Related papers: On the Efficient Global Dynamics of Newton's Metho…

We extend the algorithms of Robinson, Smyth, and McKee--Smyth to enumerate all real-rooted integer polynomials of a fixed degree, where the first few (at least three) leading coefficients are specified. Additionally, we introduce new linear…

We propose an efficient algorithm to compute the real roots of a sparse polynomial $f\in\mathbb{R}[x]$ having $k$ non-zero real-valued coefficients. It is assumed that arbitrarily good approximations of the non-zero coefficients are given…

We consider deep neural networks, in which the output of each node is a quadratic function of its inputs. Similar to other deep architectures, these networks can compactly represent any function on a finite training set. The main goal of…

For many fundamental problems in computational topology, such as unknot recognition and $3$-sphere recognition, the existence of a polynomial-time solution remains unknown. A major algorithmic tool behind some of the best known algorithms…

We provide a method for solving the roots of the general polynomial equation a[n]*x^n+a[n-1]*x^(n-1)+..+a1*x+a0=0. To do so, we express x as a powerseries of s, and calculate the first n-2 coefficients. We turn the polynomial equation into…

This paper proposes and develops a new Newton-type algorithm to solve subdifferential inclusions defined by subgradients of extended-real-valued prox-regular functions. The proposed algorithm is formulated in terms of the second-order…

We present an algorithm for isolating the roots of an arbitrary complex polynomial $p$ that also works for polynomials with multiple roots provided that the number $k$ of distinct roots is given as part of the input. It outputs $k$ pairwise…

We present an algorithm for computing discriminants and prime ideal decomposition in number fields. The algorithm is a refinement of a p-adic factorization method based on Newton polygons of higher order. The running-time and memory…

The purpose of this paper is to present three new methods for finding all simple zeros of polynomials simultaneously. First, we give a new method for finding simultaneously all simple zeros of polynomials constructed by applying the…

In the present study, we propose necessary and sufficient assumptions on the coefficients in order to only get distinct real roots of polynomials.

This paper is concerned with exact real solving of well-constrained, bivariate polynomial systems. The main problem is to isolate all common real roots in rational rectangles, and to determine their intersection multiplicities. We present…

We examine several of the normal-form multivariate polynomial rootfinding methods of Telen, Mourrain, and Van Barel and some variants of those methods. We analyze the performance of these variants in terms of their asymptotic temporal…

We analyze the performance of a variant of Newton method with quadratic regularization for solving composite convex minimization problems. At each step of our method, we choose regularization parameter proportional to a certain power of the…

Many machine learning models involve solving optimization problems. Thus, it is important to deal with a large-scale optimization problem in big data applications. Recently, subsampled Newton methods have emerged to attract much attention…

We study a variant of Newton's algorithm applied to under-determined systems of non-smooth equations. The notion of regularity employed in our work is based on Newton differentiability, which generalizes semi-smoothness. The classic notion…

Recursive maps of high order of convergence $m$ (say $m=2^{10}$ or $m=2^{20}$) induce certain monotone step functions from which one can filter relevant information needed to globally separate and compute the real roots of a function on a…

We provide an analogue of Wedderburn's factorization method for central polynomials with coefficients in an octonion division algebra, and present an algorithm for fully factoring polynomials of degree $n$ with $n$ conjugacy classes of…

A fast and weakly stable method for computing the zeros of a particular class of hypergeometric polynomials is presented. The studied hypergeometric polynomials satisfy a higher order differential equation and generalize Laguerre…

In 2015, Guth proved that if $S$ is a collection of $n$ $g$-dimensional semi-algebraic sets in $\mathbb{R}^d$ and if $D\geq 1$ is an integer, then there is a $d$-variate polynomial $P$ of degree at most $D$ so that each connected component…

This paper is triggered by the preprint "\emph{Computing Matrix Squareroot via Non Convex Local Search}" by Jain et al. (\textit{\textcolor{blue}{arXiv:1507.05854}}), which analyzes gradient-descent for computing the square root of a…