Approximate GCD in a Bernstein basis

TL;DR

This paper adapts a root-based algorithm for approximate GCD computation to polynomials in Bernstein basis, utilizing stable root-finding and bipartite matching for degree determination, with refinements for improved accuracy.

Contribution

It introduces a novel adaptation of Pan's algorithm for Bernstein basis polynomials, combining stable root-finding and bipartite matching techniques.

Findings

Effective approximate GCD computation in Bernstein basis

Enhanced stability through companion pencil method

Refinements improve accuracy and robustness

Abstract

We adapt Victor Y. Pan's root-based algorithm for finding approximate GCD to the case where the polynomials are expressed in Bernstein bases. We use the numerically stable companion pencil of Gudbj\"orn F. J\'onsson to compute the roots, and the Hopcroft-Karp bipartite matching method to find the degree of the approximate GCD. We offer some refinements to improve the process.