Some Observations on Khovanskii's Matrix Methods for extracting Roots of Polynomials

TL;DR

This paper explores Khovanskii's matrix methods for efficiently approximating roots of polynomials, providing new sequences and parameter choices that improve convergence for roots of integers and general polynomials.

Contribution

The authors extend Khovanskii's matrix approach to roots of integers and general polynomials, introducing optimal parameter choices for faster convergence and broader applicability.

Findings

Sequences converge to roots with optimized parameters.

Method applies to m-th roots of positive integers.

Extension to roots of general polynomials.

Abstract

In this article we apply a formula for the $n$-th power of a $3\times 3$ matrix (found previously by the authors) to investigate a procedure of Khovanskii's for finding the cube root of a positive integer. We show, for each positive integer $\alpha$, how to construct certain families of integer sequences such that a certain rational expression, involving the ratio of successive terms in each family, tends to $\alpha^{1/3}$. We also show how to choose the optimal value of a free parameter to get maximum speed of convergence. We apply a similar method, also due to Khovanskii, to a more general class of cubic equations, and, for each such cubic, obtain a sequence of rationals that converge to the real root of the cubic. We prove that Khovanskii's method for finding the $m$-th ($m \geq 4$) root of a positive integer works, provided a free parameter is chosen to satisfy a very simple condition. Finally, we briefly consider another procedure of Khovanskii's, which also involves $m \times m$ matrices, for approximating the root of an arbitrary polynomial of degree $m$.