Algorithmic Determination of a Large Integer in the Two-Term Machin-like Formula for π

Abstract

In our earlier publication we have shown how to compute by iteration a rational number u2,k in the two-term Machin-like formula for π of the kind π4=2k−1arctan1u1,k+arctan1u2,k,k∈Z,k≥1, where u1,k can be chosen as an integer u1,k=ak/2−ak−1 with nested radicals defined as ak=2+ak−1 and a0=0. In this work, we report an alternative method for determination of the integer u1,k. This approach is based on a simple iteration and does not require any irrational (surd) numbers from the set ak in computation of the integer u1,k. Mathematica programs validating these results are presented.

A Two-Domain MATLAB Implementation for Efficient Computation of the Voigt/Complex Error Function

“…From these examples, we can see that Hermann's ( 23), Machin's (1) and the derived (24) formulas for π belong to the same generic group, since all of them can be constructed from their generalized form (17) at different integers k, equal to 2, 3 and 4, respectively. Further, we will use Equation (24) as an example for computing digits of π.…”

An Iterative Method for Computing π by Argument Reduction of the Tangent Function

“…From these examples, we can see that Hermann's ( 23), Machin's (1) and the derived (24) formulas for π belong to the same generic group, since all of them can be constructed from their generalized form (17) at different integers k, equal to 2, 3 and 4, respectively. Further, we will use Equation (24) as an example for computing digits of π.…”

“…Therefore, the method described in our paper [22] can also be generalized to generate all two-term Machin-like formulas of kind (14), shown in Table 1 from the work [10] (see first row showing f 28 32 ). In our recent publication [24], using the two-term Machin-like Formula (9) for π, we found the following iterative formula…”

“…This equation can be employed to compute digits of the constant π with quadratic convergence (see Mathematica code provided in [24]). Motivated by recent publications [10,23,25,26] in connection to our works [21,22,24,27,28], we developed a new algorithm based on a modified version of the iterative Formula (15).…”

A Rational Approximation of the Two-Term Machin-Like Formula for π

“…In our previous publications [16] we show how the two-term Machin-like formula for π represented in form ( 8) is used to derive an iterative formula…”