Finally, Figure 9 shows the same problem reordered by
odd--even reduction. This is not a general purpose strategy for sparse
matrices, but it is often used to enhance parallelism in tridiagonal
and related systems, so we illustrate it for the sake of comparison
with more general purpose methods. In odd--even reduction (see, e.g.,
[55]), odd node numbers come before even node
numbers, and then this same renumbering is applied recursively within
each resulting subset, and so on until all nodes are numbered.
Although the resulting nonzero pattern of **A** looks superficially
different, we can see from the elimination tree that this method is
essentially equivalent to nested dissection for this type of problem.