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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 ...
It was used in the world record calculations of 2.7 trillion digits of π in December 2009, [3] 10 trillion digits in October 2011, [4] [5] 22.4 trillion digits in November 2016, [6] 31.4 trillion digits in September 2018–January 2019, [7] 50 trillion digits on January 29, 2020, [8] 62.8 trillion digits on August 14, 2021, [9] 100 trillion ...
This category presents articles pertaining to the calculation of Pi to arbitrary precision. Pages in category "Pi algorithms" The following 17 pages are in this category, out of 17 total.
The same approach can be used to calculate digits of the binary expansion of ln(2) starting from an arbitrary nth position. The number of terms in the "head" sum increases linearly with n , but the complexity of each term only increases with the logarithm of n if an efficient method of modular exponentiation is used.
The software may be obtained from the Pi-Hacks Yahoo! forum, or from Stu's Pi page. Super PI by Kanada Laboratory [101] in the University of Tokyo is the program for Microsoft Windows for runs from 16,000 to 33,550,000 digits. It can compute one million digits in 40 minutes, two million digits in 90 minutes and four million digits in 220 ...
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.
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:
BBP and BBP-inspired algorithms have been used in projects such as PiHex [5] for calculating many digits of π using distributed computing. The existence of this formula came as a surprise. It had been widely believed that computing the nth digit of π is just as hard as computing the first n digits. [1] Since its discovery, formulas of the ...