Perturbation Analysis Of Some Matrix QR Class Factorizations

The weighted QR factorization, hyperbolic QR factorization and symplectic QR factorization of matrices are the generalization of the classical QR factorization. They are among the most important tools in matrix computations and have important applications in the fields of numerical calculation and so on. In this thesis, we mainly study problems on the sensitivity analysis for these factorization.First, using the classical matrix equation approach, the refined matrix equation approach, and the matrix-vector equation approach, we first obtain the first-order perturbation bounds for the weighted QR factorization. Meanwhile, some different rigorous perturbation bounds are derived by using the method formed by the junction of the classical matrix equation approach and the refined matrix equation approach and the other method formed by the junction of the modified matrix-vector equation approach, the technique of Lyapunov majorant function and the Banach fixed point theorem.Next, we research the rigorous perturbation bounds for the hyperbolic QR factorization by using the method formed by the junction of the modified matrix-vector equation approach, the technique of Lyapunov majorant function and the Banach fixed point theorem. Moreover, the optimal first-order perturbation bounds and the normwise condition numbers are also presented.Finally, using the block matrix-vector equation approach, the technique of Lyapunov majorant function, and the Banach fixed point principle, some new rigorous perturbation bounds for the two factor of the symplectic QR factorization are derived. The rigorous perturbation bounds for R factor are tighter than the corresponding ones given in this literature. As special cases, the optimal first-order perturbation bounds are also presented.