Seventh-order derivative-free iterative method for solving nonlinear systems

Wang, Xiaofeng; Zhang, Tie; Qian, Wei; Teng, Mingyan

doi:10.1007/s11075-015-9960-2

Cited by 48 publications

(44 citation statements)

References 14 publications

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“…In this paper, the computational efficiency index (CEI) [10] is used to compare the efficiency of the iterative methods. Some discussions on the CEI can be found in [4][5][6][7]. The CEI of the iterative methods ψ i (i = 1, 2, · · · , 5) is given by…”

Section: Computational Efficiencymentioning

confidence: 99%

“…It should be noted that the Equations (2) and (3) need to compute two LU decompositions in per iteration, respectively. Some derivative-free methods are also discussed by Ezquerro et al in [6] and by Wang et al in [7,8]. The above multi-step derivative-free iterative methods can save the computing time in the High-precision computing.…”

Section: Introductionmentioning

confidence: 99%

“…Many efficient methods have been proposed for solving system of nonlinear equations, see for example [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] and the references therein. The best known method is the Steffensen method [1,2], which is given by…”

Section: Introductionmentioning

confidence: 99%

“…To reduce the computational time and improve the efficiency index of the Steffensen method, many modified high-order methods have been proposed in open literatures, see [3][4][5][6][7][8][9][10][11][12][13][14] and the references therein. Liu et al [3] obtained a fourth-order derivative-free method for solving system of nonlinear equations, which can be written as…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Two Efficient Derivative-Free Iterative Methods for Solving Nonlinear Systems

Wang

Fan

2016

Algorithms

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Abstract:In this work, two multi-step derivative-free iterative methods are presented for solving system of nonlinear equations. The new methods have high computational efficiency and low computational cost. The order of convergence of the new methods is proved by a development of an inverse first-order divided difference operator. The computational efficiency is compared with the existing methods. Numerical experiments support the theoretical results. Experimental results show that the new methods remarkably reduce the computing time in the process of high-precision computing.

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Section: Computational Efficiencymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Two Efficient Derivative-Free Iterative Methods for Solving Nonlinear Systems

Wang

Fan

2016

Algorithms

Self Cite

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“…In order to accelerate the convergence or to reduce the computational cost and function evaluation in each step of the iterative process, many efficient methods have been proposed for solving nonlinear systems, see [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] and the references therein. Cordero et al [3,4] developed some variants of Newton's method.…”

Section: Introductionmentioning

confidence: 99%

An Efficient Sixth-Order Newton-Type Method for Solving Nonlinear Systems

Wang

Yang

2017

Algorithms

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Abstract:In this paper, we present a new sixth-order iterative method for solving nonlinear systems and prove a local convergence result. The new method requires solving five linear systems per iteration. An important feature of the new method is that the LU (lower upper, also called LU factorization) decomposition of the Jacobian matrix is computed only once in each iteration. The computational efficiency index of the new method is compared to that of some known methods. Numerical results are given to show that the convergence behavior of the new method is similar to the existing methods. The new method can be applied to small-and medium-sized nonlinear systems.

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Extended convergence for a fifth‐order novel scheme free from derivatives

Behl

Argyros

Martínez

et al. 2021

Math Methods in App Sciences

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The objective in this paper is the expansion of the utilization for a fifth convergence order scheme without derivatives for finding solutions of Banach space valued equations. Conditions of the first order divided difference of the operator involved are only imposed. In this way the use of the scheme is expanded, since in earlier articles the derivatives until order four that do appear in the iterative method are required for setting convergence. Our technique also provides bounds on error distances as well as information about the location of the solutions not given in earlier works. Experiments with concrete problems complete this study.

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Seventh-order derivative-free iterative method for solving nonlinear systems

Cited by 48 publications

References 14 publications

Two Efficient Derivative-Free Iterative Methods for Solving Nonlinear Systems

Two Efficient Derivative-Free Iterative Methods for Solving Nonlinear Systems

An Efficient Sixth-Order Newton-Type Method for Solving Nonlinear Systems

Extended convergence for a fifth‐order novel scheme free from derivatives

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