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We revisit the numerical stability of the two-level orthogonal Arnoldi (TOAR)method for computing an orthonormal basis of a second--order Krylov subspaceassociated with two given matrices. We show that the computed basis is close(on certain subspace metric sense) to a basis for a second-order Krylovsubspace associated with nearby coefficient matrices, provided that the normsof the given matrices are not too large or too small. Thus, the results in thiswork provide for the first time conditions that guarantee the numericalstability of the TOAR method in computing orthonormal bases of second-orderKrylov subspaces. We also study scaling the quadratic problem for improving thenumerical stability of the TOAR procedure when the norms of the matrices aretoo large or too small. We show that in many cases the TOAR procedure appliedto scaled matrices is numerically stable when the scaling introduced by Fan,Lin and Van Dooren is used.

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