Locking or Purging a Single Eigenvalue.

The orthogonal transformations developed in the previous section will provide stable and efficient transformations needed to implement locking and purging. We shall give a somewhat detailed discussion of this deflation in complex arithmetic. However, it is clearly important from the standpoint of efficiency to be able to compute in real arithmetic if the matrix $A$ is real and non-symmetric. In this case, it is usually desirable to lock or purge complex conjugate pairs of Ritz values as a unit so that complex arithmetic never need be introduced. This can be accomplished with a block formulation (block size = 2) of the single-vector case we are about to present. The purging in this case will be a direct analog. Unfortunately, there may be numerical complications with locking a complex conjugate pair. A complete discussion would involve considerable detail and analysis. We feel these details are beyond the scope of a template presentation. They may be found in [420].