scholarly journals Higher Order Methods for Nonlinear Equations and Their Basins of Attraction

Mathematics ◽  
2016 ◽  
Vol 4 (2) ◽  
pp. 22 ◽  
Author(s):  
Kalyanasundaram Madhu ◽  
Jayakumar Jayaraman
Keyword(s):  
Nonlinear Equations ◽  
Basins Of Attraction ◽  
Higher Order ◽  
Higher Order Methods

scholarly journals Memorizing Schröder’s Method as an Efficient Strategy for Estimating Roots of Unknown Multiplicity

Mathematics ◽  
2021 ◽  
Vol 9 (20) ◽  
pp. 2570
Author(s):  
Alicia Cordero ◽  
Beny Neta ◽  
Juan R. Torregrosa
Keyword(s):  
Order Of Convergence ◽  
Iterative Scheme ◽  
Basins Of Attraction ◽  
Higher Order ◽  
Multiple Roots ◽  
Higher Order Methods ◽  
Efficient Strategy ◽  
Schröder’S Method ◽  
With Memory

In this paper, we propose, to the best of our knowledge, the first iterative scheme with memory for finding roots whose multiplicity is unknown existing in the literature. It improves the efficiency of a similar procedure without memory due to Schröder and can be considered as a seed to generate higher order methods with similar characteristics. Once its order of convergence is studied, its stability is analyzed showing its good properties, and it is compared numerically in terms of their basins of attraction with similar schemes without memory for finding multiple roots.

scholarly journals Higher-Order Root-Finding Algorithms and Their Basins of Attraction

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Amir Naseem ◽  
M. A. Rehman ◽  
Thabet Abdeljawad
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Basins Of Attraction ◽  
Higher Order ◽  
Complex Polynomials ◽  
Convergence Criteria ◽  
Root Finding ◽  
Variational Iteration ◽  
Order Root ◽  
Iteration Technique

In this paper, we proposed and analyzed three new root-finding algorithms for solving nonlinear equations in one variable. We derive these algorithms with the help of variational iteration technique. We discuss the convergence criteria of these newly developed algorithms. The dominance of the proposed algorithms is illustrated by solving several test examples and comparing them with other well-known existing iterative methods in the literature. In the end, we present the basins of attraction using some complex polynomials of different degrees to observe the fractal behavior and dynamical aspects of the proposed algorithms.

scholarly journals An Efficient Class of Traub–Steffensen-Type Methods for Computing Multiple Zeros

Axioms ◽  
2019 ◽  
Vol 8 (2) ◽  
pp. 65 ◽  
Author(s):  
Deepak Kumar ◽  
Janak Raj Sharma ◽  
Clemente Cesarano
Keyword(s):  
Iterative Methods ◽  
Basins Of Attraction ◽  
Higher Order ◽  
Order Derivative ◽  
Third Order ◽  
Higher Order Methods ◽  
Multiple Zeros ◽  
Derivative Free ◽  
Graphical Tool ◽  
Numerical Experimentation

Numerous higher-order methods with derivative evaluations are accessible in the literature for computing multiple zeros. However, higher-order methods without derivatives are very rare for multiple zeros. Encouraged by this fact, we present a family of third-order derivative-free iterative methods for multiple zeros that require only evaluations of three functions per iteration. Convergence of the proposed class is demonstrated by means of using a graphical tool, namely basins of attraction. Applicability of the methods is demonstrated through numerical experimentation on different functions that illustrates the efficient behavior. Comparison of numerical results shows that the presented iterative methods are good competitors to the existing techniques.

Higher Order Two-Point Efficient Family of Halley Type Methods for Simple Roots

2016 ◽  
Vol 13 (04) ◽  
pp. 1641016 ◽  
Author(s):  
Ramandeep Behl ◽  
S. S. Motsa
Keyword(s):  
Nonlinear Equations ◽  
Computational Cost ◽  
Basins Of Attraction ◽  
Higher Order ◽  
The Other ◽  
Sixth Order ◽  
Order Derivative ◽  
Initial Value ◽  
First Order ◽  
The Family

In this paper, we proposed a new highly efficient two-point sixth-order family of Halley type methods that do not require any second-order derivative evaluation for obtaining simple roots of nonlinear equations, numerically. In terms of computational cost, each member of the family requires two-function and two first-order derivative evaluations per iteration. On the account of the results obtained, it is found that our proposed methods are efficient and show better performance than existing sixth-order methods available in the literature. Further, it is also noted that larger basins of attraction belong to our methods as compared to the existing ones. On the other hand, the existing methods are slower and have darker basins while some of them are too sensitive upon the choice of the initial value.

scholarly journals A Characterization of the Dynamics of Schröder’s Method for Polynomials with Two Roots

2021 ◽  
Vol 5 (1) ◽  
pp. 25
Author(s):  
Víctor Galilea ◽  
José M. Gutiérrez
Keyword(s):  
Nonlinear Equations ◽  
Complex Plane ◽  
Iterative Process ◽  
Dynamical Behavior ◽  
Basins Of Attraction ◽  
Schröder’S Method

The purpose of this work is to give a first approach to the dynamical behavior of Schröder’s method, a well-known iterative process for solving nonlinear equations. In this context, we consider equations defined in the complex plane. By using topological conjugations, we characterize the basins of attraction of Schröder’s method applied to polynomials with two roots and different multiplicities. Actually, we show that these basins are half-planes or circles, depending on the multiplicities of the roots. We conclude our study with a graphical gallery that allow us to compare the basins of attraction of Newton’s and Schröder’s method applied to some given polynomials.

Sign in / Sign up

Export Citation Format

Share Document