  This article outlines our derivation of the second order perturbations to a
Schwarzschild black hole, highlighting our use of, and necessary reliance on,
computer algebra. The particular perturbation scenario that is presented here
is the case of the linear quadrapole seeding the second order quadrapole. This
problem amounts to finding the second order Zerilli wave equation, and in
particular the effective source term due to the linear quadrapole. With one
minor exception, our calculations confirm the earlier findings of Gleiser,
et.al. On route to these results we also illustrate that, with the aid of
computer algebra, the linear Schwarzschild problem can be solved in a very
direct manner (i.e., without resorting to the usual function transformations),
and it is this ``direct method'' that drives the higher order perturbation
analysis. The calculations were performed using the GRTensorII computer algebra
package, running on the Maple V platform, along with several new Maple routines
that we have written specifically for these types of problems. Although we have
chosen to consider only the ``quadrapole-quadrapole'' calculation in this
article, the GRTensor environment, with the inclusion of these new routines,
would allow this analysis to be repeated for a far more general problem. These
routines, along with Maple worksheets that reproduce our calculations, are
publicly available at the GRTensor website: www.astro.queensu.ca/~grtensor .
The interested reader is invited to download and use them to reproduce our
results and experiment.
