On the stability of the incomplete LU-factorizations and characterizations of \(H\)-matrices
An \(n\times n\) real matrix \(A= [a_{i,j}]\) is an M-matrix if \(a_{i,i}\geq 0\), if \(a_{i,j}\leq 0\) for all \(i\neq j\), and if \(A^{- 1}\geq 0\). The comparison matrix \({\mathcal M}(A)= [\alpha_{i, j}]\) of an arbitrary \(n\times n\) complex matrix \(A= [a_{i, j}]\) is defined by \(\alpha_{i, j}:= | a_{i, j}|\) if \(i= j\) and \(\alpha_{ij}:= -| a_{i, j}|\) if \(i\neq j\). An \(n\times n\) complex matrix \(A\) is an H-matrix iff \({\mathcal M}(A)\) is an M-matrix. The computational stability of incomplete LU-factorizations of H-matrices is investigated. The main results: The incomplete LU-factorization of an H-matrix \(A\) is at least as stable as the incomplete LU-factorization of its comparison matrix \({\mathcal M}(A)\). The incomplete LU-factorization of an H-matrix \(A\) is at least as stable as the complete LU-factorization of its comparison matrix \({\mathcal M}(A)\) without pivoting. If \(A\) is a Hermitian H-matrix and the Cholesky method is applied then an incomplete \(\text{LL}^*\)-factorization for \(A\) is at least as stable as the incomplete factorization of \({\mathcal M}(A)\). If \(A\) is a Hermitian H- matrix and \(A\) is symmetric then the incomplete \(\text{LL}^*\)- factorization of \(A\) is at least as stable as Cholesky's method applied to \({\mathcal M}(A)\). Some new characteristics of H-matrices in connection with their incomplete LU-factorizations are given.
scientific article; zbMATH DE number 739414