Overlapping additive and multiplicative Schwarz iterations for \(H\)-matrices
A matrix \(A=(a_{ij})\) is called an \(H\)-matrix if the matrix with the diagonal entries \(| a_{ii}| \) and the off-diagonal entries \(\,-| a_{ij}| \) is a nonsingular \(M\)-matrix. These matrices were already considered by Ostrowski in 1937. In this paper, additive and multiplicative Schwarz methods are considered for linear systems with \(H\)-matrices, and results on \(M\)-matrices or symmetric positive definite matrices are extended to them. The case of inexact local solvers is also considered. Note that \(A\) is an \(H\)-matrix iff there is a nonsingular diagonal matrix \(U\) such that \(U^{-1}AU\) is diagonally dominant. The matrix \(U\) corresponds to a rescaling, and Schwarz iterations are invariant under scaling transformations.
additive and multiplicative Schwarz methods
scientific article; zbMATH DE number 2126344