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##

Deflation

As in the case of the Hermitian band Lanczos method, the use of
multiple starting vectors necessitates a suitable *deflation*
procedure to delete linearly and almost linearly dependent vectors
in the block Krylov sequences.
We refer to §4.6.1 for a discussion
of deflation in the Hermitian case.

In the non-Hermitian case, deflation in general occurs independently
in the right and left block Krylov sequences.
An *exact deflation* in the right block Krylov
sequence (7.60) means that a vector, say ,
in the sequence (7.60) is linearly dependent on vectors
to the left of in (7.60) and that this vector
and all its -multiples are removed from (7.60).
Similarly, an *exact deflation* in the left block Krylov
sequence (7.61) means that a vector in the
sequence (7.61) is linearly dependent on previous
Krylov vectors in (7.61) and that this vector
and all its -multiples are removed from (7.61).
After exact deflations in the right block Krylov
sequence (7.60) have occurred, the remaining
nondeflated vectors of (7.60) span an -invariant
subspace.
Furthermore, the right Lanczos vectors in (7.62)
build a suitable basis for this -invariant subspace, and
all the eigenvalues of the Lanczos matrix
defined in
§7.10.2 below are also eigenvalues of .
Similarly, after exact deflations in the left block Krylov
sequence (7.61) have occurred, the remaining
nondeflated vectors of (7.61) span an -invariant
subspace.
The left Lanczos vectors in (7.62)
build a suitable basis for this -invariant subspace and
all the eigenvalues of the Lanczos matrix
are also eigenvalues of .

Of course, in finite-precision arithmetic, it is impossible to
distinguish between exactly linearly dependent and
almost linearly dependent vectors.
Therefore, in practice, almost linearly dependent vectors also
have to be detected and deleted.
In the sequel, we will refer to the process of detecting and
deleting linearly dependent and almost linearly dependent vectors
as *deflation*.

** Next:** Basic Properties
** Up:** Band Lanczos Method
** Previous:** Band Lanczos Method
** Contents**
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Susan Blackford
2000-11-20