A New Optimal Family of Schröder’s Method for Multiple Zeros

Behl, Ramandeep; Alsolami, Arwa Jeza; Pansera, Bruno Antonio; Al-Hamdan, Waleed M.; Salimi, Mehdi; Феррара, Массимилиано

doi:10.3390/math7111076

Cited by 8 publications

(7 citation statements)

References 18 publications

(28 reference statements)

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“…These can first overcome the onepoint algorithms' low efficiency index, and can also minimize the number of iterations and improve the order of convergence with multiple steps, which also lessens the computational burden in numerical work. Many researchers [20][21][22][23][24][25][26][27][28] have developed higher-order iterative techniques using the first-order derivative to locate the multiple roots of a nonlinear problem. One function and two derivative evaluations are needed per iteration for the optimal fourth-order methods established in the literature [20,21,[23][24][25]27,28].…”

Section: Introductionmentioning

confidence: 99%

“…Many researchers [20][21][22][23][24][25][26][27][28] have developed higher-order iterative techniques using the first-order derivative to locate the multiple roots of a nonlinear problem. One function and two derivative evaluations are needed per iteration for the optimal fourth-order methods established in the literature [20,21,[23][24][25]27,28]. Li et al have introduced six new fourth-order methods in [22].…”

Section: Introductionmentioning

confidence: 99%

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Numerical Solution of Nonlinear Problems with Multiple Roots Using Derivative-Free Algorithms

et al. 2023

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In the study of systems’ dynamics the presence of symmetry dramatically reduces the complexity, while in chemistry, symmetry plays a central role in the analysis of the structure, bonding, and spectroscopy of molecules. In a more general context, the principle of equivalence, a principle of local symmetry, dictated the dynamics of gravity, of space-time itself. In certain instances, especially in the presence of symmetry, we end up having to deal with an equation with multiple roots. A variety of optimal methods have been proposed in the literature for multiple roots with known multiplicity, all of which need derivative evaluations in the formulations. However, in the literature, optimal methods without derivatives are few. Motivated by this feature, here we present a novel optimal family of fourth-order methods for multiple roots with known multiplicity, which do not use any derivative. The scheme of the new iterative family consists of two steps, namely Traub-Steffensen and Traub-Steffensen-like iterations with weight factor. According to the Kung-Traub hypothesis, the new algorithms satisfy the optimality criterion. Taylor’s series expansion is used to examine order of convergence. We also demonstrate the application of new algorithms to real-life problems, i.e., Van der Waals problem, Manning problem, Planck law radiation problem, and Kepler’s problem. Furthermore, the performance comparisons have shown that the given derivative-free algorithms are competitive with existing optimal fourth-order algorithms that require derivative information.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Numerical Solution of Nonlinear Problems with Multiple Roots Using Derivative-Free Algorithms

et al. 2023

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“…Jaiswal [21] claimed to be the first to propose an optimal eighth-order method for multiple roots of unknown multiplicity. Many researchers have modified the Newton method using Schröder's approach [10] to develop new optimal methods for finding multiple roots [11][12][13][14]22,23]. However, there have been much less work done on developing methods using Traub's conceptual approach [16].…”

Section: Introductionmentioning

confidence: 99%

Modification of Newton-Househölder Method for Determining Multiple Roots of Unknown Multiplicity of Nonlinear Equations

et al. 2021

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In this study, we propose an extension of the modified Newton-Househölder methods to find multiple roots with unknown multiplicity of nonlinear equations. With four functional evaluations per iteration, the proposed method achieves an optimal eighth order of convergence. The higher the convergence order, the quicker we get to the root with a high accuracy. The numerical examples have shown that this scheme can compete with the existing methods. This scheme is also stable across all of the functions tested based on the graphical basins of attraction.

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“…For this scheme, the multiplicity of the root is not required in advance, but it involves the first-order and second-order derivative of a function at each step. Many more researchers [13][14][15][16][17][18][19][20][21][22] developed the higher order iterative schemes involving the first-order derivative to find the multiple roots of a scalar equation. The evaluation of derivatives at each step is time consuming and sometimes a difficult job for any complex problem.…”

Section: Introductionmentioning

confidence: 99%

One Parameter Optimal Derivative-Free Family to Find the Multiple Roots of Algebraic Nonlinear Equations

et al. 2020

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In this study, we construct the one parameter optimal derivative-free iterative family to find the multiple roots of an algebraic nonlinear function. Many researchers developed the higher order iterative techniques by the use of the new function evaluation or the first-order or second-order derivative of functions to evaluate the multiple roots of a nonlinear equation. However, the evaluation of the derivative at each iteration is a cumbersome task. With this motivation, we design the second-order family without the utilization of the derivative of a function and without the evaluation of the new function. The proposed family is optimal as it satisfies the convergence order of Kung and Traub’s conjecture. Here, we use one parameter a for the construction of the scheme, and for a=1, the modified Traub method is its a special case. The order of convergence is analyzed by Taylor’s series expansion. Further, the efficiency of the suggested family is explored with some numerical tests. The obtained results are found to be more efficient than earlier schemes. Moreover, the basin of attraction of the proposed and earlier schemes is also analyzed.

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A New Optimal Family of Schröder’s Method for Multiple Zeros

Cited by 8 publications

References 18 publications

Numerical Solution of Nonlinear Problems with Multiple Roots Using Derivative-Free Algorithms

Numerical Solution of Nonlinear Problems with Multiple Roots Using Derivative-Free Algorithms

Modification of Newton-Househölder Method for Determining Multiple Roots of Unknown Multiplicity of Nonlinear Equations

One Parameter Optimal Derivative-Free Family to Find the Multiple Roots of Algebraic Nonlinear Equations

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