### 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 n. Because of the regular data access, most machines can perform this algorithm efficiently by keeping three base registers and using simple offset addressing.