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The Chudnovsky algorithm is a fast method for calculating the digits of π, based on Ramanujan's π formulae.Published by the Chudnovsky brothers in 1988, [1] it was used to calculate π to a billion decimal places.
This algorithm computes π without requiring custom data types having thousands or even millions of digits. The method calculates the nth digit without calculating the first n − 1 digits and can use small, efficient data types. Fabrice Bellard found a variant of BBP, Bellard's formula, which is faster.
Start by setting [4] = = = + Then iterate + = + + = (+) + + = (+ +) + + + Then p k converges quadratically to π; that is, each iteration approximately doubles the number of correct digits.The algorithm is not self-correcting; each iteration must be performed with the desired number of correct digits for π 's final result.
The Gauss–Legendre algorithm is an algorithm to compute the digits of π. It is notable for being rapidly convergent, with only 25 iterations producing 45 million correct digits of π . However, it has some drawbacks (for example, it is computer memory -intensive) and therefore all record-breaking calculations for many years have used other ...
For the first term in the Taylor series, all digits must be processed. In the last term of the Taylor series, however, there's only one digit remaining to be processed. In all of the intervening terms, the number of digits to be processed can be approximated by linear interpolation. Thus the total is given by:
Pages in category "Pi algorithms" The following 17 pages are in this category, out of 17 total. This list may not reflect recent changes. A. ... Code of Conduct;
Machin's particular formula was used well into the computer era for calculating record numbers of digits of π, [39] but more recently other similar formulae have been used as well. For instance, Shanks and his team used the following Machin-like formula in 1961 to compute the first 100,000 digits of π : [ 39 ]
where C is the circumference of a circle, d is the diameter, and r is the radius.More generally, = where L and w are, respectively, the perimeter and the width of any curve of constant width.