High-order iterations for systems of nonlinear equations

Жанлав, Т.; Chun, Changbum; Otgondorj, Kh.; Ulziibayar, Vandandoo

doi:10.1080/00207160.2019.1652739

Cited by 17 publications

(13 citation statements)

References 19 publications

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“…cannot be handled with the analyses given previously [5][6][7] for method (4), method (5), or method (6).…”

Section: Moccari-lofti Methods (Fourth Order) [19]mentioning

confidence: 99%

“…By choosing some of the linear operators equal to the O linear operators in (3), we obtain the methods studied in [5]. Moreover, if X = R k , then we obtain the methods studied in [6,7]. In particular, the methods in [5] are of the special form…”

Section: Introductionmentioning

confidence: 99%

“…In particular, operators must have a special form to obtain the fourth, seventh or eighth order of convergence. Further specifications of operators "O" lead to well-studied methods, a few of which are listed below (other choices can be found in [6,7,9,10]): Newton method (second order) [1,4,11,12]:…”

Section: Introductionmentioning

confidence: 99%

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Generalized Three-Step Numerical Methods for Solving Equations in Banach Spaces

Argyros

Regmi

et al. 2022

Mathematics

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In this article, we propose a new methodology to construct and study generalized three-step numerical methods for solving nonlinear equations in Banach spaces. These methods are very general and include other methods already in the literature as special cases. The convergence analysis of the specialized methods is been given by assuming the existence of high-order derivatives which are not shown in these methods. Therefore, these constraints limit the applicability of the methods to equations involving operators that are sufficiently many times differentiable although the methods may converge. Moreover, the convergence is shown under a different set of conditions. Motivated by the optimization considerations and the above concerns, we present a unified convergence analysis for the generalized numerical methods relying on conditions involving only the operators appearing in the method. This is the novelty of the article. Special cases and examples are presented to conclude this article.

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“…cannot be handled with the analyses given previously [5][6][7] for method (4), method (5), or method (6).…”

Section: Moccari-lofti Methods (Fourth Order) [19]mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Generalized Three-Step Numerical Methods for Solving Equations in Banach Spaces

Argyros

Regmi

et al. 2022

Mathematics

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“…This way we obtain in practice the order of convergence without resorting to the computation of higher order derivatives appearing in the method or in the suffcient convergence criteria usually appearing in the Taylor expansions for the proofs of those results [5,6,14,20,[25][26][27][28][30][31][32][33][34] . It is worth noticing that the computation of ξ and 1 ξ uses method (2) and does not depend on Theorem 2.1 which simply guarantees convergence to x * .…”

Section: W G T T W T W G T T G T G G T T W G T T D G T W T W G T T W T W G T Tmentioning

confidence: 99%

“…The order of convergence was shown to be p+2q [32] . Numerous popular iterative methods are special cases of method (2) [2][3][4][5][6][7][8][9][10][11]14,15,[17][18][19][20][22][23][24][25][26][27][28][30][31][32][33][34] (see Section 3).…”

Section: Introductionmentioning

confidence: 99%

High Convergence Order Q-Step Methods for Solving Equations and Systems of Equations

Argyros

George²

2020

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The local convergence analysis of iterative methods is important since it demonstrates the degree of diffculty for choosing initial points. In the present study, we introduce generalized multi-step high order methods for solving nonlinear equations. The local convergence analysis is given using hypotheses only on the first derivative which actually appears in the methods in contrast to earlier works using hypotheses on higher order derivatives. This way we extend the applicability of these methods. The analysis includes computable radius of convergence as well as error bounds based on Lipschitz-type conditions not given in earlier studies. Numerical examples conclude this study.

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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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High-order iterations for systems of nonlinear equations

Cited by 17 publications

References 19 publications

Generalized Three-Step Numerical Methods for Solving Equations in Banach Spaces

Generalized Three-Step Numerical Methods for Solving Equations in Banach Spaces

High Convergence Order Q-Step Methods for Solving Equations and Systems of Equations

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

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