JULIA SETS OF GENERALIZED NEWTON'S METHOD

Fractals ◽  
2007 ◽  
Vol 15 (04) ◽  
pp. 323-336 ◽  
Author(s):  
XINGYUAN WANG ◽  
TINGTING WANG
Keyword(s):  
Newton’S Method ◽  
Iteration Method ◽  
Newton's Method ◽  
Julia Sets ◽  
Basins Of Attraction ◽  
Principal Value ◽  
Extraneous Fixed Points ◽  
Sets Theory ◽  
Steffensen Method ◽  
Generalized Newton’S Method

The Julia sets theory of generalized Newton's method is analyzed and the Julia sets of generalized Newton's method are constructed using the iteration method. From the research we find that: (1) the basins of attraction of the Julia sets of generalized Newton's method depend on the roots of the equation and their orders and also the existence of the extraneous fixed points; (2) the Steffensen method is an exception to the law given in (1); and (3) if the order of the root is decimal, then the different choice of the range of the principal value of the phase angle will cause a different evolvement of the Julia sets.

Tail-Enabled Spiraling Maneuver for Gliding Robotic Fish

2014 ◽  
Vol 136 (4) ◽  
Author(s):  
Feitian Zhang ◽  
Fumin Zhang ◽  
Xiaobo Tan
Keyword(s):  
Newton’S Method ◽  
Newton's Method ◽  
Three Dimensional ◽  
Relative Equilibria ◽  
Basins Of Attraction ◽  
Robotic Fish ◽  
Experimental Results ◽  
Underwater Gliders ◽  
Local Asymptotic Stability ◽  
Turning Radius

Gliding robotic fish, a new type of underwater robot, combines both strengths of underwater gliders and robotic fish, featuring long operation duration and high maneuverability. In this paper, we present both analytical and experimental results on a novel gliding motion, tail-enabled three-dimensional (3D) spiraling, which is well suited for sampling a water column. A dynamic model of a gliding robotic fish with a deflected tail is first established. The equations for the relative equilibria corresponding to steady-state spiraling are derived and then solved recursively using Newton's method. The region of convergence for Newton's method is examined numerically. We then establish the local asymptotic stability of the computed equilibria through Jacobian analysis and further numerically explore the basins of attraction. Experiments have been conducted on a fish-shaped miniature underwater glider with a deflected tail, where a gliding-induced 3D spiraling maneuver is confirmed. Furthermore, consistent with model predictions, experimental results have shown that the achievable turning radius of the spiraling can be as small as less than 0.4 m, demonstrating the high maneuverability.

The Application of Newton’s Method to the Problem of Elastic Stability

1972 ◽  
Vol 39 (4) ◽  
pp. 1060-1065 ◽  
Author(s):  
E. Riks
Keyword(s):  
Numerical Solution ◽  
Newton’S Method ◽  
Iteration Method ◽  
Newton's Method ◽  
Elastic Stability ◽  
Auxiliary Equation

The numerical solution of problems of elastic stability through the use of the iteration method of Newton is examined. It is found that if the equations of equilibrium are completed by a simple auxiliary equation, problems governed by a snapping condition can, in principle, always be calculated as long as the problem at hand is properly formulated. The effectiveness of the proposed procedure is demonstrated by means of an elementary example.

scholarly journals Fractal Newton basins

2006 ◽  
Vol 2006 ◽  
pp. 1-16 ◽  
Author(s):  
M. L. Sahari ◽  
I. Djellit
Keyword(s):  
Newton’S Method ◽  
Newton's Method ◽  
Julia Sets ◽  
Main Tool ◽  
Chaotic Sequence ◽  
Solutions To Equations ◽  
Main Interest ◽  
Dynamical Plane ◽  
Complex Functions ◽  
Cubic Polynomials

The dynamics of complex cubic polynomials have been studied extensively in the recent years. The main interest in this work is to focus on the Julia sets in the dynamical plane, and then is consecrated to the study of several topics in more detail. Newton's method is considered since it is the main tool for finding solutions to equations, which leads to some fantastic images when it is applied to complex functions and gives rise to a chaotic sequence.

scholarly journals A class of Newton maps with Julia sets of Lebesgue measure zero

2021 ◽  
Author(s):  
Mareike Wolff
Keyword(s):  
Lebesgue Measure ◽  
Newton’S Method ◽  
Newton's Method ◽  
Measure Zero ◽  
Julia Set ◽  
Julia Sets ◽  
Lebesgue Measure Zero ◽  
Almost Everywhere ◽  
General Conditions

AbstractLet $$g(z)=\int _0^zp(t)\exp (q(t))\,dt+c$$ g ( z ) = ∫ 0 z p ( t ) exp ( q ( t ) ) d t + c where p, q are polynomials and $$c\in {\mathbb {C}}$$ c ∈ C , and let f be the function from Newton’s method for g. We show that under suitable assumptions on the zeros of $$g''$$ g ′ ′ the Julia set of f has Lebesgue measure zero. Together with a theorem by Bergweiler, our result implies that $$f^n(z)$$ f n ( z ) converges to zeros of g almost everywhere in $${\mathbb {C}}$$ C if this is the case for each zero of $$g''$$ g ′ ′ that is not a zero of g or $$g'$$ g ′ . In order to prove our result, we establish general conditions ensuring that Julia sets have Lebesgue measure zero.

scholarly journals Increasing the interval of convergence for a generalized Newton's method of solving nonlinear equations

2016 ◽  
pp. 7-12
Author(s):  
А.Н. Громов
Keyword(s):  
Continuous Function ◽  
Nonlinear Equations ◽  
Newton’S Method ◽  
Newton's Method ◽  
The Real ◽  
Real Roots ◽  
Complex Roots ◽  
Transcendental Equations ◽  
Generalized Newton’S Method

Рассмотрен подход к построению расширения промежутка сходимости ранее предложенного обобщения метода Ньютона для решения нелинейных уравнений одного переменного. Подход основан на использовании свойства ограниченности непрерывной функции, определенной на отрезке. Доказано, что для поиска действительных корней вещественнозначного многочлена с комплексными корнями предложенный подход дает итерации с нелокальной сходимостью. Результат обобщен на случай трансцендентных уравнений. An approach to the construction of an extended interval of convergence for a previously proposed generalization of Newton's method to solve nonlinear equations of one variable. This approach is based on the boundedness of a continuous function defined on a segment. It is proved that, for the search for the real roots of a real-valued polynomial with complex roots, the proposed approach provides iterations with nonlocal convergence. This result is generalized to the case transcendental equations.

scholarly journals On the robust Newton’s method with the Mann iteration and the artistic patterns from its dynamics

2021 ◽  
Author(s):  
Krzysztof Gdawiec ◽  
Wiesław Kotarski ◽  
Agnieszka Lisowska
Keyword(s):  
Newton’S Method ◽  
Newton's Method ◽  
Basins Of Attraction ◽  
Picard Iteration ◽  
Mann Iteration ◽  
Iteration Parameter ◽  
Root Finding ◽  
Base Method ◽  
Fabric Design ◽  
Root Finding Method

AbstractThere are two main aims of this paper. The first one is to show some improvement of the robust Newton’s method (RNM) introduced recently by Kalantari. The RNM is a generalisation of the well-known Newton’s root finding method. Since the base method is undefined at critical points, the RNM allows working also at such points. In this paper, we improve the RNM method by applying the Mann iteration instead of the standard Picard iteration. This leads to an essential decrease in the number of root finding steps without visible destroying the sharp boundaries among the basins of attractions presented in polynomiographs. Furthermore, we investigate visually the dynamics of the RNM with the Mann iteration together with the basins of attraction for varying Mann’s iteration parameter with the help of polynomiographs for several polynomials. The second aim of this paper is to present the intriguing polynomiographs obtained from the dynamics of the RNM with the Mann iteration under various sequences used in this iteration. The obtained polynomiographs differ considerably from the ones obtained with the RNM and are interesting from the artistic perspective. Moreover, they can easily find applications in wallpaper or fabric design.

INFINITE BASINS OF JULIA SETS

2008 ◽  
Vol 18 (10) ◽  
pp. 3169-3173
Author(s):  
FİGEN ÇİLİNGİR
Keyword(s):  
Entire Function ◽  
Fixed Points ◽  
Newton’S Method ◽  
Newton's Method ◽  
Julia Sets ◽  
Rational Functions ◽  
Complex Parameter

The goal of this paper is to investigate the iterative behavior of a particular class of rational functions which arise from Newton's method applied to the entire function (z2 + c)eQ(z) where c is a complex parameter and Q is a nonconstant polynomial with deg(Q) ≤ 2. In particular, the basins of attracting fixed points will be described.

Influence of the multiplicity of the roots on the basins of attraction of Newton’s method

2013 ◽  
Vol 66 (3) ◽  
pp. 431-455 ◽  
Author(s):  
José M. Gutiérrez ◽  
Luis J. Hernández-Paricio ◽  
Miguel Marañón-Grandes ◽  
M. Teresa Rivas-Rodríguez
Keyword(s):  
Newton’S Method ◽  
Newton's Method ◽  
Basins Of Attraction
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