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 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 Interval and Speed of Convergence on Iterative Methods

2017 ◽  
pp. 112-114
Author(s):  
Iswarmani Adhikari
Keyword(s):  
Iterative Method ◽  
Iterative Methods ◽  
Linear Equations ◽  
Initial Guess ◽  
Similar Process ◽  
Speed Of Convergence ◽  
Non Linear ◽  
Non Linear Equations

The iterative method is a tool of solving the non-linear equations to get their approximate roots with some errors of tolerance. Repetition of the similar process is applied successively on such iterations to compute a sequence of increasingly accurate estimates of the roots. In this paper, the construction of an iterative method for solving an equation, its convergence and the determination of interval of convergence for the approximate choice of initial guess and the speed of convergence are highlighted.The Himalayan Physics Vol. 6 & 7, April 2017 (112-114)

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.

Comparative Study of Iterative Methods for Solving Non-Linear Equations

2021 ◽  
Vol 23 (07) ◽  
pp. 858-866
Author(s):  
Gauri Thakur ◽  
◽  
J.K. Saini ◽  
Keyword(s):  
Numerical Analysis ◽  
Iterative Methods ◽  
Linear Equations ◽  
Bisection Method ◽  
Practical Applications ◽  
Non Linear ◽  
False Position ◽  
Newton Raphson ◽  
Raphson Method ◽  
Non Linear Equations

In numerical analysis, methods for finding roots play a pivotal role in the field of many real and practical applications. The efficiency of numerical methods depends upon the convergence rate (how fast the particular method converges). The objective of this study is to compare the Bisection method, Newton-Raphson method, and False Position Method with their limitations and also analyze them to know which of them is more preferred. Limitations of these methods have allowed presenting the latest research in the area of iterative processes for solving non-linear equations. This paper analyzes the field of iterative methods which are developed in recent years with their future scope.

An efficient method for determining the critical modification coefficient for cylindrical worm gearing in which the worm is generated by a curve

2001 ◽  
Vol 215 (12) ◽  
pp. 1745-1755 ◽  
Author(s):  
C-K Lee ◽  
C-K Chen
Keyword(s):  
Efficient Method ◽  
Linear Equations ◽  
Contact Ratio ◽  
Non Linear ◽  
Worm Gearing ◽  
System Equations ◽  
Curvature Information ◽  
Maximum Contact ◽  
Non Linear Equations

This paper proposes an efficient method for the determination of the critical modification coefficient (CMC) for the cylindrical worm gearing in which the worm is generated by a curve. The term CMC is defined as the modification coefficient that causes the first boundary points to be just on the boundary of the area of meshing. Solved from a system of five non-linear equations, the CMC not only avoids gear undercutting, but also produces the maximum contact ratio. In developing the system equations, the equation of non-undercutting is derived directly from the curvature information of the generating curve and the parameters of relative motion, without the need to derive any partial derivatives to the equation of the generating surface. By using an equivalent generation mechanism rather than the real one, an explicit form of equation of meshing is created. The equation of meshing created can simplify the form of the equation of non-undercutting, reduce the number of non-linear equations and unknowns in the system equations, increase the speed of convergence and decrease numerical instability. Based on the proposed method, a computer program is created and applied to analyse the CMCs of four commonly used worm gearings: ZA, ZN, ZE and ZC3.

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