Complex dynamics of a sixth and seventh order family of root finding methods
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Complex dynamics of a sixth and seventh order family of root finding methods Debasis Sharma1
· Sanjaya Kumar Parhi1
Received: 17 July 2019 / Accepted: 25 May 2020 © Sociedad Española de Matemática Aplicada 2020
Abstract This paper intends to investigate the dynamical behavior of a parameter based sixth and seventh order convergent family of iterative methods in the complex plane. The fixed points of the rational function obtained from this family are described with their stability. Further to this, the stability regions for the fixed points are displayed using graphical tools. Parameter planes for independent free critical points are presented to determine stable and poor members of the family. Dynamical planes are also provided for various iterative methods of the family having important numerical properties. Keywords Iterative methods · Complex dynamics · Parameter space · Dynamical plane · Stability Mathematics Subject Classification 37F10 · 65D99 · 65H05 · 65P99 · 65Y20
1 Introduction The study of complex dynamical properties of the rational map associated with a family of iterative methods provides essential information about the stability and general convergence of its iterative schemes. In the recent past, many authors like Amat et al. [1–4], Artidiello et al. [6], Cordero et al. [13–15], Chicharro et al. [9,10], Chun et al. [11] and others [17, 18,20–22] have studied the dynamical behavior of different classes of iterative methods namely, Jarratt, Kim, Kou, King, Chebyshev-Halley, etc. Interesting numerical properties like attracting strange fixed points, periodical behavior, and some other anomalies have been found in these studies. The main interest of this article is to explore the dynamical characteristics of a uni-parametric family of iterative schemes by applying the complex dynamics tools. The study of parameter planes related to the free critical points of this family permits us to
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Debasis Sharma [email protected] Sanjaya Kumar Parhi [email protected]
1
Department of Mathematics, International Institute of Information Technology Bhubaneswar, Odisha 751003, India
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D. Sharma, S. K. Parhi
determine the good and bad numerical properties of different iterative methods associated with the family. This study gives an excellent idea of choosing a specific member of the family whose numerical behavior is stable. The sixth and seventh order convergent class of iterative schemes suggested by Cordero et al. [12] is given as follows: f (xn ) , f (xn ) ( f (xn ) + f (yn )) f (xn )2 z n = xn − θ − (1 − θ ) f (xn ) f (xn )( f (xn ) − f (yn )) f (z n ) xn+1 = z n − f [z n , yn ] + (z n − yn ) f [z n , xn , xn ] yn = xn −
(1)
The convergence order of (1) has been proved by the authors in [12]. Every member of this family has atleast sixth order of convergence and for θ = −1, the convergence order is seven. This manuscript investigates the dynamical behavior of the rational operator obtained by applying the above-mentioned family of methods on arbitrary quadratic polynomial p(z) : C →
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