24,282 research outputs found
Dynamical analysis on cubic polynomials of Damped Traub s method for approximating multiple roots
[EN] In this paper, the performance of a parametric family including Newton¿s and Traub¿s schemes on multiple roots is analyzed. The local order of convergence on nonlinear equations with multiple roots is studied as well as the dynamical behavior in terms of the damping parameter on cubic polynomials with multiple roots. The fixed and critical points, and the associated parameter plane are some of the characteristic dynamical features of the family which are obtained in this work. From the analysis of these elements we identify members of the family of methods with good numerical properties in terms of stability and efficiency both for finding the simple and multiple roots, and also other ones with very unstable behavior.This research was partially supported by Ministerio de Economia y Competitividad MTM2014-52016-C2-2-P Spain and Generalitat Valenciana PROMETEO/2016/089 SpainVázquez-Lozano, JE.; Cordero Barbero, A.; Torregrosa Sánchez, JR. (2018). Dynamical analysis on cubic polynomials of Damped Traub s method for approximating multiple roots. Applied Mathematics and Computation. 328:82-99. https://doi.org/10.1016/j.amc.2018.01.043S829932
Modified quaternion Newton methods
We revisit the quaternion Newton method for computing roots of a class of quaternion valued functions and propose modified algorithms for finding multiple roots of simple polynomials. We illustrate the performance of these new methods by presenting several numerical experiments.The research was partially supported by the Research Centre of Mathematics of the University of Minho with the Portuguese Funds from the " Fundcao para a Ciencia e a Tecnologia", through the Project PEstOE/ MAT/ UI0013/ 2014
Stable Factorization of Strictly Hurwitz Polynomials
We propose a stable factorization procedure to generate a strictly Hurwitz polynomial from a given strictly positive even polynomial. This problem typically arises in applications involving real frequency techniques. The proposed method does not require any root finding algorithm. Rather, the factorization process is directly carried out to find the solution of a set of quadratic equations in multiple variables employing Newton’s method. The selection of the starting point for the iterations is not arbitrary, and involves interrelations among the coefficients of the set of solution polynomials differing only in the signs of their roots. It is hoped that this factorization technique will provide a motivation to perform the factorization of two-variable positive function to generate scattering Hurwitz polynomials in two variables for which root finding methods are not applicable
On the speed of convergence of Newton's method for complex polynomials
We investigate Newton's method for complex polynomials of arbitrary degree
, normalized so that all their roots are in the unit disk. For each degree
, we give an explicit set of
points with the following universal property: for every normalized polynomial
of degree there are starting points in whose Newton
iterations find all the roots with a low number of iterations: if the roots are
uniformly and independently distributed, we show that with probability at least
the number of iterations for these starting points to reach all
roots with precision is . This is an improvement of an earlier result in
\cite{Schleicher}, where the number of iterations is shown to be in the worst case (allowing multiple roots)
and for
well-separated (so-called -separated) roots.
Our result is almost optimal for this kind of starting points in the sense
that the number of iterations can never be smaller than for fixed
.Comment: 13 pages, 1 figure, to appear in AMS Mathematics of Computatio
Newton's method in practice II: The iterated refinement Newton method and near-optimal complexity for finding all roots of some polynomials of very large degrees
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 () so that the observed complexity to find all roots is between
and (measuring complexity in terms of number of
Newton iterations or computing time). All computations were performed
successfully on standard desktop computers built between 2007 and 2012.Comment: 24 pages, 19 figures. Update in v2 incorporates progress on
polynomials of even higher degrees (greater than 1 billion
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