THE ARNOLDI EIGENVALUE ITERATION WITH EXACT SHIFTS CAN FAIL

The restarted Arnoldi algorithm, implemented in the ARPACK software library and MATLAB's eigs command, is among the most common means of computing select eigenvalues and eigenvectors of a large, sparse matrix. To assist convergence, a starting vector is repeatedly refined via the application of automatically constructed polynomial filters whose roots are known as "exact shifts." Though Sorensen proved the success of this procedure under mild hypotheses for Hermitian matrices, a convergence proof for the non-Hermitian case has remained elusive. The present note describes a class of examples for which the algorithm fails in the strongest possible sense; that is, the polynomial filter used to restart the iteration deflates the eigenspace one is attempting to compute.