Families of Optimal Derivative-Free Two- and Three-Point Iterative Methods for Solving Nonlinear Equations

Жанлав, Т.; Otgondorj, Kh.; Чулуунбаатар, О.

doi:10.1134/s0965542519060149

Cited by 5 publications

(6 citation statements)

References 20 publications

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“…DP can be derived from DF by substituting n n σ θ = . The following is the main result of our work [11].…”

Section: The Optimal Derivative-free Three-point Iterationsmentioning

confidence: 79%

“…Theorem 3. The iteration (4) with n τ given by (7) and with n α  given by (11) have the order of convergence eight, if the following conditions hold:…”

Section: Application Of Sufficient Convergence Condition To Derive New Df Iterationsmentioning

confidence: 99%

“…Proof. Using the Taylor expansion of smooth enough functions ( ) ( ) we obtain an expression for (11). The comparison of this expression with sufficient condition (8) gives conditions (12).…”

Section: Application Of Sufficient Convergence Condition To Derive New Df Iterationsmentioning

confidence: 99%

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Constructive Theory of Designing Optimal Eighth-Order Derivative-Free Methods for Solving Nonlinear Equations

Жанлав¹,

Otgondorj²,

Mijiddorj³

2020

AJCM

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This paper stresses the theoretical nature of constructing the optimal derivative-free iterations. We give necessary and sufficient conditions for derivative-free three-point iterations with the eighth-order of convergence. We also establish the connection of derivative-free and derivative presence three-point iterations. The use of the sufficient convergence conditions allows us to design wide class of optimal derivative-free iterations. The proposed family of iterations includes not only existing methods but also new methods with a higher order of convergence.

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“…DP can be derived from DF by substituting n n σ θ = . The following is the main result of our work [11].…”

Section: The Optimal Derivative-free Three-point Iterationsmentioning

confidence: 79%

“…Theorem 3. The iteration (4) with n τ given by (7) and with n α  given by (11) have the order of convergence eight, if the following conditions hold:…”

Section: Application Of Sufficient Convergence Condition To Derive New Df Iterationsmentioning

confidence: 99%

See 1 more Smart Citation

Constructive Theory of Designing Optimal Eighth-Order Derivative-Free Methods for Solving Nonlinear Equations

Жанлав¹,

Otgondorj²,

Mijiddorj³

2020

AJCM

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“…In the last section, we present the results of numerical experiments that confirm the theoretical conclusion about the convergence order and made a comparison with well-known methods of the same order of convergence. In this paper, we consider optimization by parameters in the two-point iterative methods [6,7].…”

Section: Introductionmentioning

confidence: 99%

Optimization by parameters in the iterative methods for solving non-linear equations

Жанлав

Otgondorj

2021

Proc. Mong. Acad. Sci.

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In this paper, we used the necessary optimality condition for parameters in a two-point iterations for solving nonlinear equations. Optimal values of these parameters fully coincide with those obtained in [6] and allow us to increase the convergence order of these iterative methods. Numerical experiments and the comparison of existing robust methods are included to confirm the theoretical results and high computational efficiency. In particular, we considered a variety of real life problems from different disciplines, e.g., Kepler’s equation of motion, Planck’s radiation law problem, in order to check the applicability and effectiveness of our proposed methods.

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“…The real constants ν 1 , ν 2 , ν 3 and ν 4 are determined such that the relation (2) holds with equality for the four functions f (x) = 1, x, x 2 , x 3 . While in [13] was used the approximation…”

Section: Introductionmentioning

confidence: 99%

Comparison of some optimal derivative-free three-point iterations

Zhanlav

Otgondorj

2020

J. Numer. Anal. Approx. Theory

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We show that the well-known Khattri et al. methods and Zheng et al. methods are identical. In passing, we propose a suitable calculation formula for Khattri et al. methods. We also show that the families of eighth-order derivative-free methods obtained in [8] include some existing methods, among them the above-mentioned ones as particular cases. We also give the sufficient convergence condition of these families. Numerical examples and comparison with some existing methods were made. In addition, the dynamical behavior of methods of these families is analyzed.

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Families of Optimal Derivative-Free Two- and Three-Point Iterative Methods for Solving Nonlinear Equations

Cited by 5 publications

References 20 publications

Constructive Theory of Designing Optimal Eighth-Order Derivative-Free Methods for Solving Nonlinear Equations

Constructive Theory of Designing Optimal Eighth-Order Derivative-Free Methods for Solving Nonlinear Equations

Optimization by parameters in the iterative methods for solving non-linear equations

Comparison of some optimal derivative-free three-point iterations

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