J.M. Gutiérrez

Geometric constructions of iterative functions to solve nonlinear equations

In this paper we present the geometrical interpretation of several iterative methods to solve a nonlinear scalar equation. In addition, we also review the extension to general Banach spaces and some computational aspects of these methods.

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.

A new semilocal convergence theorem for Newton's method

Abstract A new semilocal convergence theorem for Newtons method is established for solving a nonlinear equation F ( x ) = 0, defined in Banach spaces. It is assumed that the operator F is twice Frechet differentiable, and F ″ satisfies a Lipschitz type condition. Results on uniqueness of solution and error estimates are also given. Finally, these results are compared with those that use Kantorovich conditions.

Complex dynamics of derivative-free methods for nonlinear equations

The dynamical behavior of two iterative derivative-free schemes, Steffensen and M4 methods, is studied in case of quadratic and cubic polynomials. The parameter plane is analyzed for both procedures on quadratic polynomials. Different dynamical planes are showed when the mentioned methods are applied on particular cubic polynomials with real or complex coefficients. The property of immersion of the basins of attraction in all cases is analyzed.

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.

On some computational orders of convergence

Two variants of the Computational Order of Convergence (COC) of an iterative method for solving nonlinear equations are presented. Furthermore, the way to approximate the COC and the new variants to the local order of convergence is analyzed. The new definitions given here does not involve the unknown root. Numerical experiments using adaptive arithmetic with multiple precision and a stopping criteria are implemented without using any known root.

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.

Third-order iterative methods for operators with bounded second derivative

Abstract We analyse the classical third-order methods (Chebyshev, Halley, super-Halley) to solve a nonlnnear equation F(x) = 0, where F is an operator defined between two Banach spaces. Until now the convergence of these methods is established assuming that the second derivative F″ satisfies a Lipschitz condition. In this paper we prove, by using recurrence relations, the convergence of these and other third-order methods just assuming F″ is bounded. We show examples where our conditions are fulfilled and the classical ones fail.