Variable projection methods for approximate (greatest) common divisor computations

TL;DR

This paper introduces variable projection methods for computing approximate common divisors of polynomials, leveraging low-rank approximations and duality principles to develop efficient algorithms with practical software implementation.

Contribution

It presents novel variable projection algorithms for approximate polynomial GCD computation, connecting them to mosaic Hankel low-rank approximation and providing a software implementation.

Findings

Methods have linear complexity in polynomial degrees.

Algorithms are effective for both small and large divisor degrees.

Software implementation demonstrates practical applicability.

Abstract

We consider the problem of finding for a given $N$-tuple of polynomials (real or complex) the closest $N$-tuple that has a common divisor of degree at least $d$. Extended weighted Euclidean seminorm of the coefficients is used as a measure of closeness. Two equivalent representations of the problem are considered: (i) direct parameterization over the common divisors and quotients (image representation), and (ii) Sylvester low-rank approximation (kernel representation). We use the duality between least-squares and least-norm problems to show that (i) and (ii) are closely related to mosaic Hankel low-rank approximation. This allows us to apply to the approximate common divisor problem recent results on complexity and accuracy of computations for mosaic Hankel low-rank approximation. We develop optimization methods based on the variable projection principle both for image and kernel representation. These methods have linear complexity in the degrees of the polynomials for small and large $d$. We provide a software implementation of the developed methods, which is based on a software package for structured low-rank approximation.