Perturbation Theory for Factorizations of LU Type through Series Expansions

Component- and normwise perturbation bounds for the block LU factorization and block LDL$^*$ factorization of Hermitian matrices are presented. We also obtain, as a consequence, perturbation bounds for the usual pointwise LU, LDL$^*$, and Cholesky factorizations. Some of these latter bounds are already known, but others improve previous results. All the bounds presented are easily proved by using series expansions. Given a square matrix $A=L U$ having the LU factorization, and a perturbation $E$, the LU factors of the matrix $A+E= \widetilde{L} \widetilde{U}$ are written as two convergent series of matrices: $\widetilde{L} = \sum_{k=0}^{\infty} L_k$ and $\widetilde{U} = \sum_{k=0}^{\infty} U_k$, where $L_k = O(\|E\|^k)$, $U_k = O(\|E\|^k)$, and $L_0 = L$, $U_0 = U$. We present expressions for the matrices $L_k$ and $U_k$ in terms of $L$, $U$, and $E$. The domain and the rate of convergence of these series are studied. Simple bounds on the remainders of any order of these series are found, which significantly improve the bounds on the second-order terms existing in the literature. This is useful when first-order perturbation analysis is used.