A fixed point iteration method for the arctangent with any odd order of convergence based on sine and cosine

TL;DR

This paper introduces a fixed point iteration method for calculating the arctangent function using sine and cosine, achieving any odd order of convergence based on a parameter P.

Contribution

The paper presents a novel fixed point method for arctangent computation with customizable odd order convergence based on sine and cosine functions.

Findings

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

The method demonstrates high efficiency in practical computations.

Applicable for any positive real t and integer P.

Abstract

In this paper, we present a fixed point method for the arctangent based on sine and cosine. Let $t\in \mathbb{R}^{+}$ and $P\in \mathbb{N}$. We define: \[T\left(x\right)=x-\sum_{k=1}^{P}\,\frac{\left(-1\right)^{k-1}}{2\,k-1} \left(\frac {\sin\!\left(x\right)-t\cos\!\left(x\right)} {\cos\!\left(x\right)+t\sin\!\left(x\right)} \right)^{2\,k-1}.\] For every initial value $x_0$ sufficiently close to $\arctan\left(t\right)$, the sequence \[x_{n+1}=T\left(x_{n}\right)\;;\,n=0,1,\ldots\] is converging to $\arctan\left(t\right)$ with order of convergence exactly $\left(2\,P+1\right)$. The computational test we performed demonstrates the efficiency of the method. \selectlanguage{ngerman} \[\] \[\textbf{Zusammenfassung}\] In dieser Abhandlung stellen wir ein Fixpunktverfahren zur Berechnung des arcustangens auf Basis von sinus und cosinus vor. Es sei $t\in \mathbb{R}^{+}$ und $P\in\mathbb{N}$. Wir definieren: \[T\left(x\right)=x-\sum_{k=1}^{P}\,\frac{\left(-1\right)^{k-1}}{2\,k-1} \left(\frac {\sin\!\left(x\right)-t\cos\!\left(x\right)} {\cos\!\left(x\right)+t\sin\!\left(x\right)}\right) ^{2\,k-1}.\] F\"ur jeden Startwert $x_0$ hinreichend nahe bei $\arctan\left(t\right)$ konvergiert die Folge \[x_{n+1}=T\left(x_{n}\right)\;;\,n=0,1,\ldots\] gegen $\arctan\left(t\right)$ mit Konvergenzordnung genau $\left(2\,P+1\right)$. Anhand einer praktischen Berechnung von $\frac{\pi}{4}$ zeigen wir die Effizienz des Verfahrens. \[\text{Deutsche Version ab Seite 17}\]