A fixed-point iteration method for the number Pi with arbitrary odd order of convergence based on the sine function

TL;DR

This paper introduces a fixed-point iteration method based on the sine function for high-precision computation of pi, achieving arbitrary odd orders of convergence depending on a parameter P, with demonstrated efficiency through computational tests.

Contribution

The paper presents a novel fixed-point iteration scheme for pi that attains arbitrary odd convergence orders using a sine-based function, expanding the toolkit for high-precision calculations.

Findings

Convergence order is exactly (2P+1) for the method.

The method demonstrates high efficiency in computational tests.

Applicable for initial values close to pi.

Abstract

In this paper, we present a fixed point method for high-precision computation of number $\pi$ based on the sine function. Let $P\in \mathbb{N}$. We define the function: \[ S\left(x\right) =x+\sum_{k=1}^{P}\left(\prod_{\ell=1}^{k-1}\frac {2\,\ell-1}{2\,\ell}\right)\frac{\sin\left(x\right)^{2\,k-1}}{2\,k-1}\,. \] For every initial value $x_0$ sufficiently close to $\pi$, the sequence \[x_{n+1}=x_n+S\left(x_{n}\right)\;;\,n=0,1,\ldots\] is converging to $\pi$ with order of convergence exactly $\left(2\,P+1\right)$. The computational tests we performed demonstrate the efficiency of the method. \[\] \[\textbf{Zusammenfassung}\] In dieser Abhandlung stellen wir ein Fixpunktverfahren zur Berechnung der Kreiszahl $\pi$ auf Basis der sinus Funktion vor. Es sei $P\in \mathbb{N}$. Wir definieren die Funktion: \[ S\left(x\right) =x+\sum_{k=1}^{P}\left(\prod_{\ell=1}^{k-1}\frac {2\,\ell-1}{2\,\ell}\right)\frac{\sin\left(x\right)^{2\,k-1}}{2\,k-1}\;. \] F\"ur jeden Startwert $x_0$ hinreichend nahe bei $\pi$ konvergiert die Folge \[x_{n+1}=x_n+S\left(x_{n}\right)\;;\,n=0,1,\ldots\] gegen $\pi$ mit Konvergenzordnung genau $\left(2\,P+1\right)$. Anhand von praktischen Berechnungen zeigen wir die Effizienz des Verfahrens. \[\text{Deutsche Version ab Seite 19}\]