The feasibility of using CSI-MSVD as a preconditioner
to Krylov-based methods like Arnoldi's algorithm has also been investigated
using Matlab 4.2 implementations of CSI-MSVD in combination with
a k-step Arnoldi method (ARNUPD) as implemented in [17].
Table 3 tabulates the error reduction for approximations
to a few of the largest singular triplets of the matrix ADI (see
Table 1) when ARNUPD
is used with a starting vector that is either random or generated
by the CSI-MSVD procedure (with and ). Only
1 Arnoldi iteration is required with the CSI-MSVD starting vector
to achieve the errors reported in Table 3.
Here, ARNUPD is used with a subspace of dimension 6, and the number of
desired eigenvalues **k**=1 with **p=5** extra vectors are calculated at each step
to obtain the partial Schur decomposition for the 2-cyclic iteration matrix
**M** where

It should be noted that, since * Arnupd* tests for convergence
by ensuring that the error in the Rayleigh Quotient
is within the
user-defined tolerance * tol*, the actual error in the eigenpair
is given by and reported
in Table 3. Here, is obtained as
an eigenvalue of **H**, and where is the
eigenvector of **H** corresponding to . When
the subspace size is kept constant for a given user-defined
tolerance **tol**, starting vectors generated by the CSI-MSVD algorithm
consistently yield improved errors in the eigenpairs
computed by Arnoldi's method. Since,
the CSI-MSVD algorithm typically
requires matrix-vector multiplications to converge to
each eigenvalue [22], a considerable improvement in the
accuracy in the approximated singular triplets by
ARNUPD can be obtained at very little cost. In a more general sense,
CSI-MSVD could be considered a * preconditioning* technique for
fast generation of good starting vectors for any iterative method
used to approximate eigenpairs or singular triplets.

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

Sun May 19 11:34:27 EDT 1996