The analysis is given for the Cayley transform only, though shift-and-invert could be used as well. The reasons are threefold. First, the shift-and-invert Arnoldi method and the Cayley transform Arnoldi method produce the same Ritz vectors. Second, it makes a link with the (Jacobi) Davidson methods easier to establish. Third, the Cayley transform leads to a more natural approach to the problem.
In the Arnoldi method applied to the Cayley transform,
, we must
solve a
sequence of linear systems
By putting all the for
together in
,
in
and
in
, we obtain
Note that for a fixed and
,
depends on
and the way each of the linear systems (11.2) are solved.
When (11.2) is
solved by a preconditioner
or a stationary linear system solver, i.e.,
for
, then
is independent of
and
and so is
.
Eigenpairs are computed from
.
In Arnoldi's method, this happens by the Hessenberg matrix that arises
from the
orthogonalization of
; in the Davidson method, one uses the
projection
with
and
, e.g.,
.
contains the eigenvectors associated
with the
well-separated extreme eigenvalues of
.
This is a perturbed
, so the relation with
is
partially
lost, and accurately computing eigenpairs of
may be
difficult.
To see which parameters play a role in this perturbation,
from Theorems 4.3 and 4.4 in [323], it can be shown that
if
has distinct eigenvalues,
then for each eigenpair
of
there is an eigenpair
of
such that
This section can be concluded as follows.
The remaining question is now how to compute the eigenpairs of
or
how to exploit them for computing eigenpairs of
.
In §11.2.3 and §11.2.4,
we discuss the Rayleigh-Ritz technique; i.e.,
eigenpairs are computed from the orthogonal projection of
on the
.
In §11.2.7, the eigenpairs are computed from the
eigenpairs of
directly by use of the rational Krylov recurrence relation.
§11.2.6 presents a Lanczos algorithm that uses the
recurrence relation for the eigenvectors and the Rayleigh-Ritz projection
for the eigenvalues.
Note that and
cannot be chosen too far away from each other.
Suppose the eigenvalue
is wanted. From the conclusion given
above, the convergence is faster when
is
chosen close to
, and the computed eigenvalue can only be
accurate when
is close to
.
In theory, one could select
, which is usually used in
the Davidson method.
Since
in that case, there is a high risk of stagnation in the
latter method.
A robust selection is used in the Jacobi-Davidson method, which we
discuss
in §11.2.5.