scholarly journals Families of Rational Maps and Iterative Root-Finding Algorithms

1987 ◽  
Vol 125 (3) ◽  
pp. 467 ◽  
Author(s):  
Curt McMullen
Keyword(s):  
Rational Maps ◽  
Iterative Root ◽  
Root Finding

scholarly journals A Newton method for harmonic mappings in the plane

2019 ◽  
Vol 40 (4) ◽  
pp. 2777-2801
Author(s):  
Olivier Sète ◽  
Jan Zur
Keyword(s):  
Basins Of Attraction ◽  
Harmonic Mappings ◽  
Iterative Root ◽  
Number Of Solutions ◽  
Critical Set ◽  
Complex Formulation ◽  
Root Finding ◽  
Matlab Implementation ◽  
Root Finding Method ◽  
Finding Method

Abstract 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 spirit. For zeros close to poles of $f = h + \overline{g}$ we construct initial points for which the harmonic Newton iteration is guaranteed to converge. Moreover, we study the number of solutions of $f(z) = \eta $ close to the critical set of $f$ for certain $\eta \in \mathbb{C}$. We provide a MATLAB implementation of the method, and illustrate our results with several examples and numerical experiments, including phase plots and plots of the basins of attraction.

scholarly journals New stopping criteria for iterative root finding

2014 ◽  
Vol 1 (2) ◽  
pp. 140206 ◽  
Author(s):  
Jorgen L. Nikolajsen
Keyword(s):  
Work Load ◽  
Test Results ◽  
Iterative Root ◽  
Stopping Criteria ◽  
Root Finding

A set of simple stopping criteria is presented, which improve the efficiency of iterative root finding by terminating the iterations immediately when no further improvement of the roots is possible. The criteria use only the function evaluations already needed by the root finding procedure to which they are applied. The improved efficiency is achieved by formulating the stopping criteria in terms of fractional significant digits. Test results show that the new stopping criteria reduce the iteration work load by about one-third compared with the most efficient stopping criteria currently available. This is achieved without compromising the accuracy of the extracted roots.

scholarly journals DYNAMICS OF A FAMILY OF RATIONAL OPERATORS OF ARBITRARY DEGREE

2021 ◽  
Vol 26 (2) ◽  
pp. 188-208
Author(s):  
Beatriz Campos ◽  
Jordi Canela ◽  
Antonio Garijo ◽  
Pura Vindel
Keyword(s):  
Numerical Methods ◽  
Fourth Order ◽  
Point Of View ◽  
Rational Maps ◽  
Arbitrary Degree ◽  
Root Finding ◽  
General Family

In this paper we analyse the dynamics of a family of rational operators coming from a fourth-order family of root-finding algorithms. We first show that it may be convenient to redefine the parameters to prevent redundancies and unboundedness of problematic parameters. After reparametrization, we observe that these rational maps belong to a more general family Oa,n,k of degree n+k operators, which includes several other families of maps obtained from other numerical methods. We study the dynamics of Oa,n,k and discuss for which parameters n and k these operators would be suitable from the numerical point of view.

scholarly journals Pochhammer–Chree equation solver for dispersion correction of elastic waves in a (split) Hopkinson bar

2021 ◽  
pp. 095440622098050
Author(s):  
Hyunho Shin
Keyword(s):  
Elastic Waves ◽  
Sound Speed ◽  
Hopkinson Bar ◽  
Decimal Place ◽  
Dispersion Correction ◽  
Series Solutions ◽  
Iterative Root ◽  
Robust Algorithm ◽  
Root Finding ◽  
Split Hopkinson

A robust algorithm for solving the Bancroft version of the Pochhammer–Chree (PC) equation is developed based on the iterative root-finding process. The formulated solver not only obtains the conventional n-series solutions but also derives a new series of solutions, named m-series solutions. The n-series solutions are located on the PC function surface that relatively gradually varies in the vicinity of the roots, whereas the m-series solutions are located between two PC function surfaces with (nearly) positive and negative infinity values. The proposed solver obtains a series of sound speeds at exactly the frequencies necessary for dispersion correction, and the derived solutions are accurate to the ninth decimal place. The solver is capable of solving the PC equation up to n = 20 and m = 20 in the ranges of Poisson’s ratio ( ν) of 0.02 [Formula: see text]  ν [Formula: see text] 0.48, normalised frequency ( F) of F [Formula: see text] 30, and normalised sound speed ( C) of C [Formula: see text] 300. The developed algorithm was implemented in MATLAB®, which is available in the Supplemental Material (accessible online).

Shooting method for linear inviscid bi-global stability analysis of non-axisymmetric jets

2021 ◽  
pp. 1475472X2110054
Author(s):  
Nikhil Sohoni ◽  
Aniruddha Sinha
Keyword(s):  
Global Stability ◽  
Stability Problem ◽  
Shooting Method ◽  
Initial Guess ◽  
Iterative Root ◽  
One Dimensional ◽  
Root Finding ◽  
Axisymmetric Jets ◽  
Tracking Tasks ◽  
Good Agreement

The shooting method is commonly used to solve the linear parallel-flow stability problem for axisymmetric jets, i.e., a flow having one inhomogeneous direction. The present extension to two inhomogeneous directions – i.e., a bi-global stability problem – is motivated by inviscid non-axisymmetric jets. The azimuthal direction is Fourier transformed to obtain a set of coupled one-dimensional shooting problems that are solved by two-way integration from both radial boundaries – centreline and far field. The overall problem is formulated as one of iterative root-finding to match the solutions from the two integrations. The approach is validated against results from the well-established matrix method that discretizes the domain to obtain a matrix eigenvalue problem. We demonstrate very good agreement in two jet problems – an offset dual-stream jet, and a jet exiting from a nozzle with chevrons. A disadvantage of the shooting method is its sensitivity to the initial guess of the solution; however, this becomes an advantage when the need arises to track an eigensolution in a sweep over a problem parameter – say with increasing offset in the dual-stream jet, or with downstream distance from the nozzle exit. We demonstrate the performance of the shooting method in such tracking tasks.

NEWTON'S VERSUS HALLEY'S METHOD: A DYNAMICAL SYSTEMS APPROACH

2004 ◽  
Vol 14 (10) ◽  
pp. 3459-3475 ◽  
Author(s):  
GARETH E. ROBERTS ◽  
JEREMY HORGAN-KOBELSKI
Keyword(s):  
Dynamical Systems ◽  
Numerical Method ◽  
Systems Approach ◽  
Parameter Plane ◽  
Specific Interest ◽  
Iterative Root ◽  
Root Finding ◽  
Open Set ◽  
Cubic Polynomials ◽  
Parameter Values

We compare the iterative root-finding methods of Newton and Halley applied to cubic polynomials in the complex plane. Of specific interest are those "bad" polynomials for which a given numerical method contains an attracting cycle distinct from the roots. This implies the existence of an open set of initial guesses whose iterates do not converge to one of the roots (i.e. the numerical method fails). Searching for a set of bad parameter values leads to Mandelbrot-like sets and interesting figures in the parameter plane. We provide some analytic and geometric arguments to explain the contrasting parameter plane pictures. In particular, we show that there exists a sequence of parameter values λn for which the corresponding numerical method has a superattracting n cycle. The λn lie at the centers of a converging sequence of Mandelbrot-like sets.

On Global Aspects of Real Newton’s Method in Synthesis of Linkages

2006 ◽  
Author(s):  
Jin Xie ◽  
Kaiyin Yan ◽  
Yong Chen
Keyword(s):  
Dynamical System ◽  
Newton’S Method ◽  
Newton's Method ◽  
Global Analysis ◽  
Basins Of Attraction ◽  
Random Motion ◽  
Iterative Root ◽  
Degree Polynomial ◽  
Root Finding ◽  
Deterministic Dynamical System

Nonlinear equations arise from the synthesis of linkages. Newton’s method is one of the most accessible and easiest to implement of the iterative root-finding algorithms for these equations. As a discrete deterministic dynamical system, Newton’s method contains subsystems which have highly random motion. In a so-called chaotic zone, there is a rapid interchange between the basins of attraction for each root of the equation. Choosing initial points from such chaotic zone, one can obtain a certain number of roots or possible all of them under the Newton’s method. In this paper, how to locate the chaotic zones is addressed following the global analysis of real Newton’s method. It is show that there exist four chaotic zones for a general 4th degree polynomial. As an example, the equation derived from exact synthesis for five positions is solved.

Contrasts in the basins of attraction of structurally identical iterative root finding methods

2013 ◽  
Vol 219 (15) ◽  
pp. 7997-8008 ◽  
Author(s):  
Mário Basto ◽  
Luís P. Basto ◽  
Viriato Semiao ◽  
Francisco L. Calheiros
Keyword(s):  
Basins Of Attraction ◽  
Iterative Root ◽  
Root Finding

scholarly journals Two weighted eight-order classes of iterative root-finding methods

2014 ◽  
Vol 92 (9) ◽  
pp. 1790-1805 ◽  
Author(s):  
Santiago Artidiello ◽  
Alicia Cordero ◽  
Juan R. Torregrosa ◽  
María P. Vassileva
Keyword(s):  
Iterative Root ◽  
Root Finding
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