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Ioannis K Argyros - One of the best experts on this subject based on the ideXlab platform.
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Extended Semilocal Convergence for the Newton- Kurchatov method
Matematychni Studii, 2020Co-Authors: Halyna Yarmola, Ioannis K Argyros, Stepan ShakhnoAbstract:We provide a Semilocal analysis of the Newton-Kurchatov method for solving nonlinear equations involving a splitting of an operator. Iterative methods have a limited restricted region in general. A Convergence of this method is presented under classical Lipschitz conditions.The novelty of our paper lies in the fact that we obtain weaker sufficient Semilocal Convergence criteria and tighter error estimates than in earlier works. We find a more precise location than before where the iterates lie resulting to at least as small Lipschitz constants. Moreover, no additional computations are needed than before. Finally, we give results of numerical experiments.
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Improved Semilocal Convergence analysis in Banach space with applications to chemistry
Journal of Mathematical Chemistry, 2018Co-Authors: Ioannis K Argyros, Á. Alberto Magreñán, Elena Giménez, Í. Sarría, Juan Antonio SiciliaAbstract:We present a new Semilocal Convergence analysis for Secant methods in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. Our analysis includes the computation of the bounds on the limit points of the majorizing sequences involved. Under the same computational cost on the parameters involved our Convergence criteria are weaker and the error bounds more precise than in earlier studies. A numerical example is also presented to illustrate the theoretical results obtained in this study.
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Extending the applicability of the local and Semilocal Convergence of Newton's method
Applied Mathematics and Computation, 2017Co-Authors: Ioannis K Argyros, Á. Alberto MagreñánAbstract:We present a local as well a Semilocal Convergence analysis for Newton's method in a Banach space setting. Using the same Lipschitz constants as in earlier studies, we extend the applicability of Newton's method as follows: local case: a larger radius is given as well as more precise error estimates on the distances involved. Semilocal case: the Convergence domain is extended; the error estimates are tighter and the information on the location of the solution is at least as precise as before. Numerical examples further justify the theoretical results.
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On the Semilocal Convergence of a two step Newton method under the $\gamma-$condition
Journal of Nonlinear Analysis and Optimization: Theory & Applications, 2016Co-Authors: Ioannis K Argyros, Santhosh GeorgeAbstract:We present a Semilocal Convergence analysis of a two-step Newton method under the $\gamma-$condition in order to??approximate a locally unique solution of an equation in a Banach space setting. The Convergence criteria are??weaker than the corresponding ones in the literature even in the case of the single step Newton method\cite{14}-\cite{19}.??Numerical examplesinvolving a nonlinear integral equation where the older Convergence criteria??are not satisfied but the new Convergence criteria??are satisfied, are also presented in the paper.
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A Semilocal Convergence for a Uniparametric Family of Efficient Secant-Like Methods
Journal of Function Spaces, 2014Co-Authors: Ioannis K Argyros, Daniel González, Á. Alberto MagreñánAbstract:We present a Semilocal Convergence analysis for a uniparametric family of efficient secant-like methods (including the secant and Kurchatov method as special cases) in a Banach space setting (Ezquerro et al., 2000–2012). Using our idea of recurrent functions and tighter majorizing sequences, we provide Convergence results under the same or less computational cost than the ones of Ezquerro et al., (2013, 2010, and 2012) and Hernandez et al., (2000, 2005, and 2002) and with the following advantages: weaker sufficient Convergence conditions, tighter error estimates on the distances involved, and at least as precise information on the location of the solution. Numerical examples validating our theoretical results are also provided in this study.
Jisheng Kou - One of the best experts on this subject based on the ideXlab platform.
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Semilocal Convergence for a class of improved multi step chebyshev halley like methods under extended conditions
Journal of Fixed Point Theory and Applications, 2018Co-Authors: Xiuhua Wang, Jisheng KouAbstract:Semilocal Convergence for a class of improved multi-step Chebyshev–Halley-like methods is considered in this paper. Compared with the results for the Chebyshev method in Hernandez (J Comput Appl Math 126:131–143, 2000), the R-order of Convergence is heightened and the Holder continuity of second derivative is also relaxed. Moreover, an existence-uniqueness theorem is proved under the extended conditions.
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Semilocal Convergence for a class of improved multi-step Chebyshev–Halley-like methods under extended conditions
Journal of Fixed Point Theory and Applications, 2018Co-Authors: Xiuhua Wang, Jisheng KouAbstract:Semilocal Convergence for a class of improved multi-step Chebyshev–Halley-like methods is considered in this paper. Compared with the results for the Chebyshev method in Hernandez (J Comput Appl Math 126:131–143, 2000), the R-order of Convergence is heightened and the Holder continuity of second derivative is also relaxed. Moreover, an existence-uniqueness theorem is proved under the extended conditions.
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Analysis of Semilocal Convergence for ameliorated super-Halley methods with less computation for inversion
LMS Journal of Computation and Mathematics, 2016Co-Authors: Xiuhua Wang, Jisheng KouAbstract:In this paper, the Semilocal Convergence for ameliorated super-Halley methods in Banach spaces is considered. Different from the results in [J. M. Gutiérrez and M. A. Hernández, Comput. Math. Appl. 36 (1998) 1–8], these ameliorated methods do not need to compute a second derivative, the computation for inversion is reduced and the $R$-order is also heightened. Under a weaker condition, an existence–uniqueness theorem for the solution is proved.
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Semilocal Convergence analysis on the modifications for Chebyshev-Halley methods under generalized condition
Applied Mathematics and Computation, 2016Co-Authors: Xiuhua Wang, Jisheng KouAbstract:In this paper, we consider the Semilocal Convergence for modifications of Chebyshev-Halley methods in Banach space. Compared with the results on super-Halley method studied in reference Gutierrez and Hernandez (1998)these modified methods need less computation of inversion, the R-order is improved, and the Lipschitz continuity of second derivative is also relaxed. We prove a theorem to show existence-uniqueness of solution. The R-order for these modified methods is analyzed under generalized condition.
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Semilocal Convergence of a Class of Modified Super-Halley Methods in Banach Spaces
Journal of Optimization Theory and Applications, 2012Co-Authors: Xiuhua Wang, Jisheng KouAbstract:In this paper, we consider the Semilocal Convergence of a class of modified super-Halley methods for solving nonlinear equations in Banach spaces. The Semilocal Convergence of this class of methods is established by using recurrence relations. We construct a system of recurrence relations for the methods, and based on it, we prove an existence–uniqueness theorem that shows the R-order of the methods.
Saïd Hilout - One of the best experts on this subject based on the ideXlab platform.
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on the Semilocal Convergence of damped newton s method
Applied Mathematics and Computation, 2012Co-Authors: Ioannis K Argyros, Saïd HiloutAbstract:Abstract We establish new Semilocal Convergence results for the damped Newton’s method. Two approaches are used: the first one uses recurrent relations Ezquerro et al. (2010) and Hernandez (2000) [13] , [19] and the second concerns recurrent functions introduced by Argyros (2011) [3] . A comparison between these two methods is provided. Some values of the iteration parameters are given which are almost optimal choices of a certain accuracy and with respect to a certain polynomial. Numerical examples illustrating the theoretical results are also presented in this study.
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On the Semilocal Convergence of damped Newton’s method
Applied Mathematics and Computation, 2012Co-Authors: Ioannis K Argyros, Saïd HiloutAbstract:Abstract We establish new Semilocal Convergence results for the damped Newton’s method. Two approaches are used: the first one uses recurrent relations Ezquerro et al. (2010) and Hernandez (2000) [13] , [19] and the second concerns recurrent functions introduced by Argyros (2011) [3] . A comparison between these two methods is provided. Some values of the iteration parameters are given which are almost optimal choices of a certain accuracy and with respect to a certain polynomial. Numerical examples illustrating the theoretical results are also presented in this study.
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Semilocal Convergence conditions for the secant method, using recurrent functions
2011Co-Authors: Ioannis K Argyros, Saïd HiloutAbstract:Using our new concept of recurrent functions, we present new sufficient Convergence conditions for the secant method to a locally unique solution of a nonlinear equation in a Banach space. We combine Lipschitz and center-Lipschitz conditions on the divided difference operator to obtain the Semilocal Convergence analysis of the secant method. Our error bounds are tighter than earlier ones. Moreover, under our Convergence hypotheses, we can expand the applicability of the secant method in cases not covered before [8], [9], [12]-[14], [16], [19]-[21]. Application and examples are also provided in this study.
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extended sufficient Semilocal Convergence for the secant method
Computers & Mathematics With Applications, 2011Co-Authors: Ioannis K Argyros, Saïd HiloutAbstract:We establish new sufficient Convergence conditions for the Secant method to a locally unique solution of a nonlinear equation in a Banach space. Using our new concept of recurrent functions, and combining Lipschitz and center-Lipschitz conditions on the divided difference operator, we obtain a new Semilocal Convergence analysis of the Secant method. Moreover, our sufficient Convergence conditions expand the applicability of the Secant method in cases not covered before (Dennis, 1971 [9], Hernandez et al., 2005 [8], Laasonen, 1969 [15], Ortega and Rheinboldt, 1970 [11], Potra, 1982 [5], Potra, 1985 [7], Schmidt, 1978 [18], Yamamoto, 1987 [12], Wolfe, 1978 [19]). Numerical examples are also provided in this study.
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Semilocal Convergence of newton s method for singular systems with constant rank derivatives
Pure and Applied Mathematics, 2011Co-Authors: Ioannis K Argyros, Saïd HiloutAbstract:We provide a Semilocal Convergence result for approximating a solution of a singular system with constant rank derivatives, using Newton`s method in an Euclidean space setting. Our approach uses more precise estimates and a combination of two Lipschitz-type conditions leading to the following advantages over earlier works [13], [16], [17], [29]: tighter bounds on the distances involved, and a more precise information on the location of the solution. Numerical examples are also provided in this study.
Xiuhua Wang - One of the best experts on this subject based on the ideXlab platform.
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Semilocal Convergence for a class of improved multi step chebyshev halley like methods under extended conditions
Journal of Fixed Point Theory and Applications, 2018Co-Authors: Xiuhua Wang, Jisheng KouAbstract:Semilocal Convergence for a class of improved multi-step Chebyshev–Halley-like methods is considered in this paper. Compared with the results for the Chebyshev method in Hernandez (J Comput Appl Math 126:131–143, 2000), the R-order of Convergence is heightened and the Holder continuity of second derivative is also relaxed. Moreover, an existence-uniqueness theorem is proved under the extended conditions.
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Semilocal Convergence for a class of improved multi-step Chebyshev–Halley-like methods under extended conditions
Journal of Fixed Point Theory and Applications, 2018Co-Authors: Xiuhua Wang, Jisheng KouAbstract:Semilocal Convergence for a class of improved multi-step Chebyshev–Halley-like methods is considered in this paper. Compared with the results for the Chebyshev method in Hernandez (J Comput Appl Math 126:131–143, 2000), the R-order of Convergence is heightened and the Holder continuity of second derivative is also relaxed. Moreover, an existence-uniqueness theorem is proved under the extended conditions.
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Analysis of Semilocal Convergence for ameliorated super-Halley methods with less computation for inversion
LMS Journal of Computation and Mathematics, 2016Co-Authors: Xiuhua Wang, Jisheng KouAbstract:In this paper, the Semilocal Convergence for ameliorated super-Halley methods in Banach spaces is considered. Different from the results in [J. M. Gutiérrez and M. A. Hernández, Comput. Math. Appl. 36 (1998) 1–8], these ameliorated methods do not need to compute a second derivative, the computation for inversion is reduced and the $R$-order is also heightened. Under a weaker condition, an existence–uniqueness theorem for the solution is proved.
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Semilocal Convergence analysis on the modifications for Chebyshev-Halley methods under generalized condition
Applied Mathematics and Computation, 2016Co-Authors: Xiuhua Wang, Jisheng KouAbstract:In this paper, we consider the Semilocal Convergence for modifications of Chebyshev-Halley methods in Banach space. Compared with the results on super-Halley method studied in reference Gutierrez and Hernandez (1998)these modified methods need less computation of inversion, the R-order is improved, and the Lipschitz continuity of second derivative is also relaxed. We prove a theorem to show existence-uniqueness of solution. The R-order for these modified methods is analyzed under generalized condition.
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Semilocal Convergence of a Class of Modified Super-Halley Methods in Banach Spaces
Journal of Optimization Theory and Applications, 2012Co-Authors: Xiuhua Wang, Jisheng KouAbstract:In this paper, we consider the Semilocal Convergence of a class of modified super-Halley methods for solving nonlinear equations in Banach spaces. The Semilocal Convergence of this class of methods is established by using recurrence relations. We construct a system of recurrence relations for the methods, and based on it, we prove an existence–uniqueness theorem that shows the R-order of the methods.
Himanshu Kumar - One of the best experts on this subject based on the ideXlab platform.
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On Semilocal Convergence of three-step Kurchatov method under weak condition
Arabian Journal of Mathematics, 2021Co-Authors: Himanshu KumarAbstract:The purpose of this paper to establish the Semilocal Convergence analysis of three-step Kurchatov method under weaker conditions in Banach spaces. We construct the recurrence relations under the assumption that involved first-order divided difference operators satisfy the $$\omega $$ ω condition. Theorems are given for the existence-uniqueness balls enclosing the unique solution. The application of the iterative method is shown by solving nonlinear system of equations and nonlinear Hammerstein-type integral equations. It illustrates the theoretical development of this study.
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On Semilocal Convergence of two step Kurchatov method
International Journal of Computer Mathematics, 2018Co-Authors: Himanshu Kumar, P. K. ParidaAbstract:ABSTRACTIn this article we present a new Semilocal Convergence analysis for the two step Kurchatov method by using recurrence relations under Lipschitz type conditions on first-order divided difference operator. The main advantage of this iterative method is that it does not require to evaluate any Frechet derivative but it includes extra parameters in the first-order divided difference in order to ensure a good approximation to the first derivative in each iteration. The detailed study of the domain of parameters of the method has been carried out and the applicability of the proposed Convergence analysis is illustrated by solving some numerical examples. It has been concluded that the present method converges more rapidly than the one step Kurchatov method and two step Secant method.
