The Experts below are selected from a list of 2445 Experts worldwide ranked by ideXlab platform
P. K. Parida - One of the best experts on this subject based on the ideXlab platform.
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Local convergence analysis for Chebyshev’s method
Journal of Applied Mathematics and Computing, 2019Co-Authors: Chandni Kumari, P. K. ParidaAbstract:In this work, we are working to present a local convergence analysis for Chebyshev’s method by using Majorizing Sequence. The given method is a third order iterative process, used in order to approximate a zero of an nonlinear operator equation in a Banach space. Here we are using a new type of majorant conditions to prove the convergence. We will also try to establish relations between this majorant conditions with results of based on Kantorovich-type and Smale-type assumptions.
Ioannis K Argyros - One of the best experts on this subject based on the ideXlab platform.
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A New Semi-local Convergence Analysis of the Secant Method
International Journal of Applied and Computational Mathematics, 2017Co-Authors: Ioannis K Argyros, Ekaterina NathansonAbstract:We provide a new semi-local convergence analysis for the secant method in a Banach space setting. Argyros and other authors have analyzed the method using a Lipschitz condition and a simple center Lipschitz condition. However, the secant method has two starting vectors \(u_0\), \(u_{-1}\), and it makes sense to analyze it using a mixed center-Lipschitz condition based on both vectors. The direct analysis of the Majorizing Sequence employed in this paper can be used to obtain weaker convergence conditions than those used in earlier studies. A numerical example is given to further justify the theoretical results.
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Expanding the Applicability of High-Order Traub-Type Iterative Procedures
Journal of Optimization Theory and Applications, 2014Co-Authors: Sergio Amat, Ioannis K Argyros, Sonia Busquier, Saïd HiloutAbstract:We propose a collection of hybrid methods combining Newton’s method with frozen derivatives and a family of high-order iterative schemes. We present semilocal convergence results for this collection on a Banach space setting. Using a more precise Majorizing Sequence and under the same or weaker convergence conditions than the ones in earlier studies, we expand the applicability of these iterative procedures.
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An improved convergence analysis of a one-step intermediate Newton iterative scheme for nonlinear equations
Journal of Applied Mathematics and Computing, 2012Co-Authors: Ioannis K Argyros, Livinus U. UkoAbstract:We revisit a one-step intermediate Newton method for the iterative computation of a zero of the sum of two nonlinear operators that was analyzed by Uko and Velásquez (Rev. Colomb. Mat. 35:21–27, 2001 ). By utilizing weaker hypotheses of the Zabrejko-Nguen kind and a modified Majorizing Sequence we perform a semilocal convergence analysis which yields finer error bounds and more precise information on the location of the solution that the ones obtained in Rev. Colomb. Mat. 35:21–27, 2001 . This error analysis is obtained at the same computational cost as the analogous results of Uko and Velásquez (Rev. Colomb. Mat. 35:21–27, 2001 ). We also give two generalizations of the well-known Kantorovich theorem on the solvability of nonlinear equations and the convergence of Newton’s method. Finally, we provide a numerical example to illustrate the predicted-by-theory performance of the Newton iterates involved in this paper.
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Solving equations using Newton's method under weak conditions on Banach spaces with a convergence structure
2008Co-Authors: Ioannis K ArgyrosAbstract:We provide new semilocal results for Newton's method on Banach spaces with a convergence structure. Using more precise Majorizing Sequence we show that, under weaker convergence conditions than before, we can obtain finer error bounds on the distances involved and a more precise information on the location of the solution.
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A semilocal convergence analysis for the method of tangent parabolas
2005Co-Authors: Ioannis K ArgyrosAbstract:We present a semilocal convergence analysis for the method of tangent parabolas (Euler-Chebyshev) using a combination of Lipschitz and center Lipschitz conditions on the Frechet derivatives involved. This way we produce a Majorizing Sequence which converges under weaker conditions than before. The error bounds obtained are more precise and the information of the location of the solution better than in earlier results.
Chandni Kumari - One of the best experts on this subject based on the ideXlab platform.
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Local convergence analysis for Chebyshev’s method
Journal of Applied Mathematics and Computing, 2019Co-Authors: Chandni Kumari, P. K. ParidaAbstract:In this work, we are working to present a local convergence analysis for Chebyshev’s method by using Majorizing Sequence. The given method is a third order iterative process, used in order to approximate a zero of an nonlinear operator equation in a Banach space. Here we are using a new type of majorant conditions to prove the convergence. We will also try to establish relations between this majorant conditions with results of based on Kantorovich-type and Smale-type assumptions.
Atef Ibrahim Elmahdy - One of the best experts on this subject based on the ideXlab platform.
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An Iteratively Regularized Projection Method for Nonlinear Ill-posed Problems
2015Co-Authors: Santhosh George, Atef Ibrahim ElmahdyAbstract:An iterative regularization method in the setting of a finite dimen-sional subspace Xh of the real Hilbert space X has been considered for obtaining stable approximate solution to nonlinear ill-posed oper-ator equations F (x) = y where F: D(F) ⊆ X − → X is a nonlinear monotone operator on X. We assume that only a noisy data yδ with ‖y − yδ ‖ ≤ δ are available. Under the assumption that the Fréchet derivative F ′ of F is Lipschitz continuous, a choice of the regularization parameter using an adaptive selection of the parameter and a stopping rule for the iteration index using a Majorizing Sequence are presented. We prove that under a general source condition on x0 − x̂, the error ‖xh,δn,α − x̂ ‖ between the regularized approximation xh,δn,α, (xh,δ0,α: = Phx0 where Ph is an orthogonal projection on to Xh) and the solution x ̂ is of optimal order. The results of computational experiments are provided which shows the reliability of our method
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on the method of lavrentiev regularization and Majorizing Sequence for solving ill posed problems with monotone operators
Applied mathematical sciences, 2015Co-Authors: Atef Ibrahim Elmahdy, I A AlabdiAbstract:In this paper we consider the Lavrentiev regularization method for obtaining stable approximate solution to nonlinear ill-posed operator equations F (x) = y where F : D(F ) X ! X is a nonlinear monotone operator dened on a real Hilbert space X. We assume that only a noisy data y 2 X withky y k are available. Under the assumption that F is Lipschitz continuous, the iteration x n; converges to the unique solution x of the equation F (x) + (x x0) = y (x0 := x 0; ). It is known that (Tautanhahn (2002)) x converges to the solution ^ of F (x) = y: The convergence analysis and the stopping rule are based on a suitably constructed Majorizing Sequence. Under a general source condition on x0 ^ x we proved that the errorkx n; ^
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CONVERGENCE ANALYSIS OF AN ITERATIVE METHOD FOR NONLINEAR OPERATOR EQUATIONS USING A Majorizing Sequence
International Journal of Mathematical Archive, 2011Co-Authors: Atef Ibrahim ElmahdyAbstract:In this paper we consider the Lavrentiev regularization method for obtaining stable approximate solution to nonlinear ill-posed operator equations where is a nonlinear monotone operator and is a real Hilbert space. Under the assumption that is Lipschitz continuous, the iteration converges to the unique solution of the equation .
Santhosh George - One of the best experts on this subject based on the ideXlab platform.
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An Iteratively Regularized Projection Method for Nonlinear Ill-posed Problems
2015Co-Authors: Santhosh George, Atef Ibrahim ElmahdyAbstract:An iterative regularization method in the setting of a finite dimen-sional subspace Xh of the real Hilbert space X has been considered for obtaining stable approximate solution to nonlinear ill-posed oper-ator equations F (x) = y where F: D(F) ⊆ X − → X is a nonlinear monotone operator on X. We assume that only a noisy data yδ with ‖y − yδ ‖ ≤ δ are available. Under the assumption that the Fréchet derivative F ′ of F is Lipschitz continuous, a choice of the regularization parameter using an adaptive selection of the parameter and a stopping rule for the iteration index using a Majorizing Sequence are presented. We prove that under a general source condition on x0 − x̂, the error ‖xh,δn,α − x̂ ‖ between the regularized approximation xh,δn,α, (xh,δ0,α: = Phx0 where Ph is an orthogonal projection on to Xh) and the solution x ̂ is of optimal order. The results of computational experiments are provided which shows the reliability of our method
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Newton-Type Iteration for Tikhonov Regularization of Nonlinear Ill-Posed Problems
Hindawi Limited, 2013Co-Authors: Santhosh GeorgeAbstract:Recently in the work of George, 2010, we considered a modified Gauss-Newton method for approximate solution of a nonlinear ill-posed operator equation F(x)=y, where F:D(F)⊆X→Y is a nonlinear operator between the Hilbert spaces X and Y. The analysis in George, 2010 was carried out using a Majorizing Sequence. In this paper, we consider also the modified Gauss-Newton method, but the convergence analysis and the error estimate are obtained by analyzing the odd and even terms of the Sequence separately. We use the adaptive method in the work of Pereverzev and Schock, 2005 for choosing the regularization parameter. The optimality of this method is proved under a general source condition. A numerical example of nonlinear integral equation shows the performance of this procedure
