  A recently developed linear algebraic method for the computation of
perturbation expansion coefficients to large order is applied to the problem of
a hydrogenic atom in a magnetic field. We take as the zeroth order
approximation the $D \rightarrow \infty$ limit, where $D$ is the number of
spatial dimensions. In this pseudoclassical limit, the wavefunction is
localized at the minimum of an effective potential surface. A perturbation
expansion, corresponding to harmonic oscillations about this minimum and higher
order anharmonic correction terms, is then developed in inverse powers of
$(D-1)$ about this limit, to 30th order. To demonstrate the implicit
parallelism of this method, which is crucial if it is to be successfully
applied to problems with many degrees of freedom, we describe and analyze a
particular implementation on massively parallel Connection Machine systems
(CM-2 and CM-5). After presenting performance results, we conclude with a
discussion of the prospects for extending this method to larger systems.
