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On the convergence of Broyden's method in Hilbert spaces
Argyros, I.K.,Cho, Y.J.,Khattri, S.K. Elsevier [etc.] 2014 Applied Mathematics and Computation Vol.242 No.-
In this paper, we present a new semilocal convergence analysis for an inverse free Broyden's method in a Hilbert space setting. In the analysis, we apply our new idea of recurrent functions concepts of divided differences of order one and Lipschitz/center-Lipschitz conditions on the operator involved. Our analysis extends the applicability of Broyden's method in cases not covered before. Finally, we give an example to illustrate the main result in this paper.
Argyros, Ioannis K.,George, Santhosh Chungcheong Mathematical Society 2015 충청수학회지 Vol.28 No.4
We present a local convergence analysis of some Newton-like methods of R-order at least three in order to approximate a solution of a nonlinear equation in a Banach space. Our sufficient convergence conditions involve only hypotheses on the first and second $Fr{\acute{e}}chet$-derivative of the operator involved. These conditions are weaker that the corresponding ones given by Hernandez, Romero [10] and others [1], [4]-[9] requiring hypotheses up to the third $Fr{\acute{e}}chet$ derivative. Numerical examples are also provided in this study.
THE CONVERGENCE BALL OF INEXACT NEWTON-LIKE METHOD IN BANACH SPACE UNDER WEAK LIPSHITZ CONDITION
Argyros, Ioannis K.,George, Santhosh Chungcheong Mathematical Society 2015 충청수학회지 Vol.28 No.1
We present a local convergence analysis for inexact Newton-like method in a Banach space under weaker Lipschitz condition. The convergence ball is enlarged and the estimates on the error distances are more precise under the same computational cost as in earlier studies such as [6, 7, 11, 18]. Some special cases are considered and applications for solving nonlinear systems using the Newton-arithmetic mean method are improved with the new convergence technique.
ON THE CONVERGENCE OF NEWTON'S METHOD FOR SET VALUED MAPS UNDER WEAK CONDITIONS
Argyros, Ioannis K. The Kangwon-Kyungki Mathematical Society 2012 한국수학논문집 Vol.20 No.1
We provide a convergence analysis of Newton's method for set valued maps under center H$\ddot{o}$lder continuity conditions on the Fr$\acute{e}$chet derivative of the operator involved. This approach extends the applicability of earlier works [4,5,7].
LOCAL CONVERGENCE OF NEWTON-LIKE METHODS FOR GENERALIZED EQUATIONS
Argyros, Ioannis K. The Youngnam Mathematical Society 2009 East Asian mathematical journal Vol.25 No.4
We provide a local convergence analysis for Newton-like methods for the solution of generalized equations in a Banach space setting. Using some ideas of ours introduced in [2] for nonlinear equations we show that under weaker hypotheses and computational cost than in [7] a larger convergence radius and finer error bounds on the distances involved can be obtained.
ON THE "TERRA INCOGNITA" FOR THE NEWTON-KANTROVICH METHOD WITH APPLICATIONS
Argyros, Ioannis Konstantinos,Cho, Yeol Je,George, Santhosh Korean Mathematical Society 2014 대한수학회지 Vol.51 No.2
In this paper, we use Newton's method to approximate a locally unique solution of an equation in Banach spaces and introduce recurrent functions to provide a weaker semilocal convergence analysis for Newton's method than before [1]-[13], in some interesting cases, provided that the Fr$\acute{e}$chet-derivative of the operator involved is p-H$\ddot{o}$lder continuous (p${\in}$(0, 1]). Numerical examples involving two boundary value problems are also provided.
IMPROVED CONVERGENCE RESULTS FOR GENERALIZED EQUATIONS
Argyros, Ioannis K. The Youngnam Mathematical Society Korea 2008 East Asian mathematical journal Vol.24 No.2
We revisit the study of finding solutions of equations containing a differentiable and a continuous term on a Banach space setting [1]-[5], [9]-[11]. Using more precise majorizing sequences than before [9]-[11], we provide a semilocal convergence analysis for the generalized Newton's method as well the generalized modified Newton's method. It turns out that under the same or even weaker hypotheses: finer error estimates on the distances involved, and an at least as precise information on the location of the solution can be obtained. The above benefits are obtained under the same computational cost.
A SEMILOCAL CONVERGENCE ANALYSIS FOR A CERTAIN CLASS OF MODIFIED NEWTON PROCESSES
Argyros, Ioannis K. The Youngnam Mathematical Society Korea 2008 East Asian mathematical journal Vol.24 No.2
An error analysis introduced in [1], [2] is utilized in combination with nondiscrete mathematical induction to provide a finer than before [3]-[7], [11] semilocal convergence analysis for a certain class of modified Newton processes.
ON THE SOLUTION OF NONLINEAR EQUATIONS CONTAINING A NON-DIFFERENTIABLE TERM
Argyros, Ioannis K. The Youngnam Mathematical Society Korea 2008 East Asian mathematical journal Vol.24 No.3
We approximate a locally unique solution of a nonlinear operator equation containing a non-differentiable operator in a Banach space setting using Newton's method. Sufficient conditions for the semilocal convergence of Newton's method in this case have been given by several authors using mainly increasing sequences [1]-[6]. Here, we use center as well as Lipschitz conditions and decreasing majorizing sequences to obtain new sufficient convergence conditions weaker than before in many interesting cases. Numerical examples where our results apply to solve equations but earlier ones cannot [2], [5], [6] are also provided in this study.
LOCAL CONVERGENCE OF THE SECANT METHOD UPPER $H{\ddot{O}}LDER$ CONTINUOUS DIVIDED DIFFERENCES
Argyros, Ioannis K. The Youngnam Mathematical Society Korea 2008 East Asian mathematical journal Vol.24 No.1
The semilocal convergence of the secant method under $H{\ddot{o}}lder$ continuous divided differences in a Banach space setting for solving nonlinear equations has been examined by us in [3]. The local convergence was recently examined in [4]. Motivated by optimization considerations and using the same hypotheses but more precise estimates than in [4] we provide a local convergence analysis with the following advantages: larger radius of convergence and finer error estimates on the distances involved. The results can be used for projection methods, to develop the cheapest possible mesh refinement strategies and to solve equations involving autonomous differential equations [1], [4], [7], [8].