Reading.
Get started

Library / Literature & Fiction

Catalan's Constant [Ramanujan's Formula]

by Greg Fee

Pages
about 15
Category
Literature & Fiction
Language
English
Reading time
17 min at 250 wpm
Start readingThe whole book, as cards or one word at a time. 9 min at 450 wpm in Read mode.

From the opening

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.

inline void constcatalan (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.isapproxzero ()) { // 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; } }

Time required: 1 day 8 hour 15 min 15 sec 55 hsec

Catalan constant is: sum((-1)**(n+1)/(2*n-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.

The series used is by putting x--> -1/2 . In other words the formula used is: the ordinary formula for Catalan

Computation of Catalan's constant using Ramanujan's Formula, by Greg Fee, ACM 1990, Proceedings of the ISAAC conference, 1990, p. 157.

A short sample. Read the whole book

More to read

Public-domain text from Project Gutenberg, Standard Ebooks or a similar open source. Copyright differs between countries. Rights holder? See our takedown process.