  A method for practical realization of the inverse scattering transform method
for the Korteweg-de Vries equation is proposed. It is based on analytical
representations for Jost solutions and for integral kernels of transformation
operators obtained recently by the authors. The representations have the form
of functional series in which the first coefficient plays a crucial role both
in solving the direct scattering and the inverse scattering problems. The
direct scattering problem reduces to computation of a number of the
coefficients following a simple recurrent integration procedure with a
posterior calculation of scattering data by well known formulas. The inverse
scattering problem reduces to a system of linear algebraic equations from which
the first component of the solution vector leads to the recovery of the
potential. We prove the applicability of the finite section method to the
system of linear algebraic equations and discuss numerical aspects of the
proposed method. Numerical examples are given, which reveal the accuracy and
speed of the method.
