A root finding method with arbitrary order of convergence

TL;DR

This paper introduces a polynomial-based root-finding method that achieves arbitrary order of convergence for calculating real M-th roots, demonstrated to be efficient through computational tests.

Contribution

It presents a novel polynomial evaluation method for computing M-th roots with customizable convergence order, improving upon existing techniques.

Findings

Method achieves high convergence order with polynomial evaluation.

Computational tests confirm efficiency and accuracy.

Applicable for real positive roots with arbitrary precision.

Abstract

Let $a\in \mathbb{R}^{+}\backslash\left\{0\right\}$ and $M\in\mathbb{N}$. We consider the equation $t^M-a=0$, which is equivalent to $1-\frac{t^M}{a}=0\,.$ The real solution is $\sqrt[M]{a}$. In this publication, we present a method that enables the calculation of $\sqrt[M]{a}$ with arbitrary order of convergence using only polynomials. We define the fixed point function \[ F\left(x\right) =\prod_{\ell=1}^{P}\left(1+\frac{1}{\ell\cdot M}\right) \int\limits_{0}^{x}\!\left(1-{\frac{{t}^{M}}{a}}\right)^{P}{\rm d}t =\sum\limits_{k=0}^{P}\frac{\left(-1\right)^{\,k}}{a^{\,k}}\cdot\binom{P}{k}\cdot\frac{x^{\,k\,\cdot M+1}}{k\,\cdot M+1} \] This is a polynomial of degree $\left(P\cdot M+1\right)$ with $\left(P+1\right)$ terms. The calculation of $\sqrt[M]{a}$ is thus reduced to a polynomial evaluation. The computational tests we performed demonstrate the efficiency of the method. -- Es sei $a\in \mathbb{R}^{+}\backslash\left\{0\right\}$ und $M\in\mathbb{N}$. Vorgelegt ist die Gleichung $t^M-a=0$, die \"aquivalent zu $1-\frac{t^M}{a}=0$ ist. Die reelle L\"osung hiervon ist $\sqrt[M]{a}$. In dieser Ver\"offentlichung stellen wir ein Verfahren vor, das die Berechnung von $\sqrt[M]{a}$ mit beliebiger Konvergenzordnung erm\"oglicht und nur Polynome verwendet. Wir definieren die Fixpunktfunktion \[F\left(x\right) =\prod_{\ell=1}^{P}\left(1+\frac{1}{\ell\cdot M}\right) \int\limits_{0}^{x}\!\left(1-{\frac{{t}^{M}}{a}}\right)^{P}{\rm d}t =\sum\limits_{k=0}^{P}\frac{\left(-1\right)^{\,k}}{a^{\,k}}\cdot\binom{P}{k}\cdot\frac{x^{\,k\,\cdot M+1}}{k\,\cdot M+1} \] Das ist ein Polynom vom Grad $\left(P\cdot M+1\right)$ mit $\left(P+1\right)$ Summanden. Anhand ausgew\"ahlter Beispiele von Wurzelberechnungen zeigen wir die Effizienz des Verfahrens.