Algorithm

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 $\hat{v}_j=0$ occurs, then the algorithm has fully exhausted the Krylov sequence generated by $A$ and $b$ and thus termination is natural. In fact, in this case, the Lanczos vectors generated so far span an $A$-invariant subspace, and all eigenvalues of the Lanczos tridiagonal matrix are also eigenvalues of $A$.

(5)

In practice, one also needs to stop if a so-called near breakdown, i.e., $\beta_j\approx 0$, occurs. A look-ahead version of the algorithm remedies both exact breakdowns, i.e., $\beta_j = 0$, and near breakdowns; see, e.g., [178,180].

(11)

To test for convergence, the eigenvalues $\theta_i^{(j)}$, $i=1,2,\ldots,j$, of the complex symmetric tridiagonal matrix $T_j$ are computed, and the algorithm is stopped if some of the $\theta_i^{(j)}$'s are good enough approximations to the desired eigenvalues of $A$.