  We consider right and left formulations of the inverse scattering problem for
the Zakharov-Shabat system and the corresponding integral
Gelfand-Levitan-Marchenko equations. Both formulations are helpful for
numerical solving of the inverse scattering problem, which we perform using the
previously developed Toeplitz Inner Bordering (TIB) algorithm. First, we
establish general relations between the right and left scattering coefficients.
Here, along with the known results, we introduce a relation between the left
and right norming coefficients for the N-soliton solution. Then we propose an
auxiliary kernel of the left Gelfand-Levitan-Marchenko equations, which allows
one to solve the right scattering problem numerically. We generalize the TIB
algorithm, initially proposed in the left formulation, to the right scattering
problem case with the obtained formulas. The test runs of the TIB algorithm
illustrate our results reconstructing the various nonsymmetrical potentials
from their right scattering data.
