  We introduce an accurate and efficient algebraic technique for the
computation of the vibrational spectra of triatomic molecules, of both linear
and bent equilibrium geometry. The full three-dimensional potential energy
surface (PES), which can be based on entirely {\it ab initio} data, is
parameterized as a product Morse-cosine expansion, expressed in bond-angle
internal coordinates, and includes explicit interactions among the local modes.
We describe the stretching degrees of freedom in the framework of a Morse-type
expansion on a suitable algebraic basis, which provides exact analytical
expressions for the elements of a sparse Hamiltonian matrix. Likewise, we use a
cosine power expansion on a spherical harmonics basis for the bending degree of
freedom. The resulting matrix representation in the product space is very
sparse and vibrational levels and eigenfunctions can be obtained by efficient
diagonalization techniques. We apply this method to carbonyl sulfide OCS,
hydrogen cyanide HCN, water H$_2$O, and nitrogen dioxide NO$_2$. When we base
our calculations on high-quality PESs tuned to the experimental data, the
computed spectra are in very good agreement with the observed band origins.
