  The aim of this article is further development of the theory of linear
difference equations with constant coefficients. We present a new algorithm for
calculating the solution to the Cauchy problem for a three-dimensional
difference equation with constant coefficients in a parallelepiped at the point
using the coefficients of the difference equation and Cauchy data. The
implemented algorithm is the next significant achievement in a series of
articles justifying the Apanovich and Leinartas' theorems about the solvability
and well-posedness of the Cauchy problem. We also use methods of computer
algebra since the three-dimensional case usually demands extended calculations.
