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Basic Theory
As for the GHEP, we want to avoid
transformation of
to a standard eigenproblem.
This would require the solution
of some linear system, involving and/or , per iteration step.
Furthermore, for stability reasons we want to work with orthonormal
transformations, and for this reason our goal is to compute Schur vectors
for the pencil , rather than eigenvectors. This leads to
a generalization of the JacobiDavidson algorithm, in which we compute
a partial Schur form for the pencil. This algorithm can be interpreted
as a subspace iteration variant of the QZ algorithm. A consequence of
this approach is that we have to work with search and test spaces that
are different.
With
, the generalized eigenproblem
is equivalent to the eigenproblem

(221) 
and we denote a generalized eigenvalue of the matrix pair
as a pair
. This approach is preferred, because
underflow or overflow for
in finite
precision arithmetic may occur when and/or are
zero or close to zero, in which case the pair is still well defined
[328], [425, Chap. VI], [376].
A partial generalized Schur
form of dimension for a matrix pair is the
decomposition

(222) 
where and are orthogonal by matrices and
and are upper triangular by matrices. A column
of is referred to as a generalized Schur vector, and we
refer to a pair
, with
as a generalized Schur pair.
It follows that if
is a generalized eigenpair
of then
is a
generalized eigenpair of .
We will now describe a JacobiDavidson method for the generalized
eigenproblem (8.8).
From the relations (8.9) we conclude that
which suggests that we should follow a PetrovGalerkin condition for
the construction of reduced systems.
In each step the approximate eigenvector
is selected from a search subspace
. We require
that the residual
is orthogonal to some other
wellchosen test subspace
:

(223) 
Both subspaces are of the same dimension, say . Equation
(8.10) leads to the projected eigenproblem

(224) 
The pencil
can be reduced
by the QZ algorithm (see §8.2) to generalized Schur form
(note that this is an dimensional problem).
This leads to orthogonal by matrices and
and upper triangular by matrices
and , such that

(225) 
The decomposition can be reordered such that the first column of
and the entries of and represent
the wanted Petrov solution [172].
With
and
,
,
the Petrov vector is defined
as
for the associated generalized
Petrov value . In our
algorithm we will construct orthogonal
matrices and , so that also .
With the decomposition in (8.12), we
construct an approximate partial generalized Schur form (cf.
(8.9)): approximates a , and
approximates the associated . Since
(cf. (8.9)), this
suggests to choose such that
coincides
with
.
With the weights and
we can influence the convergence
of the Petrov values. If we want
eigenpair approximations for eigenvalues
close to a target ,
then the choice
is very effective [172], especially if we want to compute
eigenvalues in the interior of the spectrum of .
We will call the Petrov approximations for this choice
the harmonic Petrov eigenpairs.
The JacobiDavidson correction equation for
the component for
the pencil
becomes

(226) 
with
and
.
It can be shown that if (8.13)
is solved exactly, the convergence to
the generalized eigenvalue will be quadratic;
see [408, Theorem 3.2].
Usually, this correction equation is solved
only approximately, for
instance, with a (preconditioned) iterative
solver. The obtained vector
is used for the expansion of and
is used for the
expansion of . For both spaces we work with orthonormal bases.
Therefore, the new columns are orthonormalized with respect to the
current basis by a modified GramSchmidt orthogonalization process
(see §4.7.1).
It can be shown that, with the above choices for and ,

(227) 
The relation between the partial generalized Schur
form for the given large problem and the
complete generalized Schur form for
the reduced problem (8.11) via
right vectors () is
similar to the relation via left vectors
(). This can also
be exploited for restarts.
Next: Deflation and Restart
Up: JacobiDavidson Method G. Sleijpen and
Previous: JacobiDavidson Method G. Sleijpen and
Contents
Index
Susan Blackford
20001120