  Spectral integration was deployed by Orszag and co-workers (1977, 1980, 1981)
to obtain stable and efficient solvers for the incompressible Navier-Stokes
equation in rectangular geometries. Two methods in current use for channel flow
and plane Couette flow, namely, Kleiser-Schumann (1980) and Kim-Moin-Moser
(1977), rely on the same technique. In its current form, the technique of
spectral integration, as applied to the Navier-Stokes equations, is dominated
by rounding errors at higher Reynolds numbers which would otherwise be within
reach. In this article, we derive a number of versions of spectral integration
and explicate their properties, with a view to extending the Kleiser-Schumann
and Kim-Moin-Moser algorithms to higher Reynolds numbers. More specifically, we
show how spectral integration matrices that are banded, but bordered by dense
rows, can be reduced to purely banded matrices. Key properties, such as the
accuracy of spectral integration even when Green's functions are not resolved
by the underlying grid, the accuracy of spectral integration in spite of
ill-conditioning of underlying linear systems, and the accuracy of derivatives,
are thoroughly explained.
