If the matrix 
is banded with bandwidth that is fairly constant
from row to row,
then it is worthwhile to take advantage of this
structure in the storage scheme by storing subdiagonals of the matrix
in consecutive locations. Not only can we eliminate the vector
identifying the column and row, we can pack the nonzero elements in such a
way as to make the matrix-vector product more efficient.
This storage scheme is particularly useful if the matrix arises from
a finite element or finite difference discretization on a tensor
product grid.
We say that the matrix 
is
banded if there are nonnegative constants 
, 
, called the left and
right halfbandwidth, such
that 
only if 
. In this case, we can
allocate for the matrix an array val(1:n,-p:q).
The declaration with reversed dimensions (-p:q,n) corresponds to the
LINPACK band format [73], which unlike CDS,
does not allow for an
efficiently vectorizable matrix-vector multiplication if 
is small.
Usually, band formats involve storing some zeros. The CDS
format may even contain some array elements that do not
correspond to matrix elements at all.
Consider the nonsymmetric matrix
defined by
Using the CDS format, we
store this matrix in an array of dimension (6,-1:1) using
the mapping
Hence, the rows of the val(:,:) array are
.
Notice the two zeros corresponding to non-existing matrix elements.
A generalization of the CDS format more suitable for manipulating
general sparse matrices on vector supercomputers is discussed by
Melhem in [154]. This variant of CDS uses a stripe data structure to store the matrix . This structure is
more efficient in storage in the case of varying bandwidth, but it
makes the matrix-vector product slightly more expensive, as it
involves a gather operation.
As defined in [154],
a stripe in the 
matrix is a set of positions
, where 
and 
is a strictly increasing function.
Specifically, if 
and 
are in 
,
then
When computing the
matrix-vector product 
using stripes, each
element of in stripe 
is multiplied
with both 
and 
and these products are
accumulated in 
and 
, respectively. For
the nonsymmetric matrix defined by
the 
stripes of the matrix stored
in the rows of the val(:,:) array would be
.
