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##

Arnoldi Method with Inexact Cayley Transform

In this paragraph, we discuss the case of the inexact Cayley transform
within
a Galerkin projection framework.
On each iteration, a Ritz value is chosen as zero.
We restrict ourselves to the case where the linear system
(11.2) is
solved by a preconditioner or a stationary linear system solver such
that
for
.
Thus we can use the Arnoldi method to build the Krylov space of
.

The disadvantage of this method is that an update of
requires a complete restart of the Arnoldi process.

The asymptotic convergence towards, e.g., , is governed by the
separation of the eigenvalues of
.
As for the SI, the separation improves when the pole
is closer to .

The separation of the eigenvalues of
the exact transformation improves when is very close to the Ritz value
, but the corresponding linear systems are more
difficult to solve than for a a bit away from the spectrum.
In [323], the linear systems are solved with preconditioners such as
incomplete factorizations and multigrid.
The numerical examples show faster convergence of
Algorithm 11.1 when lies a bit away from the
spectrum.

** Next:** Davidson Method
** Up:** Inexact Methods K. Meerbergen
** Previous:** Inexact Matrix Transformations
** Contents**
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Susan Blackford
2000-11-20