Computational Methods in Nonlinear Analysis

Argyros, Ioannis K.; Hilout, Saïd

doi:10.1142/8475

Cited by 150 publications

(226 citation statements)

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“…holds in general and L L0 can be arbitrarily large [3,8,4]. The conditions involving ∥F (x 0 )∥ and q in (C 5 ) and (C 6 ) can be solved for ∥F (x 0 )∥ and q.…”

Section: Semilocal Convergencementioning

confidence: 99%

“…holds in general and L l0 can be arbitrarily large [3,4,8]. The quadratic polynomials in (H 4 ) have a positive root by (20) and since the coefficients of t and t 2 are positive.…”

Section: Local Convergencementioning

confidence: 99%

“…Many problems from Computational Sciences and other disciplines can be brought in a form similar to equation (1) using Mathematical Modeling [3,4,5]. For example in data fitting, we have X = Y = R m , m is the number of parameters and observations.…”

Section: Introductionmentioning

confidence: 99%

“…That is why the solution methods for these equations are usually iterative. In particular, the practice of Numerical Analysis for finding such solutions is essentially connected to Newton-like methods [3,4,6,7]. The study about convergence matter of iterative procedures is usually centered on two types: semilocal and local convergence analysis.…”

Section: Introductionmentioning

confidence: 99%

“…In the semilocal case, we provide estimates on the smallness of ∥F (x 0 )∥ as well as computable estimates for ∥x n − x * ∥. Notice that such estimates were not given in [1,2] for Damped Newton-like method (4).…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

On the convergence of a damped Newton-like method with modified right hand side vector

Argyros

Cordero

Magreñán

et al. 2015

Applied Mathematics and Computation

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We present a convergence analysis for a Damped Newton-like method with modified right-hand side vector in order to approximate a locally unique solution of a nonlinear equation in a Banach spaces setting. In the special case when the method is defined on R m , our method provides computable error estimates based on the initial data. Such estimates were not given in relevant studies such as [1,2]. Numerical examples further validating the theoretical results are also presented in this study.

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“…holds in general and L L0 can be arbitrarily large [3,8,4]. The conditions involving ∥F (x 0 )∥ and q in (C 5 ) and (C 6 ) can be solved for ∥F (x 0 )∥ and q.…”

Section: Semilocal Convergencementioning

confidence: 99%

“…holds in general and L l0 can be arbitrarily large [3,4,8]. The quadratic polynomials in (H 4 ) have a positive root by (20) and since the coefficients of t and t 2 are positive.…”

Section: Local Convergencementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On the convergence of a damped Newton-like method with modified right hand side vector

Argyros

Cordero

Magreñán

et al. 2015

Applied Mathematics and Computation

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On the Chandrasekhar integral equation

Hernández‐Verón

Martínez

Singh

2021

Comp and Math Methods

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This study is devoted to solve the Chandrasekhar integral equation that it is used for modeling problems in theory of radiative transfer in a plane‐parallel atmosphere, and others research areas like the kinetic theory of gases, neutron transport, traffic model, the queuing theory among others. First of all, we transform the Chandrasekhar integral equation into a nonlinear Hammerstein‐type integral equation with the corresponding Nemystkii operator and the proper nonseparable kernel. Them, we approximate the kernel in order to apply an iterative scheme. This procedure it is solved in two different ways. First one, we solve a nonlinear equation with separable kernel and define an adequate nonlinear operator between Banach spaces that approximates the first problem. Second one, we introduce an approximation for the inverse of the Fréchet derivative that appears in the Newton's iterative scheme for solving nonlinear equations. Finally, we perform a numerical experiment in order to compare our results with previous ones published showing that are competitive.

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About a fixed‐point‐type transformation to solve quadratic matrix equations using the Krasnoselskij method

Hernández‐Verón

Romero

2022

Math Methods in App Sciences

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In this paper, we study the simplest quadratic matrix equation: (X) = X 2 +BX + C = 0. We transform this equation into an equivalent fixed-point equation, and based on it, we construct the Krasnoselskij method. From this transformation, we can obtain iterative schemes more accurate than the successive approximation method. Moreover, under suitable conditions, we establish different results for the existence and localization of a solution for this equation with the Krasnoselskij method. Finally, we see numerically that the predictor-corrector iterative scheme, with the Krasnoselskij method as a predictor and the Newton method as corrector method, can improve the numerical application of the Newton method when approximating a solution of the quadratic matrix equation.

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Computational Methods in Nonlinear Analysis

Cited by 150 publications

References 0 publications

On the convergence of a damped Newton-like method with modified right hand side vector

On the convergence of a damped Newton-like method with modified right hand side vector

On the Chandrasekhar integral equation

About a fixed‐point‐type transformation to solve quadratic matrix equations using the Krasnoselskij method

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