In Introduction we discuss the history of the continued fraction and of its generalizations. 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 §3 we compare the geometric interpretations of the multidimensional generalizations of the continued fraction given by Klein, by Voronoi and by author (see preprint no. 86/2003). In §4 we improve an algorithm of computation of a generalization of the covex continued fraction given by the author (preprint no. 10/2004). In §5 we improve an algorithm for ordering three points on a plane.

Mathematical problems and theory of numerical methods