The Numerical Factorization of Polynomials
نویسندگان
چکیده
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.
منابع مشابه
Random Polynomials over Finite Fields: Statistics and Algorithms
Polynomials appear in many research articles of Philippe Flajolet. Here we concentrate only in papers where polynomials play a crucial role. These involve his studies of the shape of random polynomials over finite fields, the use of these results in the analysis of algorithms for the factorization of polynomials over finite fields, and the relation between the decomposition into irreducibles of...
متن کاملAn Algorithm for Approximate Factorization of Bivariate Polynomials
In this paper, we propose a new numerical method for factoring approximate bivariate polynomials over C. The method relies on Ruppert matrix and singular value decomposition. We also design a new reliable way to compute the approximate GCD of bivariate polynomials with floating-point coefficients. The algorithm has been implemented in Maple 9. The experiments show that the algorithms are very e...
متن کاملGeneralized numerical ranges of matrix polynomials
In this paper, we introduce the notions of C-numerical range and C-spectrum of matrix polynomials. Some algebraic and geometrical properties are investigated. We also study the relationship between the C-numerical range of a matrix polynomial and the joint C-numerical range of its coefficients.
متن کاملDeterministically Factoring Sparse Polynomials into Multilinear Factors and Sums of Univariate Polynomials
We present the first efficient deterministic algorithm for factoring sparse polynomials that split into multilinear factors and sums of univariate polynomials. Our result makes partial progress towards the resolution of the classical question posed by von zur Gathen and Kaltofen in [6] to devise an efficient deterministic algorithm for factoring (general) sparse polynomials. We achieve our goal...
متن کاملHigher rank numerical ranges of rectangular matrix polynomials
In this paper, the notion of rank-k numerical range of rectangular complex matrix polynomials are introduced. Some algebraic and geometrical properties are investigated. Moreover, for ϵ > 0; the notion of Birkhoff-James approximate orthogonality sets for ϵ-higher rank numerical ranges of rectangular matrix polynomials is also introduced and studied. The proposed denitions yield a natural genera...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- Foundations of Computational Mathematics
دوره 17 شماره
صفحات -
تاریخ انتشار 2017