A New Derivative-Free Method to Solve Nonlinear Equations

Neta, Beny

doi:10.3390/math9060583

Cited by 16 publications

(6 citation statements)

References 13 publications

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“…The qualitative performance of different iterative schemes designed for solving nonlinear equations with multiple roots has been studied by different authors (see, for example, Reference [17][18][19]). It has been made by using discrete complex dynamics, as all these schemes are without memory.…”

Section: Qualitative Study Of the Proposed Iterative Methods With Memory For Multiple Rootsmentioning

confidence: 99%

Memorizing Schröder’s Method as an Efficient Strategy for Estimating Roots of Unknown Multiplicity

2021

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In this paper, we propose, to the best of our knowledge, the first iterative scheme with memory for finding roots whose multiplicity is unknown existing in the literature. It improves the efficiency of a similar procedure without memory due to Schröder and can be considered as a seed to generate higher order methods with similar characteristics. Once its order of convergence is studied, its stability is analyzed showing its good properties, and it is compared numerically in terms of their basins of attraction with similar schemes without memory for finding multiple roots.

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Section: Qualitative Study Of the Proposed Iterative Methods With Memory For Multiple Rootsmentioning

confidence: 99%

Memorizing Schröder’s Method as an Efficient Strategy for Estimating Roots of Unknown Multiplicity

2021

Self Cite

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“…Also, when the derivative value is zero, the application of these methods is not possible. The exploration of nonlinear equation solving has also led to the formulation of derivative-free methods with memory, as showcased by B. Neta [32], who utilized Traub's method and Newton's method. Furthermore, Chanu et al [33] proposed optimal memoryless techniques of fourth and eighth orders, extending them to incorporate memory.…”

Section: Introductionmentioning

confidence: 99%

Derivative-Free Families of With- and Without-Memory Iterative Methods for Solving Nonlinear Equations and Their Engineering Applications

Sharma,

Panday,

Mittal

et al. 2023

Mathematics

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In this paper, we propose a new fifth-order family of derivative-free iterative methods for solving nonlinear equations. Numerous iterative schemes found in the existing literature either exhibit divergence or fail to work when the function derivative is zero. However, the proposed family of methods successfully works even in such scenarios. We extended this idea to memory-based iterative methods by utilizing self-accelerating parameters derived from the current and previous approximations. As a result, we increased the convergence order from five to ten without requiring additional function evaluations. Analytical proofs of the proposed family of derivative-free methods, both with and without memory, are provided. Furthermore, numerical experimentation on diverse problems reveals the effectiveness and good performance of the proposed methods when compared with well-known existing methods.

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“…First we start a short literature review of existing derivative free methods. Various derivative free algorithms of different order for solution of nonlinear equations introduced by [5][6][7][8][9][10] which are based on different interpolating technique and two new Chebyshev-Halley type derivative-free methods have been presented for numerical solution nonlinear equations. Both families require just three and four functional evaluations, respectively, to reach optimum fourth and eighth order of convergence [11,12] Several techniques have been developed that are based on combinations of bisection, regula falsi, and parabolic interpolation named combined bracketing methods for solution of the nonlinear equations.…”

Section: Introductionmentioning

confidence: 99%

A Novel Two Point Optimal Derivative free Method for Numerical Solution of Nonlinear Algebraic, Transcendental Equations and Application Problems using Weight Function

Jamali,

Kalhoro,

Shaikh

et al. 2022

VFAST trans. math.

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It’s a big challenge for researchers to locate the root of nonlinear equations with minimum cost, lot of methods are already exist in literature to find root but their cost are very high In this regard we introduce a two-step fourth order method by using weight function. And proposed method is optimal and derivative free for solution of nonlinear algebraic and transcendental and application problems. MATLAB, Mathematica and Maple software are used to solve the convergence and numerical problems of proposed and their counterpart methods.

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A New Derivative-Free Method to Solve Nonlinear Equations

Cited by 16 publications

References 13 publications

Memorizing Schröder’s Method as an Efficient Strategy for Estimating Roots of Unknown Multiplicity

Memorizing Schröder’s Method as an Efficient Strategy for Estimating Roots of Unknown Multiplicity

Derivative-Free Families of With- and Without-Memory Iterative Methods for Solving Nonlinear Equations and Their Engineering Applications

A Novel Two Point Optimal Derivative free Method for Numerical Solution of Nonlinear Algebraic, Transcendental Equations and Application Problems using Weight Function

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