Re: Pi

[Karim BELABAS <Karim.Belabas@math.u-psud.fr>]

On Tue, 14 Jan 2003, Bill Allombert wrote:

> On Sun, Jan 12, 2003 at 06:07:00PM +0100, Ralf Stephan wrote:
> > The manual says (p.69) a copy of Pi is created on the heap for quicker
> > computations next time the function is called. This does not seem
> > to be true in 2.2.4 as with both \p 12000 and 12200, Pi needs about
> > one second here (I know it's fast enough but the manual says it).
>
> If it is your question, there is no incremental algorithm to compute Pi
> implemented.  Computing Pi at 12200 d.p does not use the knowledge of the first
> 12000 d.p.
>
> I do not know if it would make sense to do that.

It definitely would, using the log((1+i) / 2) expansion. I don't think it
would have a major effect on running times, but if somebody wants to try it...

Since the GMP kernel offers assymptotically fast multiplication for the basic
PARI types (t_INT natively, and t_REAL via your recent hack), it will even be
possible to compute a huge number of digits with gp  [ losing constant
factors compared to special purpose implementations, due to the various
overheads, but still in decent time ].

Decimal output would probably take infinite time though... [ still uses a
quadratic method, see convi/confrac ]

Also, a different Pi formula needs to be implemented [ the current one
was chosen so that almost all multiplication/divisions involve a single
precision operand, which is not at all what we want now ! ].

Cheers,

    Karim.

P.S: I'll try to add GMP support to Configure "real soon now" [have a series of
lectures to prepare first...]
-- 
Karim Belabas                    Tel: (+33) (0)1 69 15 57 48
Dép. de Mathématiques, Bât. 425  Fax: (+33) (0)1 69 15 60 19
Université Paris-Sud             Email: Karim.Belabas@math.u-psud.fr
F-91405 Orsay (France)           http://www.math.u-psud.fr/~belabas/
--
PARI/GP Home Page: http://www.parigp-home.de/