The Numerical Factorization of Polynomials

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.

The Computation of Multiple Roots of a Bernstein Basis Polynomial

“…A multiple root of a polynomial is ill-conditioned because a perturbation in its coefficients causes it to break up into simple roots, and standard methods for their computation yield unsatisfactory results because they may return simple roots. This has led to the development of new methods for the computation of multiple roots of a polynomial, many of which involve greatest common divisor (GCD) computations and polynomial deconvolutions [6,16,20,23,26,28,29,30,31]. Both these operations are ill-conditioned, and thus simple methods for these calculations yield bad results.…”

“…, r, are polynomials such that the degree of z i (y) is less than or equal to the degree of f i (y). The polynomials z i (y) that constrain (28) to have an exact solution are not unique, but uniqueness is imposed by selecting the polynomials z i (y) of minimum norm. This leads to a least squares equality problem, which can be solved by the QR decomposition.…”

Geometric modeling and regularization of algebraic problems

“…Modeling the factorization problem for polynomials including multivariate cases is given in [23] where the proof of the Factorization Manifold Theorem can be substantially simplified by citing Theorem 1 rather than essentially mirroring its proof.…”

“…Specifically taylored to the application of solving ill-posed algebraic problems in this paper, we prove a weak but sufficient version of the tubular neighborhood theorem for complex analytic manifolds in Euclidean spaces using the techniques of nonlinear least squares. The theorem and the proof fills a gap in the regularization theory of solving ill-posed algebraic problems and complete the works of numerical factorization [23,25] and numerical greatest common divisors of polynomials [30,26].…”

“…The geometric modeling and regularization from this perspective lead to robust algorithms such as those in accurate computation of multiple roots [24], greatest common divisors [16,26], polynomial factorizations [23,25], defective eigenvalue problems [27] and singular linear systems [29]. These algorithms are implemented in our software package NAClab [31].…”