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Numerical Methods for Solving Linearized Problems

As discussed in §9.2.2, one can use either the generalized eigenvalue problem (9.4) with (9.5) or the generalized eigenvalue problem (9.9) with (9.10) for solving the corresponding quadratic eigenvalue problem (9.1).

If all matrices $M$, $C$, and $K$ are Hermitian and $M$ is positive definite, as in the special case (9.2), then the decision comes down to choosing either intrinsically non-Hermitian generalized eigenvalue problem (9.4) and (9.5), with a Hermitian positive definite $B$ matrix, or a generalized eigenvalue problem (9.9) and (9.10), where both $A$ and $B$ matrices are Hermitian but neither of them will be positive definite.

Numerical methods discussed in Chapter 8 can be used for solving these generalized ``linear'' eigenvalue problems. For example, in the MSC/NASTRAN [274], the non-Hermitian formulation (9.4) with (9.5) is used.

The symmetric indefinite Lanczos method discussed in §8.6 is specifically targeted for the generalized symmetric indefinite eigenvalue problem (9.9). The potential trouble is that in the symmetric indefinite Lanczos method, the basis vectors are orthogonal with respect to an indefinite inner product. Therefore, these basis vectors may not be linearly independent and the algorithm may break down and be numerically unstable. Nevertheless, this is often an attractive way to solve the original QEP because of potential savings in memory requirements and floating point operations. See §8.6 for further details.


next up previous contents index
Next: Jacobi-Davidson Method Up: Quadratic Eigenvalue Problems Z. Bai, Previous: QEP with Cayley Transform.   Contents   Index
Susan Blackford 2000-11-20