M.A. Hernández

An acceleration of Newton's method: Super-Halley method

From a study of the convexity we give an acceleration for Newtons method and obtain a new third order method. Then we use this method for solving non-linear equations in Banach spaces, establishing conditions on convergence, existence and uniqueness of solution, as well as error estimates

Recurrence relations for the super-Halley method

In this paper we give sufficient conditions in order to assure the convergence of the super-Halley method in Banach spaces. We use a system of recurrence relations analogous to those given in the classical Newton-Kantorovich theorem, or those given for Chebyshev and Halley methods by different authors. (~) 1998 Elsevier Science Ltd. All rights reserved.

Chebyshev's Approximation Algorithms and Applications

Abstract We introduce a new family of multipoint methods to approximate a solution of a nonlinear operator equation in Banach spaces. An existence-uniqueness theorem and error estimates are provided for these iterations using a technique based on a new system of recurrence relations. To finish, we apply the results obtained to some nonlinear integral equations of the Fredholm type.

Modification of the Kantorovich assumptions for semilocal convergence of the Chebyshev method

This study obtains two semilocal convergence results for the well-known Chebyshev method, which is a third-order iterative process. The hypotheses required are modifications to the normal Kantorovich ones. The results obtained are applied to the reduction of nonlinear integral equations of the Fredholm type and first kind.

Secant-like methods for solving nonlinear integral equations of the Hammerstein type

We consider a one-parametric family of secant-type iterations for solving nonlinear equations in Banach spaces. We establish a semilocal convergence result for these iterations by means of a technique based on a new system of recurrence relations. This result is then applied to obtain existence and uniqueness results for nonlinear integral equations of the Hammerstein type. We also present a numerical example where the solution of a particular Hammerstein integral equation is approximated by different secant-type methods.

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.

ACCESSIBILITY OF SOLUTIONS BY NEWTON'S METHOD

We give new conditions for the convergence of Newtons method in Banach spaces, in terms of the degree of logarithmic convexity. These conditions guarantee the convergence of Newton sequence in cases where the hypothesis of Kantorovich theorem are not verified, as we show in some examples.

Semilocal convergence of the secant method under mild convergence conditions of differentiability

Abstract In this work, we obtain a semilocal convergence result for the secant method in Banach spaces under mild convergence conditions. We consider a condition for divided differences which generalizes those usual ones, i.e., Lipschitz continuous and Holder continuous conditions. Also, we obtain a result for uniqueness of solutions.

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.

An optimization of Chebyshev's method

From Chebyshevs method, new third-order multipoint iterations are constructed with their efficiency close to that of Newtons method and the same region of accessibility.