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##

Methods with Preconditioned Inner Iterations

We note that there is no analog of the inverse power method
in our class of preconditioned eigensolvers, as it would
require exact solving linear systems
with the matrix .
However, if such systems are solved using a
preconditioned iterative method as inner iterations,
then the method of inner/outer iterations falls into
the class of preconditioned eigensolvers.
As an example, we consider here the most
straightforward version of the RQI,
applied to the pencil ; see §4.3.

It is common to use MINRES
as a choice of the inner iterative solver, as the system matrix
is indefinite.
When only a fixed
number of inner iterations is used,
we cannot assume that
the equation
is solved any accurately, therefore, well-known convergence
properties of the RQIs,
like cubic convergence, cannot be used directly
to establish convergence of the actual inner/outer
iterative method. Similarly, the standard perturbation theory cannot
be easily applied
unless an assumption is made that the number of inner
steps is large enough.

The matrix
of
the equation is ill-conditioned not only because
is ill-conditioned, but also because
is getting closer to an eigenvalue
of the pencil . It is common to
try to improve the condition number of the matrix
by projecting out the subspace
using regularization
or/and choosing an indefinite preconditioner .
It is not evident, however, whether an improvement of convergence
of inner iterations is beneficial to
convergence of the actual inner/outer
iterative method with only a few inner steps.

There are several similar methods with analogous
properties, e.g., different versions of Newton's
method for finding eigenvectors as stationary
vectors of the Rayleigh quotient (e.g., [461,468])
or truncated rational
Krylov methods (e.g., [378,291])
or inexact homotopy methods (e.g., [309,235,469,310]);
see also the previous section.

The Jacobi-Davidson method [411], described
in §5.6 for a generalized symmetric eigenproblem,
is one of the most famous in this class.
The inexact Jacobi-Davidson method with preconditioned
system solver as inner iteration does
satisfy our definition of a preconditioned eigensolver.

The convergence behavior of such methods with a small
number of inner iterations is not well understood.

** Next:** Preconditioned Conjugate Gradient Methods
** Up:** Preconditioned Eigensolvers A. Knyazev
** Previous:** Davidson Method
** Contents**
** Index**
Susan Blackford
2000-11-20