scholarly journals On iterative techniques for estimating all roots of nonlinear equation and its system with application in differential equation

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Mudassir Shams ◽  
Naila Rafiq ◽  
Nasreen Kausar ◽  
Praveen Agarwal ◽  
Choonkil Park ◽  
...  
Keyword(s):  
Nonlinear Equation ◽  
Iterative Methods ◽  
Nonlinear Equations ◽  
Basin Of Attraction ◽  
Numerical Test ◽  
Computational Cost ◽  
System Of Nonlinear Equations ◽  
Single Root ◽  
Root Finding

AbstractIn this article, we construct a family of iterative methods for finding a single root of nonlinear equation and then generalize this family of iterative methods for determining all roots of nonlinear equations simultaneously. Further we extend this family of root estimating methods for solving a system of nonlinear equations. Convergence analysis shows that the order of convergence is 3 in case of the single root finding method as well as for the system of nonlinear equations and is 5 for simultaneous determination of all distinct and multiple roots of a nonlinear equation. The computational cost, basin of attraction, efficiency, log of residual and numerical test examples show that the newly constructed methods are more efficient as compared to the existing methods in literature.

scholarly journals Study of Dynamical Behavior and Stability of Iterative Methods for Nonlinear Equation with Applications in Engineering

2020 ◽  
Vol 2020 ◽  
pp. 1-20
Author(s):  
Naila Rafiq ◽  
Saima Akram ◽  
Nazir Ahmad Mir ◽  
Mudassir Shams
Keyword(s):  
Nonlinear Equation ◽  
Iterative Methods ◽  
Dynamical Behavior ◽  
Numerical Test ◽  
Computational Cost ◽  
Basins Of Attraction ◽  
Single Root ◽  
Root Finding ◽  
The Stability

In this article, we first construct a family of optimal 2-step iterative methods for finding a single root of the nonlinear equation using the procedure of weight function. We then extend these methods for determining all roots simultaneously. Convergence analysis is presented for both cases to show that the order of convergence is 4 in case of the single-root finding method and is 6 for simultaneous determination of all distinct as well as multiple roots of a nonlinear equation. The dynamical behavior is presented to analyze the stability of fixed and critical points of the rational operator of one-point iterative methods. The computational cost, basins of attraction, efficiency, log of the residual, and numerical test examples show that the newly constructed methods are more efficient as compared with the existing methods in the literature.

scholarly journals On Efficient Iterative Numerical Methods for Simultaneous Determination of all Roots of Non-Linear Function

2020 ◽  
Author(s):  
Mudassir Shams ◽  
Nazir Mir ◽  
Naila Rafiq
Keyword(s):  
Simultaneous Determination ◽  
Iterative Methods ◽  
Linear Equations ◽  
Basin Of Attraction ◽  
Numerical Test ◽  
Computational Cost ◽  
Single Root ◽  
Non Linear ◽  
Non Linear Equations

We construct a family of two-step optimal fourth order iterative methods for finding single root of non-linear equations. We generalize these methods to simultaneous iterative methods for determining all the distinct as well as multiple roots of single variable non-linear equations. Convergence analysis is present for both cases to show that the order of convergence is four in case of single root finding method and is twelve for simultaneous determination of all roots of non-linear equation. The computational cost, Basin of attraction, efficiency, log of residual and numerical test examples shows, the newly constructed methods are more efficient as compared to the existing methods in literature.

scholarly journals On Dynamics of Iterative Techniques for Nonlinear Equation with Applications in Engineering

2020 ◽  
Vol 2020 ◽  
pp. 1-17
Author(s):  
Mudassir Shams ◽  
Nazir Ahmad Mir ◽  
Naila Rafiq ◽  
A. Othman Almatroud ◽  
Saima Akram
Keyword(s):  
Nonlinear Equation ◽  
Numerical Test ◽  
Computational Cost ◽  
Basins Of Attraction ◽  
Optimal Order ◽  
Multiple Roots ◽  
Optimal Order Of Convergence ◽  
Single Root ◽  
Root Finding

In this article, we construct an optimal family of iterative methods for finding the single root and then extend this family for determining all the distinct as well as multiple roots of single-variable nonlinear equations simultaneously. Convergence analysis is presented for both the cases to show that the optimal order of convergence is 4 in the case of single root finding methods and 6 for simultaneous determination of all distinct as well as multiple roots of a nonlinear equation. The computational cost, basins of attraction, efficiency, log of residual, and numerical test examples show that the newly constructed methods are more efficient as compared to the existing methods in the literature.

Accelerated iterative methods for finding solutions of a system of nonlinear equations

2007 ◽  
Vol 190 (2) ◽  
pp. 1815-1823 ◽  
Author(s):  
Miquel Grau-Sánchez ◽  
Josep M. Peris ◽  
José M. Gutiérrez
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
System Of Nonlinear Equations

Efficient Family of Sixth-Order Methods for Nonlinear Models with Its Dynamics

2019 ◽  
Vol 16 (05) ◽  
pp. 1840008
Author(s):  
Ramandeep Behl ◽  
Prashanth Maroju ◽  
S. S. Motsa
Keyword(s):  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Complex Dynamics ◽  
Nonlinear Models ◽  
Computational Cost ◽  
Sixth Order ◽  
System Of Nonlinear Equations ◽  
Van Der Pol ◽  
Numerical Work ◽  
The Family

In this study, we design a new efficient family of sixth-order iterative methods for solving scalar as well as system of nonlinear equations. The main beauty of the proposed family is that we have to calculate only one inverse of the Jacobian matrix in the case of nonlinear system which reduces the computational cost. The convergence properties are fully investigated along with two main theorems describing their order of convergence. By using complex dynamics tools, its stability is analyzed, showing stable members of the family. From this study, we intend to have more information about these methods in order to detect those with best stability properties. In addition, we also presented a numerical work which confirms the order of convergence of the proposed family is well deduced for scalar, as well as system of nonlinear equations. Further, we have also shown the implementation of the proposed techniques on real world problems like Van der Pol equation, Hammerstein integral equation, etc.

scholarly journals A Family of Multiple-Root Finding Iterative Methods Based on Weight Functions

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2194
Author(s):  
Francisco I. Chicharro ◽  
Rafael A. Contreras ◽  
Neus Garrido
Keyword(s):  
Weight Function ◽  
Iterative Methods ◽  
Numerical Test ◽  
Multiple Root ◽  
Initial Guess ◽  
Dynamical Analysis ◽  
Weight Functions ◽  
Root Finding ◽  
Wide Range ◽  
The Family

A straightforward family of one-point multiple-root iterative methods is introduced. The family is generated using the technique of weight functions. The order of convergence of the family is determined in its convergence analysis, which shows the constraints that the weight function must satisfy to achieve order three. In this sense, a family of iterative methods can be obtained with a suitable design of the weight function. That is, an iterative algorithm that depends on one or more parameters is designed. This family of iterative methods, starting with proper initial estimations, generates a sequence of approximations to the solution of a problem. A dynamical analysis is also included in the manuscript to study the long-term behavior of the family depending on the parameter value and the initial guess considered. This analysis reveals the good properties of the family for a wide range of values of the parameter. In addition, a numerical test on academic and engineering multiple-root functions is performed.

scholarly journals Higher-Order Root-Finding Algorithms and Their Basins of Attraction

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Amir Naseem ◽  
M. A. Rehman ◽  
Thabet Abdeljawad
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Basins Of Attraction ◽  
Higher Order ◽  
Complex Polynomials ◽  
Convergence Criteria ◽  
Root Finding ◽  
Variational Iteration ◽  
Order Root ◽  
Iteration Technique

In this paper, we proposed and analyzed three new root-finding algorithms for solving nonlinear equations in one variable. We derive these algorithms with the help of variational iteration technique. We discuss the convergence criteria of these newly developed algorithms. The dominance of the proposed algorithms is illustrated by solving several test examples and comparing them with other well-known existing iterative methods in the literature. In the end, we present the basins of attraction using some complex polynomials of different degrees to observe the fractal behavior and dynamical aspects of the proposed algorithms.

Determination of the Most Dangerous Meshing Point for Modified Hourglass Worm Drives

2011 ◽  
Author(s):  
Yaping Zhao ◽  
Tianchao Wu
Keyword(s):  
Nonlinear Equations ◽  
Slight Modification ◽  
Second Step ◽  
System Of Nonlinear Equations ◽  
Type Ii ◽  
Severe Condition ◽  
Worm Gearing ◽  
Curvature Interference ◽  
Worm Drive

A kind of modified hourglass worm drives, which is frequently called the type II worm gearing for short, has various favorable meshing features. Nevertheless, its sole shortcoming is the undercutting of the worm wheel. In the condition of adopting slight modification, this problem can be overcome due to the removal of a part of one sub-conjugate area containing the curvature interference limit line. In order to measure the effect of the avoidance of undercutting, a strategy to determine the meshing point in the most severe condition is proposed for a type II worm drive. The presented strategy can be divided into two steps. The first step is to establish a system of nonlinear equations in five variables in accordance with the theory of gearing. The second step is to solve the procured nonlinear equations by numerical iterative method to ascertain the meshing point required. A numerical example is presented to verify the validity and feasibility of the proposed scheme.

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