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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.
Though the BBP formula can directly calculate the value of any given digit of π with less computational effort than formulas that must calculate all intervening digits, BBP remains linearithmic (()), whereby successively larger values of n require increasingly more time to calculate; that is, the "further out" a digit is, the longer it ...
The formula is a special case of the Euler–Boole summation formula for alternating series, providing yet another example of a convergence acceleration technique that can be applied to the Leibniz series. In 1992, Jonathan Borwein and Mark Limber used the first thousand Euler numbers to calculate π to 5,263 decimal places with the Leibniz ...
An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition, [8] as explained here. This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms.
Madhava's correction term is a mathematical expression attributed to Madhava of Sangamagrama (c. 1340 – c. 1425), the founder of the Kerala school of astronomy and mathematics, that can be used to give a better approximation to the value of the mathematical constant π (pi) than the partial sum approximation obtained by truncating the Madhava–Leibniz infinite series for π.
An m-1-term, σ-approximated summation for a series of period T can be written as follows: = + = () [ + ()], in terms of the normalized sinc function: = . and are the typical Fourier Series coefficients, and p, a non negative parameter, determines the amount of smoothening applied, where higher values of p further reduce the ...
At about the same time, the Egyptian Rhind Mathematical Papyrus (dated to the Second Intermediate Period, c. 1600 BCE, although stated to be a copy of an older, Middle Kingdom text) implies an approximation of π as 256 ⁄ 81 ≈ 3.16 (accurate to 0.6 percent) by calculating the area of a circle via approximation with the octagon. [5] [12]
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.