General Framework of Preconditioning

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:

Thus, we have a complete description of a general preconditioned eigensolver for the pencil , as shown below:

THE PRECONDITIONED EIGENSOLVER FOR

- Start: select .
- Iterate steps to compute .
- Compute .

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:

where and are -dimensional subspaces. Such a method is easier to analyze theoretically; see an example of a preconditioned subspace iterations based on the power method with shift in [148,149], but may converge slower in practice.

In (11.11),
the iterative subspace is defined as a span

of individual vectors . It is common to use (11.11) recursively by combining it with a procedure for selecting individual vectors in as initial vectors for the next recursive step. The Rayleigh-Ritz method is the usual choice for such a procedure. One step of the recursion is shown below:

THE BLOCK PRECONDITIONED EIGENSOLVER FOR

- Start: select
.
- Iterate steps to compute
.
- Use the Rayleigh-Ritz method for the pencil in the subspace , to compute the Ritz values and the corresponding Ritz vectors .

In the following sections, we consider particular examples of preconditioned eigensolvers.