scholarly journals Erratum to: “A new proof of interval extension of the classic Ostrowski’s method and its modified method for computing the enclosure solutions of nonlinear equations” by “Tahereh Eftekhari”

2014 ◽  
Vol 69 (1) ◽  
pp. 167-167
Author(s):  
Tahereh Eftekhari
Keyword(s):  
Nonlinear Equations ◽  
Modified Method ◽  
Interval Extension ◽  
Ostrowski’S Method

scholarly journals An Efficient Numerical Method for a Class of Nonlinear Volterra Integro-Differential Equations

2018 ◽  
Vol 2018 ◽  
pp. 1-7 ◽  
Author(s):  
M. H. Daliri Birjandi ◽  
J. Saberi-Nadjafi ◽  
A. Ghorbani
Keyword(s):  
Differential Equations ◽  
Convergence Rate ◽  
Numerical Method ◽  
Collocation Method ◽  
Nonlinear Equations ◽  
Iteration Method ◽  
Spectral Collocation Method ◽  
Spectral Collocation ◽  
Modified Method ◽  
Computational Work

We investigate an efficient numerical method for solving a class of nonlinear Volterra integro-differential equations, which is a combination of the parametric iteration method and the spectral collocation method. The implementation of the modified method is demonstrated by solving several nonlinear Volterra integro-differential equations. The results reveal that the developed method is easy to implement and avoids the additional computational work. Furthermore, the method is a promising approximate tool to solve this class of nonlinear equations and provides us with a convenient way to control and modify the convergence rate of the solution.

scholarly journals Modifications of the continuation method for the solution of systems of nonlinear equations

1979 ◽  
Vol 2 (2) ◽  
pp. 299-308
Author(s):  
G. R. Lindfield ◽  
D. C. Simpson
Keyword(s):  
Nonlinear Equations ◽  
Continuation Method ◽  
Systems Of Nonlinear Equations ◽  
Test Problems ◽  
System Of Nonlinear Equations ◽  
Modified Method ◽  
Wide Range

Modifications are proposed to the Davidenko-Broyden algorithm for the solution of a system of nonlinear equations. The aim of the modifications is to reduce the overall number of function evaluations by limiting the number of function evaluations for any one subproblem. To do this alterations are made to the strategy used in determining the subproblems to be solved. The modifications are compared with other methods for a wide range of test problems, and are shown to significantly reduce the number of function evaluations for the difficult problems. For the easier problems the modified method is equivalent to the Davidenko-Broyden algorithm.

scholarly journals Efficacy of Optimal Methods for Nonlinear Equations with Chemical Engineering Applications

2019 ◽  
Vol 2019 ◽  
pp. 1-11 ◽  
Author(s):  
Obadah Said Solaiman ◽  
Ishak Hashim
Keyword(s):  
Nonlinear Equations ◽  
Chemical Engineering ◽  
High Efficiency ◽  
Binary Solution ◽  
Real Life ◽  
Chemical Reactor ◽  
Efficiency Index ◽  
Optimal Methods ◽  
Azeotropic Point ◽  
Modified Method

In this study, we propose a modified predictor-corrector Newton-Halley (MPCNH) method for solving nonlinear equations. The proposed sixteenth-order MPCNH is free of second derivatives and has a high efficiency index. The convergence analysis of the modified method is discussed. Different problems were tested to demonstrate the applicability of the proposed method. Some are real life problems such as a chemical equilibrium problem (conversion in a chemical reactor), azeotropic point of a binary solution, and volume from van der Waals equation. Several comparisons with other optimal and nonoptimal iterative techniques of equal order are presented to show the efficiency of the modified method and to clarify the question, are the optimal methods always good for solving nonlinear equations?

scholarly journals Dichotomy. Dichotomy? Dichotomy! Basic provisions, problems of terminology and inspection analysis of the method of dichotomy

2020 ◽  
pp. 93-110
Author(s):  
Andrey Maistrenko ◽  
◽  
Konstantin Maistrenko ◽  
Anatoliy Svetlakov ◽  
◽  
...  
Keyword(s):  
Convergence Rate ◽  
Automatic Control ◽  
Control Systems ◽  
Nonlinear Equations ◽  
Approximate Solutions ◽  
Distinctive Features ◽  
Automatic Control Systems ◽  
Modified Method ◽  
Scalar Equations ◽  
Low Rate

When creating modern systems of automatic control of various processes and objects operating in real time, very often one has to face the problem of solving various kinds of nonlinear scalar equations. Currently, there are a number of computational methods and algorithms for its solution, one of which is the dichotomy method. This method has a number of advantages in comparison with other known methods for solving nonlinear equations, but at present it has not found wide practical use. The main reason for its low popularity is the low rate of convergence of the sequence of approximate solutions and a large amount of computation required to obtain sufficiently accurate solutions. The purpose of the study is to consider in detail distinctive features of the dichotomy method and show the preference of its use in comparison with other known methods. We propose a modified version of the dichotomy method that allows one to obtain more rapidly converging sequences of approximate solutions to nonlinear scalar equations and requires significantly fewer computations required to obtain solutions with the desired accuracy. By solving a number of specific nonlinear equations, it is possible to illustrate the higher convergence rate of the sequence of approximate solutions calculated using the modified dichotomy method and, thereby, to substantiate the advantage of the new method for its use in creating various automatic control and regulation systems. Based on the results obtained a modification of the method for segment bisection is proposed. It has all the main advantages of the modified method. The research results can be used in the development of modern automatic control systems for various technological processes and objects.

scholarly journals An Optimal Derivative-Free Ostrowski’s Scheme for Multiple Roots of Nonlinear Equations

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1809 ◽  
Author(s):  
Ramandeep Behl ◽  
Samaher Khalaf Alharbi ◽  
Fouad Othman Mallawi ◽  
Mehdi Salimi
Keyword(s):  
Nonlinear Equations ◽  
Real Life ◽  
Higher Order ◽  
Multiple Roots ◽  
Continuous Stirred Tank Reactor ◽  
Computational Mathematics ◽  
Life Problems ◽  
Derivative Free ◽  
Continuous Stirred Tank ◽  
Ostrowski’S Method

Finding higher-order optimal derivative-free methods for multiple roots (m≥2) of nonlinear expressions is one of the most fascinating and difficult problems in the area of numerical analysis and Computational mathematics. In this study, we introduce a new fourth order optimal family of Ostrowski’s method without derivatives for multiple roots of nonlinear equations. Initially the convergence analysis is performed for particular values of multiple roots—afterwards it concludes in general form. Moreover, the applicability and comparison demonstrated on three real life problems (e.g., Continuous stirred tank reactor (CSTR), Plank’s radiation and Van der Waals equation of state) and two standard academic examples that contain the clustering of roots and higher-order multiplicity (m=100) problems, with existing methods. Finally, we observe from the computational results that our methods consume the lowest CPU timing as compared to the existing ones. This illustrates the theoretical outcomes to a great extent of this study.

scholarly journals Dichotomy. Dichotomy? Dichotomy! A modified method of dichotomy for solution of nonlinear scalar equations and some results of its investigation

2021 ◽  
pp. 85-102
Author(s):  
Andrey Maistrenko ◽  
◽  
Konstantin Maistrenko ◽  
Anatoliy Svetlakov ◽  
◽  
...  
Keyword(s):  
Automatic Control ◽  
Rate Of Convergence ◽  
Control Systems ◽  
Nonlinear Equations ◽  
Approximate Solutions ◽  
Practical Significance ◽  
Automatic Control Systems ◽  
Modified Method ◽  
Proposed Modification ◽  
Scalar Equations

Introduction: when creating modern automatic control systems for various processes and objects operating in real time, very often one has to face the problem of solving various kinds of nonlinear scalar equations. In the first part of this work entitled “Dichotomy. Dichotomy? Dichotomy!: fundamentals, terminology problems and inspection analysis of the dichotomy method”, a modified version of the dichotomy method was proposed, which has all the main advantages of the modified method. This method has a number of advantages in comparison with other methods for solving nonlinear equations, but at present it has not found wide practical use. The main reason for its low popularity is a low rate of convergence of the sequence of approximate solutions, and a large amount of computation required to obtain sufficiently accurate solutions. Purpose of the study: to propose a modified version of the dichotomy method, which allows one to obtain more rapidly converging sequences of approximate solutions to nonlinear scalar equations and requires significantly less computations required to obtain solutions with the desired accuracy, to illustrate, a higher convergence rate of the sequence of approximate solutions calculated using the modified dichotomy method by solving a number of specific nonlinear equations and, thereby, to substantiate the advantage of the new method for its use in creating various automatic control and regulation systems. Results: a modification of the method for dividing a segment in half is proposed, which has all the main advantages of the modified method. The results of solving 4 nonlinear equations are presented illustrating a higher rate of convergence of solutions calculated using the proposed modification. Practical significance: the research results can be used in the development of modern automatic control systems for various technological processes and objects.

scholarly journals A New Optimal Eighth-Order Ostrowski-Type Family of Iterative Methods for Solving Nonlinear Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Taher Lotfi ◽  
Tahereh Eftekhari
Keyword(s):  
Weight Function ◽  
Iterative Methods ◽  
Nonlinear Equations ◽  
Efficiency Index ◽  
Eighth Order ◽  
New Class ◽  
First Derivative ◽  
New Methods ◽  
New Family ◽  
Ostrowski’S Method

Based on Ostrowski's method, a new family of eighth-order iterative methods for solving nonlinear equations by using weight function methods is presented. Per iteration the new methods require three evaluations of the function and one evaluation of its first derivative. Therefore, this family of methods has the efficiency index which equals 1.682. Kung and Traub conjectured that a multipoint iteration without memory based on n evaluations could achieve optimal convergence order 2n−1. Thus, we provide a new class which agrees with the conjecture of Kung-Traub for n=4. Numerical comparisons are made to show the performance of the presented methods.

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