N. Romero

A modified Chebyshev’s iterative method with at least sixth order of convergence

This paper is devoted to the construction and analysis of a high order variant of the classical Chebyshev method. The method has order of convergence at least six for simple roots. The extension to system of equations and its semilocal convergence for nonlinear equations are presented. Finally, an application to well-known algebraic Riccati equation is considered.

Dynamics of a new family of iterative processes for quadratic polynomials

In this work we show the presence of the well-known Catalan numbers in the study of the convergence and the dynamical behavior of a family of iterative methods for solving nonlinear equations. In fact, we introduce a family of methods, depending on a parameter m@?N@?{0}. These methods reach the order of convergence m+2 when they are applied to quadratic polynomials with different roots. Newtons and Chebyshevs methods appear as particular choices of the family appear for m=0 and m=1, respectively. We make both analytical and graphical studies of these methods, which give rise to rational functions defined in the extended complex plane. Firstly, we prove that the coefficients of the aforementioned family of iterative processes can be written in terms of the Catalan numbers. Secondly, we make an incursion into its dynamical behavior. In fact, we show that the rational maps related to these methods can be written in terms of the entries of the Catalan triangle. Next we analyze its general convergence, by including some computer plots showing the intricate structure of the Universal Julia sets associated with the methods.

On the semilocal convergence of Newton-Kantorovich method under center-Lipschitz conditions

In this work we study Newtons method for solving nonlinear equations with operators defined between two Banach spaces. Together with the classical Kantorovich theory, we consider a center-Lipschitz condition for the Frechet derivative of the involved operator. This fact allow us to obtain a majorizing sequence for the sequence defined in Banach spaces and to give conditions for the convergence. In this way, we obtain a generalization of Kantorovichs theorem that improves the values of the universal constant that appears in it as well as the radius where the solution is located and where it is unique. Finally we illustrate the main theoretical result by means of some examples.

Attracting cycles for the relaxed Newton's method

We study the relaxed Newtons method applied to polynomials. In particular, we give a technique such that for any n>=2, we may construct a polynomial so that when the method is applied to a polynomial, the resulting rational function has an attracting cycle of period n. We show that when we use the method to extract radicals, the set consisting of the points at which the method fails to converge to the roots of the polynomial p(z)=z^m-c (this set includes the Julia set) has zero Lebesgue measure. Consequently, iterate sequences under the relaxed Newtons method converge to the roots of the preceding polynomial with probability one.

On a two-step relaxed Newton-type method

We propose a new two-step relaxed Newton-type method for the approximation of nonlinear equations in Banach spaces. The method is free of any bilinear operator. Moreover, in each iteration, we only approximate an associated linear system. We analyze its semilocal convergence under @w-conditioned divided differences. Finally, we include several practical advantages of the method.

On Iterative Methods with Accelerated Convergence for Solving Systems of Nonlinear Equations

We present a modified method for solving nonlinear systems of equations with order of convergence higher than other competitive methods. We generalize also the efficiency index used in the one-dimensional case to several variables. Finally, we show some numerical examples, where the theoretical results obtained in this paper are applied.

On Steffensen's method on Banach spaces

We present a modification of Steffensens method as a predictor-corrector iterative method, so that we can use Steffensens method to approximate a solution of a nonlinear equation in Banach spaces from the same starting points from which Newtons method converges. We study the semilocal convergence of the predictor-corrector method by using the majorant principle. We illustrate the method with an application to a discrete problem.

Dynamics of a higher-order family of iterative methods

We study the dynamics of a higher-order family of iterative methods for solving non-linear equations. We show that these iterative root-finding methods are generally convergent when extracting radicals. We examine the Julia sets of these methods with particular polynomials. The examination takes place in the complex plane.

Semilocal convergence by using recurrence relations for a fifth-order method in Banach spaces

In this paper, a semilocal convergence result in Banach spaces of an efficient fifth-order method is analyzed. Recurrence relations are used in order to prove this convergence, and some a priori error bounds are found. This scheme is finally used to estimate the solution of an integral equation and so, the theoretical results are numerically checked. We use this example to show the better efficiency of the current method compared with other existing ones, including Newtons scheme.

An extension of Gander's result for quadratic equations

In the study of iterative methods with high order of convergence, Gander provides a general expression for iterative methods with order of convergence at least three in the scalar case. Taking into account an extension of this result, we define a family of iterations in Banach spaces with R-order of convergence at least four for quadratic equations.