Locking.

Because of the different rates of convergence of each of the approximate eigenvalues computed by the subspace iteration, it is common practice to extract them one at a time and perform a type of deflation. Thus, as soon as the first eigenvector has converged there is no need to continue to multiply it by $A$ in the subsequent iterations. Indeed, we can freeze this vector and work only with the vectors $v_2, \ldots,v_m$. However, we will still need to perform the subsequent orthogonalizations with respect to the frozen vector $v_1$ whenever such orthogonalizations are needed. The term used for this strategy is locking; that is, we do not further attempt to improve the locked approximation for $v_1$.

The following algorithm describes a practical subspace iteration with deflation (locking) for computing the $\nev$ dominant eigenvalues.

We now describe some implementation details.

(1)

The initial starting matrix $Z$ should be constructed to be dominant in eigenvector directions of interest in order to accelerate convergence. When no such information is known a priori, a random matrix is as good a choice as any other.

(4)

The iteration parameter $\iter$ should be chosen to minimize orthonormalization cost while maintaining a reasonable amount of numerical accuracy. The amplification factor $(\lambda_1/\lambda_p)^{\iter}$, where the eigenvalues $\lambda_i$ are ordered in decreasing absolute values, gives the loss of accuracy. Rutishauser [381] plays it safe and allows an amplification factor of only $10$, losing one decimal, while Stewart and Jennings [426] let the algorithm run to $\eps^{-1/2}$, half the machine accuracy, but not to more than 10 iterations.