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#### Invert QEP.

For most iterative methods for solving a generalized eigenvalue problem, the formulation (9.4), with either (9.5) or with (9.10), is suitable if one wants to determine a few of the exterior eigenvalues and eigenvectors. If one wants to compute some of the smallest (in modulus) eigenvalues and eigenvectors, then the obvious transformation is , and, after multiplying the QEP (9.1) with , we obtain the invert QEP:
 (256)

Here it is assumed that is not an eigenvalue of the original QEP (9.1), i.e., that is nonsingular.

The QEP for the triplet can be linearized as discussed in §9.2.2, for instance, as (9.4) with (9.5), where interchanged with . We can reformulate this generalized linearized eigenproblem in terms of , instead of , which leads to

 (257)

where
 (258)

Note that from the factorization

we know that the pencil is equivalent to

Since , we conclude that the matrix pencil is regular if and only if the quadratic matrix polynomial is regular and the eigenvalues of the original QEP (9.1) coincide with the eigenvalues of the matrix pencil .

For the special case (9.2), we may formulate the generalized eigenvalue problem , with

 (259)

In this case, both matrices are Hermitian, but indefinite. Linearization with (9.15) results after left multiplication of (9.14) with a block-diagonal matrix . Therefore, if , then the pencil is regular if and only if the quadratic matrix polynomial is regular.

Next: Shifted QEP. Up: Spectral Transformations for QEP Previous: Spectral Transformations for QEP   Contents   Index
Susan Blackford 2000-11-20