About one non linear generalization of the compression reflection principle

Abstract

Generalize method of iteration is proposed, presented as a combination of the classical iteration and proportional division methods, on which the conditions of Boltsano-Cochy theorems are satisfied. Аn evidence of the proposed algorithm’s convergence is brought. Originally, the compression reflection operator as a function from one variable is considered. A theorem about the convergence of this iteration and its evidence are brought. Both cases of stage and spiral iterations are considered. A geometrical interpretation of a convergence of a generalize method of iteration is brought, the case of stage and spiral iterations are considered. The formula for the non linear generalize compression reflection operator as a function from one variable is obtained. A formulation of the non linear generalizes reflections as a function from some variables is also exposed. On the exercises of some examples of the transcendent equations systems the advantage of a generalize method of iteration on respect to the known methods a view point as of the width of region as of the speed of convergence was showed.

Keywords: generalize method of iteration; non linear generalize reflection; compressing
original; convergence; immovable point; transcendent equations systems.