  Solutions obtained by the quasilinearization method (QLM) are compared with
the WKB solutions. While the WKB method generates an expansion in powers of h,
the quasilinearization method (QLM) approaches the solution of the nonlinear
equation obtained by casting the Schroedinger equation into the Riccati form by
approximating nonlinear terms by a sequence of linear ones. It does not rely on
the existence of any kind of smallness parameter. It also, unlike the WKB,
displays no unphysical turning point singularities. It is shown that both
energies and wave functions obtained in the first QLM iteration are accurate to
a few parts of the percent. Since the first QLM iterate is represented by the
closed expression it allows to estimate analytically and precisely the role of
different parameters, and influence of their variation on the properties of the
quantum systems. The next iterates display very fast quadratic convergence so
that accuracy of energies and wave functions obtained after a few iterations is
extremely high, reaching 20 significant figures for the energy of the sixth
iterate. It is therefore demonstrated that the QLM method could be preferable
over the usual WKB method.
