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###

CDS Matrix-Vector Product

If the matrix is
stored in CDS format, it is still possible to
perform a matrix-vector product by either rows or columns, but this
does not take advantage of the CDS format. The idea
is to make a change in coordinates in the doubly nested loop. Replacing
we get

With the index in the inner loop we see that the
expression accesses the th diagonal of the matrix
(where the main diagonal has number 0).
The algorithm will now have a doubly nested loop with the outer loop
enumerating the diagonals `diag=-p,q` with and the
(nonnegative) numbers of diagonals to the left and right of the main
diagonal. The bounds for the inner loop follow from the requirement
that

The algorithm becomes
for i = 1, n
y(i) = 0
end;
for diag = -diag_left, diag_right
for loc = max(1,1-diag), min(n,n-diag)
y(loc) = y(loc) + val(loc,diag) * x(loc+diag)
end;
end;

The transpose matrix-vector product is a minor variation of the
algorithm above. Using the update formula

we obtain
for i = 1, n
y(i) = 0
end;
for diag = -diag_right, diag_left
for loc = max(1,1-diag), min(n,n-diag)
y(loc) = y(loc) + val(loc+diag, -diag) * x(loc+diag)
end;
end;

The memory access for the CDS-based matrix-vector product
(or ) is
three vectors per inner iteration. On the other hand, there is no
indirect addressing, and the algorithm is vectorizable with vector
lengths of essentially the matrix order . Because of the regular data
access, most machines can perform this algorithm
efficiently by keeping three base registers and using simple offset
addressing.

** Next:** Fast Matrix-Vector Multiplication for
** Up:** Sparse BLAS
** Previous:** CRS Matrix-Vector Product
** Contents**
** Index**
Susan Blackford
2000-11-20