The numerical factorization of polynomials

TL;DR

This paper introduces a regularized approach to polynomial factorization that is stable under coefficient perturbations, providing theoretical guarantees and an efficient algorithm for accurate numerical factorization.

Contribution

It formulates a geometric framework for numerical polynomial factorization, establishing existence, uniqueness, stability, and convergence results, and presents a practical MATLAB algorithm.

Findings

Proves Lipschitz continuity and stability of numerical factorization.

Develops a robust algorithm capable of handling perturbed coefficients.

Demonstrates accurate polynomial factorization with floating point arithmetic.

Abstract

Polynomial factorization in conventional sense is an ill-posed problem due to its discontinuity with respect to coefficient perturbations, making it a challenge for numerical computation using empirical data. As a regularization, this paper formulates the notion of numerical factorization based on the geometry of polynomial spaces and the stratification of factorization manifolds. Furthermore, this paper establishes the existence, uniqueness, Lipschitz continuity, condition number, and convergence of the numerical factorization to the underlying exact factorization, leading to a robust and efficient algorithm with a MATLAB implementation capable of accurate polynomial factorizations using floating point arithmetic even if the coefficients are perturbed.