Recall that Algorithm 7.17 produces a complex symmetric
tridiagonal matrix and that, in order to obtain approximate
eigenvalues of
, at least some of the eigenvalues of
need to be computed.
Using the QR algorithm for the task of computing all eigenvalues
of
does not exploit the tridiagonal structure of
.
Indeed, after one step of the QR algorithm, in general, the
tridiagonal structure of
will have been destroyed and one
obtains a full upper Hessenberg matrix, just as in the case of
QR for general NHEPs.
Cullum and Willoughby [91,92,94] developed a
QL procedure for complex symmetric tridiagonal matrices
that exploits this special structure.
The algorithm is based on factorizations of complex symmetric
tridiagonal matrices into a complex orthogonal matrix
and a
lower triangular matrix
.
However, since
is not unitary in general, carefully chosen
heuristics are needed in order to monitor and maintain numerical
stability; we refer the reader to [94] for a
detailed description of the QL procedure.
If the complex symmetric Lanczos method is run with look-ahead,
then the complex symmetric tridiagonal structure of the Lanczos
matrix will be destroyed as soon as a look-ahead step
occurs.
The matrix
of Lanczos vectors
is then no longer complex orthogonal, but instead
is a complex symmetric block-diagonal matrix, where the sizes of the
diagonal blocks correspond to the lengths of the look-ahead steps.
In particular, a
``block'' occurs whenever
a regular Lanczos step (without look-ahead) is possible, and each ``true''
block of size bigger than
corresponds to a look-ahead step.
Furthermore, instead of (7.101), one now has the relation