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
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 , 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
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
where a group of vectors
is computed simultaneously:
the iterative subspace is defined as a span
THE BLOCK PRECONDITIONED EIGENSOLVER FOR
In the following sections, we consider particular examples of preconditioned eigensolvers.