Sonia Busquier

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.

Dynamics of a family of third-order iterative methods that do not require using second derivatives

The purpose of this article is to present results that amount to a description of the conjugacy classes of three third-order root-finding iterative methods that do not require the use of second derivatives for their formulation, for complex polynomials of degrees two, three and four. For degrees two and three, a full description of the conjugacy classes is accomplished, in each case, by a one-parameter family of polynomials. This is done in such a way that, when one applies one of these three root-finding iterative methods to the elements of these parametrized families, a family of iterative methods is obtained, in such a way that its dynamics represents, up to conjugacy, the dynamics of the corresponding iterative root-finding method applied to any complex polynomial having the same degree. For degree four, analogous partial results are obtained.

Adaptive Approximation of Nonlinear Operators

Abstract A multiresolution transform corresponding to interpolatory techniques is used for fast application of second order Taylors approximations. In designing this algorithm we apply data compression to the linear and the bilinear forms that appear on the approximation. Analysis of the error is performed. Finally, some numerical results are presented.

A class of quasi-Newton generalized Steffensen methods on Banach spaces

We consider a class of generalized Steffensen iterations procedure for solving nonlinear equations on Banach spaces without any derivative. We establish the convergence under the Kantarovich-Ostrowskis conditions. The majorizing sequence will be a Newtons type sequence, thus the convergence can have better properties. Finally, a numerical comparation with the classical methods is presented.

Reducing Chaos and Bifurcations in Newton-Type Methods

We study the dynamics of some Newton-type iterative methods when they are applied of polynomials degrees two and three. The methods are free of high-order derivatives which are the main limitation of the classical high-order iterative schemes. The iterative schemes consist of several steps of damped Newtons method with the same derivative. We introduce a damping factor in order to reduce the bad zones of convergence. The conclusion is that the damped schemes become real alternative to the classical Newton-type method since both chaos and bifurcations of the original schemes are reduced. Therefore, the new schemes can be utilized to obtain good starting points for the original schemes.

On a Steffensen’s type method and its behavior for semismooth equations

Abstract A generalization of Steffensen’s method is proposed. Our goal is to obtain similar convergence as Newton’s method, but without to evaluate any derivative and without to have stability problems. Convergence analysis and numerical results for semismooth equations are presented.

On the local convergence of secant-type methods

In this article, we carry out a local convergence study for Secant-type methods. Our goal is to enlarge the radius of convergence, without increasing the necessary hypothesis. Finally, some numerical tests and comparisons with early results are analyzed. E-mail: sonia.busquier@upct.es E-mail: sergio.amat@upct.esIn this article, we carry out a local convergence study for Secant-type methods. Our goal is to enlarge the radius of convergence, without increasing the necessary hypothesis. Finally, some numerical tests and comparisons with early results are analyzed. E-mail: sonia.busquier@upct.es E-mail: sergio.amat@upct.es

On a higher order Secant method

In this paper we carry out a modification of the classical Secant method. Our goal is to accelerate the convergence (until order two), but without using any derivative. Two convergence theorems are presented. Finally, some numerical experiments are analyzed.

On two families of high order Newton type methods

Abstract We study a general class of high order Newton type methods. The schemes consist of the application of several steps of Newton type methods with frozen derivatives. We are interested to improve the order of convergence in each sub-step. In particular, we should finish the computation after some stop criteria and before the full computation of the current approximation. We prove that only two sequences of parameters can be derived verifying these properties. One corresponds to a very well known family and the other is a little (but not natural) modification. Finally, we study some dynamical aspects of these families in order to find differences. Surprisingly, the less natural family seems to have a simpler dynamic.

A fast Chebyshev's method for quadratic equations

A multiresolution transform corresponding to interpolatory technique is used for fast application of Chebyshevs method. In designing this algorithm we apply data compression to the linear and the bilinear forms that appear on the method. A convergence theorem is performed. Finally, some numerical results are presented.