- calculation of the CSI-iterate using Equations (26) and (27),
- calculation of the new moments for the current iterate, and
- updating the bidiagonal matrix and approximating the eigenvalues of the two-cyclic iteration matrix through the QR-iteration.

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.

Michael W. Berry (berry@cs.utk.edu)

Sun May 19 11:34:27 EDT 1996