  A new iterative method is developed to numerically calculate the periodic,
matched beam envelope solution of the coupled Kapchinskij-Vladimirskij (KV)
equations describing the transverse evolution of a beam in a periodic, linear
focusing lattice of arbitrary complexity. Implementation of the method is
straightforward. It is highly convergent and can be applied to all usual
parameterizations of the matched envelope solutions. The method is applicable
to all classes of linear focusing lattices without skew couplings, and also
applies to all physically achievable system parameters -- including where the
matched beam envelope is strongly unstable. Example applications are presented
for periodic solenoidal and quadrupole focusing lattices. Convergence
properties are summarized over a wide range of system parameters.
