On the Convex Feasibility Problem

The convergence of the projection algorithm for solving the convex feasibility problem for a family of closed convex sets, is in connection with the regularity properties of the family. In the paper [18] are pointed out four cases of such a family depending of the two characteristics: the emptiness and boudedness of the intersection of the family. The case four (the interior of the intersection is empty and the intersection itself is bounded) is unsolved. In this paper we give a (partial) answer for the case four: in the case of two closed convex sets in R3 the regularity property holds.Comment: 14 pages, exposed on 5th International Conference "Actualities and Perspectives on Hardware and Software" - APHS2009, Timisoara, Romani

Sharp estimation of local convergence radius for the Picard iteration

We investigate the local convergence radius of a general Picard iteration in the frame of a real Hilbert space. We propose a new algorithm to estimate the local convergence radius. Numerical experiments show that the proposed procedure gives sharp estimation (i.e., close to or even identical with the best one) for several well known or recent iterative methods and for various nonlinear mappings. Particularly, we applied the proposed algorithm for classical Newton method, for multi-step Newton method (in particular for third-order Potra-Ptak method) and for fifth-order "M5" method. We present also a new formula to estimate the local convergence radius for multi-step Newton method

On the problem of starting points for iterative methods

We propose an algorithm to find a starting point for iterative methods. Numerical experiments show empirically that the algorithm provides starting points for different iterative methods (like Newton method and its variants) with low computational cost

Thresholds of the Inner Steps in Multi-Step Newton Method

We investigate the efficiency of multi-step Newton method (the classical Newton method in which the first derivative is re-evaluated periodically after m steps) for solving nonlinear equations, F ( x ) = 0 , F : D ⊆ R n → R n . We highlight the following property of multi-step Newton method with respect to some other Newton-type method: for a given n, there exist thresholds of m, that is an interval ( m i , m s ) , such that for m inside of this interval, the efficiency index of multi-step Newton method is better than that of other Newton-type method. We also search for optimal values of m

convergence in norm of modified Krasnoselski-Mann iterations for fixed points of demicontractive mappings

International audienc

Parallel Variants of Broyden’s Method

In this paper we investigate some parallel variants of Broyden’s method and, for the basic variant, we present its convergence properties. The main result is that the behavior of the considered parallel Broyden’s variants is comparable with the classical parallel Newton method, and significantly better than the parallel Cimmino method, both for linear and nonlinear cases. The considered variants are also compared with two more recently proposed parallel Broyden’s method. Some numerical experiments are presented to illustrate the advantages and limits of the proposed algorithms