Arnoldi Iteration - Implicitly Restarted Arnoldi Method (IRAM)

Implicitly Restarted Arnoldi Method (IRAM)

Due to practical storage consideration, common implementations of Arnoldi methods typically restart after some number of iterations. One major innovation in restarting was due to Lehoucq and Sorensen who proposed the Implicitly Restarted Arnoldi Method. They also implemented the algorithm in a freely available software package called ARPACK. This has spurred a number of other variations including Implicitly Restarted Lanczos method. It also influenced how other restarted methods are analyzed. Theoretical results have shown that convergence improves with an increase in the Krylov subspace dimension n. However, an a-priori value of n which would lead to optimal convergence is not known. Recently a dynamic switching strategy has been proposed which fluctuates the dimension n before each restarts and thus leads to acceleration in the rate of convergence.