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# Catalan's Constant [Ramanujan's Formula]By: Greg Fee ### First Page:

Catalan's Constant [Ramanujan's Formula]

Catalan constant to 300000 digits computed on September 29, 1996 by 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, accelerated by an Euler transform. The algorithm was implemented using the LiDIA library for computational number theory and it is part of the multiprecision 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 = 2i1

// 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

Catalan constant is: sum(( 1)(n1)/(2n 1)2,n=1..infinity) also known under the name beta(2), see ?catalan in Maple for more details.

The previous record was 200000 digits, also from Thomas Papanikolaou and before that: 100000 digits was due to Greg Fee and Simon Plouffe on August 14, 1996, by using a SGI r10000 Power Challenge with 194 Mhz in 5.63 hours and the standard implementation of Catalan on MapleV, Release 4. (which uses Greg's idea).

Euler Tranform: References, Abramowitz and Stegun, formula 3.6.27 page 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 other words the formula used is: the ordinary formula for Catalan

sum(( 1)(n1)/(2n1)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

.91596559417721901505460351493238411077414937428167213426649811962176301977625 476947935651292611510624857442261919619957903589880332585905943159473748115840 699533202877331946051903872747816408786590902470648415216300022872764094238825 995774150881639747025248201156070764488380787337048990086477511322599713434074 854075532307685653357680958352602193823239508007206803557610482357339423191498 298361899770690364041808621794110191753274314997823397610551224779530324875371 878665828082360570225594194818097535097113157126158042427236364398500173828759 779765306837009298087388749561089365977194096872684444166804621624339864838916 280448281506273022742073884311722182721904722558705319086857354234985394983099 191159673884645086151524996242370437451777372351775440708538464401321748392999 947572446199754961975870640074748707014909376788730458699798606448749746438720 623851371239273630499850353922392878797906336440323547845358519277777872709060 830319943013323167124761587097924554791190921262018548039639342434956537596739 494354730014385180705051250748861328564129344959502298722983162894816461622573 989476231819542006607188142759497559958983637303767533853381354503127681724011 814072153468831683568168639327293677586673925839540618033387830687064901433486 017298106992179956530958187157911553956036... Continue reading book >>

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