A root finding method with arbitrary order of convergence
Alois Schiessl
TL;DR
This work introduces a polynomial fixed-point scheme for computing the Mth root $\sqrt[M]{a}$ of a positive real number $a$ with arbitrary order of convergence. By defining $f(t)=1-t^M/a$ and constructing the fixed-point function $F(x)=\prod_{ell=1}^{P}(1+1/(ell M))\int_{0}^{x} f(t)^P dt$, which is a polynomial of degree $P M+1$, the authors show that $\sqrt[M]{a}$ is a fixed point of $F$ and that $F^{(k)}(\sqrt[M]{a})=0$ for $1\le k\le P$ while $F^{(P+1)}(\sqrt[M]{a})$ is nonzero. Using Banach's fixed-point theorem, the iteration $x_{n+1}=F(x_n)$ converges to $\sqrt[M]{a}$ with order $P+1$, with an explicit asymptotic error constant. The method reduces root finding to polynomial evaluation, enabling high-order convergence while relying solely on polynomial arithmetic; the paper presents explicit polynomials (up to order 5) and demonstrates practical results, including a cubic-root example and a million-digit computation of $\sqrt{2}$ using moderate iteration counts. The approach offers a scalable, exact-division-free framework for high-precision root computations with tunable convergence order, suitable for efficient implementation on standard hardware.
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 äquivalent zu $1-\frac{t^M}{a}=0$ ist. Die reelle Lösung hiervon ist $\sqrt[M]{a}$. In dieser Veröffentlichung stellen wir ein Verfahren vor, das die Berechnung von $\sqrt[M]{a}$ mit beliebiger Konvergenzordnung ermöglicht 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 ausgewählter Beispiele von Wurzelberechnungen zeigen wir die Effizienz des Verfahrens.