Study of a High Order Family: Local Convergence and Dynamics

The study of the dynamics and the analysis of local convergence of an iterative method, when approximating a locally unique solution of a nonlinear equation, is presented in this article. We obtain convergence using a center-Lipschitz condition where the ball radii are greater than previous studies. We investigate the dynamics of the method. To validate the theoretical results obtained, a real-world application related to chemistry is provided.

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Local Convergence and the Dynamics of a Two-Step Newton-Like Method

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127416300123 ◽

2016 ◽

Vol 26 (05) ◽

pp. 1630012 ◽

Cited By ~ 7

Author(s):

Ioannis K. Argyros ◽

Á. Alberto Magreñán

Keyword(s):

Nonlinear Equation ◽

Convergence Analysis ◽

Unique Solution ◽

Local Convergence ◽

Multiplicity One ◽

Local Convergence Analysis

We present the local convergence analysis and the study of the dynamics of a two-step Newton-like method in order to approximate a locally unique solution of multiplicity one of a nonlinear equation.

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Convergence and Dynamics of a Higher-Order Method

Symmetry ◽

10.3390/sym12030420 ◽

2020 ◽

Vol 12 (3) ◽

pp. 420 ◽

Cited By ~ 2

Author(s):

Alejandro Moysi ◽

Ioannis K. Argyros ◽

Samundra Regmi ◽

Daniel González ◽

Á. Alberto Magreñán ◽

...

Keyword(s):

Nonlinear Equation ◽

Iterative Method ◽

Higher Order ◽

High Order ◽

Local Form ◽

Order Method ◽

First Derivative ◽

The Family ◽

Higher Order Method

Solving problems in various disciplines such as biology, chemistry, economics, medicine, physics, and engineering, to mention a few, reduces to solving an equation. Its solution is one of the greatest challenges. It involves some iterative method generating a sequence approximating the solution. That is why, in this work, we analyze the convergence in a local form for an iterative method with a high order to find the solution of a nonlinear equation. We extend the applicability of previous results using only the first derivative that actually appears in the method. This is in contrast to either works using a derivative higher than one, or ones not in this method. Moreover, we consider the dynamics of some members of the family in order to see the existing differences between them.

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Multistep High-Order Methods for Nonlinear Equations Using Padé-Like Approximants

Discrete Dynamics in Nature and Society ◽

10.1155/2017/3204652 ◽

2017 ◽

Vol 2017 ◽

pp. 1-6

Author(s):

Alicia Cordero ◽

José L. Hueso ◽

Eulalia Martínez ◽

Juan R. Torregrosa

Keyword(s):

Nonlinear Equation ◽

Iterative Methods ◽

Nonlinear Equations ◽

High Order ◽

High Order Methods ◽

Optimal Methods ◽

Numerical Tests ◽

Theoretical Results

We present new high-order optimal iterative methods for solving a nonlinear equation, f(x)=0, by using Padé-like approximants. We compose optimal methods of order 4 with Newton’s step and substitute the derivative by using an appropriate rational approximant, getting optimal methods of order 8. In the same way, increasing the degree of the approximant, we obtain optimal methods of order 16. We also perform different numerical tests that confirm the theoretical results.

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Convergence of a Two-Step Iterative Method for Nondifferentiable Operators in Banach Spaces

Complexity ◽

10.1155/2018/7352780 ◽

2018 ◽

Vol 2018 ◽

pp. 1-11

Author(s):

Abhimanyu Kumar ◽

Dharmendra K. Gupta ◽

Eulalia Martínez ◽

Sukhjit Singh

Keyword(s):

Iterative Method ◽

Banach Spaces ◽

Local Convergence ◽

Recurrence Relations ◽

Semilocal Convergence ◽

Convergence Domain ◽

Differentiability Condition ◽

Type Integral ◽

Weaker Conditions ◽

Theoretical Results

The semilocal and local convergence analyses of a two-step iterative method for nonlinear nondifferentiable operators are described in Banach spaces. The recurrence relations are derived under weaker conditions on the operator. For semilocal convergence, the domain of the parameters is obtained to ensure guaranteed convergence under suitable initial approximations. The applicability of local convergence is extended as the differentiability condition on the involved operator is avoided. The region of accessibility and a way to enlarge the convergence domain are provided. Theorems are given for the existence-uniqueness balls enclosing the unique solution. Finally, some numerical examples including nonlinear Hammerstein type integral equations are worked out to validate the theoretical results.

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Inverse Numerical Iterative Technique for Finding all Roots of Nonlinear Equations with Engineering Applications

Journal of Mathematics ◽

10.1155/2021/6643514 ◽

2021 ◽

Vol 2021 ◽

pp. 1-10

Author(s):

Mudassir Shams ◽

Naila Rafiq ◽

Babar Ahmad ◽

Nazir Ahmad Mir

Keyword(s):

Lower Bound ◽

Nonlinear Equation ◽

Iterative Method ◽

Local Convergence ◽

Nonlinear Models ◽

Numerical Test ◽

Convergence Order ◽

Iterative Technique ◽

Simultaneous Methods ◽

Engineering Sciences

We introduce here a new two-step derivate-free inverse simultaneous iterative method for estimating all roots of nonlinear equation. It is proved that convergence order of the newly constructed method is four. Lower bound of the convergence order is determined using Mathematica and verified with theoretical local convergence order of the method introduced. Some nonlinear models which are taken from physical and engineering sciences as numerical test examples to demonstrate the performance and efficiency of the newly constructed modified inverse simultaneous methods as compared to classical methods existing in literature are presented. Dynamical planes and residual graphs are drawn using MATLAB to elaborate efficiency, robustness, and authentication in its domain.

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Iterative Hermitian R-conjugate solutions to general coupled sylvester matrix equations

Filomat ◽

10.2298/fil1707061l ◽

2017 ◽

Vol 31 (7) ◽

pp. 2061-2072 ◽

Cited By ~ 3

Author(s):

Sheng-Kun Li

Keyword(s):

Iterative Method ◽

Convergence Analysis ◽

Unique Solution ◽

Orthogonal Matrix ◽

Matrix Group ◽

Matrix Equations ◽

Numerical Examples ◽

Sylvester Matrix ◽

Sylvester Matrix Equations ◽

Theoretical Results

For a given symmetric orthogonal matrix R, i.e., RT = R, R2 = I, a matrix A ? Cnxn is termed Hermitian R-conjugate matrix if A = AH, RAR = ?. In this paper, an iterative method is constructed for finding the Hermitian R-conjugate solutions of general coupled Sylvester matrix equations. Convergence analysis shows that when the considered matrix equations have a unique solution group then the proposed method is always convergent for any initial Hermitian R-conjugate matrix group under a loose restriction on the convergent factor. Furthermore, the optimal convergent factor is derived. Finally, two numerical examples are given to demonstrate the theoretical results and effectiveness.

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