scholarly journals Generating Optimal Eighth Order Methods for Computing Multiple Roots

Symmetry ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 1947
Author(s):  
Deepak Kumar ◽  
Sunil Kumar ◽  
Janak Raj Sharma ◽  
Matteo d’Amore
Keyword(s):  
Real Life ◽  
Nonlinear Function ◽  
Basins Of Attraction ◽  
Computational Time ◽  
Multiple Roots ◽  
Time Stability ◽  
Academic Problems ◽  
Eighth Order ◽  
Multiple Zeros ◽  
New Family

There are a few optimal eighth order methods in literature for computing multiple zeros of a nonlinear function. Therefore, in this work our main focus is on developing a new family of optimal eighth order iterative methods for multiple zeros. The applicability of proposed methods is demonstrated on some real life and academic problems that illustrate the efficient convergence behavior. It is shown that the newly developed schemes are able to compete with other methods in terms of numerical error, convergence and computational time. Stability is also demonstrated by means of a pictorial tool, namely, basins of attraction that have the fractal-like shapes along the borders through which basins are symmetric.

scholarly journals Some Real-Life Applications of a Newly Constructed Derivative Free Iterative Scheme

Symmetry ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 239 ◽  
Author(s):  
Ramandeep Behl ◽  
M. Salimi ◽  
M. Ferrara ◽  
S. Sharifi ◽  
Samaher Alharbi
Keyword(s):  
Iterative Methods ◽  
Chemical Engineering ◽  
Flame Temperature ◽  
Real Life ◽  
Chemical Reactor ◽  
Basins Of Attraction ◽  
Eighth Order ◽  
Present Scheme ◽  
New Family ◽  
Derivative Free

In this study, we present a new higher-order scheme without memory for simple zeros which has two major advantages. The first one is that each member of our scheme is derivative free and the second one is that the present scheme is capable of producing many new optimal family of eighth-order methods from every 4-order optimal derivative free scheme (available in the literature) whose first substep employs a Steffensen or a Steffensen-like method. In addition, the theoretical and computational properties of the present scheme are fully investigated along with the main theorem, which demonstrates the convergence order and asymptotic error constant. Moreover, the effectiveness of our scheme is tested on several real-life problems like Van der Waal’s, fractional transformation in a chemical reactor, chemical engineering, adiabatic flame temperature, etc. In comparison with the existing robust techniques, the iterative methods in the new family perform better in the considered test examples. The study of dynamics on the proposed iterative methods also confirms this fact via basins of attraction applied to a number of test functions.

scholarly journals An Efficient Family of Optimal Eighth-Order Multiple Root Finders

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 310 ◽  
Author(s):  
Fiza Zafar ◽  
Alicia Cordero ◽  
Juan Torregrosa
Keyword(s):  
Life Science ◽  
Real Life ◽  
Multiple Root ◽  
Dynamical Analysis ◽  
Multiple Roots ◽  
Eighth Order ◽  
Science And Engineering ◽  
New Family ◽  
Modified Newton’S Method ◽  
Numerical Performance

Finding a repeated zero for a nonlinear equation f ( x ) = 0 , f : I ⊆ R → R has always been of much interest and attention due to its wide applications in many fields of science and engineering. Modified Newton’s method is usually applied to solve this kind of problems. Keeping in view that very few optimal higher-order convergent methods exist for multiple roots, we present a new family of optimal eighth-order convergent iterative methods for multiple roots with known multiplicity involving a multivariate weight function. The numerical performance of the proposed methods is analyzed extensively along with the basins of attractions. Real life models from life science, engineering, and physics are considered for the sake of comparison. The numerical experiments and dynamical analysis show that our proposed methods are efficient for determining multiple roots of nonlinear equations.

scholarly journals Development of Optimal Eighth Order Derivative-Free Methods for Multiple Roots of Nonlinear Equations

Symmetry ◽  
2019 ◽  
Vol 11 (6) ◽  
pp. 766 ◽  
Author(s):  
Janak Raj Sharma ◽  
Sunil Kumar ◽  
Ioannis K. Argyros
Keyword(s):  
Basins Of Attraction ◽  
Higher Order ◽  
Order Derivative ◽  
Multiple Roots ◽  
Nonlinear Functions ◽  
Eighth Order ◽  
Multiple Zeros ◽  
Derivative Free ◽  
Graphical Tool ◽  
Derivative Free Methods

A number of higher order iterative methods with derivative evaluations are developed in literature for computing multiple zeros. However, higher order methods without derivative for multiple zeros are difficult to obtain and hence such methods are rare in literature. Motivated by this fact, we present a family of eighth order derivative-free methods for computing multiple zeros. Per iteration the methods require only four function evaluations, therefore, these are optimal in the sense of Kung-Traub conjecture. Stability of the proposed class is demonstrated by means of using a graphical tool, namely, basins of attraction. Boundaries of the basins are fractal like shapes through which basins are symmetric. Applicability of the methods is demonstrated on different nonlinear functions which illustrates the efficient convergence behavior. Comparison of the numerical results shows that the new derivative-free methods are good competitors to the existing optimal eighth-order techniques which require derivative evaluations.

scholarly journals A Family of Optimal Eighth Order Iteration Functions for Multiple Roots and Its Dynamics

2021 ◽  
Vol 2021 ◽  
pp. 1-18
Author(s):  
Saima Akram ◽  
Faiza Akram ◽  
Moin-ud-Din Junjua ◽  
Misbah Arshad ◽  
Tariq Afzal
Keyword(s):  
Numerical Test ◽  
Basins Of Attraction ◽  
Dynamical Analysis ◽  
Weight Functions ◽  
Multiple Roots ◽  
Order Convergence ◽  
Eighth Order ◽  
New Family ◽  
Special Cases ◽  
Iteration Functions

In this manuscript, we present a new general family of optimal iterative methods for finding multiple roots of nonlinear equations with known multiplicity using weight functions. An extensive convergence analysis is presented to verify the optimal eighth order convergence of the new family. Some special cases of the family are also presented which require only three functions and one derivative evaluation at each iteration to reach optimal eighth order convergence. A variety of numerical test functions along with some real-world problems such as beam designing model and Van der Waals’ equation of state are presented to ensure that the newly developed family efficiently competes with the other existing methods. The dynamical analysis of the proposed methods is also presented to validate the theoretical results by using graphical tools, termed as the basins of attraction.

scholarly journals Efficient Four-Parametric with-and-without-Memory Iterative Methods Possessing High Efficiency Indices

2018 ◽  
Vol 2018 ◽  
pp. 1-12 ◽  
Author(s):  
Alicia Cordero ◽  
Moin-ud-Din Junjua ◽  
Juan R. Torregrosa ◽  
Nusrat Yasmin ◽  
Fiza Zafar
Keyword(s):  
Iterative Methods ◽  
High Efficiency ◽  
Nonlinear Function ◽  
Efficiency Index ◽  
Simple Zero ◽  
Eighth Order ◽  
New Family ◽  
Derivative Free ◽  
With Memory ◽  
Theoretical Results

We construct a family of derivative-free optimal iterative methods without memory to approximate a simple zero of a nonlinear function. Error analysis demonstrates that the without-memory class has eighth-order convergence and is extendable to with-memory class. The extension of new family to the with-memory one is also presented which attains the convergence order 15.5156 and a very high efficiency index 15.51561/4≈1.9847. Some particular schemes of the with-memory family are also described. Numerical examples and some dynamical aspects of the new schemes are given to support theoretical results.

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 A stable class of modified Newton-like methods for multiple roots and their dynamics

2020 ◽  
Vol 21 (6) ◽  
pp. 603-621
Author(s):  
Munish Kansal ◽  
Alicia Cordero ◽  
Juan R. Torregrosa ◽  
Sonia Bhalla
Keyword(s):  
Real Life ◽  
Chemical Reactor ◽  
Function Technique ◽  
Continuous Stirred Tank Reactor ◽  
Parallel Plates ◽  
Nonlinear Functions ◽  
Eighth Order ◽  
Multiple Zeros ◽  
Life Problems ◽  
Global Carbon

AbstractThere have appeared in the literature a lot of optimal eighth-order iterative methods for approximating simple zeros of nonlinear functions. Although, the similar ideas can be extended for the case of multiple zeros but the main drawback is that the order of convergence and computational efficiency reduce dramatically. Therefore, in order to retain the accuracy and convergence order, several optimal and non-optimal modifications have been proposed in the literature. But, as far as we know, there are limited number of optimal eighth-order methods that can handle the case of multiple zeros. With this aim, a wide general class of optimal eighth-order methods for multiple zeros with known multiplicity is brought forward, which is based on weight function technique involving function-to-function ratio. An extensive convergence analysis is demonstrated to establish the eighth-order of the developed methods. The numerical experiments considered the superiority of the new methods for solving concrete variety of real life problems coming from different disciplines such as trajectory of an electron in the air gap between two parallel plates, the fractional conversion in a chemical reactor, continuous stirred tank reactor problem, Planck’s radiation law problem, which calculates the energy density within an isothermal blackbody and the problem arising from global carbon dioxide model in ocean chemistry, in comparison with methods of similar characteristics appeared in the literature.

An Optimal Eighth-Order Scheme for Multiple Zeros of Univariate Functions

2019 ◽  
Vol 16 (04) ◽  
pp. 1843002 ◽  
Author(s):  
Ramandeep Behl ◽  
Fiza Zafar ◽  
Ali Saleh Alshormani ◽  
Moin-Ud-Din Junjua ◽  
Nusrat Yasmin
Keyword(s):  
Order Of Convergence ◽  
Real Life ◽  
Eighth Order ◽  
Multiple Zeros ◽  
Iterative Schemes ◽  
Life Problems ◽  
Order Scheme ◽  
Multiplicity Formula ◽  
First Time ◽  
Better Than

We construct an optimal eighth-order scheme which will work for multiple zeros with multiplicity [Formula: see text], for the first time. Earlier, the maximum convergence order of multi-point iterative schemes was six for multiple zeros in the available literature. So, the main contribution of this study is to present a new higher-order and as well as optimal scheme for multiple zeros for the first time. In addition, we present an extensive convergence analysis with the main theorem which confirms theoretically eighth-order convergence of the proposed scheme. Moreover, we consider several real life problems which contain simple as well as multiple zeros in order to compare with the existing robust iterative schemes. Finally, we conclude on the basis of obtained numerical results that our iterative methods perform far better than the existing methods in terms of residual error, computational order of convergence and difference between the two consecutive iterations.

scholarly journals An Iteration Function Having Optimal Eighth-Order of Convergence for Multiple Roots and Local Convergence

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1419
Author(s):  
Ramandeep Behl ◽  
Ioannis K. Argyros ◽  
Michael Argyros ◽  
Mehdi Salimi ◽  
Arwa Jeza Alsolami
Keyword(s):  
Quantum Physics ◽  
Real Life ◽  
Chemical Reactor ◽  
Theoretical Models ◽  
Eighth Order ◽  
Multiple Zeros ◽  
Stirred Reactor ◽  
Life Problems ◽  
Special Cases ◽  
Iteration Functions

In the study of dynamics of physical systems an important role is played by symmetry principles. As an example in classical physics, symmetry plays a role in quantum physics, turbulence and similar theoretical models. We end up having to deal with an equation whose solution we desire to be in a closed form. But obtaining a solution in such form is achieved only in special cases. Hence, we resort to iterative schemes. There is where the novelty of our study lies, as well as our motivation for writing it. We have a very limited literature with eighth-order convergent iteration functions that can handle multiple zeros m≥1. Therefore, we suggest an eighth-order scheme for multiple zeros having optimal convergence along with fast convergence and uncomplicated structure. We develop an extensive convergence study in the main theorem that illustrates eighth-order convergence of our scheme. Finally, the applicability and comparison was illustrated on real life problems, e.g., Van der Waal’s equation of state, Chemical reactor with fractional conversion, continuous stirred reactor and multi-factor problems, etc., with existing schemes. These examples further show the superiority of our schemes over the earlier ones.

scholarly journals Optimal One-Point Iterative Function Free from Derivatives for Multiple Roots

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 709 ◽  
Author(s):  
Deepak Kumar ◽  
Janak Raj Sharma ◽  
Ioannis K. Argyros
Keyword(s):  
Complex Geometry ◽  
Basins Of Attraction ◽  
Second Order ◽  
Multiple Roots ◽  
Single Variable ◽  
Optimal Method ◽  
Multiple Zeros ◽  
Derivative Free ◽  
Derivative Free Methods ◽  
The Stability

We suggest a derivative-free optimal method of second order which is a new version of a modification of Newton’s method for achieving the multiple zeros of nonlinear single variable functions. Iterative methods without derivatives for multiple zeros are not easy to obtain, and hence such methods are rare in literature. Inspired by this fact, we worked on a family of optimal second order derivative-free methods for multiple zeros that require only two function evaluations per iteration. The stability of the methods was validated through complex geometry by drawing basins of attraction. Moreover, applicability of the methods is demonstrated herein on different functions. The study of numerical results shows that the new derivative-free methods are good alternatives to the existing optimal second-order techniques that require derivative calculations.

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