  By defining two important terms called basic perturbation vectors and
obtaining their linear bounds, we obtain the linear componentwise perturbation
bounds for unitary factors and upper triangular factors of the generalized
Schur decomposition. The perturbation bounds for the diagonal elements of the
upper triangular factors and the generalized invariant subspace are also
derived. From the former, we present an upper bound and a condition number of
the generalized eigenvalue. Furthermore, with numerical iterative method, the
nonlinear componentwise perturbation bounds of the generalized Schur
decomposition are also provided. Numerical examples are given to test the
obtained bounds. Among them, we compare our upper bound and condition number of
the generalized eigenvalue with their counterparts given in the literature.
Numerical results show that they are very close to each other but our results
don't contain the information on the left and right generalized eigenvectors.
