Multidimensional generalization of iterative methods for solving nonlinear problems by means of weight-function procedure

Artidiello, Santiago; Cordero, Alicia; Torregrosa, Juan R.; Vassileva, María P.

doi:10.1016/j.amc.2015.07.024

Cited by 23 publications

(25 citation statements)

References 19 publications

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“…From them, we choose their best expressions (8) and (14,15) (for t 1 � − 9/4 and s 2 � 9/8), respectively, denoted by AS and HS.…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…Recently, Sharma et al [14] proposed fourth-order and sixorder iterative methods based on weighted-Newton iteration. Very recently, Artidiello et al [15] provided fourthorder methods based on the weight function approach. Some researchers have also used the approaches like quadrature formulae, Adomian polynomial, divided difference approach for constructing iterative schemes to solve nonlinear systems.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A New High-Order and Efficient Family of Iterative Techniques for Nonlinear Models

Behl

Martínez

2020

Complexity

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In this paper, we want to construct a new high-order and efficient iterative technique for solving a system of nonlinear equations. For this purpose, we extend the earlier scalar scheme [16] to a system of nonlinear equations preserving the same convergence order. Moreover, by adding one more additional step, we obtain minimum 5th-order convergence for every value of a free parameter, θ∈ℝ, and for θ=−1, the method reaches maximum 6-order convergence. We present an extensive convergence analysis of our scheme. The analytical discussion of the work is upheld by performing numerical experiments on some applied science problems and a large system of nonlinear equations. Furthermore, numerical results demonstrate the validity and reliability of the suggested methods.

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“…From them, we choose their best expressions (8) and (14,15) (for t 1 � − 9/4 and s 2 � 9/8), respectively, denoted by AS and HS.…”

Section: Numerical Experimentsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A New High-Order and Efficient Family of Iterative Techniques for Nonlinear Models

Behl

Martínez

2020

Complexity

View full text Add to dashboard Cite

show abstract

“…In order to properly describe the Taylor development of the matrix weight function, we recall the denotation defined by Artidiello et al in [19]: Let X = R n×n denote the Banach space of real square matrices of size n × n, then the function H : X → X can be defined such that the Fréchet derivative satisfies…”

Section: Design and Convergence Analysis Of The Proposed Classmentioning

confidence: 99%

“…Among others, Sharma et al in [18] designed a scheme with fourth-order of convergence by using this procedure and, more recently, Artidiello et al constructed in [19,20] several classes of high-order schemes by means of matrix weight functions.…”

Section: Introductionmentioning

confidence: 99%

A New Three-Step Class of Iterative Methods for Solving Nonlinear Systems

2019

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In this work, a new class of iterative methods for solving nonlinear equations is presented and also its extension for nonlinear systems of equations. This family is developed by using a scalar and matrix weight function procedure, respectively, getting sixth-order of convergence in both cases. Several numerical examples are given to illustrate the efficiency and performance of the proposed methods.

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“…Recently, Sharma et al [31] proposed fourth and six-order iterative methods based on weighted-Newton iteration. Very recently, Artidiello et al [5] proposed fourth-order methods based on the weight function approach. Different researchers have used quadrature formulae, Adomian polynomial, divided difference approach, ... for constructing iterative schemes to solve nonlinear systems.…”

Section: Introductionmentioning

confidence: 99%

Stable high-order iterative methods for solving nonlinear models

Behl

Cordero

Motsa

et al. 2017

Applied Mathematics and Computation

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There are several problems of pure and applied science which can be studied in the unified framework of the scalar and vectorial nonlinear equations. In this paper, we propose a sixth-order family of Jarratt type methods for solving nonlinear equations. Further, we extend this family to the multidimensional case preserving the order of convergence. Their theoretical and computational properties are fully investigated along with two main theorems describing the order of convergence. We use complex dynamics techniques in order to select, among the elements of this class of iterative methods, those more stable. This process is made by analyzing the conjugacy class, calculating the fixed and critical points and getting conclusions from parameter and dynamical planes. For the implementation of the proposed schemes for system of nonlinear equations, we consider some applied science problems namely, Van der Pol problem, kinematic syntheses, etc. Further, we compare them with existing sixth-order methods to check the validity of the theoretical results. From the numerical experiments, we find that our proposed schemes perform better than the existing ones. Further, we also consider a variety of nonlinear equations to check the performance of the proposed methods for scalar equations.

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Multidimensional generalization of iterative methods for solving nonlinear problems by means of weight-function procedure

Cited by 23 publications

References 19 publications

A New High-Order and Efficient Family of Iterative Techniques for Nonlinear Models

A New High-Order and Efficient Family of Iterative Techniques for Nonlinear Models

A New Three-Step Class of Iterative Methods for Solving Nonlinear Systems

Stable high-order iterative methods for solving nonlinear models

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