scholarly journals A New Family of Fourth-Order Optimal Iterative Schemes and Remark on Kung and Traub’s Conjecture

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Chein-Shan Liu ◽  
Tsung-Lin Lee
Keyword(s):  
Weight Function ◽  
Fourth Order ◽  
Iterative Scheme ◽  
Convergence Order ◽  
Function Technique ◽  
Iterative Schemes ◽  
Optimal Convergence ◽  
New Family ◽  
Kung And Traub’S Conjecture

Kung and Traub conjectured that a multipoint iterative scheme without memory based on m evaluations of functions has an optimal convergence order p = 2 m − 1 . In the paper, we first prove that the two-step fourth-order optimal iterative schemes of the same class have a common feature including a same term in the error equations, resorting on the conjecture of Kung and Traub. Based on the error equations, we derive a constantly weighting algorithm obtained from the combination of two iterative schemes, which converges faster than the departed ones. Then, a new family of fourth-order optimal iterative schemes is developed by using a new weight function technique, which needs three evaluations of functions and whose convergence order is proved to be p = 2 3 − 1 = 4 .

scholarly journals A New Optimal Family of Schröder’s Method for Multiple Zeros

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1076 ◽  
Author(s):  
Ramandeep Behl ◽  
Arwa Jeza Alsolami ◽  
Bruno Antonio Pansera ◽  
Waleed M. Al-Hamdan ◽  
Mehdi Salimi ◽  
...  
Keyword(s):  
Fourth Order ◽  
High Order ◽  
Numerical Results ◽  
Convergence Order ◽  
Optimal Variant ◽  
Multiple Zeros ◽  
Optimal Convergence ◽  
New Family ◽  
Schröder’S Method

Here, we suggest a high-order optimal variant/modification of Schröder’s method for obtaining the multiple zeros of nonlinear uni-variate functions. Based on quadratically convergent Schröder’s method, we derive the new family of fourth -order multi-point methods having optimal convergence order. Additionally, we discuss the theoretical convergence order and the properties of the new scheme. The main finding of the present work is that one can develop several new and some classical existing methods by adjusting one of the parameters. Numerical results are given to illustrate the execution of our multi-point methods. We observed that our schemes are equally competent to other existing methods.

scholarly journals AN OPTIMAL FAMILY OF EIGHTH-ORDER ITERATIVE METHODS WITH AN INVERSE INTERPOLATORY RATIONAL FUNCTION ERROR CORRECTOR FOR NONLINEAR EQUATIONS

2017 ◽  
Vol 22 (3) ◽  
pp. 321-336 ◽  
Author(s):  
Young I. Kim ◽  
Ramandeep Behl ◽  
Sandile S. Motsa
Keyword(s):  
Rational Function ◽  
Fourth Order ◽  
Numerical Experiments ◽  
Iterative Scheme ◽  
Basins Of Attraction ◽  
Theoretical Development ◽  
Eighth Order ◽  
Main Motivation ◽  
Iterative Schemes ◽  
Computational Properties

The main motivation of this study is to propose an optimal scheme with an inverse interpolatory rational function error corrector in a general way that can be applied to any existing optimal multi-point fourth-order iterative scheme whose first sub step employs Newton’s method to further produce optimal eighth-order iterative schemes. In addition, we also discussed the theoretical and computational properties of our scheme. Variety of concrete numerical experiments and basins of attraction are extensively treated to confirm the theoretical development.

A new class of three-point methods with optimal convergence order eight and its dynamics

2014 ◽  
Vol 68 (2) ◽  
pp. 261-288 ◽  
Author(s):  
Taher Lotfi ◽  
Somayeh Sharifi ◽  
Mehdi Salimi ◽  
Stefan Siegmund
Keyword(s):  
Convergence Order ◽  
Optimal Convergence ◽  
New Class

scholarly journals Convergence of Infinite Family of Multivalued Quasi-Nonexpansive Mappings Using Multistep Iterative Processes

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Renu Chugh ◽  
Sanjay Kumar
Keyword(s):  
Real Banach Space ◽  
Iterative Scheme ◽  
Nonexpansive Mappings ◽  
Infinite Family ◽  
Finite Family ◽  
Countable Family ◽  
Uniformly Convex ◽  
Iterative Schemes ◽  
Strong And Weak Convergence ◽  
Convergence Results

We prove strong and weak convergence results using multistep iterative sequences for countable family of multivalued quasi-nonexpansive mappings by using some conditions in uniformly convex real Banach space. The results presented extended and improved the corresponding result of Zhang et al. (2013), Bunyawat and Suantai (2012), and some others from finite family, one countable family, and two countable families tok-number of countable families of multivalued quasi-nonexpansive mappings. Also we used a numerical example in C++ computational programs to prove that the iterative scheme we used has better rate of convergence than other existing iterative schemes.

Mixed Schemes for Fourth-Order DIV Equations

2019 ◽  
Vol 19 (2) ◽  
pp. 341-357
Author(s):  
Ronghong Fan ◽  
Yanru Liu ◽  
Shuo Zhang
Keyword(s):  
Theoretical Analysis ◽  
Fourth Order ◽  
Convergence Rates ◽  
Mixed Formulation ◽  
Optimal Convergence ◽  
Mixed Formulations ◽  
Optimal Convergence Rates

AbstractIn this paper, stable mixed formulations are designed and analyzed for the quad div problems under two frameworks presented in [23] and [22], respectively. Analogue discretizations are given with respect to the mixed formulation, and optimal convergence rates are observed, which confirm the theoretical analysis.

scholarly journals A New Class of Iterative Processes for Solving Nonlinear Systems by Using One Divided Differences Operator

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 776 ◽  
Author(s):  
Alicia Cordero ◽  
Cristina Jordán ◽  
Esther Sanabria ◽  
Juan R. Torregrosa
Keyword(s):  
Nonlinear Systems ◽  
Boundary Problem ◽  
Fourth Order ◽  
Difference Operator ◽  
Divided Difference ◽  
Order Convergence ◽  
New Class ◽  
Numerical Tests ◽  
New Family ◽  
Theoretical Results

In this manuscript, a new family of Jacobian-free iterative methods for solving nonlinear systems is presented. The fourth-order convergence for all the elements of the class is established, proving, in addition, that one element of this family has order five. The proposed methods have four steps and, in all of them, the same divided difference operator appears. Numerical problems, including systems of academic interest and the system resulting from the discretization of the boundary problem described by Fisher’s equation, are shown to compare the performance of the proposed schemes with other known ones. The numerical tests are in concordance with the theoretical results.

scholarly journals Fixed Point Root-Finding Methods of Fourth-Order of Convergence

Symmetry ◽  
2019 ◽  
Vol 11 (6) ◽  
pp. 769 ◽  
Author(s):  
Alicia Cordero ◽  
Lucía Guasp ◽  
Juan R. Torregrosa
Keyword(s):  
Weight Function ◽  
Dynamical Behavior ◽  
Nonlinear Problems ◽  
Dynamical Analysis ◽  
Function Technique ◽  
Weight Functions ◽  
Good Tool ◽  
Quadratic Polynomials ◽  
Root Finding ◽  
The Stability

In this manuscript, by using the weight-function technique, a new class of iterative methods for solving nonlinear problems is constructed, which includes many known schemes that can be obtained by choosing different weight functions. This weight function, depending on two different evaluations of the derivative, is the unique difference between the two steps of each method, which is unusual. As it is proven that all the members of the class are optimal methods in the sense of Kung-Traub’s conjecture, the dynamical analysis is a good tool to determine the best elements of the family in terms of stability. Therefore, the dynamical behavior of this class on quadratic polynomials is studied in this work. We analyze the stability of the presented family from the multipliers of the fixed points and critical points, along with their associated parameter planes. In addition, this study enables us to select the members of the class with good stability properties.

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