Mixed forward-backward stability of the two-level orthogonal Arnoldi method for quadratic problems
Provides the first theoretical stability guarantees for the TOAR method, benefiting researchers using this algorithm for quadratic eigenvalue problems.
The paper proves that the two-level orthogonal Arnoldi (TOAR) method for computing orthonormal bases of second-order Krylov subspaces is numerically stable under certain norm conditions, and shows that scaling can improve stability when norms are extreme.
We revisit the numerical stability of the two-level orthogonal Arnoldi (TOAR) method for computing an orthonormal basis of a second--order Krylov subspace associated with two given matrices. We show that the computed basis is close (on certain subspace metric sense) to a basis for a second-order Krylov subspace associated with nearby coefficient matrices, provided that the norms of the given matrices are not too large or too small. Thus, the results in this work provide for the first time conditions that guarantee the numerical stability of the TOAR method in computing orthonormal bases of second-order Krylov subspaces. We also study scaling the quadratic problem for improving the numerical stability of the TOAR procedure when the norms of the matrices are too large or too small. We show that in many cases the TOAR procedure applied to scaled matrices is numerically stable when the scaling introduced by Fan, Lin and Van Dooren is used.