  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.
