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

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

Advances in Difference Equations ◽

10.1186/s13662-021-03636-x ◽

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.

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

Mathematical Problems in Engineering ◽

10.1155/2020/5853296 ◽

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.

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

10.20944/preprints202002.0410.v1 ◽

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.

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Optimal Fourth, Eighth and Sixteenth Order Methods by Using Divided Difference Techniques and Their Basins of Attraction and Its Application

Mathematics ◽

10.3390/math7040322 ◽

2019 ◽

Vol 7 (4) ◽

pp. 322 ◽

Cited By ~ 1

Author(s):

Yanlin Tao ◽

Kalyanasundaram Madhu

Keyword(s):

Dynamical Behavior ◽

Computational Cost ◽

Basins Of Attraction ◽

Order Method ◽

Eighth Order ◽

Projectile Motion ◽

First Derivative ◽

Iteration Functions ◽

Optimal Launch Angle ◽

Fourth Order Method

The principal objective of this work is to propose a fourth, eighth and sixteenth order scheme for solving a nonlinear equation. In terms of computational cost, per iteration, the fourth order method uses two evaluations of the function and one evaluation of the first derivative; the eighth order method uses three evaluations of the function and one evaluation of the first derivative; and sixteenth order method uses four evaluations of the function and one evaluation of the first derivative. So these all the methods have satisfied the Kung-Traub optimality conjecture. In addition, the theoretical convergence properties of our schemes are fully explored with the help of the main theorem that demonstrates the convergence order. The performance and effectiveness of our optimal iteration functions are compared with the existing competitors on some standard academic problems. The conjugacy maps of the presented method and other existing eighth order methods are discussed, and their basins of attraction are also given to demonstrate their dynamical behavior in the complex plane. We apply the new scheme to find the optimal launch angle in a projectile motion problem and Planck’s radiation law problem as an application.

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Fixed Point Root-Finding Methods of Fourth-Order of Convergence

Symmetry ◽

10.3390/sym11060769 ◽

2019 ◽

Vol 11 (6) ◽

pp. 769 ◽

Cited By ~ 2

Author(s):

Alicia Cordero ◽

Lucía Guasp ◽

Juan R. Torregrosa

Keyword(s):

Weight Function ◽

Dynamical Behavior ◽

Nonlinear Problems ◽

Dynamical Analysis ◽

Function Technique ◽

Weight Functions ◽

Good Tool ◽

Quadratic Polynomials ◽

Root Finding ◽

The Stability

In this manuscript, by using the weight-function technique, a new class of iterative methods for solving nonlinear problems is constructed, which includes many known schemes that can be obtained by choosing different weight functions. This weight function, depending on two different evaluations of the derivative, is the unique difference between the two steps of each method, which is unusual. As it is proven that all the members of the class are optimal methods in the sense of Kung-Traub’s conjecture, the dynamical analysis is a good tool to determine the best elements of the family in terms of stability. Therefore, the dynamical behavior of this class on quadratic polynomials is studied in this work. We analyze the stability of the presented family from the multipliers of the fixed points and critical points, along with their associated parameter planes. In addition, this study enables us to select the members of the class with good stability properties.

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Determination of Protein Structural Ensembles by Hybrid-Resolution SAXS Restrained Molecular Dynamics.

10.26434/chemrxiv.8798285.v1 ◽

2019 ◽

Author(s):

Cristina Paissoni ◽

Alexander Jussupow ◽

Carlo Camilloni

Keyword(s):

Dynamical Behavior ◽

Computational Cost ◽

Three Dimensional ◽

Md Simulations ◽

Coarse Grained ◽

Conformational Ensemble ◽

Biomolecular Systems ◽

Conformational Ensembles ◽

Independent Experimental Data

<div><div><div><p>SAXS experiments provide low-resolution but valuable information about the dynamics of biomolecular systems, which could be ideally integrated in MD simulations to accurately determine conformational ensembles of flexible proteins. The applicability of this strategy is hampered by the high computational cost required to calculate scattering intensities from three-dimensional structures. We previously presented a metainference-based hybrid resolution method that makes atomistic SAXS-restrained MD simulation feasible by adopting a coarse-grained approach to efficiently back-calculate scattering intensities; here, we extend this technique, applying it in the framework of multiple-replica simulations with the aim to investigate the dynamical behavior of flexible biomolecules. The efficacy of the method is assessed on the K63-diubiquitin multi-domain protein, showing that inclusion of SAXS-restraints is effective in generating reliable and heterogenous conformational ensemble, also improving the agreement with independent experimental data.</p></div></div></div>

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

Mathematics ◽

10.3390/math8122194 ◽

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.

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

Journal of Mathematics ◽

10.1155/2020/5070363 ◽

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.

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Basins of Attraction for Various Steffensen-Type Methods

Journal of Applied Mathematics ◽

10.1155/2014/539707 ◽

2014 ◽

Vol 2014 ◽

pp. 1-17 ◽

Cited By ~ 17

Author(s):

Alicia Cordero ◽

Fazlollah Soleymani ◽

Juan R. Torregrosa ◽

Stanford Shateyi

Keyword(s):

Computer Algebra System ◽

Dynamical Behavior ◽

Basin Of Attraction ◽

Basins Of Attraction ◽

Numerical Tests ◽

Derivative Free ◽

Iteration Functions ◽

The Stability ◽

Programming Package ◽

The Basin Of Attraction

The dynamical behavior of different Steffensen-type methods is analyzed. We check the chaotic behaviors alongside the convergence radii (understood as the wideness of the basin of attraction) needed by Steffensen-type methods, that is, derivative-free iteration functions, to converge to a root and compare the results using different numerical tests. We will conclude that the convergence radii (and the stability) of Steffensen-type methods are improved by increasing the convergence order. The computer programming package MATHEMATICAprovides a powerful but easy environment for all aspects of numerics. This paper puts on show one of the application of this computer algebra system in finding fixed points of iteration functions.

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An Optimal Fourth Order Iterative Method for Solving Nonlinear Equations and Its Dynamics

Journal of Complex Analysis ◽

10.1155/2015/259167 ◽

2015 ◽

Vol 2015 ◽

pp. 1-9 ◽

Cited By ~ 2

Author(s):

Rajni Sharma ◽

Ashu Bahl

Keyword(s):

Fourth Order ◽

Dynamical Behavior ◽

Computational Cost ◽

Efficiency Index ◽

Basins Of Attraction ◽

Optimal Order ◽

Extraneous Fixed Points ◽

Fourth Order Method ◽

Jarratt Method ◽

Better Than

We present a new fourth order method for finding simple roots of a nonlinear equation f(x)=0. In terms of computational cost, per iteration the method uses one evaluation of the function and two evaluations of its first derivative. Therefore, the method has optimal order with efficiency index 1.587 which is better than efficiency index 1.414 of Newton method and the same with Jarratt method and King’s family. Numerical examples are given to support that the method thus obtained is competitive with other similar robust methods. The conjugacy maps and extraneous fixed points of the presented method and other existing fourth order methods are discussed, and their basins of attraction are also given to demonstrate their dynamical behavior in the complex plane.

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