** Next:** Preconditioned Eigensolvers A. Knyazev
** Up:** Inexact Methods K. Meerbergen
** Previous:** Example 11.2.3.
** Contents**
** Index**

##

Inexact Shift-and-Invert

Recall from §11.2.1 that with exact arithmetic and
exact linear system solvers, the SI and the
Cayley
transform produce the same Krylov spaces and the same Ritz pairs.
We have shown in the previous sections by theoretical arguments and
numerical
examples that the Cayley transform is a good choice when the linear
systems
are solved with a relative error tolerance smaller than 1.
When the Cayley transform is used, the linear system

is solved, where
.
This linear system can be written as

So, with
and
, we
obtain the
shift-and-invert linear system

where
.
In other words, we can use the SI, but instead of
a relative residual tolerance, we should use a tolerance that is proportional
to the norm of the Ritz residual,
.

Susan Blackford
2000-11-20