New rigorous perturbation bounds for the LU and QR factorizations

Combining the modified matrix-vector equation approach with the technique of Lyapunov majorant function and the Banach fixed point principle, we obtain new rigorous perturbation bounds for the LU and QR factorizations with normwise or componentwise perturbations in the given matrix, where the componentwise perturbations have the form of backward errors resulting from the standard factorization algorithms. Each of the new rigorous perturbation bounds is a rigorous version of the first-order perturbation bound derived by the matrix-vector equation approach in the literature, and we present their explicit expressions. These bounds improve the results given by Chang and Stehl\'{e} [SIAM Journal on Matrix Analysis and Applications 2010; 31:2841--2859]. Moreover, we derive new tighter first-order perturbation bounds including two optimal ones for the LU factorization, and provide the explicit expressions of the optimal first-order perturbation bounds for the LU and QR factorizations.