Sergio Amat

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.

Tensor product multiresolution analysis with error control for compact image representation

A class of multiresolution representations based on nonlinear prediction is studied in the multivariate context based on tensor product strategies. In contrast to standard linear wavelet transforms, these representations cannot be thought of as a change of basis, and the error induced by thresholding or quantizing the coefficients requires a different analysis. We propose specific error control algorithms which ensure a prescribed accuracy in various norms when performing such operations on the coefficients. These algorithms are compared with standard thresholding, for synthetic and real images.

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.

Analysis of a New Nonlinear Subdivision Scheme. Applications in Image Processing

A nonlinear multiresolution scheme within Hartens framework is presented, based on a new nonlinear, centered piecewise polynomial interpolation technique. Analytical properties of the resulting subdivision scheme, such as convergence, smoothness, and stability, are studied. The stability and the compression properties of the associated multiresolution transform are demonstrated on several numerical experiments on images.

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.

On the approximation of derivatives using divided difference operators preserving the local convergence order of iterative methods

A development of an inverse first-order divided difference operator for functions of several variables is presented. Two generalized derivative-free algorithms built up from Ostrowskis method for solving systems of nonlinear equations are written and analyzed. A direct computation of the local order of convergence for these variants of Ostrowskis method is given. In order to preserve the local order of convergence, any divided difference operator is not valid. Two counterexamples of computation of a classical divided difference operator without preserving the order are presented. A rigorous study to know a priori if the new method will preserve the order of the original modified method is presented. The conclusion is that this fact does not depend on the method but on the systems of equations and if the associated divided difference verifies a particular condition. A new divided difference operator solving this problem is proposed. Furthermore, a computation that approximates the order of convergence is generated for the examples and it confirms in a numerical way that the order of the methods is well deduced. This study can be applied directly to other Newtons type methods where derivatives are approximated by divided differences.

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