An extension of the class of matrices arising in the numerical solutions of PDEs
Chengyi Zhang, Shuanghua Luo, Jicheng Li, Fengmin Xu · Electronic Journal of Linear Algebra · 2012
This paper studies block matrices A = [A ij ] ∈ C km×km , where every block A ij ∈ C k×k for i, j ∈ m = {1, 2, . . ., m} and A ii is non-Hermitian positive definite for all i ∈ m .Such a matrix is called an extended H-matrix if its block comparison matrix is a generalized M -matrix.Matrices of this type are an extension of generalized M -matrices proposed by Elsner and Mehrmann [L.Elsner and V. Mehrmann.Convergence of block iterative methods for linear systems arising in the numerical solution of Euler equations.Numer.Math., 59:541-559, 1991.]and generalized H-matrices by Nabben [R.Nabben.On a class of matrices which arise in the numerical solution of Euler equations.Numer.Math., 63:411-431, 1992.].This paper also discusses some properties including positive definiteness and invariance under block Gaussian elimination of a subclass of extended H-matrices, especially, convergence of some block iterative methods for linear systems with such a subclass of extended H-matrices.Furthermore, the incomplete LDU -factorization of these matrices is investigated and applied to establish some convergent results on some iterative methods.Finally, this paper generalizes theory on generalized H-matrices and answers the open problem proposed by R. Nabben.