  We present an algorithm to solve $- \lap u - f(x,u) = g$ with Dirichlet
boundary conditions in a bounded domain $\Omega$. The nonlinearities are
non-resonant and have finite spectral interaction: no eigenvalue of $-\lap_D$
is an endpoint of $\bar{\partial_2f(\Omega,\RR)}$, which in turn only contains
a finite number of eigenvalues. The algorithm is based in ideas used by Berger
and Podolak to provide a geometric proof of the Ambrosetti-Prodi theorem and
advances work by Smiley and Chun for the same problem.
