Incomplete Factorization of Singular M -Matrices
John J. Buoni · SIAM Journal on Algebraic and Discrete Methods · 1986
In 1981, Varga and Cai characterized those M-matrices A (perhaps singular) which admit a factorization into M-matrices L and $U ( A = LU )$ where L is required to be a nonsingular and lower triangular M-matrix and U is required to be an upper triangular M-matrix, a result that was first proved by Fiedler and Ptak (1962) in the case when A is nonsingular. Because this factorization may, as a result of fill-in, produce a lower triangular matrix which is considerably less sparse than A, one attempts to control the fill-in of the factorization of A by means of a graph. This method leads to the concept of incomplete factorizations of A. Meijerink and van der Vorst (1977).who have shown that incomplete factorizations of nonsingular M-matrices are possible, while Manteufiel (1980) has extended this result to the H-matrix case. The purpose of this paper is to give a condition on a singular M-matrix which guarantees the incomplete factorization of a singular M-matrix.