A family of Newton-type methods with seventh and eighth-order of convergence for solving systems of nonlinear equations

Жанлав, Т.; Mijiddorj, Renchin-Ochir; Otgondorj, Khuder

doi:10.15672/hujms.1061471

Cited by 6 publications

(2 citation statements)

References 25 publications

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“…Another family of iterative methods, featuring seventh and eighth-order Newton-type schemes, is proposed by [28] for solving nonlinear equations. Applications to diverse problems and comparisons with established methods are conducted to assess their effectiveness.…”

Section: Introductionmentioning

confidence: 99%

A three step seventh order iterative method for solution nonlinear equation using Lagrange interpolation technique

Jamali,

Lakho,

Kalhoro

et al. 2024

VFAST trans. math.

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This research paper comprehensively presents the derivation of a seventh-order iteration scheme designed to obtain simple roots of nonlinear equations through the utilization of Lagrange interpolation technique. The scheme is characterized by the requirement for three function evaluations and one evaluation of the first derivative in each iteration. A detailed convergence analysis is also carried out to assess the efficacy of the proposed method. Additionally, the paper includes comprehensive numerical experiments aimed at confirming the theoretical results and illustrating the competitive performance of the derived iteration scheme.

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Section: Introductionmentioning

confidence: 99%

A three step seventh order iterative method for solution nonlinear equation using Lagrange interpolation technique

Jamali,

Lakho,

Kalhoro

et al. 2024

VFAST trans. math.

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“…Another line of research is based on Steffensen's method for solving nonlinear systems, following which some seventh-order derivative-free schemes were designed [7][8][9][10][11]. It is evident that while efforts are being made by the researchers to enhance the order of convergence of an iterative approach, most of the time, this results in an increase in the computational cost per iteration, for example, the seventh-and eighth-order methods developed recently [12][13][14][15][16]. Therefore, even when we create new iterative techniques, we ought to make an effort to minimize the computing expense.…”

Section: Introductionmentioning

confidence: 99%

A Novel Higher-Order Numerical Scheme for System of Nonlinear Load Flow Equations

Zafar,

Cordero,

Maryam

et al. 2024

Algorithms

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Power flow problems can be solved in a variety of ways by using the Newton–Raphson approach. The nonlinear power flow equations depend upon voltages Vi and phase angle δ. An electrical power system is obtained by taking the partial derivatives of load flow equations which contain active and reactive powers. In this paper, we present an efficient seventh-order iterative scheme to obtain the solutions of nonlinear system of equations, with only three steps in its formulation. Then, we illustrate the computational cost for different operations such as matrix–matrix multiplication, matrix–vector multiplication, and LU-decomposition, which is then used to calculate the cost of our proposed method and is compared with the cost of already seventh-order methods. Furthermore, we elucidate the applicability of our newly developed scheme in an electrical power system. The two-bus, three-bus, and four-bus power flow problems are then solved by using load flow equations that describe the applicability of the new schemes.

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A novel families of higher‐order multistep iterative methods for solving nonlinear systems

Saheya,

Zhanlav,

Mijiddorj

2023

Math Methods in App Sciences

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In this paper, we propose the first time fifth‐ and sixth‐order two‐step schemes. Based on the proposed two‐step methods, we develop three‐step iterative methods with the ninth‐ and tenth‐order of convergence. We also suggest two parametric families of these schemes. The obtained results are extended to ‐step iterations. The main advantages of the proposed methods are that they are higher order of convergence and utilize only one matrix inversion per iteration which makes these methods computationally more efficient. Numerical experiments are performed to confirm the theoretical results concerning the order of convergence and computational efficiency.

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A family of Newton-type methods with seventh and eighth-order of convergence for solving systems of nonlinear equations

Cited by 6 publications

References 25 publications

A three step seventh order iterative method for solution nonlinear equation using Lagrange interpolation technique

A three step seventh order iterative method for solution nonlinear equation using Lagrange interpolation technique

A Novel Higher-Order Numerical Scheme for System of Nonlinear Load Flow Equations

A novel families of higher‐order multistep iterative methods for solving nonlinear systems

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