Should this happen? : sage: f._mul_fateman(f) ERROR: An unexpected error occurred while tokenizing input The following traceback may be corrupted or invalid The error message is: ('EOF in multi-line statement', (22841, 0))
ERROR: An unexpected error occurred while tokenizing input The following traceback may be corrupted or invalid The error message is: ('EOF in multi-line statement', (7, 0)) --------------------------------------------------------------------------- OverflowError Traceback (most recent call last) /home/wbhart/<ipython console> in <module>() /usr/local/sage/sage-4.4/local/lib/python2.6/site-packages/sage/rings/ polynomial/polynomial_element.so in sage.rings.polynomial.polynomial_element.Polynomial._mul_fateman (sage/ rings/polynomial/polynomial_element.c:15643)() /usr/local/sage/sage-4.4/local/lib/python2.6/site-packages/sage/rings/ polynomial/polynomial_fateman.pyc in _mul_fateman_mul(f, g) 81 z_poly_f_list = z_poly_f.coeffs() 82 z_poly_g_list = z_poly_g.coeffs() ---> 83 padding = _mul_fateman_to_int2(z_poly_f_list,z_poly_g_list) 84 85 n_f = z_poly_f(1<<padding) /usr/local/sage/sage-4.4/local/lib/python2.6/site-packages/sage/rings/ polynomial/polynomial_fateman.pyc in _mul_fateman_to_int2(f_list, g_list) 24 max_coeff_g = max([abs(i) for i in g_list]) 25 b = (1+min(len(f_list),len(g_list)))*max_coeff_f*max_coeff_g ---> 26 return int(pyceil(pylog(b,2))) 27 28 def _mul_fateman_to_poly(number,padding): OverflowError: cannot convert float infinity to integer Bill. On Apr 29, 7:47 pm, Bill Hart <goodwillh...@googlemail.com> wrote: > Being a little more precise about it. Here are the Sage timings > performed on the same machine as I did the FLINT timings: > > sage: f=(x+1)^1000 > sage: timeit("f*f") > 5 loops, best of 3: 87.5 ms per loop > sage: timeit("f._mul_karatsuba(f)") > 5 loops, best of 3: 370 ms per loop > > flint2 (with FHT): 5.5 ms > > sage: R.<x> = RR[] > sage: f=(x+1)^10000 > sage: timeit("f*f") > 5 loops, best of 3: 8.74 s per loop > sage: timeit("f._mul_karatsuba(f)") > 5 loops, best of 3: 17.3 s per loop > > flint2 (with FHT): 0.255s > > Bill. > > On Apr 29, 5:45 pm, Bill Hart <goodwillh...@googlemail.com> wrote: > > > > > I made the speedup I mentioned to the Fast Hartley Transform code in > > flint2. > > > Again I pick on the benchmarks Robert gave: > > > > sage: f = (x+1)^1000 > > > sage: timeit("f*f") > > > 5 loops, best of 3: 142 ms per loop > > > sage: timeit("f._mul_karatsuba(f)") > > > 5 loops, best of 3: 655 ms per loop > > > That time is down to 5.5 ms with the FHT code in FLINT. > > > (If we change that to f = (x+1)^10000 then the time to compute f*f is > > now about 255 ms.) > > > Bill. > > > On Apr 28, 8:01 pm, Bill Hart <goodwillh...@googlemail.com> wrote: > > > > I coded up a basic classical multiplication (no attempt whatsoever to > > > prevent cancellation). Fortunately the result of the classical and > > > Hartley multiplication routines agree (up to some precision loss). > > > > I picked on the timings Robert gave: > > > > sage: f = (x+1)^1000 > > > sage: timeit("f*f") > > > 5 loops, best of 3: 142 ms per loop > > > sage: timeit("f._mul_karatsuba(f)") > > > 5 loops, best of 3: 655 ms per loop > > > > So with the classical algorithm this multiplication takes about 16 ms. > > > With the Fast Hartley Transform it currently takes 26 ms. > > > > If we change that to (1+x)^10000 then the times are 1400 ms and 610 ms > > > respectively. > > > > The FHT should be much faster after I make an important speedup some > > > time today or tomorrow. It should comfotably win in that first > > > benchmark. > > > > Bill. > > > > On Apr 28, 6:01 pm, Bill Hart <goodwillh...@googlemail.com> wrote: > > > > > Rader-Brenner is less stable because it uses purely imaginary twiddle > > > > factors whose absolute value is considerably bigger than 1. It has > > > > been largely discarded in favour of the split-radix method which uses > > > > less real multiplications and additions anyhow. > > > > > It's very interesting to see an implementation of it, because it is > > > > not described in very many places! > > > > > Bill. > > > > > On Apr 28, 5:10 pm, Bill Hart <goodwillh...@googlemail.com> wrote: > > > > > > A very nice looking library, but it uses Karatsuba and Toom Cook, the > > > > > very things we decided we unstable. I also read that the Rader-Brenner > > > > > FFT is very unstable. > > > > > > Bill. > > > > > > On Apr 28, 2:14 pm, YannLC <yannlaiglecha...@gmail.com> wrote: > > > > > > > This might be of interest: > > > > > > > from the people of MPFR, > > > > > > > "Mpfrcx is a library for the arithmetic of univariate polynomials > > > > > > over > > > > > > arbitrary precision real (Mpfr) or complex (Mpc) numbers, without > > > > > > control on the rounding. For the time being, only the few functions > > > > > > needed to implement the floating point approach to complex > > > > > > multiplication are implemented. On the other hand, these comprise > > > > > > asymptotically fast multiplication routines such as Toom–Cook and > > > > > > the > > > > > > FFT." > > > > > > >http://www.multiprecision.org/index.php?prog=mpfrcx > > > > > > > On Apr 28, 3:53 am, Bill Hart <goodwillh...@googlemail.com> wrote: > > > > > > > > You can access the code here: > > > > > > > >http://selmer.warwick.ac.uk/gitweb/flint2.git?a=shortlog;h=refs/heads... > > > > > > > > You need the latest MPIR release candidate, the latest MPFR and > > > > > > > either > > > > > > > an x86 or x86_64 machine to run it. > > > > > > > > set the liib and include paths in the top level makefile, set the > > > > > > > LD_LIBRARY_PATH's in flint_env, then do: > > > > > > > > source flint_env > > > > > > > make library > > > > > > > make check MOD=mpfr_poly > > > > > > > > Most of the relevant code is in the mpfr_poly directory. > > > > > > > > If it doesn't work for you, please understand this is development > > > > > > > code > > > > > > > only, therefore probably broken in many ways. > > > > > > > > Information on the Fast Hartley Transform can be found in Joerg > > > > > > > Arndt's fxtbook: > > > > > > > >http://www.jjj.de/fxt/fxtbook.pdf > > > > > > > > Bill. > > > > > > > > On Apr 28, 2:45 am, Bill Hart <goodwillh...@googlemail.com> wrote: > > > > > > > > > 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 > > ... > > read more » -- 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