Dear all, > Date: Wed, 6 Jan 2010 06:51:33 -0800 > From: William Stein <[email protected]> > > On Wed, Jan 6, 2010 at 5:58 AM, John Cremona <[email protected]> wrote: > > According to this: > > > > http://news.bbc.co.uk/1/hi/technology/8442255.stm > > > > someone has just computed pi to 2.7 trillion digits on a "desktop > > computer". The article does not mention software. > > > > How well would Sage do? > > Sage just uses MPFR, so this is really a question for Paul Zimmerman, > who I've cc'd.
as far as I know, Fabrice Bellard has implemented all his software from scratch (please correct me if needed). The main issue is that he had to implement a multiple-precision library with partial results saved on disk, due to the huge precision (2.7e12 digits represent more than 1Tb). In principle, Sage (through MPFR) implements the best known algorithm for Pi (AGM iteration, see Section 5.1 in http://www.mpfr.org/algorithms.pdf). However the AGM iteration is based on the square root, which is itself (at least with the GMP implementation) based on the division, which currently (in GMP) has complexity O(M(n) log n) instead of the best possible O(M(n)), where M(n) is the time to multiply two n-bits numbers. Also GMP (and thus MPFR since it depends on GMP) does not (automatically) do partial computations on disk. Paul Zimmermann
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