Preconditioned methods are designed to handle
the case when the only operation we can perform with
matrices and
of the pencil
is multiplication of a vector by
and
.
To accelerate the convergence, we introduce a
preconditioner
. It is also common to call the inverse
the preconditioner; see, e.g., the previous section.
Applying the preconditioner
to a vector
usually involves solving a linear system
.
In many engineering applications, preconditioned iterative
solvers for linear systems are already available,
and efficient preconditioners
are constructed.
We shall show that the same preconditioner
can be used to solve
an eigenvalue problem
and
.
Moreover, existing
codes for the system
can often be just slightly modified to
solve the partial eigenvalue problem with
.
We will assume that the preconditioner is
symmetric positive definite.
As
is also symmetric positive definite, there exist
positive constants
such that
Indefinite preconditioners for symmetric eigenproblems are also possible, but not recommended. Iterative methods for nonsymmetric problems, e.g., based on minimization of the residual, should generally be used when the preconditioner is indefinite, which may increase computational costs considerably.
We will not assume that
the preconditioner commutes with
, or
, despite
the fact that
such an assumption would greatly simplify the theory
of preconditioned methods.
We first define, following [268], a preconditioned
single-vector iterative solver for the pencil
,
as a generalized polynomial method of the following kind:
We only need to choose a polynomial,
either a priori or during the process of iterations,
and use a recursive formula which leads to
an iterative scheme. For
an approximation
to an eigenvalue of the pencil
for a given eigenvector
approximation
, the Rayleigh quotient
(11.8) is typically used:
THE PRECONDITIONED EIGENSOLVER FOR
With and
, we can
obtain a similar algorithm for
.
The polynomials can be chosen in a special way
to force convergence of
to an eigenvector other than
the one corresponding to an extreme eigenvalue.
Similarly, we define general preconditioned
block-iterative methods,
where a group of vectors
,
is computed simultaneously:
In (11.11),
the iterative subspace is defined as a span
THE BLOCK PRECONDITIONED EIGENSOLVER FOR
In the following sections, we consider particular examples of preconditioned eigensolvers.