Research Articles

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Browsing Research Articles by Author "Rajni Sharma"

Now showing 1 - 9 of 9
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    A NEW SEVENTH-ORDER SCHEME FOR FINDING MULTIPLE ROOT OF NONLINEAR EQUATIONS WITH IMPROVED EFFICIENCY
    (Bull. Cal. Math. Soc., 116, (6) 813–826, 2024) Ashu Bahl; Ranjita Guglani; Rajni Sharma; Neeru Bala
    In this contribution, a new seventh-order iterative scheme is presented for finding multiple roots of nonlinear equations. For numerical experimentation, various nonlinear equations arousing in different scientific fields have been considered. The efficiency of proposed scheme is tested in comparison with some already existing methods. Graphical comparison is also given using basins of attraction.
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    A Novel Family of Multiple Root Finders with Optimal Eighth-Order Convergence and their Basins of Attraction
    (International Conference of Numerical Analysis and Applied Mathematics AIP Conf. Proc., 2024) Rajni Sharma; Ashu Bahl; Ranjita Guglani
    In this contribution, a novel scheme of multiple root finders for nonlinear equations with the univariate and bivariate weight functions is proposed. The basic requirement of the presented scheme is the known value of ’m’ (multiplicity of the root). Analysis of convergence is given to show that the proposed family achieves eighth-order convergence. Numerical experimentation is done to demonstrate accuracy and computational efficiency of the scheme. In addition, the obtained results show that the proposed scheme has an edge-over other considered eighth-order methods
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    An Optimal Fourth Order Iterative Method for Solving Nonlinear Equations and Its Dynamics
    (Hindawi Publishing Corporation,Journal of Complex Analysis, 2015) Rajni Sharma; Ashu Bahl
    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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    An optimal fourth order weighted-Newton method for computing multiple roots and basin attractors for various methods
    (Pelagia Research Library Advances in Applied Science Research, 2016, 7(4):47-54, 2016) Ashu Bahl; Rajni Sharma
    In this paper, we present an optimal fourth order method for finding multiple roots of a nonlinear equation f(x)=0. In terms of computational cost, the method uses one evaluation of the function and two evaluations of its first derivative per iteration. Therefore, the method has optimal order with efficiency index 1.587 which is better than efficiency indices 1.414 of Newton method, 1.442 of Halley’s method and 1.414 of Neta-Johnson method. Numerical examples are given to support that the method thus obtained is competitive with other similar robust methods. The basins of attraction of the proposed method are presented and compared with other existing methods.
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    Formulation and convergence analysis of an efficient higher order iterative scheme
    (Asia Pacific Academy of Science Pte. Ltd., 2024) Ranjita Guglani; Ashu Bahl; Rajni Sharma
    This contribution presents a highly efficient three-step iterative scheme. The proposed scheme is different in itself by achieving seventh-order convergence. The scheme is very useful for equations of nonlinear nature having multiple roots. The Taylor series expansion is employed to rigorously analyze the convergence of the presented scheme. That the scheme is effective and robust can be fit through a variety of examples from different fields. Numerical experimentation demonstrates the scheme’s rapid and reliable convergence to the true root and comparing its performance against existing techniques in the literature. Additionally, basins of attraction are visualized to offer a clear, comparative view of how different methods perform with varying initial guesses. The results show that this new scheme consistently compete well over other methods. This makes it a powerful tool for solving complex equations.
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    General Family of Third Order Methods for Multiple Roots of Nonlinear Equations and Basin Attractors for Various Methods
    (Hindawi Publishing Corporation Advances in Numerical Analysis Volume 2014, Article ID 963878, 8 pages, 2014) Rajni Sharma; Ashu Bahl
    A general scheme of third order convergence is described for finding multiple roots of nonlinear equations. The proposed scheme requires one evaluation of 𝑓, 𝑓󸀠 , and 𝑓󸀠󸀠 each per iteration and contains several known one-point third order methods for finding multiple roots, as particular cases. Numerical examples are included to confirm the theoretical results and demonstrate convergence behavior of the proposed methods. In the end, we provide the basins of attraction for some methods to observe their dynamics in the complex plane.
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    One parameter fifth order family of iterative methods for solving nonlinear Equations and Basins of Attraction
    (NeuroQuantology Volume 20 Issue 11 Page 8973-8986, 2022) Anshu; Rajni Sharma; Ashu Bahl
    In this study, we develop one-parameter family of fifth order iterative scheme for finding simple zeros of nonlinear equations. In terms of computational cost, this proposed scheme involves four evaluations of the given function and its first-order derivative per iteration. Various numerical examples are demonstrated to compare the performance of developed scheme with existing schemes of same order. To verify how well our scheme works in practise, we apply them to solve the Van der Waals equation of state. Moreover, the dynamics of the proposed method is demonstrated and compare it with other existing fifth order iterative methods.
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    Optimaleighth-order multiple root finding iterative methods using bivariate weight function
    (ElsevierB.V., 2023) Rajni Sharma; Ashu Bahl; Ranjita Guglani
    Inthiscontribution, anovel eighth-order scheme ispresentedfor solvingnonlinearequations withmultipleroots.Theproposedschemecomprisesof threestepswiththemodifiedNewton methodas its first stepfollowedbytwoweightedNewtonsteps involvingoneunivariateand onebivariatefunctionrespectively.Analysisofconvergenceconfirmsthatthepresentedscheme obtains optimal computational order of convergence. The efficiencyof presented scheme is comparednumericallywith recent eighth-ordermethods. Functions like populationgrowth problem,Newton’sbeamproblem, etc.,havebeenconsideredfornumerical experimentation. For the comparative study in the complex plane, we employed the concept of basins of attraction
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    RESOLVING NONLINEAR PHYSICS PROBLEMS WITH AN EFFICIENT SEVENTH ORDER ITERATIVE APPROACH
    (South East Asian J. of Mathematics and Mathematical Sciences Vol. 20, No. 2 (2024), pp. 269-282, 2024) Ashu Bahl; Ranjita Guglani; Rajni Sharma
    Our research introduces a novel seventh-order iterative method specifically designed to address nonlinear equations having multiple roots. Inspired by the pioneering work of Sharma et al. (2019), our approach represents a significant advancement in computational techniques for solving complex mathematical problems. Through rigorous convergence analysis, we establish that our proposed method achieves seventh-order convergence. To evaluate its efficacy, we conduct extensive numerical experiments utilizing a range of nonlinear equations encountered in applied physics domains, including Planck’s Law, electron trajectory problems, and Newton’s beam designing problem. Our findings reveal that the suggested method consistently outperforms other existing techniques of similar nature available in the literature. Notably, our method demonstrates exceptional convergence behavior even in challenging scenarios involving multiple roots, indicating its suitability for solving complex problems encountered in applied physics and related fields. This superiority is evidenced by its ability to efficiently converge to solutions even in scenarios involving multiple roots. The practical implications of our research extend to various fields reliant on nonlinear equation.