Round-off error due to finite precision arithmetic will cloud the situation, as usual. It is clear that the goal is to prevent components in from corrupting the vectors Thus to begin, the starting vector should be of the form If a final approximate eigenvector has components in they may be purged by replacing and then normalizing. To see the effect of this, note thatand all components in which are of the form will have been purged. This final application of may be done implicitly in two ways. One is to note that if with then and the approximate eigenvector is replaced with the improved approximation where . This correction was originally suggested by Ericsson and Ruhe  as a mean of performing a formal step of the power method with The residual error of the computed Ritz vector with respect to the original problem is
Another recent suggestion due to Meerbergen and Spence is to use implicit restarting with a zero shift . Recall that implicit restarting with zero shifts is equivalent to starting the -Arnoldi process with a starting vector of and all the resulting Ritz vectors will be multiplied by as well. After applying the implicit shifts to , the leading submatrix of order will provide the updated Ritz values. No additional explicit matrix-vector products with are required.
The ability to apply zero shifts (i.e. to multiply by implicitly) is very important when has zero eigenvalues. If thenThus, in order to completely eradicate components from one must multiply by where is equal to the dimension of the largest Jordan block corresponding to a zero eigenvalue of . Eigenvector purification by implicit restarting may be incorporated into ARPACK at a future date.
Spectral transformations were studied extensively by Ericsson and Ruhe  and the first eigenvector purification strategy was developed in . Shift and invert techniques play an essential role in the block Lanczos code developed by Grimes, Lewis, and Simon and the many nuances of this technique in practical applications are discussed thoroughly in . The development presented here and the eigenvector purification through implicit restarting is due to Meerbergen and Spence .