  The decomposition method which makes the parallel solution of the
block-tridiagonal matrix systems possible is presented. The performance of the
method is analytically estimated based on the number of elementary
multiplicative operations for its parallel and serial parts. The computational
speedup with respect to the conventional sequential Thomas algorithm is
assessed for various types of the application of the method. It is observed
that the maximum of the analytical speedup for a given number of blocks on the
diagonal is achieved at some finite number of parallel processors. The values
of the parameters required to reach the maximum computational speedup are
obtained. The benchmark calculations show a good agreement of analytical
estimations of the computational speedup and practically achieved results. The
application of the method is illustrated by employing the decomposition method
to the matrix system originated from a boundary value problem for the
two-dimensional integro-differential Faddeev equations. The block-tridiagonal
structure of the matrix arises from the proper discretization scheme including
the finite-differences over the first coordinate and spline approximation over
the second one. The application of the decomposition method for parallelization
of solving the matrix system reduces the overall time of calculation up to 10
times.
