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Re: [Maxima-discuss] e, pi, gamma, log(2), also mpfr, also sympy
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From: Richard F. <fa...@be...> - 2014-09-21 16:10:23
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Thanks for the history. This is the kind of thing I would like to see in the file, nothidden in the source-code-control comments or entirely missing... the original methods were the ones I wrote in 1974. An efficient computation is usually not very important since a given run will probably call the program rarely. Once a high-precision value is computed, it is cached and all lower-precision requests are done by rounding down. On the other hand, people sometimes compare systems by metrics like "How long does it take to compute the 100,000 digit value of pi?" in Mathematica 9, Timing[N[Pi,10000]] takes 0.0312 seconds. (the second time takes 0. seconds). while in maxima 5.31.2, gcl 2.6.8 fpprec:10000 takes 0.03 seconds (the second time takes 0.0 seconds) then bfloat(%pi)$ takes 0.22 seconds (the second time takes 0.0 seconds) So obviously Mathematica 9 is 7 times better than Maxima :) I don't know what the underlying bignum integer arithmetic is on the two systems, and whether GCL or Mathematica is using GMP arithmetic, and whether if it is, if it is using GMP arithmetic tuned to the Pentium 4 I am using (tuned asm code runs several times faster than genericC). and how it compares to MFPR's code, which includes the following — Function: int*mpfr_const_log2*(mpfr_t rop, mpfr_rnd_t rnd) — Function: int*mpfr_const_pi*(mpfr_t rop, mpfr_rnd_t rnd) — Function: int*mpfr_const_euler*(mpfr_t rop, mpfr_rnd_t rnd) — Function: int*mpfr_const_catalan*(mpfr_t rop, mpfr_rnd_t rnd) The code for computing pi in mpfr can be found here https://github.com/dschatzberg/mpfr/blob/master/src/const_pi.c but there is no useful comment there describing the method. And the time would be heavily dependent, I think, on the GMP version, anyway. In general I defer to people who devote large parts of their waking hours to optimizing these procedures, and so loading mpfr into Maxima as the back-end for all of bigfloats would be an interesting prospect. I have, from time to time, loaded mpfr into lisp for experiments, like can you multiply polynomials fast by chunking the coefficients into very long integers. Except that only some Lisps allow loading foreign libraries, so not all Maximas (esp. GCL versions) can do it. An alternative to loading MPFR is to encourage Volker to do a really nice job. Thanks, Volker! There is a lot of interest in some circles in "experimental mathematics" around computing lots of digits of pi... see http://en.wikipedia.org/wiki/Bailey%E2%80%93Borwein%E2%80%93Plouffe_formula I do not know if the splitting formula is the very fastest... I found this reference http://www.craig-wood.com/nick/articles/pi-chudnovsky/ which is not the most erudite discussion, but he claims his new code is faster than "BPP". However most of the time is spent in computing the sqrt of an integer. Oh, while I'm talking about foreign-language libraries, there's a bunch of people using python developing sympy. It might be plausible to link the sympy library to Maxima, if there are features in it that are of interest. I think that much of thesympy development to date has been to (mostly naively) duplicate superficial features of Maxima, ... superficial because they may depend on proceduresthat are less effective and run slower. Eventually there may be situations wherethe python procedure goes further and/or faster. I wouldn't expect python generallyto be faster than compiled lisp, but particular procedures might have better algorithms, and who knows if python will eventually become "fast". There is, for example, nothing to prevent someone including more features in the sympy programs for "solve" or "integrate". Although my initial impression is that sympy authors are fully capable of making bad design choices that they would not make if they studied the literature or the "competition". RJF On 9/21/2014 8:19 AM, Volker van Nek wrote: > Hendrik, some Details on e and pi. > > The computation of e and pi is done by the functions fpe1 resp. fppi1 > which are - as Richard Fateman already posted - in src/float.lisp . > > Before 2007 e was computed by it's Taylor series and pi by the formula > pi = 6*asin(1/2) and the asin-series. > > In 2007 I replaced the asin-formula by the Chudnovsky-formula and for > the computation of e I implemented a mechanism that combines a lot of > Taylor summands to one. This mechanism uses the fact that division is > expensive and multiplication of large integers is much cheaper. > At that time I didn't know binary splitting. Binary splitting is also a > mechanism to combine Taylor summands and also uses large integer > multiplication but in contrast to my approach it is a recursive divide > and conquer algorithm and provides better timings for high precisions. > > Very soon I will replace the current versions by the binary splitting > variants (which are attached to my first mail in this thread). > > Volker van Nek > > Am 21.09.2014 09:59, schrieb Jan Hendrik Mueller: >> Maybe I overlooked that: What was the "old" way maxima computed e and pi? >> Is there a possibility to look at these codes for non developers? >> bw >> Jan >> >>> Am 20.09.2014 um 22:53 schrieb Raymond Toy <toy...@gm...>: >>> >>> >>> >>>> On Sat, Sep 20, 2014 at 2:58 AM, Volker van Nek <vol...@gm...> wrote: >>>> Am 19.09.2014 20:35, schrieb Raymond Toy: >>>>>>>>>> "Volker" == Volker van Nek <vol...@gm...> writes: >>>>> Volker> [ I also implemented exp(x), log(x), sin(x) and friends for bigfloat x. >>>>> Volker> In contrast to fpe1, .. these implementions are still some kind of raw >>>>> Volker> material and need more error analysis. And of course it's a lot more >>>>> Volker> difficult to merge them into Maxima. My idea is to put them into >>>>> Volker> /share/contrib for now. ] >>>>> >>>>> Is there something wrong with the current implementations of these >>>>> functions? >>>>> >>>>> Ray >>>>> >>>> Ray, as I wrote in my last sentence I do not feel any pressure to >>>> replace the current functions in src. >>>> >>>> After coding the binary splitting versions for the constants I was >>>> curious to check out the other algorithms described in this paper. >>>> >>>> I thought it is worth to share the implementation of these algorithms. >>>> Perhaps someone can use them or is simply interested to study them. >>>> >>>> The binary splitting variants are very fast and in some far future these >>>> implementations might be worth to consider. At the moment I don't want >>>> to risk breaking anything in src for the sake of speed advantages. >>> Ok. I haven't looked at your code yet, but if they're significantly faster and not (much) less accurate, we should consider adding these. I'm sure we can come up with a nice set of tests to cover the code to show that it's good. >>>> Volker >>>> >>>>> ------------------------------------------------------------------------------ >>>>> Slashdot TV. Video for Nerds. Stuff that Matters. >>>>> http://pubads.g.doubleclick.net/gampad/clk?id=160591471&iu=/4140/ostg.clktrk >>>>> _______________________________________________ >>>>> Maxima-discuss mailing list >>>>> Max...@li... >>>>> https://lists.sourceforge.net/lists/listinfo/maxima-discuss >>>>> >>>> >>>> ------------------------------------------------------------------------------ >>>> Slashdot TV. Video for Nerds. Stuff that Matters. >>>> http://pubads.g.doubleclick.net/gampad/clk?id=160591471&iu=/4140/ostg.clktrk >>>> _______________________________________________ >>>> Maxima-discuss mailing list >>>> Max...@li... >>>> https://lists.sourceforge.net/lists/listinfo/maxima-discuss >>> >>> >>> -- >>> Ray >>> ------------------------------------------------------------------------------ >>> Slashdot TV. Video for Nerds. Stuff that Matters. >>> http://pubads.g.doubleclick.net/gampad/clk?id=160591471&iu=/4140/ostg.clktrk >>> _______________________________________________ >>> Maxima-discuss mailing list >>> Max...@li... >>> https://lists.sourceforge.net/lists/listinfo/maxima-discuss > > > ------------------------------------------------------------------------------ > Slashdot TV. Video for Nerds. Stuff that Matters. > http://pubads.g.doubleclick.net/gampad/clk?id=160591471&iu=/4140/ostg.clktrk > _______________________________________________ > Maxima-discuss mailing list > Max...@li... > https://lists.sourceforge.net/lists/listinfo/maxima-discuss |
