Related papers: From Approximate Factorization to Root Isolation w…
Factorization of polynomials is one of the foundations of symbolic computation. Its applications arise in numerous branches of mathematics and other sciences. However, the present advanced programming languages such as C++ and J++, do not…
We present a practical implementation based on Newton's method to find all roots of several families of complex polynomials of degrees exceeding one billion ($10^9$) so that the observed complexity to find all roots is between $O(d\ln d)$…
In this paper we show a new way of constructing deterministic polynomial-time approximation algorithms for computing complex-valued evaluations of a large class of graph polynomials on bounded degree graphs. In particular, our approach…
In this paper we develop a new method which is a generalization of the Obreshkoff -Ehrlich method for the cases of algebraic, trigonometric and exponential polynomials. This method has a cubic rate of convergence. It is efficient from the…
We investigate Newton's method as a root finder for complex polynomials of arbitrary degree. While polynomial root finding continues to be one of the fundamental tasks of computing, with essential use in all areas of theoretical…
The absolute separation of a polynomial is the minimum nonzero difference between the absolute values of its roots. In the case of polynomials with integer coefficients, it can be bounded from below in terms of the degree and the height…
In this paper we investigate the following related problems: (A) the separation of $p$-adic roots of integer polynomials of a fixed degree and bounded height; and (B) counting integer polynomials of a fixed degree and bounded height with…
We present a new algorithm for solving the real roots of a bivariate polynomial system $\Sigma=\{f(x,y),g(x,y)\}$ with a finite number of solutions by using a zero-matching method. The method is based on a lower bound for bivariate…
Approximate computing has shown to provide new ways to improve performance and power consumption of error-resilient applications. While many of these applications can be found in image processing, data classification or machine learning, we…
We present the submatrix method, a highly parallelizable method for the approximate calculation of inverse p-th roots of large sparse symmetric matrices which are required in different scientific applications. We follow the idea of…
In this paper we provide a new method to certify that a nearby polynomial system has a singular isolated root with a prescribed multiplicity structure. More precisely, given a polynomial system f $=(f\_1, \ldots, f\_N)\in C[x\_1, \ldots,…
We present a modification of Newton's method to restore quadratic convergence for isolated singular solutions of polynomial systems. Our method is symbolic-numeric: we produce a new polynomial system which has the original multiple solution…
Existing algorithms for isolating real solutions of zero-dimensional polynomial systems do not compute the multiplicities of the solutions. In this paper, we define in a natural way the multiplicity of solutions of zero-dimensional…
This paper presents two new constructions related to singular solutions of polynomial systems. The first is a new deflation method for an isolated singular root. This construc-tion uses a single linear differential form defined from the…
A new version of the Graeffe algorithm for finding all the roots of univariate complex polynomials is proposed. It is obtained from the classical algorithm by a process analogous to renormalization of dynamical systems. This iteration is…
Rook polynomials are a powerful tool in the theory of restricted permutations. It is known that the rook polynomial of any board can be computed recursively, using a cell decomposition technique of Riordan. In this paper, we give a new…
Motivated by a connection with the factorization of multivariate polynomials, we study integral convex polytopes and their integral decompositions in the sense of the Minkowski sum. We first show that deciding decomposability of integral…
We propose a numerical analysis of a simplified version of the previous paper "Multiplicity hunting and approximating multiple roots of polynomial systems" written by the two authors.
An algorithm for computing an analytic function of a matrix $A$ is described. The algorithm is intended for the case where $A$ has some close eigenvalues, and clusters (subsets) of close eigenvalues are separated from each other. This…
This paper is concerned with certifying that a given point is near an exact root of an overdetermined or singular polynomial system with rational coefficients. The difficulty lies in the fact that consistency of overdetermined systems is…