Derivative free iterative methods with memory having higher R-order of convergence

2020 ◽  
Vol 21 (6) ◽  
pp. 641-648
Author(s):  
Pankaj Jain ◽  
Prem Bahadur Chand
Keyword(s):  
Iterative Methods ◽  
Order Of Convergence ◽  
Function Evaluation ◽  
Optimal Methods ◽  
New Methods ◽  
Derivative Free ◽  
Iteration Index ◽  
Order Four ◽  
With Memory ◽  
Theoretical Results

AbstractWe derive two iterative methods with memory for approximating a simple root of any nonlinear equation. For this purpose, we take two optimal methods without memory of order four and eight and convert them into the methods with memory without increasing any further function evaluation. These methods involve a self-accelerator (parameter) that depends upon the iteration index to increase the order of the optimal methods. Consequently, the efficiency of the new methods is considerably high as compared to the methods without memory. Some numerical examples are provided in support of the theoretical results.

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.

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 Solving Nondifferentiable Nonlinear Equations by New Steffensen-Type Iterative Methods with Memory

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
J. P. Jaiswal
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Efficiency Index ◽  
The Self ◽  
Function Evaluation ◽  
Eighth Order ◽  
Derivative Free ◽  
With Memory

It is attempted to present two derivative-free Steffensen-type methods with memory for solving nonlinear equations. By making use of a suitable self-accelerator parameter in the existing optimal fourth- and eighth-order without memory methods, the order of convergence has been increased without any extra function evaluation. Therefore, its efficiency index is also increased, which is the main contribution of this paper. The self-accelerator parameters are estimated using Newton’s interpolation. To show applicability of the proposed methods, some numerical illustrations are presented.

scholarly journals Efficient Four-Parametric with-and-without-Memory Iterative Methods Possessing High Efficiency Indices

2018 ◽  
Vol 2018 ◽  
pp. 1-12 ◽  
Author(s):  
Alicia Cordero ◽  
Moin-ud-Din Junjua ◽  
Juan R. Torregrosa ◽  
Nusrat Yasmin ◽  
Fiza Zafar
Keyword(s):  
Iterative Methods ◽  
High Efficiency ◽  
Nonlinear Function ◽  
Efficiency Index ◽  
Simple Zero ◽  
Eighth Order ◽  
New Family ◽  
Derivative Free ◽  
With Memory ◽  
Theoretical Results

We construct a family of derivative-free optimal iterative methods without memory to approximate a simple zero of a nonlinear function. Error analysis demonstrates that the without-memory class has eighth-order convergence and is extendable to with-memory class. The extension of new family to the with-memory one is also presented which attains the convergence order 15.5156 and a very high efficiency index 15.51561/4≈1.9847. Some particular schemes of the with-memory family are also described. Numerical examples and some dynamical aspects of the new schemes are given to support theoretical results.

scholarly journals Optimal High-Order Methods for Solving Nonlinear Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
S. Artidiello ◽  
A. Cordero ◽  
Juan R. Torregrosa ◽  
M. P. Vassileva
Keyword(s):  
Weight Function ◽  
Iterative Methods ◽  
Nonlinear Equations ◽  
Order Of Convergence ◽  
High Order ◽  
Function Technique ◽  
Numerical Tests ◽  
New Methods ◽  
Theoretical Results ◽  
Made In

A class of optimal iterative methods for solving nonlinear equations is extended up to sixteenth-order of convergence. We design them by using the weight function technique, with functions of three variables. Some numerical tests are made in order to confirm the theoretical results and to compare the new methods with other known ones.

Design and Analysis of a New Class of Derivative-Free Optimal Order Methods for Nonlinear Equations

2018 ◽  
Vol 15 (03) ◽  
pp. 1850010 ◽  
Author(s):  
Janak Raj Sharma ◽  
Ioannis K. Argyros ◽  
Deepak Kumar
Keyword(s):  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Optimal Order ◽  
Optimal Order Of Convergence ◽  
Numerical Examples ◽  
New Class ◽  
New Methods ◽  
Derivative Free ◽  
Local Convergence Analysis ◽  
Theoretical Results

We develop a general class of derivative free iterative methods with optimal order of convergence in the sense of Kung–Traub hypothesis for solving nonlinear equations. The methods possess very simple design, which makes them easy to remember and hence easy to implement. The Methodology is based on quadratically convergent Traub–Steffensen scheme and further developed by using Padé approximation. Local convergence analysis is provided to show that the iterations are locally well defined and convergent. Numerical examples are provided to confirm the theoretical results and to show the good performance of new methods.

scholarly journals Derivative-Free King’s Scheme for Multiple Zeros of Nonlinear Functions

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1242
Author(s):  
Ramandeep Behl ◽  
Sonia Bhalla ◽  
Eulalia Martínez ◽  
Majed Aali Alsulami
Keyword(s):  
Iterative Methods ◽  
Fourth Order ◽  
Order Of Convergence ◽  
Real Life ◽  
Multiple Roots ◽  
Computational Results ◽  
Nonlinear Functions ◽  
First Order ◽  
Life Problems ◽  
Derivative Free

There is no doubt that the fourth-order King’s family is one of the important ones among its counterparts. However, it has two major problems: the first one is the calculation of the first-order derivative; secondly, it has a linear order of convergence in the case of multiple roots. In order to improve these complications, we suggested a new King’s family of iterative methods. The main features of our scheme are the optimal convergence order, being free from derivatives, and working for multiple roots (m≥2). In addition, we proposed a main theorem that illustrated the fourth order of convergence. It also satisfied the optimal Kung–Traub conjecture of iterative methods without memory. We compared our scheme with the latest iterative methods of the same order of convergence on several real-life problems. In accordance with the computational results, we concluded that our method showed superior behavior compared to the existing 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 On a Derivative-Free Variant of King’s Family with Memory

2015 ◽  
Vol 2015 ◽  
pp. 1-5
Author(s):  
M. Sharifi ◽  
S. Karimi Vanani ◽  
F. Khaksar Haghani ◽  
M. Arab ◽  
S. Shateyi
Keyword(s):  
Nonlinear Equations ◽  
Numerical Experiments ◽  
Order Of Convergence ◽  
Derivative Free ◽  
With Memory

The aim of this paper is to construct a method with memory according to King’s family of methods without memory for nonlinear equations. It is proved that the proposed method possesses higherR-order of convergence using the same number of functional evaluations as King’s family. Numerical experiments are given to illustrate the performance of the constructed scheme.

scholarly journals Memory in a New Variant of King’s Family for Solving Nonlinear Systems

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1251
Author(s):  
Munish Kansal ◽  
Alicia Cordero ◽  
Sonia Bhalla ◽  
Juan R. Torregrosa
Keyword(s):  
Nonlinear Systems ◽  
Convergence Analysis ◽  
Numerical Experiments ◽  
Order Of Convergence ◽  
Recent Literature ◽  
Divided Difference ◽  
Difference Operators ◽  
New Variant ◽  
Order Four ◽  
With Memory

In the recent literature, very few high-order Jacobian-free methods with memory for solving nonlinear systems appear. In this paper, we introduce a new variant of King’s family with order four to solve nonlinear systems along with its convergence analysis. The proposed family requires two divided difference operators and to compute only one inverse of a matrix per iteration. Furthermore, we have extended the proposed scheme up to the sixth-order of convergence with two additional functional evaluations. In addition, these schemes are further extended to methods with memory. We illustrate their applicability by performing numerical experiments on a wide variety of practical problems, even big-sized. It is observed that these methods produce approximations of greater accuracy and are more efficient in practice, compared with the existing methods.

Sign in / Sign up

Export Citation Format

Share Document