Real root refinements for univariate polynomial equations

TL;DR

This paper introduces two new algorithms, LZ1 and LZ2, for efficiently refining real roots of univariate polynomials by combining Newton's and secant methods, achieving quadratic and cubic convergence.

Contribution

The paper presents novel root refinement algorithms that improve speed and accuracy using floating-point interval methods and are faster than existing Maple functions.

Findings

Algorithms achieve quadratic and cubic convergence.

Methods outperform Maple15's RefineBox in speed.

Effective on benchmark polynomials.

Abstract

Real root finding of polynomial equations is a basic problem in computer algebra. This task is usually divided into two parts: isolation and refinement. In this paper, we propose two algorithms LZ1 and LZ2 to refine real roots of univariate polynomial equations. Our algorithms combine Newton's method and the secant method to bound the unique solution in an interval of a monotonic convex isolation (MCI) of a polynomial, and have quadratic and cubic convergence rates, respectively. To avoid the swell of coefficients and speed up the computation, we implement the two algorithms by using the floating-point interval method in Maple15 with the package intpakX. Experiments show that our methods are effective and much faster than the function RefineBox in the software Maple15 on benchmark polynomials.