Mixed forward-backward stability of the two-level orthogonal Arnoldi method for quadratic problems

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 to a basis for a second-order Krylov subspace associated with nearby matrices, provided that the norms of the given matrices are not too large or too small. Thus, we provide 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.