Efficient derivative-free variants of Hansen-Patrick’s family with memory for solving nonlinear equations

Kansal, Munish; Kanwar, V.; Bhatia, Saurabh

doi:10.1007/s11075-016-0127-6

Cited by 11 publications

(3 citation statements)

References 24 publications

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“…Beberapa peneliti yang telah memodifikasi metode Hansen-Patrick dengan mereduksi turunan kedua menggunakan berbagai pendekatan, diantaranya: deret Taylor [9], tafsiran fungsi [10], dan tafsiran jumlah dua metode iterasi [11,12]. Selanjutnya, pada artikel ini dilakukan modifikasi metode iterasi Hansen-Patrick yang diberikan pada Persamaan (4).…”

Section: Pendahuluanunclassified

Modifikasi Metode Hansen-Patrick Dengan Orde Konvergensi Optimal Untuk Menyelesaikan Persamaan Nonlinier

Wartono

2022

KDMA

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The Hansen-Patrick method is a third-order iterative method used to solve nonlinear equation. The method requires three evaluation of functions and has an efficiency index 31/3 » 1,4224. This study discusses a modification of the Hansen-Patrick method using the second order Taylor series. The second derivative is reduced using hyperbolic function with one parameter h. The aim of modification is to improve the convergence order of the Hansen-Patrick’s method. Based on the convergence analysis, the method has a fourth-order of convergence and envolve three evaluation of functions. So, its efficiency index is 41/3 » 1,5874. Numerical simulation is given to illustrate performance of the iterative method using six real functions. The performance of the iterative method include : a computational order of convergence, the number of iteration, evaluation of function, absolute error, and value of function, will be compared with Newton’s method, Halley’s method, Newton-Steffensen’s method, and Hansen-Patrick method. The numerical simulation shows that the performance of the method better than others

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Section: Pendahuluanunclassified

Modifikasi Metode Hansen-Patrick Dengan Orde Konvergensi Optimal Untuk Menyelesaikan Persamaan Nonlinier

Wartono

2022

KDMA

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“…Recently, the research and development of multi-point iterative algorithms have recently attracted more attention to obtain higher order of convergence with decrease in computational cost. Nowadays, lot of attention is given to multi-point iterative methods with memory that employs one or more self accelerated parameters, e.g, Abdullah et al [1], Choubey and Jaiswal [7], Choubey et al [10], Kansal et al [20], and Wang and Zhang [54], etc. to accelearte order of convergence without any extra function evaluations.…”

Section: Introductionmentioning

confidence: 99%

Two Novel With and Without Memory Multi-Point Iterative Methods for Solving Non-Linear Equations

Abdullah,

Choubey,

Dara

2024

Comm. Math. Appl.

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In this work, two new iterative methods are proposed for finding simple roots of non-linear equations. The new methods are the modifications of the existing work proposed by Rafiullah and Jabeen (New eighth and sixteenth order iterative methods to solve nonlinear equations, International Journal of Applied and Computational Mathematics 3 (2017), 2467 -2476). The first method obtained is of fifth-order two-step with memory method while the second scheme is three-point eight-order optimal without memory method. Firstly, the Hermite interpolation polynomial is employed to eliminate the first derivative. To maintain order, the conversion to a memory scheme was accomplished by introducing self-accelerated parameters, all without requiring any new function evaluations. Additionally, the Gauss quadrature approach was incorporated for the first derivative, aiming to attain optimal eighth-order convergence. In particular, the efficiency index is increased from 1.4953 to 1.7099 and 1.5157 to 1.6817 for fifth-and eighth-orders respectively. Some real-life applicationbased problems, such as Kepler's equation, an ocean engineering problem, Planck's radiation law, a blood rheology model, and the charge between two parallel plates were presented to validate and demonstrate the superiority of the proposed scheme. Another benefit of the proposed scheme is on the restriction of the Newton's method that f ′ (v) ̸ = 0 can be eliminated close to the root.

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“…The R-order of convergence of the iterative methods (1.9) with memory is at least 7. Kansal et al's method [11] with memory as follows:…”

Section: Introductionmentioning

confidence: 99%

Creating a new two-step recursive memory method with eight-order based on Kung and Traub’s method

Torkashvand

Momenzadeh

Lotf

2020

Proyecciones (Antofagasta)

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Efficient derivative-free variants of Hansen-Patrick’s family with memory for solving nonlinear equations

Cited by 11 publications

References 24 publications

Modifikasi Metode Hansen-Patrick Dengan Orde Konvergensi Optimal Untuk Menyelesaikan Persamaan Nonlinier

Modifikasi Metode Hansen-Patrick Dengan Orde Konvergensi Optimal Untuk Menyelesaikan Persamaan Nonlinier

Two Novel With and Without Memory Multi-Point Iterative Methods for Solving Non-Linear Equations

Creating a new two-step recursive memory method with eight-order based on Kung and Traub’s method

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