scholarly journals Two Iterative Methods with Memory Constructed by the Method of Inverse Interpolation and Their Dynamics

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1080
Author(s):  
Xiaofeng Wang ◽  
Mingming Zhu
Keyword(s):  
Iterative Method ◽  
Iterative Methods ◽  
Matrix Method ◽  
Basins Of Attraction ◽  
Convergence Order ◽  
Numerical Computations ◽  
Numerical Comparisons ◽  
New Methods ◽  
Inverse Interpolation ◽  
With Memory

In this paper, we obtain two iterative methods with memory by using inverse interpolation. Firstly, using three function evaluations, we present a two-step iterative method with memory, which has the convergence order 4.5616. Secondly, a three-step iterative method of order 10.1311 is obtained, which requires four function evaluations per iteration. Herzberger’s matrix method is used to prove the convergence order of new methods. Finally, numerical comparisons are made with some known methods by using the basins of attraction and through numerical computations to demonstrate the efficiency and the performance of the presented methods.

scholarly journals On Some Iterative Methods with Memory and High Efficiency Index for Solving Nonlinear Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Tahereh Eftekhari
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Order Of Convergence ◽  
High Efficiency ◽  
Efficiency Index ◽  
Order Convergence ◽  
Eighth Order ◽  
Additional Function ◽  
Numerical Comparisons ◽  
With Memory

Based on iterative methods without memory of eighth-order convergence proposed by Thukral (2012), some iterative methods with memory and high efficiency index are presented. We show that the order of convergence is increased without any additional function evaluations. Numerical comparisons are made to show the performance of the presented methods.

scholarly journals Ball Convergence of an Efficient Eighth Order Iterative Method Under Weak Conditions

Mathematics ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 260 ◽  
Author(s):  
Janak Sharma ◽  
Ioannis Argyros ◽  
Sunil Kumar
Keyword(s):  
Iterative Method ◽  
Iterative Methods ◽  
Higher Order ◽  
High Order ◽  
Convergence Order ◽  
Eighth Order ◽  
First Derivative ◽  
Ball Convergence ◽  
Derivatives Of

The convergence order of numerous iterative methods is obtained using derivatives of a higher order, although these derivatives are not involved in the methods. Therefore, these methods cannot be used to solve equations with functions that do not have such high-order derivatives, since their convergence is not guaranteed. The convergence in this paper is shown, relying only on the first derivative. That is how we expand the applicability of some popular methods.

scholarly journals A Family of Fourteenth-Order Convergent Iterative Methods for Solving Nonlinear Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Fiza Zafar ◽  
Gulshan Bibi
Keyword(s):  
Iterative Method ◽  
Iterative Methods ◽  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Efficiency Index ◽  
Numerical Examples ◽  
New Class ◽  
First Derivative ◽  
New Methods ◽  
Specific Step

We present a family of fourteenth-order convergent iterative methods for solving nonlinear equations involving a specific step which when combined with any two-step iterative method raises the convergence order by n+10, if n is the order of convergence of the two-step iterative method. This new class include four evaluations of function and one evaluation of the first derivative per iteration. Therefore, the efficiency index of this family is 141/5 =1.695218203. Several numerical examples are given to show that the new methods of this family are comparable with the existing methods.

SOME NEWTON-TYPE ITERATIVE METHODS WITH AND WITHOUT MEMORY FOR SOLVING NONLINEAR EQUATIONS

2014 ◽  
Vol 11 (05) ◽  
pp. 1350078 ◽  
Author(s):  
XIAOFENG WANG ◽  
TIE ZHANG
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Type Method ◽  
New Methods ◽  
Hermite Interpolating Polynomial ◽  
With Memory ◽  
Theoretical Results ◽  
Very High ◽  
High Computational Efficiency

In this paper, we present some three-point Newton-type iterative methods without memory for solving nonlinear equations by using undetermined coefficients method. The order of convergence of the new methods without memory is eight requiring the evaluations of three functions and one first-order derivative in per full iteration. Hence, the new methods are optimal according to Kung and Traubs conjecture. Based on the presented methods without memory, we present two families of Newton-type iterative methods with memory. Further accelerations of convergence speed are obtained by using a self-accelerating parameter. This self-accelerating parameter is calculated by the Hermite interpolating polynomial and is applied to improve the order of convergence of the Newton-type method. The corresponding R-order of convergence is increased from 8 to 9, [Formula: see text] and 10. The increase of convergence order is attained without any additional calculations so that the two families of the methods with memory possess a very high computational efficiency. Numerical examples are demonstrated to confirm theoretical results.

scholarly journals Modified Newton method to determine multiple zeros of nonlinear equations

2016 ◽  
Vol 11 (10) ◽  
pp. 5774-5780
Author(s):  
Rajinder Thukral
Keyword(s):  
Iterative Method ◽  
Nonlinear Equations ◽  
Newton Method ◽  
New Method ◽  
Convergence Order ◽  
Multiple Zeros ◽  
Numerical Comparisons ◽  
Modified Newton Method ◽  
Iteration Methods ◽  
Better Than

New one-point iterative method for solving nonlinear equations is constructed.  It is proved that the new method has the convergence order of three. Per iteration the new method requires two evaluations of the function.  Kung and Traub conjectured that the multipoint iteration methods, without memory based on n evaluations, could achieve maximum convergence order2n-1  but, the new method produces convergence order of three, which is better than expected maximum convergence order of two.  Hence, we demonstrate that the conjecture fails for a particular set of nonlinear equations. Numerical comparisons are included to demonstrate exceptional convergence speed of the proposed method using only a few function evaluations.

scholarly journals On two new families of iterative methods for solving nonlinear equations with optimal order

2011 ◽  
Vol 5 (1) ◽  
pp. 93-109 ◽  
Author(s):  
M. Heydari ◽  
S.M. Hosseini ◽  
G.B. Loghmani
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Efficiency Index ◽  
Convergence Order ◽  
Optimal Order ◽  
Eighth Order ◽  
Optimal Convergence ◽  
Numerical Comparisons ◽  
First Derivative ◽  
The Third

In this paper, two new families of eighth-order iterative methods for solving nonlinear equations is presented. These methods are developed by combining a class of optimal two-point methods and a modified Newton?s method in the third step. Per iteration the presented methods require three evaluations of the function and one evaluation of its first derivative and therefore have the efficiency index equal to 1:682. Kung and Traub conjectured that a multipoint iteration without memory based on n evaluations could achieve optimal convergence order 2n?1. Thus the new families of eighth-order methods agrees with the conjecture of Kung-Traub for the case n = 4. Numerical comparisons are made with several other existing methods to show the performance of the presented methods.

Derivative free iterative methods with memory of arbitrary high convergence order

2013 ◽  
Vol 67 (3) ◽  
pp. 565-580 ◽  
Author(s):  
Gustavo Fernández-Torres
Keyword(s):  
Iterative Methods ◽  
Convergence Order ◽  
Derivative Free ◽  
With Memory

scholarly journals Higher-Order Derivative-Free Iterative Methods for Solving Nonlinear Equations and Their Basins of Attraction

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1052 ◽  
Author(s):  
Jian Li ◽  
Xiaomeng Wang ◽  
Kalyanasundaram Madhu
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Complex Plane ◽  
Dynamical Behavior ◽  
Efficiency Index ◽  
Basins Of Attraction ◽  
Type Method ◽  
New Methods ◽  
Derivative Free ◽  
Higher Order Derivative

Based on the Steffensen-type method, we develop fourth-, eighth-, and sixteenth-order algorithms for solving one-variable equations. The new methods are fourth-, eighth-, and sixteenth-order converging and require at each iteration three, four, and five function evaluations, respectively. Therefore, all these algorithms are optimal in the sense of Kung–Traub conjecture; the new schemes have an efficiency index of 1.587, 1.682, and 1.741, respectively. We have given convergence analyses of the proposed methods and also given comparisons with already established known schemes having the same convergence order, demonstrating the efficiency of the present techniques numerically. We also studied basins of attraction to demonstrate their dynamical behavior in the complex plane.

scholarly journals Stability Analysis of Jacobian-Free Newton’s Iterative Method

Algorithms ◽  
2019 ◽  
Vol 12 (11) ◽  
pp. 236
Author(s):  
Abdolreza Amiri ◽  
Alicia Cordero ◽  
Mohammad Taghi Darvishi ◽  
Juan R. Torregrosa
Keyword(s):  
Stability Analysis ◽  
Iterative Method ◽  
Iterative Methods ◽  
Critical Points ◽  
Jacobian Matrix ◽  
Basins Of Attraction ◽  
Dynamical Analysis ◽  
Newton’S Iterative Method ◽  
Term Stability ◽  
Derivative Free

It is well known that scalar iterative methods with derivatives are highly more stable than their derivative-free partners, understanding the term stability as a measure of the wideness of the set of converging initial estimations. In multivariate case, multidimensional dynamical analysis allows us to afford this task and it is made on different Jacobian-free variants of Newton’s method, whose estimations of the Jacobian matrix have increasing order. The respective basins of attraction and the number of fixed and critical points give us valuable information in this sense.

scholarly journals Derivative-Free Iterative Methods with Some Kurchatov-Type Accelerating Parameters for Solving Nonlinear Systems

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 943
Author(s):  
Xiaofeng Wang ◽  
Yingfanghua Jin ◽  
Yali Zhao
Keyword(s):  
Nonlinear Systems ◽  
Iterative Methods ◽  
Numerical Experiments ◽  
Order Of Convergence ◽  
Nonlinear Ordinary Differential Equations ◽  
New Methods ◽  
Derivative Free ◽  
Initial Scheme ◽  
With Memory ◽  
Theoretical Results

Some Kurchatov-type accelerating parameters are used to construct some derivative-free iterative methods with memory for solving nonlinear systems. New iterative methods are developed from an initial scheme without memory with order of convergence three. New methods have the convergence order 2+5≈4.236 and 5, respectively. The application of new methods can solve standard nonlinear systems and nonlinear ordinary differential equations (ODEs) in numerical experiments. Numerical results support the theoretical results.

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