In Introductions we discuss the history of the continued fraction and of its generalizations. In Part I authors propose a new generalization of the continued fraction that gives periodicity for cubic irrationalities with positive discriminant. In Part II we propose a new generalization giving periodicity for cubic irrationalities with negative discriminant. We consider the simultaneous rational approximations of a number and its square. At first we describe the structure of the best integer approximations in homogeneous coordinates when three or two real forms are given. After that we propose an algorithm to compute the approximations. Examples of computations are given as well.

Mathematical problems and theory of numerical methods