Componentwise perturbation analysis for the generalized Schur decomposition

Abstract

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 need to calculate the left and right eigenvectors.