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Deflation.

The partial generalized Schur form can be obtained in a number of successive steps. Suppose that we have the partial generalized Schur form and . We want to expand this partial generalized Schur form with the new right Schur vector and the left Schur vector to

and

The new generalized Schur pair satisfies

or, since ,

The vectors and can be computed from

Hence, the generalized Schur pair is an eigenpair of the deflated matrix pair
 (228)

This eigenproblem can be solved again with the Jacobi-Davidson process that we have outlined in §8.4.1. In that process we construct vectors that are orthogonal to and vectors that are orthogonal to . This simplifies the computation of the interaction matrices and , associated with the deflated operators:
 (229)

and and can be simply computed as and , respectively.

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Susan Blackford 2000-11-20