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##

Orthogonal Deflating Transformation

We shall utilize a special orthogonal transformation to implement the
deflation schemes mentioned above. The deflation schemes are related to an eigenvector
associated with a Ritz value that is to be deflated (either locked or purged).
Given a vector of unit length, the algorithm shown in
Algorithm 4.9 computes an
orthogonal matrix such that (hence ).
This orthogonal matrix has a very special form and may be written as
Q = R + y e_1^*, with
R e_1 = 0 , R^* y = 0,
where is upper triangular. It may also be written as
Q = L + y g^* , with
L e_1 = 0 , L^* y = e_1 - g,
where is lower triangular
and
.
Here we assume that
.
Now, consider the matrix . The
substitutions
,
from (4.24) and (4.23)
and the facts
and will give
Q^* T Q &=& Q^* T ( R + y e_1^*)

&=& (L^* + g y^*) T R + e_1 e_1^*

&=& L^*TR + g y^*T R + e_1 e_1^*.
Since both and are upper triangular, it follows
that is upper Hessenberg
with the first row and the first column each being zero
due to
. Also,
.
From this we conclude that must also be symmetric and hence
tridiagonal. Therefore, we see that
is of the form

where is symmetric and tridiagonal.
It should be noted that, as computed by Algorithm 4.9,
will have componentwise relative
errors on the order of machine precision with no
element growth.

**Subsections**

** Next:** Locking or Purging a
** Up:** Implicitly Restarted Lanczos Method
** Previous:** Deflation and Stopping Rules
** Contents**
** Index**
Susan Blackford
2000-11-20