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.

Publication language:english

Research direction:

Mathematical problems and theory of numerical methods