[gambit-list] Computing pi with parallel gambit
Bradley Lucier
lucier at math.purdue.edu
Sun Nov 27 15:08:49 EST 2016
Marc:
Because of accumulated roundoff error in IEEE floating-point arithmetic,
Gambit's FFT-based bignum multiplication algorithm is limited to
multiplying two numbers each of which has <= 2^{29} bits (on a 64-bit
machine).
Multiplying larger numbers falls back on Karatsuba multiplication, which
cuts the size of the multiplicands roughly in half in each step, until
you get back down to numbers of <= 2^{29} bits again, in which case FFT
is used again.
So, in fact, to compute a billion decimal digits of pi, we rely on
Karatsuba to do the largest multiplications (and divisions and square
roots, since they all reduce to multiplication).
On a machine with enough memory, all the "sub-multiplications" in
Karatsuba can be done in parallel. I haven't figured out how much
memory is needed; it takes a few GB of temporary RAM to multiply the
largest numbers with the FFT algorithm.
I believe it took about 16GB total to calculate a billion decimal digits
of pi using the current serial algorithm.
Once parallel gambit is usable, computing pi to a billion (or even 10
billion, who knows) digits by parallelizing the larger Karatsuba
multiplications might be a good test of the system.
Brad
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