Explicit Restarts

As was mentioned earlier, the standard implementations of the Arnoldi method are limited by their high storage and computational requirements as $m$ increases. Suppose that we are interested in only one eigenvalue/eigenvector of $A$, namely, the eigenvalue of largest real part of $A$. Then one way to circumvent the difficulty is to restart the algorithm. After a run with $m$ Arnoldi vectors, we compute the approximate eigenvector and use it as an initial vector for the next run with the Arnoldi method. This process, which is the simplest of this kind, is iterated to convergence to compute one eigenpair. For computing other eigenpairs, and for improving the efficiency of the process, a number of strategies have been developed, which are somewhat related. These include deflation procedures briefly discussed in the next section, and the implicit restarting strategy described in §7.6.