Linear systems with \(\Omega\)-diagonally dominant matrices and related ones (Q1801454)

Let \(I\) be the set of real compact intervals,\(A\in I^{n\times n}\) be an interval matrix, and \(b\in I^ n\) be an interval vector. Let \(\langle A\rangle\) be a comparison matrix of \(A\), defined by \(\langle A\rangle=- | A_{ij}|\) if \(i\neq j\) and \(\langle A\rangle_{ii}=\min\{| a_{ii}|\): \(a_{ii}\in A_{ii}\}\) with \(i,j=1,\dots,n\). If the interval Gaussian algorithm is applied to the system of equations, \(Ax=b\), the result is denoted by \(x_ G\) if it exists. The paper establishes sufficient conditions for the existence of \(x_ G\) which are applicable to \(A\), when the comparison matrix, \(\langle A\rangle\), is diagonally dominant. The conditions are rather general and cover interval matrices the comparison matrices of which are strictly diagonally dominant, irreducibly diagonally dominant, or \(\Omega\)- diagonally dominant.