  This paper is part of a long term program to Cauchy-characteristic matching
(CCM) codes as investigative tools in numerical relativity. The approach has
two distinct features: (i) it dispenses with an outer boundary condition and
replaces this with matching conditions at an interface between the Cauchy and
characteristic regions, and (ii) by employing a compactified coordinate, it
proves possible to generate global solutions. In this paper CCM is applied to
an exact two-parameter family of cylindrically symmetric vacuum solutions
possessing both gravitational degrees of freedom due to Piran, Safier and Katz.
This requires a modification of the previously constructed CCM cylindrical code
because, even after using Geroch decomposition to factor out the $z$-direction,
the family is not asymptotically flat. The key equations in the characteristic
regime turn out to be regular singular in nature.
