomials of length 512 with 100 floating point bits precision per coefficient takes 17s. I think I can make it take about half that time (at least), but that is a job for tomorrow. I haven't looked yet to see how much precision is lost heuristically. Bill. On Apr 27, 11:49 pm, Bill Hart <goodwillh...@googlemail.com> wrote: > Well, I coded up a mini mpfr_poly module and Fast Hartley Transform in > flint2 using mpfr's as coefficients. > > It's hard to know if it is doing the right thing, there's no test code > yet. But it compiles and doesn't segfault. :-) > > A little bit more work required to turn it into a convolution, but as > far as I know, no inverse transform required, as it is its own > inverse. Maybe by tomorrow I'll have a fast convolution and we'll be > able to answer some of these questions heuristically. > > It won't be super efficient, no truncation, no radix 4, no special > case for double precision coefficients, no cache friendly matrix > fourier stuff, etc, but I did reuse the twiddle factors as much as > possible, I think. > > Bill. > > On Apr 27, 5:18 pm, Bill Hart <goodwillh...@googlemail.com> wrote: > > > > > > > Also see pages 252 and following of Polynomial and Matrix > > Computations, Volume 1 by Dario Bini and Victor Pan, which seems to > > answer your question in detail. > > > Bill. > > > On Apr 27, 4:57 pm, Bill Hart <goodwillh...@googlemail.com> wrote: > > > > Numerical stability is not something I have any experience with. I am > > > not sure if is is equivalent in some sense to the loss of precision > > > which occurs when doing FFT arithmetic using a floating point FFT. > > > > The issue seems to be accumulation of "numerical noise". There are > > > proven bounds on how many bits are lost to this as the computation > > > proceeds, but the general consensus has been that for "random > > > coefficients" the numerical noise is much, much less than the proven > > > bounds would indicate. > > > > Zimmermann, Gaudry, Kruppa suggest that the following paper contains > > > information about proven bounds for the floating point FFT: > > > > Percival, C. Rapid multiplication modulo the sum > > > and difference of highly composite numbers. Math. > > > Comp. 72, 241 (2003), 387–395 > > > > Paul Zimmermann would almost certainly know what the current State of > > > the Art is. > > > > Similar stability issues arise in matrix arithmetic with floating > > > point coefficient. It seems that one can always prove something when > > > you know something about the successive minima. But I am not too sure > > > what the equivalent for an FFT would be. You might be into the area of > > > research rather than something that is known, i.e. it might just be > > > conventional wisdom that the FFT behaves well. > > > > The people to ask about this are definitely Joris van der Hoeven and > > > Andreas Enge. If you find a very nice algorithm, let me know and we'll > > > implement it in flint2, which should have a type for polynomials with > > > floating point coefficients (using MPFR). > > > > Bill. > > > > On Apr 27, 2:45 pm, Jason Grout <jason-s...@creativetrax.com> wrote: > > > > > On 04/26/2010 10:54 PM, Robert Bradshaw wrote: > > > > > > I should comment on this, as I wrote the code and comments in > > > > > question. > > > > > There actually is a fair amount of research out there on stable > > > > > multiplication of polynomials over the real numbers, but (if I > > > > > remember > > > > > right, it was a while ago) there were some results to the effect that > > > > > no > > > > > asymptotically fast algorithm has good stability properties over all > > > > > possible coefficients. > > > > > If you have a moment and if you remember, could you point out a > > > > reference or two? I searched for several hours before posting, and > > > > couldn't find anything that seemed to address the question of stability > > > > satisfactorily. > > > > > > Of course, in practice one often runs into "well > > > > > behavied" polynomials in R[x]. (In particular, most of the interest > > > > > is, > > > > > understandably, about truncated power series, though there are other > > > > > uses.) The obvious thing to do is a (non-discrete) FFT. What I was > > > > > thinking about implementing (though I never got back to it) is > > > > > something > > > > > that works like this: > > > > > > 1) Choose some a and rescale f(x) -> f(alpha*x) so all coefficients > > > > > have > > > > > roughly the same size. > > > > > 2) Multiply the resulting polynomial over Z using the insanely fast > > > > > code > > > > > in FLINT. The resulting answer will be exact product of the truncated > > > > > input, and the precision loss (conversely, required precision for no > > > > > loss) was fairly easy to work out if I remember right . > > > > > 3) Cast the result back into R[x]. > > > > > Thanks! > > > > > Jason > > > > > -- > > > > 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 > > > 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 > > 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 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
