Finding polynomial roots by dynamical systems – a case study

04/07/2020
by   Sergey Shemyakov, et al.
0

We investigate two well known dynamical systems that are designed to find roots of univariate polynomials by iteration: the methods known by Newton and by Ehrlich-Aberth. Both are known to have found all roots of high degree polynomials with good complexity. Our goal is to determine in which cases which of the two algorithms is more efficient. We come to the conclusion that Newton is faster when the polynomials are given by recursion so they can be evaluated in logarithmic time with respect to the degree, or when all the roots are all near the boundary of their convex hull. Conversely, Ehrlich-Aberth has the advantage when no fast evaluation of the polynomials is available, and when roots are in the interior of the convex hull of other roots.

READ FULL TEXT
research
12/07/2022

Generalized Companion Subresultants of Several Univariate Polynomials in Newton Basis

In this paper, the concept of companion subresultant for polynomials in ...
research
04/09/2020

The Weierstrass root finder is not generally convergent

Finding roots of univariate polynomials is one of the fundamental tasks ...
research
05/11/2021

Sums of Separable and Quadratic Polynomials

We study separable plus quadratic (SPQ) polynomials, i.e., polynomials t...
research
07/30/2019

Computing Approximate Common Factors of Matrix Polynomials

Computation of (approximate) polynomials common factors is an important ...
research
05/30/2017

Fast Computation of the Roots of Polynomials Over the Ring of Power Series

We give an algorithm for computing all roots of polynomials over a univa...
research
01/30/2020

The Lagrangian remainder of Taylor's series, distinguishes O(f(x)) time complexities to polynomials or not

The purpose of this letter is to investigate the time complexity consequ...
research
10/24/2018

Complexity, combinatorial positivity, and Newton polytopes

The Nonvanishing Problem asks if a coefficient of a polynomial is nonzer...

Please sign up or login with your details

Forgot password? Click here to reset