Efficient computation of pi by the Newton - Raphson iteration and a two-term Machin-like formula

TL;DR

This paper introduces an efficient method for computing pi using Newton-Raphson iteration applied to a two-term Machin-like formula with small arctangent arguments, improving computational speed despite large rational numbers involved.

Contribution

It demonstrates how to effectively resolve computational challenges in Machin-like formulas for pi by applying Newton-Raphson iteration, enhancing calculation efficiency.

Findings

Newton-Raphson method accelerates pi computation

Effective handling of large rational numbers in formulas

Improved convergence in Machin-like formula calculations

Abstract

In our recent publication we have proposed a new methodology for determination of the two-term Machin-like formula for pi with small arguments of the arctangent function of kind $$ \frac{\pi }{4} = {2^{k - 1}}\arctan \left( {\frac{1}{{{\beta_1}}}} \right) + \arctan \left( {\frac{1}{{{\beta_2}}}} \right), $$ where $k$ and ${\beta_1}$ are some integers and ${\beta_2}$ is a rational number, dependent upon ${\beta_1}$ and $k$. Although ${1/\left|\beta_2\right|}$ may be significantly smaller than ${1/\beta_1}$, the large numbers in the numerator and denominator of $\beta_2$ decelerate the computation. In this work we show how this problem can be effectively resolved by the Newton--Raphson iteration method.