A Quartically Convergent Jarratt-Type Method for Nonlinear System of Equations

Ghorbanzadeh, Mohammad; Soleymani, Fazlollah

doi:10.3390/a8030415

Cited by 5 publications

(3 citation statements)

References 17 publications

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“…where α is a real free parameter. Finally, Ghorbanzadeh and Soleymani presented in [11] an iterative method solving nonlinear equations or nonlinear systems, which is a particular case of our scheme using the weight function…”

Section: Convergence and Stabilitymentioning

confidence: 99%

Fixed Point Root-Finding Methods of Fourth-Order of Convergence

2019

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In this manuscript, by using the weight-function technique, a new class of iterative methods for solving nonlinear problems is constructed, which includes many known schemes that can be obtained by choosing different weight functions. This weight function, depending on two different evaluations of the derivative, is the unique difference between the two steps of each method, which is unusual. As it is proven that all the members of the class are optimal methods in the sense of Kung-Traub's conjecture, the dynamical analysis is a good tool to determine the best elements of the family in terms of stability. Therefore, the dynamical behavior of this class on quadratic polynomials is studied in this work. We analyze the stability of the presented family from the multipliers of the fixed points and critical points, along with their associated parameter planes. In addition, this study enables us to select the members of the class with good stability properties.

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Section: Convergence and Stabilitymentioning

confidence: 99%

Fixed Point Root-Finding Methods of Fourth-Order of Convergence

2019

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“…Then, a new fourth-order Jarratt-type method for solving systems of non-linear equations is proposed. The local convergence order of this method is proven (Ghorbanzadeh & Soleymani, 2015). A generalization of a family of methods for solving non-linear equations with unknown multiplicities to the system of non-linear equations is also proposed by using a non-zero multi-variable auxiliary function.…”

Section: Introductionmentioning

confidence: 99%

The Solution of Non-Linear Equations System Containing Interpolation Functions by Relaxing the Newton Method

Rokhman

Wahyudi

Hendarto

2022

ComTech

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Many world phenomena lead to nonlinear equations systems. For some applications, the non-linear equations which construct the non-linear equations system are interpolation functions. However, the interpolation functions are usually not represented as mathematical expressions but as computer programs in specific programming languages. The research proposed using the relaxed Newton method to solve the non-linear equations system that contained interpolation functions. The interpolation functions were represented in the R programming language. Then, the experiment used the Spline interpolation function to construct a two-dimensional non-linear equations system. Eleven initial guesses, maximum of ten-time iterations, and 10-7 precision were applied. The solution of the non-linear equations system and the iteration needed on each initial guess were observed. The experiment shows that the proposed method works well for solving the non-linear equations system constructed by Spline interpolation functions. By observing the initial guesses used in the experiment, there are four possible results: true solution, spurious solution, false solution, and no solution. Applying 11 initial guesses have five initial guesses resulting in true solutions, one initial guess in spurious solution, three initial guesses in false solutions, and one initial guess in no solution. The discussions imply that this method can be generalized to the three-dimensional non-linear equations system or higher dimensions.

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“…Recently, authors in [1] designed an efficient two-step bi-parametric iterative method with memory (CM) possessing seventh R -order of convergence (4) where N j ( l ) stands for Newton's interpolatory polynomial of j th order passing through j + 1 nodes at the point l .…”

Section: Introductory Notesmentioning

confidence: 99%