A Global Property of Restarted FOM Algorithm

Abstract

In \u00a0this \u00a0paper \u00a0an \u00a0interesting \u00a0property \u00a0of \u00a0the \u00a0restarted \u00a0FOM \u00a0algorithm \u00a0for \u00a0solving \u00a0large nonsymmetric \u00a0linear \u00a0systems \u00a0is \u00a0presented \u00a0and \u00a0studied. \u00a0By \u00a0establishing \u00a0a \u00a0relationship \u00a0between \u00a0the convergence of its residual vectors and the convergence of Ritz values in the Arnoldi procedure, it is shown that some important information of previous FOM(m) cycles may be saved by the iteration approximates at the time of restarting, with which the FOM(m) cycles can complement one another harmoniously in reducing the \u00a0iteration \u00a0residual. \u00a0Based \u00a0on \u00a0the \u00a0study \u00a0of \u00a0FOM(m), \u00a0two \u00a0polynomial \u00a0preconditioning \u00a0techniques \u00a0are proposed; \u00a0one \u00a0is \u00a0for \u00a0solving \u00a0nonsymmetric \u00a0linear \u00a0systems \u00a0and \u00a0another \u00a0is \u00a0for \u00a0forming \u00a0an \u00a0effective \u00a0starting vector in the restarted Arnoldi method for solving nonsymmetric eigenvalue problems.