Three-step iterative methods for numerical solution of systems of nonlinear equations

Dehghan, Mehdi; Shirilord, Akbar

doi:10.1007/s00366-020-01072-1

Cited by 6 publications

(2 citation statements)

References 22 publications

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“…Since analytical methods are not always suitable for finding solutions to equations, various numerical methods for solving nonlinear problems have been presented by scientists from different branches of science and engineering. Iterative methods are one of the best and most efficient methods to obtain solutions to nonlinear problems in mathematics [3][4][5][6][7][8].…”

Section: Introductionmentioning

confidence: 99%

A family of iterative methods to solve nonlinear problems with applications in fractional differential equations

Erfanifar,

Hajarian,

Sayevand

2023

Math Methods in App Sciences

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In this work, first, a family of fourth‐order methods is proposed to solve nonlinear equations. The methods satisfy the Kung‐Traub optimality conjecture. By developing the methods into memory methods, their efficiency indices are increased. Then, the methods are extended to the multi‐step methods for finding the solutions to systems of problems. The formula for the order of convergence of the multi‐step iterative methods is , where is the step number of the methods. It is clear that computing the Jacobian matrix derivative evaluation and its inversion are expensive; therefore, we compute them only once in every cycle of the methods. The important feature of these multi‐step methods is their high‐efficiency index. Numerical examples that confirm the theoretical results are performed. In applications, some nonlinear problems related to the numerical approximation of fractional differential equations (FDEs) are constructed and solved by the proposed methods.

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Section: Introductionmentioning

confidence: 99%

A family of iterative methods to solve nonlinear problems with applications in fractional differential equations

Erfanifar,

Hajarian,

Sayevand

2023

Math Methods in App Sciences

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“…While some methods give good accuracy but take a large number of iterations [24]. On the other hand, some methods provide high accuracy in short iteration numbers but the computational complexity is immense which takes huge time to compute [25,26,27]. For this consequence, it is inevitable to compare these iterative methods from various aspects.…”

Section: Introductionmentioning

confidence: 99%

Efficient Family of Iterative Methods for Solving Nonlinear Simultaneous Equations: A Comparative Study

Ali¹,

Datta²,

Kamrujjaman³

2021

JAMC

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The solution of a system of nonlinear equations is presumably one of the most common but difficult features in numerical analysis in the sense of different aspirations for instance; high accuracy, minimum computation time, a small number of iterations along with less computational cost. In this study, four distinct iterative methods are presented for solving a system of nonlinear equations such as Broyden's method (BM), Optimal fourth-order method (OFOM), Optimal sixth order method (OSOM), and Homotopy continuation method (HCM). Detail formulations are explained along with the solution procedure. In addition, the rate of convergence and computational complexities are also explained within the formulations. Furthermore, to demonstrate and compare the efficiency of these methods, we solve two real-world practical nonlinear models. However, the results are presented numerically in a tabular form and the approximate results are compared with the exact and approximate solutions of other existing iterative methods. After analyzing the results, we conclude which method is better in what aspects.

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High Efficient Iterative Methods with Scalar Parameter Coefficients for Systems of Nonlinear Equations

Zhanlav,

Otgondorj

2024

J Math Sci

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Three-step iterative methods for numerical solution of systems of nonlinear equations

Cited by 6 publications

References 22 publications

A family of iterative methods to solve nonlinear problems with applications in fractional differential equations

A family of iterative methods to solve nonlinear problems with applications in fractional differential equations

Efficient Family of Iterative Methods for Solving Nonlinear Simultaneous Equations: A Comparative Study

High Efficient Iterative Methods with Scalar Parameter Coefficients for Systems of Nonlinear Equations

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