Produced by Greg Fee
Catalan's Constant [Ramanujan's Formula]
Catalan constant to 300000 digits computed on September 29, 1996by using a Sun Ultra-Sparc in 1 day 8 hour 15 min 15 sec 55 hsec.The algorithm used is the standard series for Catalan, acceleratedby an Euler transform. The algorithm was implemented using the LiDIAlibrary for computational number theory and it is part of themultiprecision floating-point arithmetic of the package.
LiDIA is available from
ftp://crypt1.cs.uni-sb.de/pub/systems/LiDIA/LiDIA-1.2.1.tgz http://www-jb.cs.uni-sb.de/LiDIA/linkhtml/lidia/lidia.html
The implementation of the algorithm is:
inline void
const_catalan (bigfloat & y)
{
bigfloat p;
bigfloat t;
int i = 1, j = 3; // j = 2*i+1
// y = t = p = 1/2 divide (y, 1, 2); t.assign (y); p.assign (y);
// while t is greater than the desired accuracy
while (!t.is_approx_zero ())
{
// do
// p = p * (i/j);
// t = (t * i + p) / j;
// y = y + t;
// i++; j+=2;
multiply (p, p, i);
divide (p, p, j);
multiply (t, t, i);
add (t, t, p);
divide (t, t, j);
add (y, y, t);
i++;
j += 2;
}
}
Here is the output of the program:
Calculating Catalan's constant to 300000 decimals
Time required: 1 day 8 hour 15 min 15 sec 55 hsec
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Additional REFERENCES:
Catalan constant is: sum((-1)**(n+1)/(2*n-1)**2,n=1..infinity) also knownunder the name beta(2), see ?catalan in Maple for more details.
The previous record was 200000 digits, also from Thomas Papanikolaouand before that: 100000 digitswas due to Greg Fee and Simon Plouffe on August 14, 1996,by using a SGI r10000 Power Challenge with 194 Mhz in 5.63 hoursand the standard implementation of Catalan on MapleV, Release 4.(which uses Greg's idea).
Euler Tranform: References, Abramowitz and Stegun, formula 3.6.27page 16 in Handbook of Mathematical Functions and Tables, Dover 1964.
Ramanujan Notebooks, part I formula 34.1 of page 293.
The series used is by putting x—> -1/2 . In otherwords the formula used is: the ordinary formula for Catalan
sum((-1)**(n+1)/(2*n+1)**2,n=0..infinity)
and then you apply the Euler Transform to it.
Computation of Catalan's constant using Ramanujan's Formula, by
Greg Fee, ACM 1990, Proceedings of the ISAAC conference, 1990, p. 157.
Catalan constant to 300000 digits
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.915965594177219015054603514932384110774149374281672134266498119621763019776254769479356512926115106248574422619196199579035898803325859059431594737481158406995332028773319460519038727478164087865909024706484152163000228727640942388259957741508816397470252482011560707644883807873370489900864775113225997134340748540755323076856533576809583526021938232395080072068035576104823573394231914982983618997706903640418086217941101917532743149978233976105512247795303248753718786658280823605702255941948180975350971131571261580424272363643985001738287597797