Re: Closer and closer you move to the Cantorian cliff

From: tinyurl.com/uh3t (rem642b_at_Yahoo.Com)
Date: 03/27/05 

Date: Sat, 26 Mar 2005 15:50:20 -0800

> From: "Michel Hack" <hack@watson.ibm.com>
> You need to catch up with new developments in the last 10 years.

Hmm, maybe so. The last I knew before, the best way to compute PI was
to first compute some inverse of PI using parallel arith/geo mean, then
invert that using Newton's method or Taylor series or somesuch. The
arith/geo mean method could be trimmed to yield just the last bunch of
bits you want of 1/PI instead of all of the bits to that point, if
that's all you wanted, but that didn't help you get the bits (or
digits) or PI itself.

> Google for BBP algorithm (Bailey, Borwein and Plouffe) which generates
> the nth hexadecimal digit (hence, trivially, binary digit) of Pi (and
> other "polylogarithmic" transcendental numbers) directly and
> efficiently, without having to compute all preceding bits.

Hmmm, looking... indeed, good old partial-fraction decomposition, but
for relatively prime integers instead of relatively prime polynomials
multiplied in denominator. It's funny how pfd is such an old technique
for integral calculus and number theory, yet it wasn't until 1996 that
somebody realized it could be applied usefully to some power series. I
guess it's another idea that seems so obvious after you see it, like
relativity and fractals.

Relevant Pages

  • Re: Closer and closer you move to the Cantorian cliff
    ... Hmm, maybe so. ... to first compute some inverse of PI using parallel arith/geo mean, ... invert that using Newton's method or Taylor series or somesuch. ... for relatively prime integers instead of relatively prime polynomials ...
    (sci.logic)
  • Re: Closer and closer you move to the Cantorian cliff
    ... Hmm, maybe so. ... to first compute some inverse of PI using parallel arith/geo mean, ... invert that using Newton's method or Taylor series or somesuch. ... for relatively prime integers instead of relatively prime polynomials ...
    (sci.math)