  We present a new method for obtaining matched solutions of the rms envelope
equations. In this approach, the envelope equations are first expressed in
Hamiltonian form. The Hamiltonian defines a nonlinear mapping, $\cal M$, and
for periodic transport systems the fixed points of the one-period map are the
matched envelopes. Expanding the Hamiltonian around a fiducial trajectory one
obtains a linear map, $M$, that describes trajectories (rms envelopes) near the
fiducial trajectory. Using $\cal M$ and $M$ we construct a contraction mapping
that can be used to obtain the matched envelopes. The algorithm is
quadratically convergent. Using the zero-current matched parameters as starting
values, the contraction mapping typically converges in a few to several
iterations. Since our approach uses numerical integration to obtain all the
mappings, it includes the effects of nonidealized, $z$-dependent transverse and
longitudinal focusing fields. We present numerical examples including finding a
matched beam in a quadrupole channel with rf bunchers.
