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ON THE CONVERGENCE OF NEWTON'S METHOD AND LOCALLY HÖLDERIAN INVERSES OF OPERATORS
Ioannis K. Argyros 한국수학교육학회 2009 純粹 및 應用數學 Vol.16 No.1
A semilocal convergence analysis is provided for Newton's method in a Banach space. The inverses of the operators involved are only locally HÖlderian. We make use of a point-based approximation and center-HÖlderian hypotheses for the inverses of the operators involved. Such an approach can be used to approximate solutions of equations involving nonsmooth operators.
APPROXIMATING SOLUTIONS OF EQUATIONSBY COMBINING NEWTON-LIKE METHODS
Ioannis K. Argyros 한국수학교육학회 2008 純粹 및 應用數學 Vol.15 No.1
In cases su±cient conditions for the semilocal convergence of Newton- like methods are violated, we start with a modi¯ed Newton-like method (whose weaker convergence conditions hold) until we stop at a certain ¯nite step. Then using as a starting guess the point found above we show convergence of the Newton- like method to a locally unique solution of a nonlinear operator equation in a Banach space setting. A numerical example is also provided.
On a Quadratically Convergent Iterative Method Using Divided Differencesof Order One
Ioannis K. Argyros 한국수학교육학회 2007 純粹 및 應用數學 Vol.14 No.3
We introduce a new two-point iterative method to approximate solu-tions of nonlinear operator equations. The method uses only divided dierences oforder one, and two previous iterates. However in contrast to the Secant methodwhich is of order 1.618..., our method is of order two. A local and a semilocalconvergence analysis is provided based on the majorizing principle. Finally themonotone convergence of the method is explored on partially ordered topologicalspaces. Numerical examples are also provided where our results compare favorablyto earlier ones [1], [4], [5], [19].
ON THE CONVERGENCE OF NEWTON'S METHOD ANDLOCALLY H?OLDERIAN OPERATORS
Ioannis K. Argyros 한국수학교육학회 2008 純粹 및 應用數學 Vol.15 No.2
A semilocal convergence analysis is provided for Newton's method ina Banach space setting. The operators involved are only locally Holderian. We make use of a point-based approximation and center-Holderian hypotheses. This approach can be used to approximate solutions of equations involving nonsmooth operators.
An Extension of the Contraction Mapping Theorem
Ioannis K. Argyros 한국수학교육학회 2007 純粹 및 應用數學 Vol.14 No.4
An extension of the contraction mapping theorem is provided in a Ba- nach space setting to approximate fixed points of operator equations. Our approach is justified by numerical examples where our results apply whereas the classical contraction mapping principle cannot.
CONVERGENCE THEOREMS FOR NEWTON'S AND MODIFIED NEWTON'S METHODS
Ioannis K. Argyros 한국수학교육학회 2009 純粹 및 應用數學 Vol.16 No.4
In this study we are concerned with the problem of approximating a locally unique solution of an equation in a Banach space setting using Newton's and modi¯ed Newton's methods. We provide weaker convergence conditions for both methods than before [5]-[7]. Then, we combine Newton's with the modi¯ed Newton's method to approximate locally unique solutions of operator equations. Finer error estimates, a larger convergence domain, and a more precise information on the location of the solution are obtained under the same or weaker hypotheses than before [5]-[7]. The results obtained here improve our earlier ones reported in [4]. Numerical examples are also provided.
CONCERNING THE RADII OF CONVERGENCE FOR ACERTAIN CLASS OF NEWTON-LIKE METHODS
Ioannis K. Argyros 한국수학교육학회 2008 純粹 및 應用數學 Vol.15 No.1
Local convergence results for three Newton-like methods in Banach space are provided. A comparison is given between the three convergence radii. Then we show that using the largest convergence radius we can pick an initial guess from with we start the corresponding iteration. It turns out that after a ¯nite number of steps we can always use the iterate found as the starting guess for a faster method, since this iterate will be inside the convergence domain of the new method.