Особенности применения алгоритма Вейерштрасса для возвратных алгебраических уравнений

Abstract

В статье доказаны необходимые и достаточные условия расположения всех корней такого класса уравнений на окружности комплексной плоскости = Application of Weierstrass algorithm of simultaneous establishment of all roots of an algebraic equation of arbitrary degree for the class of reverse algebraic equations, for which the problem of the choice of primary nearing vector is solved fully, is studied. Geometric nature of the characteristic of the reverse is found out, convenient analytical formulas for the fourth and sixth degree equation roots, which contain not only radicals, are obtained; comparison with the method of Nikiporets functions is made. Numerous examples are given which confirm convergence of the corresponding iteration process with «almost» arbitrary choice of the primary nearing vector. Such behaviour of the iteration process is explained by the system of invariants, i.e. the system of surfaces on which points of iteration process are located. Necessary and sufficient conditions of the location of all roots of such class equations on the circumference of a complex surface are proved.