Approximate GCD in Lagrange bases

TL;DR

This paper presents an algorithm for computing approximate GCDs of polynomials expressed in Lagrange bases, leveraging rootfinding via companion matrices without basis conversion, thus maintaining good numerical conditioning.

Contribution

The paper introduces a novel method to find approximate GCDs directly in Lagrange bases, avoiding basis change and preserving numerical stability.

Findings

Rootfinding via companion matrices is effective for Lagrange basis polynomials.

The method maintains good conditioning by avoiding basis conversion.

Computational cost is primarily due to eigenvalue computations, cubic in polynomial degree.

Abstract

For a pair of polynomials with real or complex coefficients, given in any particular basis, the problem of finding their GCD is known to be ill-posed. An answer is still desired for many applications, however. Hence, looking for a GCD of so-called approximate polynomials where this term explicitly denotes small uncertainties in the coefficients has received significant attention in the field of hybrid symbolic-numeric computation. In this paper we give an algorithm, based on one of Victor Ya. Pan, to find an approximate GCD for a pair of approximate polynomials given in a Lagrange basis. More precisely, we suppose that these polynomials are given by their approximate values at distinct known points. We first find each of their roots by using a Lagrange basis companion matrix for each polynomial, cluster the roots of each polynomial to identify multiple roots, and then "marry" the two polynomials to find their GCD. At no point do we change to the monomial basis, thus preserving the good conditioning properties of the original Lagrange basis. We discuss advantages and drawbacks of this method. The computational cost is dominated by the rootfinding step; unless special-purpose eigenvalue algorithms are used, the cost is cubic in the degrees of the polynomials. In principle, this cost could be reduced but we do not do so here.