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JacobiDavidson Method with Cayley Transform
The JacobiDavidson method is discussed in §7.12.
The JacobiDavidson method differs from the Davidson method in that the
linear
system to be solved is projected onto the space orthogonal to the current
Ritz vector.
This leads to the solution of the correction equation

(276) 
where
is a Ritz pair on iteration .
(Note that usually, a minus sign is put in front of the residual in the
righthand side.)
Assume that is orthogonal to , i.e.,
.
When an inexact solver is used, we have a residual that satisfies
Note that the projection in front of is dropped since is
assumed orthogonal to .
We can rewrite this into
where
.
In words, the solution of the correction equation is obtained by
the
action of the Cayley transform
to the most
recent
Ritz vector.
Example 11.2.1 in [411] shows that tends to zero on
convergence.
Both pole and zero of the Cayley transform lie close to the desired
eigenvalue.
This meets the conditions for good matching between
eigenvectors of and
,
motivated at the end of §11.2.2.
The following observation is a bit funny.
Since
,
when , we have from
that
So, the pole of the Cayley transform is the RayleighRitz quotient of
and the zero is the
harmonic RayleighRitz quotient with target .
Next: Preconditioned Lanczos Method
Up: Inexact Methods K. Meerbergen
Previous: Example 11.2.2.
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Susan Blackford
20001120