Related papers: Dynamics of Newton-like root finding methods
We present an iterative root finding method for harmonic mappings in the complex plane, which is a generalization of Newton's method for analytic functions. The complex formulation of the method allows an analysis in a complex variables…
The $p$-adic Newton polygon is a visual tool that encodes information about the roots and factorization of a polynomial relative to a prime $p$. In this article, we investigate how the Newton polygon changes under polynomial composition. If…
Univariate polynomial root-finding is a classical subject, still important for modern computing. Frequently one seeks just the real roots of a polynomial with real coefficients. They can be approximated at a low computational cost if the…
Considering the L-function of exponential sums associated to a polynomial over a finite field F_q, Deligne proved that a reciprocal root's p-adic order is a rational number in the interval [0, 1]. Based on hypergeometric theory, in this…
We discuss a recursive family of iterative methods for the numerical approximation of roots of nonlinear functions in one variable. These methods are based on Newton-Cotes closed quadrature rules. We prove that when a quadrature rule with…
Using the interplay between topological, combinatorial, and geometric properties of polynomials and analytic results (primarily the covering structure and distortion estimates), we analyze a path-lifting method for finding approximate…
Univariate polynomial root-finding is both classical and important for modern computing. Frequently one seeks just the real roots of a polynomial with real coefficients. They can be approximated at a low computational cost if the polynomial…
We start with a random polynomial $P^{N}(z)$ of degree $N$ with independent coefficients. We then consider a new polynomial $P_{t}^{N}$ obtained by $\lceil Nt\rceil$ applications of a fractional differential operator of the form $z^{a}…
The Newton-Raphson method is a fundamental root-finding technique with numerous applications in physics. In this study, we propose a parameterized variant of the Newton-Raphson method, inspired by principles from physics. Through analytical…
The Newton-Raphson (N-R) method is useful to find the roots of a polynomial of degree n. However, this method is limited since it diverges for the case in which polynomials only have complex roots if a real initial condition is taken. In…
Iterating Newton's method symbolically for the general quadratic yields a rational function, the numerator and denominator of which are polynomials with highly composite coefficients.
Finding a Z-eigenpair of a symmetric tensor is equivalent to finding a KKT point of a sphere constrained minimization problem. Based on this equivalency, in this paper, we first propose a class of iterative methods to get a Z-eigenpair of a…
The purpose of this paper is to study some properties of the Newton maps associated to real quintic polynomials. First using the Tschirnhaus transformation, we reduce the study of Newton's method for general quintic polynomials to the case…
A new method of root finding is formulated that uses a numerical iterative process involving three points. A given function y = f(x) whose roots are desired is fitted and approximated by a polynomial function of the form P(x)= a(x-b)^N that…
We give an anecdotal discussion of the problem of searching for polynomials with all roots on the unit circle, whose coefficients are rational numbers subject to certain congruence conditions. We illustrate with an example from a…
We study the Cebyshev-Halley methods applied to the family of polynomials $f_{n,c}(z)=z^n+c$, for $n\ge 2$ and $c\in \mathbb{C}^{*}$. We prove the existence of parameters such that the immediate basins of attraction corresponding to the…
We introduce a new algorithm denoted DSC2 to isolate the real roots of a univariate square-free polynomial f with integer coefficients. The algorithm iteratively subdivides an initial interval which is known to contain all real roots of f.…
This paper investigates the cost of solving systems of sparse polynomial equations by homotopy continuation. First, a space of systems of $n$-variate polynomial equations is specified through $n$ monomial bases. The natural locus for the…
We devise a simple but remarkably accurate iterative routine for calculating the roots of a polynomial of any degree. We demonstrate that our results have significant improvement in accuracy over those obtained by methods used in popular…
An intriguing and unexpected result for students learning numerical analysis is that Newton's method, applied to the simple polynomial z^3 - 1 = 0 in the complex plane, leads to intricately interwoven basins of attraction of the roots. As…