Recall that a * band matrix* is one which is nonzero only on certain
diagonals. We say a matrix **A** has * lower bandwidth ***lb** if
for **i>j+lb**, and * upper bandwidth ***ub** if
for **i<j-ub**. If **lb** and/or **ub** are large, then the
techniques of the Section 6 are still applicable,
and the LAPACK routines for band matrices (` sgbsv` and ` spbsv`)
have been optimized for this situation [37,21].
Different techniques are needed when the bandwidth is small.
The reason is that proportionately less and less parallel
work is available in updating the trailing submatrix, which is where
we use the Level 3 BLAS. In the limiting case of tridiagonal
matrices, the parallel algorithm derived as in section 6.1
and the standard serial algorithm are nearly identical.

The problem of solving banded linear systems with a narrow band has been
studied by many authors, see for instance the references in
[9,38].
We will only sketch some of the main ideas and we will do so for
rather simple problems. The reader should keep in mind that these ideas
can easily be generalized for more complicated situations, and many
have appeared in the literature.