Numerical Examples

We present several examples which illustrate some of the features of the symmetric indefinite Lanczos procedure described above. In particular, the examples below illustrate three points:

• When $B$ is singular, premultiplying the starting vector by $H^{-1} B$ provides a dramatic improvement in the quality of the computed Ritz pairs.
• When $B$ is singular, the postprocessing step described in equation (8.27) may further improve the Ritz pairs.
• Partial reorthogonalization provides substantial savings over complete reorthogonalization.

In each of our example problems, the symmetric indefinite eigenvalue problem stems from the linearization of a quadratic eigenvalue problem which arises in structural dynamics. The matrices come from one of two finite element models generated by the structural engineering package MSC/NASTRAN [274]. All of the examples were run on a Sun UltraSPARC with a 336 MHz processor using a simple MATLAB implementation of Algorithm 8.4. Ritz values were accepted as ``converged'' based on the error bound derived in §8.6.3. The bound was found to be pessimistic in most cases, and a reliable tighter bound is still an open research topic.

The first model is an acoustics problem representing a speaker box. Two sets of matrices have been generated from the model. In the first set, the linearized problem $\{A,B\}$ has order 1076. In a smaller second set of matrices, modal reduction has been employed and the associated matrices $\{A,B\}$ have order 668.

The second model represents a shaft on bearing supports with a translational viscous damper attached at the midpoint of the shaft. By controlling the mesh size, two sets of matrices have been produced for this model. With a finer mesh, the order of $\{A,B\}$ is 800, while a coarser mesh yields matrices $\{A,B\}$ with order 160. In each case, the resulting $B$ matrix is singular with this model.