Sergio Amat

Miquel Grau-Sánchez, Miquel Noguera, Sergio Amat

A development of an inverse first-order divided difference operator for functions of several variables is presented. Two generalized derivative-free algorithms builded up from Ostrowski's method for solving systems of nonlinear equations are written and analyzed. A direct computation of the local order of convergence for these variants of Ostrowski's 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 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.

Sergio Amat, Karine Dadourian, Jacques Liandrat

This paper is devoted to the presentation and the analysis of a new nonlinear subdivision scheme eliminating the Gibbs oscillations close to discontinuities. Its convergence, stability and order of approximation are analyzed. It is proved that this schemes converges towards limit functions of Hölder regularity index larger than 1.192. Numerical estimates provide an Hölder regularity index of 2.438. Up to our knowledge, this scheme is the first one that achieves simultaneously the control of the Gibbs phenomenon and regularity index larger than 1 for its limit functions.