logcf: An Efficient Tool for Real Root Isolation

TL;DR

This paper introduces logcf, a C++ software tool that improves real root isolation of univariate polynomials using continued fractions, achieving significant speedups over existing methods and software.

Contribution

The paper presents a new algorithm for computing an upper bound of positive roots, enhancing the efficiency of continued fractions-based real root isolation methods.

Findings

logcf is 2-3 times faster than Mathematica's RootIntervals

logcf outperforms CF, a leading open software for root isolation

logcf is significantly faster than numerical methods Sleeve and eigensolve for real-root-only polynomials

Abstract

This paper revisits an algorithm for isolating real roots of univariate polynomials based on continued fractions. It follows the work of Vincent, Uspen- sky, Collins and Akritas, Johnson and Krandick. We use some tricks, especially a new algorithm for computing an upper bound of positive roots. In this way, the algorithm of isolating real roots is improved. The complexity of our method for computing an upper bound of positive roots is O(n log(u+1)) where u is the optimal upper bound satisfying Theorem 3 and n is the degree of the polynomial. Our method has been implemented as a software package logcf using C++ language. For many benchmarks logcf is two or three times faster than the function RootIntervals of Mathematica. And it is much faster than another continued fractions based software CF, which seems to be one of the fastest available open software for exact real root isolation. For those benchmarks which have only real roots, logcf is much faster than Sleeve and eigensolve which are based on numerical computation.