  We treat the calculation of gravitational radiation using the mixed
timelike-null initial value formulation of general relativity. The
determination of an exterior radiative solution is based on boundary values on
a timelike worldtube $\Gamma$ and on characteristic data on an outgoing null
cone emanating from an initial cross-section of $\Gamma$. We present the
details of a 3-dimensional computational algorithm which evolves this initial
data on a numerical grid, which is compactified to include future null infinity
as finite grid points. A code implementing this algorithm is calibrated in the
quasispherical regime. We consider the application of this procedure to the
extraction of waveforms at infinity from an interior Cauchy evolution, which
provides the boundary data on $\Gamma$. This is a first step towards
Cauchy-characteristic matching in which the data flow at the boundary $\Gamma$
is two-way, with the Cauchy and characteristic computations providing exact
boundary values for each other. We describe strategies for implementing
matching and show that for small target error it is much more computationally
efficient than alternative methods.
