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  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
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