Figure 2 shows the dependencies involved in the steps of
the above outlined procedure. The pipelined nature of the computation
indicates that Steps 1, 2, and
3 described could be carried out concurrently. For example,
the computation of the anti-diagonal elements 
(shown in the box labeled PHI in
Figure 2) could be overlapped with the computation of the
iterates 
and 
. Also, when the bidiagonal
matrix has been updated with the elements 
and 
(the box labeled GAMMA) by using Equation (25),
the approximation of eigenvalues through the bidiagonal-QR iteration
could be done in parallel with the computation of the next anti-diagonal
elements (
, k+l=8 or k+l=10). Thus, the three functional
components MATVEC, PHI, and GAMMA could be executed on three
different processors. Information about the two-cyclic matrix M of
Equation (2) is required only for the computations in MATVEC, so
that the implementations of PHI and GAMMA is independent of the
format used for storing the matrix.
