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Orthogonal Projection Methods.
Let be an complex matrix and be an
dimensional subspace of and consider the eigenvalue
problem of finding
belonging to and belonging to
such that

(7) 
An orthogonal projection technique onto the subspace seeks an
approximate eigenpair
to the above
problem, with in and in . This approximate
eigenpair is obtained by imposing the
following Galerkin condition:

(8) 
or, equivalently,

(9) 
In order to translate this into a matrix problem,
assume that an orthonormal basis
of is available. Denote by the matrix with column vectors
, i.e.,
.
Because we seek a , it can be written as

(10) 
Then, equation eq:4.16 becomes
Therefore, and
must satisfy

(11) 
with
B_m = V^ A V .
The approximate eigenvalues
resulting from the projection process are
all the eigenvalues of the matrix . The associated eigenvectors
are the vectors in which is an eigenvector of associated
with .
This procedure for numerically computing the Galerkin
approximations to the eigenvalues/eigenvectors of is known as the
RayleighRitz procedure.
RAYLEIGHRITZ PROCEDURE
 Compute an orthonormal basis
of the subspace .
Let
.
 Compute
.
 Compute the eigenvalues of and select the desired ones
, where (for
instance the largest ones).
 Compute the eigenvectors
, of
associated with
, and the
corresponding approximate eigenvectors of ,
.
In implementations of this approach, one often does not compute the eigenpairs
of for each set of generated basis vectors.
The values computed from this procedure are referred to as
Ritz values and the vectors are the associated
Ritz vectors.
The numerical solution of the eigenvalue problem in steps 3
and 4 can be treated by standard library subroutines such as those in
LAPACK [12].
Another important note is that in step 4 one can replace
eigenvectors by Schur vectors to get approximate Schur vectors
instead of approximate eigenvectors.
Schur vectors can be obtained in a numerically stable way and,
in general, eigenvectors are more sensitive to rounding errors
than are Schur vectors.
Next: Oblique Projection Methods.
Up: Basic Ideas Y. Saad
Previous: Basic Ideas Y. Saad
Contents
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Susan Blackford
20001120