The acceleration of matrix power methods by cyclic variations of the shift parameter

AbstractThe traditional matrix power method converges very slowly when the dominat eigenvalues have weak relative separation. In the present paper we show that the convergence rate can be greatly accelerated by incorporating systematic cyclic displacements within the power iteration. Analytic convergence results, together with test-case numerical examples, confirm that computation times can be dramatically reduced, typically by factors of 20 and more. Indeed the cyclic shift technique works most effectively on those problems which are most recalcitrant to the traditional power method.We also apply the technique to accelerating the simultaneous determination of several dominant eigenmodes by the block power method. Our results are shown to provide a simple strategy for isolating the dominant modes of self-adjoint differential operators, by explicit time-integration methods