1.2NAOct 11, 2011
On the approximation of derivatives using divided difference operators preserving the local convergence order of iterative methodsMiquel 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.
1.2NAOct 7, 2008
Analysis of a class of non linear subdivision schemes and associated multi-resolution transformsS. Amat, K. Dadourian, J. Liandrat
This paper is devoted to the convergence and stability analysis of a class of nonlinear subdivision schemes and associated multi-resolution transforms. These schemes are defined as a perturbation of a linear subdivision scheme. Assuming a contractivity property, stability and convergence are derived. These results are then applied to various schemes such as uncentered interpolatory linear scheme, WENO scheme [13], Power-P scheme [16] and a non linear scheme using local spherical coordinates [18].
1.2NADec 13, 2008
On a C2-nonlinear subdivision scheme avoiding Gibbs oscillationsSergio 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.
1.2NAJun 17, 2015
On a Moser-Steffensen type method for nonlinear systems of equationsS. Amat, M. Grau-Sanchez, M. A. Hernandez-Veron et al.
This paper is devoted to the construction and analysis of a Moser-Steffensen iterative scheme. The method has quadratic convergence without evaluating any derivative nor inverse operator. We present a complete study of the order of convergence for systems of equations, hypotheses ensuring the local convergence and finally we focus our attention to its numerical behavior. The conclusion is that the method improves the applicability of both Newton and Steffensen methods having the same order of convergence.