  The SOAR method for computing an orthonormal basis of a second-order Krylov
subspace can be numerically unstable (see Lu et al. (2016)). In the Two-level
orthogonal Arnoldi(TOAR) procedure, an alternative to SOAR, the problem of
instability had circumvented. A stability analysis of the second-order Krylov
subspace's orthonormal basis in TOAR with respect to the coefficient matrices
of a quadratic problem remain open; see Lu et al. (2016). This paper proposes
the Improved-TOAR method(I-TOAR) and solves the said open problem for I-TOAR.
