A complete statement of the complex symmetric Lanczos algorithm (without look-ahead) is as follows.

Next, we comment on a few of the steps of Algorithm 7.17.

**(3)**- If occurs, then the algorithm has fully
exhausted the Krylov sequence generated by and and
thus termination is natural.
In fact, in this case, the Lanczos
vectors generated so far span an -invariant subspace,
and all eigenvalues of
the Lanczos tridiagonal matrix are also eigenvalues of .
**(5)**- In practice, one also needs to stop if a so-called
*near breakdown*, i.e., , occurs. A look-ahead version of the algorithm remedies both exact breakdowns, i.e., , and near breakdowns; see, e.g., [178,180]. **(11)**- To test for convergence, the eigenvalues
,
, of the complex
symmetric tridiagonal matrix are computed, and the
algorithm is stopped if some of the
's are
good enough approximations to the desired eigenvalues of .