The first example demonstrates the importance of the choice of starting vector
when the matrix is singular. The matrices used in this example come from
the shaft model using a fine mesh, and the order of the matrices
is 800. A shift of was used and complete
reorthogonalization was employed.
The following table illustrates one ill effect which may result when the Lanczos
vectors do not belong to the range of .
The symmetric indefinite Lanczos procedure was executed twice with an initial
vector of all ones. In the first run, the initial vector is preprocessed by
multiplying it by , while in the second run, no
preprocessing is used. The preprocessing step dramatically improves
the quality of the Ritz pairs, as measured by the residuals
by ensuring that the Lanczos vectors, and hence the Ritz vectors, belong to
the range of .