  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.
