  In a recent study we have obtained correction terms to the large N asymptotic
expansions of the eigenvalue density for the Gaussian unitary and Laguerre
unitary ensembles of random N by N matrices, both in the bulk and at the soft
edge of the spectrum. In the present study these results are used to similarly
analyze the eigenvalue density for Gaussian and Laguerre random matrix
ensembles with orthogonal and symplectic symmetry. As in the case of unitary
symmetry, a matching is exhibited between the asymptotic expansion of the bulk
density, expanded about the edge, and the asymptotic expansion of the edge
density, expanded into the bulk. In addition, aspects of the asymptotic
expansion of the smoothed density, which involves delta functions at the
endpoints of the support, are interpreted microscopically.
