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## Jacobi-Davidson Method with Cayley Transform

The Jacobi-Davidson method is discussed in §7.12. The Jacobi-Davidson 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 right-hand 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 Rayleigh-Ritz quotient of and the zero is the harmonic Rayleigh-Ritz quotient with target .

Next: Preconditioned Lanczos Method Up: Inexact Methods   K. Meerbergen Previous: Example 11.2.2.   Contents   Index
Susan Blackford 2000-11-20