A Truncated RQ - Iteration forLarge Scale Eigenvalue CalculationsD

We introduce a new Krylov subspace iteration for large scale eigen-value problems that is able to accelerate the convergence through an inexact (iterative) solution to a shift-invert equation. The method also takes full advantage of an exact solution when it is possible to apply a sparse direct method to solve the shift-invert equations. We call this new iteration the Truncated RQ iteration (TRQ). It is based upon a recursion that develops in the leading k columns of the implicitly shifted RQ iteration for dense matrices. Inverse-iteration-like convergence to a partial Schur decomposition occurs in the leading k columns of the updated basis vectors and Hessenberg matrices. The TRQ iteration is competitive with the Rational Krylov Method of Ruhe when the shift-invert equations can be solved directly and with the Jacobi-Davidson Method of Sleijpen and Van der Vorst when these equations are solved inexactly with a preconditioned iterative method. The TRQ iteration is related to both of these but is derived directly from the RQ iteration and thus inherits the convergence properties of that method. Existing RQ deeation strategies may be employed directly in the TRQ iteration.