Suppose that and
are symmetric and that we have a positive
definite
preconditioner
for
, i.e.,
.
We can use the Arnoldi method applied to
for computing
eigenvalues near
.
(Again, we use a Ritz value
as zero.)
This leads to the recurrence relation
Let satisfy
and define the Ritz vector
.
This vector is obtained from the Lanczos (Arnoldi) recurrence relation,
so
not from a Galerkin projection.
The Ritz value, on the other hand, can be computed via the Rayleigh
quotient, i.e.,
Recall that the Lanczos recurrence relation for the inexact Cayley transform
is
Morgan and Scott [336] proved the convergence for the following
algorithm, for computing eigenvalues of
, i.e., with
.
Note that if , then the preconditioner must not be a too-good
approximation to
; otherwise
.
The value of
may differ for each
.
Morgan and Scott suggest the stopping test
, which is cheap within the
Lanczos method. It is shown in [336] that
if both
and
are uniformly bounded in norm,
converges to an eigenvalue of
.
Moreover, the asymptotic convergence is quadratic.