A Truncated RQ Iteration for Large Scale Eigenvalue Calculations

• Authors: D. C. Sorensen;C. Yang
• Venue: SIAM Journal on Matrix Analysis and Applications
• Year: 1998

We introduce a new Krylov subspace iteration for large scale eigenvalue 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 exact solutions when they can be obtained with sparse direct method. We call this new iteration the truncated RQ (TRQ) iteration. 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 deflation strategies may be employed directly in the TRQ iteration.