Implementation of Locking and Purging

Although the basic deflation ideas of locking and purging are conceptually straightforward, there are still a number of implementation details and various strategies one might consider. In the end, some ad-hoc decisions will have to be made. The deflation strategy adopted here is somewhat conservative. However, the performance appears to be quite reasonable in practice. In [420], computational examples demonstrate that very crude tolerances can be specified without missing multiple or clustered eigenvalues.

The implementation details become quite involved and it is difficult to convey the ideas by displaying code. Instead, we shall indicate the main ideas of the strategy with a fairly high-level verbal description.

Implementation notes:

(1)

Initially, we work with an $(m = k+p)$-step Lanczos factorization and apply $p$ shifts per each implicit restart. Each time a Ritz value is locked, it is advantageous to decrease the effective value of $p$ by 1. This allows the polynomial filter to have a larger relative magnitude on the Ritz value that is most likely to converge next. (See §4.5.4.) Of course, this must be limited to avoid lowering the degree of the filter so much that it becomes ineffective. If $k_o$ is the number of locked Ritz values, the IRLM iteration takes place in columns $k_o + 1,\dots, m$ of $V$, working within an $m-k_o$ length Lanczos factorization. The effective value of $k$ becomes $m-k_o - p_o$ and the effective value of $p$ becomes $p_o = p - k_o$. This has two important effects. The rate of convergence as described in §4.5.4 is increased and the amount of work per implicit restart is decreased.

(2)

One might also wish to purge all converged but unwanted Ritz pairs at this stage.

(3)

The purpose of introducing the random start vector here is to greatly increase the likelihood of components in directions of wanted eigenvectors that have not yet been found.

(4)

This ad-hoc stopping strategy is reasonable if there is no opportunity to factor $A$. However, there is no ultimate assurance that the $k$ wanted eigenvalues have all been found (especially in the case of clustered or multiple eigenvalues). If $A$ (or better yet $A-\sigma I$ ) can be factored into a symmetric indefinite factorization then convergence may be greatly enhanced through spectral transformation and one will have an inertia count available. That will give a more reliable verification that all $k$ wanted eigenvalues have been found. See the discussion in §4.4.3.