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##

Preconditioned Lanczos Method

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

Since is positive definite, we have that is a valid inner
product.
When we use the inner product, , in the Arnoldi method
rather than
the standard one, , we have
and
,
so
is a symmetric matrix.
This implies that the Arnoldi method with the inner product reduces
to
the Lanczos method with inner product applied to the nonsymmetric
matrix
. From earlier discussions,
we conclude that the preconditioned Lanczos method
with inner product has similar convergence behavior as the
spectral transformation Lanczos method applied to a perturbed problem.
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.,

which is cheap to compute, e.g. when is a diagonal matrix.
Recall that the Lanczos recurrence relation for the inexact Cayley transform
is

The recurrence relation can also be written in the form

We know that for a Ritz pair ,
*can* be
small,
when
.
So, if is close to an eigenvector of
and if
is close to , then is close to
an eigenvector of
and so the Ritz value is close to an
eigenvalue of
.
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.

** Next:** Inexact Rational Krylov Method
** Up:** Inexact Methods K. Meerbergen
** Previous:** Jacobi-Davidson Method with Cayley
** Contents**
** Index**
Susan Blackford
2000-11-20