Real Root Isolation of Polynomial Equations Based on Hybrid Computation

TL;DR

This paper introduces a hybrid computational algorithm combining homotopy continuation and interval methods for efficient real root isolation of polynomial equations, outperforming traditional symbolic approaches.

Contribution

The paper presents a novel hybrid algorithm that integrates homotopy continuation, Kantorovich theorem, and Krawczyk interval iteration for improved real root isolation.

Findings

More efficient than traditional symbolic methods

Successfully applied to various benchmark problems

Provides accurate real root isolation boxes

Abstract

A new algorithm for real root isolation of polynomial equations based on hybrid computation is presented in this paper. Firstly, the approximate (complex) zeros of the given polynomial equations are obtained via homotopy continuation method. Then, for each approximate zero, an initial box relying on the Kantorovich theorem is constructed, which contains the corresponding accurate zero. Finally, the Krawczyk interval iteration with interval arithmetic is applied to the initial boxes so as to check whether or not the corresponding approximate zeros are real and to obtain the real root isolation boxes. Meanwhile, an empirical construction of initial box is provided for higher performance. Our experiments on many benchmarks show that the new hybrid method is more efficient, compared with the traditional symbolic approaches.