The numerical greatest common divisor of univariate polynomials

TL;DR

This paper develops a regularization-based approach for numerically computing the greatest common divisor of univariate polynomials, ensuring robustness and accuracy even with perturbed data through convergence analysis and a detailed algorithm.

Contribution

It introduces a regularization theory and a blackbox algorithm that address ill-posedness in numerical GCD computation, providing rigorous convergence and stability guarantees.

Findings

The regularized GCD is strongly well-posed.

The algorithm achieves accurate GCD computation with floating point arithmetic.

Numerical experiments demonstrate robustness and precision.

Abstract

This paper presents a regularization theory for numerical computation of polynomial greatest common divisors and a convergence analysis, along with a detailed description of a blackbox-type algorithm. The root of the ill-posedness in conventional GCD computation is identified by its geometry where polynomials form differentiable manifolds entangled in a stratification structure. With a proper regularization, the numerical GCD is proved to be strongly well-posed. Most importantly, the numerical GCD solves the problem of finding the GCD accurately using floating point arithmetic even if the data are perturbed. A sensitivity measurement, error bounds at each computing stage, and the overall convergence are established rigorously. The computing results of selected test examples show that the algorithm and software appear to be robust and accurate.