scholarly journals Dynamical Techniques for Analyzing Iterative Schemes with Memory

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-13 ◽  
Author(s):  
Neha Choubey ◽  
A. Cordero ◽  
J. P. Jaiswal ◽  
J. R. Torregrosa
Keyword(s):  
Multidimensional Analysis ◽  
Iterative Schemes ◽  
Derivative Free ◽  
Small Domain ◽  
Hermite Interpolating Polynomial ◽  
Seventh Order ◽  
The Stability ◽  
With Memory ◽  
Theoretical Results ◽  
Made In

We construct a new biparametric three-point method with memory to highly improve the computational efficiency of its original partner, without adding functional evaluations. In this way, through different estimations of self-accelerating parameters, we have modified an existing seventh-order method. The parameters have been defined by Hermite interpolating polynomial that allows the accelerating effect. In particular, the R-order of the proposed iterative method with memory is increased from seven to ten. A real multidimensional analysis of the stability of this method with memory is made, in order to study its dependence on the initial estimations. Taking into account that usually iterative methods with memory are more stable than their derivative-free partners and the obtained results in this study, the behavior of this scheme shows to be excellent, but for a small domain. Numerical examples and comparison are also provided, confirming the theoretical results.

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 A family of two-point methods with memory for solving nonlinear equations

2011 ◽  
Vol 5 (2) ◽  
pp. 298-317 ◽  
Author(s):  
Miodrag Petkovic ◽  
Jovana Dzunic ◽  
Ljiljana Petkovic
Keyword(s):  
Nonlinear Equations ◽  
Free Parameter ◽  
Convergence Order ◽  
Optimal Order ◽  
Additional Function ◽  
Numerical Examples ◽  
Derivative Free ◽  
With Memory ◽  
Theoretical Results ◽  
High Computational Efficiency

An efficient family of two-point derivative free methods with memory for solving nonlinear equations is presented. It is proved that the convergence order of the proposed family is increased from 4 to at least 2 + ?6 ? 4.45, 5, 1/2 (5 + ?33) ? 5.37 and 6, depending on the accelerating technique. The increase of convergence order is attained using a suitable accelerating technique by varying a free parameter in each iteration. The improvement of convergence rate is achieved without any additional function evaluations meaning that the proposed methods with memory are very efficient. Moreover, the presented methods are more efficient than all existing methods known in literature in the class of two-point methods and three-point methods of optimal order eight. Numerical examples and the comparison with the existing two-point methods are included to confirm theoretical results and high computational efficiency. 2010 Mathematics Subject Classification. 65H05

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 King-Type Derivative-Free Iterative Families: Real and Memory Dynamics

Complexity ◽  
2017 ◽  
Vol 2017 ◽  
pp. 1-15 ◽  
Author(s):  
F. I. Chicharro ◽  
A. Cordero ◽  
J. R. Torregrosa ◽  
M. P. Vassileva
Keyword(s):  
Quadratic Polynomials ◽  
The Real ◽  
Iterative Schemes ◽  
Derivative Free ◽  
Parametric Plane ◽  
Real Behavior ◽  
Order Four ◽  
Real Stability ◽  
With Memory ◽  
Parametric Class

A biparametric family of derivative-free optimal iterative methods of order four, for solving nonlinear equations, is presented. From the error equation of this class, different families of iterative schemes with memory can be designed increasing the order of convergence up to six. The real stability analysis of the biparametric family without memory is made on quadratic polynomials, finding areas in the parametric plane with good performance. Moreover, in order to study the real behavior of the parametric class with memory, we associate it with a discrete multidimensional dynamical system. By analyzing the fixed and critical points of its vectorial rational function, we can select those methods with best stability properties.

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.

scholarly journals Convergence and Stability of a Parametric Class of Iterative Schemes for Solving Nonlinear Systems

Mathematics ◽  
2021 ◽  
Vol 9 (1) ◽  
pp. 86
Author(s):  
Alicia Cordero ◽  
Eva G. Villalba ◽  
Juan R. Torregrosa ◽  
Paula Triguero-Navarro
Keyword(s):  
Nonlinear Systems ◽  
Dynamical Behavior ◽  
Order Convergence ◽  
Hammerstein Integral Equation ◽  
Iterative Schemes ◽  
The Third ◽  
The Family ◽  
The Stability ◽  
Theoretical Results ◽  
Parametric Class

A new parametric class of iterative schemes for solving nonlinear systems is designed. The third- or fourth-order convergence, depending on the values of the parameter being proven. The analysis of the dynamical behavior of this class in the context of scalar nonlinear equations is presented. This study gives us important information about the stability and reliability of the members of the family. The numerical results obtained by applying different elements of the family for solving the Hammerstein integral equation and the Fisher’s equation confirm the theoretical results.

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

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
T. Lotfi ◽  
F. Soleymani ◽  
Z. Noori ◽  
A. Kılıçman ◽  
F. Khaksar Haghani
Keyword(s):  
Nonlinear Equation ◽  
High Efficiency ◽  
Efficiency Index ◽  
Simple Zero ◽  
Numerical Examples ◽  
Derivative Free ◽  
Derivative Free Methods ◽  
With Memory ◽  
Theoretical Results ◽  
High Computational Efficiency

Two families of derivative-free methods without memory for approximating a simple zero of a nonlinear equation are presented. The proposed schemes have an accelerator parameter with the property that it can increase the convergence rate without any new functional evaluations. In this way, we construct a method with memory that increases considerably efficiency index from81/4≈1.681to121/4≈1.861. Numerical examples and comparison with the existing methods are included to confirm theoretical results and high computational efficiency.

scholarly journals An Efficient Class of Traub-Steffensen-Like Seventh Order Multiple-Root Solvers with Applications

Symmetry ◽  
2019 ◽  
Vol 11 (4) ◽  
pp. 518 ◽  
Author(s):  
Janak Raj Sharma ◽  
Deepak Kumar ◽  
Ioannis K. Argyros
Keyword(s):  
Complex Geometry ◽  
Multiple Root ◽  
Basins Of Attraction ◽  
Higher Order ◽  
Simple Approach ◽  
New Family ◽  
Derivative Free ◽  
Seventh Order ◽  
Higher Order Derivative ◽  
The Stability

Many higher order multiple-root solvers that require derivative evaluations are available in literature. Contrary to this, higher order multiple-root solvers without derivatives are difficult to obtain, and therefore, such techniques are yet to be achieved. Motivated by this fact, we focus on developing a new family of higher order derivative-free solvers for computing multiple zeros by using a simple approach. The stability of the techniques is checked through complex geometry shown by drawing basins of attraction. Applicability is demonstrated on practical problems, which illustrates the efficient convergence behavior. Moreover, the comparison of numerical results shows that the proposed derivative-free techniques are good competitors of the existing techniques that require derivative evaluations in the iteration.

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.

Multidimensional analysis of the state of Russian studies on the education issues over 1993–2016

2018 ◽  
pp. 50-62
Author(s):  
Дмитрий Рубвальтер ◽  
Dmitry Rubvalter ◽  
Александр Либкинд ◽  
Alexander Libkind ◽  
Валентина Маркусова ◽  
...  
Keyword(s):  
Web Of Science ◽  
The State ◽  
Multidimensional Analysis ◽  
Russian Regions ◽  
Global Flow ◽  
Educational Issues ◽  
Foreign Countries ◽  
High Level ◽  
The Web ◽  
Made In

A multidimensional analysis of the state of Russian studies on the education issues over 1993–2016 was carried out based on the materials of the data contained in the Web of Science (SSCI, A & HCI and SCI-E databases). There were determined the dynamics and trends of a number of relevant indicators, such as the number of Russian publications by year, the share of these publications in the global flow of publications on education issues, the dynamics of the share of publications made in co-authorship with foreign colleagues, etc. A number of distributions of Russian publications on educational issues was compiled and analyzed: by journals, by Russian regions and cities, by organizations and authors of the publications. It was found that most of these distributions were characterized by a high level of non-uniformity. A list of journals (125 titles) in which Russian works on education issues had been published was compiled. Russian organizations (308) and domestic researchers (about two thousand) engaged in studying the issues of education were identified. It was discovered that more than 200 organizations and about 400 academicians from 60 foreign countries had participated in Russian studies on the education issues.

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