Derivative-Free King’s Scheme for Multiple Zeros of Nonlinear Functions

Behl, Ramandeep; Bhalla, Sonia; Martínez, Eulalia; Alsulami, Majed Aali

doi:10.3390/math9111242

Cited by 7 publications

(4 citation statements)

References 23 publications

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“…Behl et al. (2021a) method:where μ t = x t + γ Θ( x t ), yt=)(normalΘ(zt)normalΘ(μt)1m, st=)(normalΘ(zt)normalΘ(xt)1m and γ = 0.5.…”

Section: Numerical Experimentsmentioning

confidence: 99%

An optimal derivative-free King's family for multiple zeros and its dynamics

Rani

Kansal

2022

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PurposeThe purpose of this article is to develop and analyze a new derivative-free class of higher-order iterative methods for locating multiple roots numerically.Design/methodology/approachThe scheme is generated by using King-type iterative methods. By employing the Traub-Steffensen technique, the proposed class is designed into the derivative-free family.FindingsThe proposed class requires three functional evaluations at each stage of computation to attain fourth-order convergency. Moreover, it can be observed that the theoretical convergency results of family are symmetrical for particular cases of multiplicity of zeros. This further motivates the authors to present the result in general, which confirms the convergency order of the methods. It is also worth mentioning that the authors can obtain already existing methods as particular cases of the family for some suitable choice of free disposable parameters. Finally, the authors include a wide variety of benchmark problems like van der Waals's equation, Planck's radiation law and clustered root problem. The numerical comparisons are included with several existing algorithms to confirm the applicability and effectiveness of the proposed methods.Originality/valueThe numerical results demonstrate that the proposed scheme performs better than the existing methods in terms of CPU timing and absolute residual errors.

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“…Behl et al. (2021a) method:where μ t = x t + γ Θ( x t ), yt=)(normalΘ(zt)normalΘ(μt)1m, st=)(normalΘ(zt)normalΘ(xt)1m and γ = 0.5.…”

Section: Numerical Experimentsmentioning

confidence: 99%

An optimal derivative-free King's family for multiple zeros and its dynamics

Rani

Kansal

2022

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“…As for the convergence criteria, it was required that the distance of two consecutive approximations (δ) for the zero was less than 10 −100 . We display the number of iterations (IT), the approximate zero u n and the functional value f (u n ) in Tables (1)(2)(3)(4)(5). The computational order of convergence (COC) (see [8]) is computed to check the behaviour of the proposed methods for presented examples and given by: We use x o = −1.40 as an initial guess for the computer programs in this example.…”

Section: Numerical Examples and Applicationsmentioning

confidence: 99%

Solution of nonlinear equations using three point Gaussian quadrature formula and decomposition technique

Sana

Noor

2021

Punjab Univ. j. math.

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The problem of solving nonlinear equations (real or complex) is a nontrivial task in many areas of science and engineering. Usually, the analytic methods for such equations are not directly affordable and require an iterative approach for getting an approximate solution. Keeping in view the above facts, we suggest and analyze some new iterative methods for solving nonlinear equation of the form f(u) = 0 by using the decomposition technique coupled with a system of equations and threepoints Gaussian quadrature formula. We also determine the convergence order of our proposed iterative methods. Some test examples are given to endorse and validate the performance of new methods as compared to previously well-known methods.

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“…Kansal et al [7] and Kumar et al [8] gave second order iterative schemes to find repeated roots of nonlinear equations. Sharma et al [17], Kumar et al [9,10], Behl et al [2] and Rani and Kansal [13] proposed a fourth-order root finding methods for multiple roots. Qudsi et al [11,12] presented three step sixth order iterative methods for finding the multiple roots.…”

Section: Introductionmentioning

confidence: 99%

An optimal eighth order derivative free multiple root finding numerical method and applications to chemistry

et al. 2022

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In this paper, we present an optimal eighth order derivative-free family of methods for multiple roots which is based on the first order divided difference and weight functions. This iterative method is a three step method with the first step as Traub–Steffensen iteration and the next two taken as Traub–Steffensen-like iteration with four functional evaluations per iteration. We compare our proposed method with the recent derivative-free methods using some chemical engineering problems modelled as nonlinear equations with simple and multiple roots. Stability of the presented family of methods is demonstrated by using the graphical tool known as basins of attraction.

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Derivative-Free King’s Scheme for Multiple Zeros of Nonlinear Functions

Cited by 7 publications

References 23 publications

An optimal derivative-free King's family for multiple zeros and its dynamics

An optimal derivative-free King's family for multiple zeros and its dynamics

Solution of nonlinear equations using three point Gaussian quadrature formula and decomposition technique

An optimal eighth order derivative free multiple root finding numerical method and applications to chemistry

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