scholarly journals UNA NUEVA FORMA DEL TEOREMA DE KANTOROVICH PARA EL ME´ TODO DE NEWTON

2017 ◽  
Vol 23 (1) ◽  
pp. 79
Author(s):  
Leopoldo Paredes Soria ◽  
Pedro Canales García
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Unique Solution ◽  
Newton’S Method ◽  
Convergence Theorem ◽  
Newton's Method ◽  
Convergence Condition ◽  
Palabras Clave ◽  
Lipschitz Conditions

Una nueva forma de convergencia de tipo Kantorovich para el me´todo de Newton es establecido para aproximarse localmente a una solucio´n u´nica de la ecuacio´n F (x) = 0 definido sobre un espacio de Banach. Se asume que el operador F es dos veces diferenciable Fre´chet, y que Fr, F rr satisface las condiciones de Lipschitz. Nuestra condicio´n de convergencia difiere de los me´todos conocidos y por lo tanto tiene un valor teo´rico y pra´ctico Palabras clave.-Operador lineal, Diferenciable Fre´chet, Sucesio´n convergente, Unicidad. ABSTRACTA new Kantorovich-type convergence theorem for Newton’s method is established for approximating a locally unique solution of an equation F (x) = 0 defined on a Banach space. It is assumed that the operator F is twice Fre´chet differentiable, and that Fr, F rr satisfy Lipschitz conditions. Our convergence condition differs from earlier ones and therefore it has theoretical and practical value. Keywords.-Linear operator, Differentiable Fre´chet, Convergent succession, Uniqueness.

scholarly journals Unified Local Convergence for Newton’s Method and Uniqueness of the Solution of Equations under Generalized Conditions in a Banach Space

Mathematics ◽  
2019 ◽  
Vol 7 (5) ◽  
pp. 463 ◽  
Author(s):  
Ioannis K. Argyros ◽  
Ángel Alberto Magreñán ◽  
Lara Orcos ◽  
Íñigo Sarría
Keyword(s):  
Banach Space ◽  
Newton’S Method ◽  
Newton's Method ◽  
Convergence Region ◽  
Precise Information ◽  
Solution Of Equations ◽  
Lipschitz Constants ◽  
Uniqueness Of The Solution ◽  
Generalized Newton ◽  
Theoretical Results

Under the hypotheses that a function and its Fréchet derivative satisfy some generalized Newton–Mysovskii conditions, precise estimates on the radii of the convergence balls of Newton’s method, and of the uniqueness ball for the solution of the equations, are given for Banach space-valued operators. Some of the existing results are improved with the advantages of larger convergence region, tighter error estimates on the distances involved, and at-least-as-precise information on the location of the solution. These advantages are obtained using the same functions and Lipschitz constants as in earlier studies. Numerical examples are used to test the theoretical results.

scholarly journals Unified Semi-Local Convergence for k—Step Iterative Methods with Flexible and Frozen Linear Operator

Mathematics ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 233 ◽  
Author(s):  
Ioannis Argyros ◽  
Santhosh George
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Iterative Method ◽  
Convergence Analysis ◽  
Iterative Methods ◽  
Local Convergence ◽  
Small Region ◽  
Convergence Region ◽  
Local Convergence Analysis ◽  
Lipschitz Conditions

The aim of this article is to present a unified semi-local convergence analysis for a k-step iterative method containing the inverse of a flexible and frozen linear operator for Banach space valued operators. Special choices of the linear operator reduce the method to the Newton-type, Newton’s, or Stirling’s, or Steffensen’s, or other methods. The analysis is based on center, as well as Lipschitz conditions and our idea of the restricted convergence region. This idea defines an at least as small region containing the iterates as before and consequently also a tighter convergence analysis.

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