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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.
To calculate 16 n−k mod (8k + 1) quickly and efficiently, the modular exponentiation algorithm is done at the same loop level, not nested. When its running 16x product becomes greater than one, the modulus is taken, just as for the running total in each sum. Now to complete the calculation, this must be applied to each of the four sums in turn.
So we have the series expansion (common factors have been canceled from numerators and denominators): = (+ + + + + +). (After cancellation the numerator and denominator values in OEIS: A092676 and OEIS: A092677 respectively; without cancellation the numerator terms are values in OEIS: A002067.)
Algorithms for calculating variance play a major role in computational statistics.A key difficulty in the design of good algorithms for this problem is that formulas for the variance may involve sums of squares, which can lead to numerical instability as well as to arithmetic overflow when dealing with large values.
A series of N measurements z n = e iθ n drawn from a wrapped normal distribution may be used to estimate certain parameters of the distribution. The average of the series z is defined as ¯ = = and its expectation value will be just the first moment:
For example, it is used to equate a probability for a random variable with the Lebesgue-Stieltjes integral typically associated with computing the probability: = for all in the Borel σ-algebra on , where () is the cumulative distribution function for , defined on , while is a probability measure, defined on a σ-algebra of subsets of some ...
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 ...
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.