In Introduction we discuss the history of the continued fraction and of its generalisations. In §1 we compare the geometric interpretations of the continued fraction given by Klein, by Voronoi and by author, and define the convex continued fraction. In §2 we propose an algorithm of computation of the convex continued fraction. In §S3 we compare the geometric interpretations of the multidimensional generalisations of the continued fraction given by Klein, by Voronoi and by author (see preprint no. 86/2003). In §4 we propose an algorithm of computation of a generalisation of the covex continued fraction. In §5 we compare points of Klein, of Voronoi and of the author in two-dimensional and three-dimensional cases.

Mathematical problems and theory of numerical methods