A unifying local–semilocal convergence analysis and applications for two-point Newton-like methods in Banach space

Argyros, Ioannis K.

doi:10.1016/j.jmaa.2004.04.008

Cited by 230 publications

(220 citation statements)

References 22 publications

(61 reference statements)

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“…A semilocal convergence analysis for general secant-type methods was given in general by J. E. Dennis [15]. Bosarge and Falb [10], Dennis [11], Potra [21][22][23], Argyros [5][6][7][8][9], Hernández et al [15] and others [14], [19], [27], have provided sufficient convergence conditions for the secant method based on Lipschitz-type conditions on δF .…”

Section: Introductionmentioning

confidence: 99%

Expanding the Applicability of Secant Method With Applications

Magreñán¹,

Argyros²

2015

Bulletin of the Korean Mathematical Society

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Abstract. We present new sufficient convergence criteria for the convergence of the secant-method to a locally unique solution of a nonlinear equation in a Banach space. Our idea uses Lipschitz and center-Lipschitz instead of just Lipschitz conditions in the convergence analysis. The new convergence criteria can always be weaker than the corresponding ones in earlier studies. Numerical examples are also provided in this study to solve equations in cases not possible before.

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Section: Introductionmentioning

confidence: 99%

Expanding the Applicability of Secant Method With Applications

Magreñán¹,

Argyros²

2015

Bulletin of the Korean Mathematical Society

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“…An important extension of the Kantorovich theorem was obtained recently by Argyros [1], [2], who used a combination of Lipschitz and center-Lipschitz conditions in place of the Lipschitz conditions used by Kantorovich. In the present paper, we will formulate and prove an extension of the Kantorovich theorem for the generalized equation (1.3). The depth and scope of this theorem is such that when we specialize it to nonlinear operator equations we get results that are weaker than the Kantorovich theorem.…”

Section: Introductionmentioning

confidence: 99%

On the solution of generalized equations and variational inequalities

Argyros

Hilout

2011

Cubo

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Uko and Argyros provided in [18] a Kantorovich-type theorem on the existence and uniqueness of the solution of a generalized equation of the form f (u)+g(u) ∋ 0, where f is a Fréchet-differentiable function, and g is a maximal monotone operator defined on a Hilbert space. The sufficient convergence conditions are weaker than the corresponding ones given in the literature for the Kantorovich theorem on a Hilbert space. However, the convergence was shown to be only linear.In this study, we show under the same conditions, the quadratic instead of the linear convergenve of the generalized Newton iteration involved. RESUMENUko y Argyros estudian en [18] un teorema tipo-Kantorovich en el existencia y unicidad de la solución de una ecuación generalizada de la forma f (u) + g(u) ∋ 0, donde f es una función Fréchet-diferenciable, y g es un operador monotono máximo definido en un espacio de Hilbert. Las condiciones de convergencia suficientes son más débiles que los correspondientemente dadas en la literatura para el teorema de Kantorovich en un espacio de Hilbert. Sin embargo, la convergencia ha demostrado ser sólo lineal.En este estudio, mostramos en las mismas condiciones, la ecuación cuadrática en lugar de la lineal convergente de la iteración generalizada de Newton involucradas.

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“…There is an extensive literature on the local as well as the semilocal convergence analysis of Newton-type methods under various conditions in the more general setting when X and Y are Banach spaces [1]- [10].…”

Section: Introductionmentioning

confidence: 99%

“…Recently, we have successfully used in [1]- [3] a combination of Lipschitz and center-Lipschitz conditions instead of only Lipschitz conditions as in to provide a finer local and semilocal convergence analysis for Newton-type methods, when F is an isomorphism. The main idea is derived from the observation that more precise upper bounds on the norms k F 0 (x) −1 F 0 (x ) k can be obtained if the needed center-Lipschitz condition is used:…”

Section: Introductionmentioning

confidence: 99%

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On the Gauss-Newton method for solving equations

Argyros

Hitlout

2012

Proyecciones (Antofagasta)

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We use a combination of the center-Lipschitz condition with the Lipschitz condition condition on the Fréchet-derivative of the operator involved to provide a semilocal convergence analysis of the GaussNewton method to a solution of an equation. Using more precise estimates on the distances involved, under weaker hypotheses, and under the same computational cost, we provide an analysis of the GaussNewton method with the following advantages over the corresponding results in [8]: larger convergence domain; finer error estimates on the distances involved, and an at least as precise information on the location of the solution AMS Subject Classification. 65F20, 65G99, 65H10, 49M15.

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A unifying local–semilocal convergence analysis and applications for two-point Newton-like methods in Banach space

Cited by 230 publications

References 22 publications

Expanding the Applicability of Secant Method With Applications

Expanding the Applicability of Secant Method With Applications

On the solution of generalized equations and variational inequalities

On the Gauss-Newton method for solving equations

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