  We consider the geometric generalization of ordinary continued fraction to
the multidimensional case introduced by F. Klein in 1895. A multidimensional
periodic continued fraction is the union of sails with some special group
acting freely on these sails. This group transposes the faces. In this article,
we present a method of constructing "approximate" fundamental domains of
algebraic multidimensional continued fractions and an algorithm testing whether
this domain is indeed fundamental or not. We give some polynomial estimates on
number of the operations for the algorithm. In conclusion we present an example
of fundamental domains calculation for a two-dimensional series of
two-dimensional periodic continued fractions.
