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## 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 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 .     Next: Solving the Reduced Eigenvalue Up: Lanczos Method for Complex Previous: Properties of the Algorithm   Contents   Index
Susan Blackford 2000-11-20