scholarly journals A cubically convergent class of root finding iterative methods

2014 ◽  
Vol 7 (1) ◽  
pp. 17-23
Author(s):  
N. Rezaei A. ◽  
Esmaeili H.
Keyword(s):  
Iterative Methods ◽  
Root Finding

scholarly journals Recursive Elucidation of Polynomial Congruences Using Root-Finding Numerical Techniques

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
M. Khalid Mahmood ◽  
Farooq Ahmad
Keyword(s):  
Iterative Methods ◽  
Logarithmic Scale ◽  
Numerical Techniques ◽  
Root Finding ◽  
Polynomial Congruences ◽  
Polynomial Congruence

In this paper we put forward a family of algorithms for lifting solutions of a polynomial congruencemod pto polynomial congruencemod pk. For this purpose, root-finding iterative methods are employed for solving polynomial congruences of the formaxn≡b(mod pk),k≥1,wherea,b,andn>0are integers which are not divisible by an odd primep. It is shown that the algorithms suggested in this paper drastically reduce the complexity for such computations to a logarithmic scale. The efficacy of the proposed technique for solving negative exponent equations of the formax-n≡b(mod pk)has also been addressed.

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 An Optimal Eighth-Order Family of Iterative Methods for Multiple Roots

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 672 ◽  
Author(s):  
Saima Akram ◽  
Fiza Zafar ◽  
Nusrat Yasmin
Keyword(s):  
Weight Function ◽  
Iterative Methods ◽  
Multiple Root ◽  
Multiple Roots ◽  
Order Convergence ◽  
Eighth Order ◽  
Root Finding ◽  
New Family ◽  
Two Parameters ◽  
Theoretical Results

In this paper, we introduce a new family of efficient and optimal iterative methods for finding multiple roots of nonlinear equations with known multiplicity ( m ≥ 1 ) . We use the weight function approach involving one and two parameters to develop the new family. A comprehensive convergence analysis is studied to demonstrate the optimal eighth-order convergence of the suggested scheme. Finally, numerical and dynamical tests are presented, which validates the theoretical results formulated in this paper and illustrates that the suggested family is efficient among the domain of multiple root finding methods.

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.

scholarly journals Rootfinding : An Iterative Tool to Solve Different Problems

2015 ◽  
Vol 5 ◽  
pp. 121-125
Author(s):  
Iswarmani Adhikari
Keyword(s):  
Numerical Analysis ◽  
Iterative Methods ◽  
Linear Equations ◽  
Secant Method ◽  
Root Finding ◽  
Non Linear ◽  
Iteration Methods ◽  
False Position ◽  
The Common ◽  
Non Linear Equations

The aim of this paper is to apply the iteration methods for the solution of non-linear equations. Among the various root finding techniques, two of the common iterative methods Regula-falsi (false position) and the Secant method are used in two different problems to show the applications of numerical analysis in different fields. The Himalayan Physics Vol. 5, No. 5, Nov. 2014 Page: 121-125

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.

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