Subproper and regular splittings for restricted rectangular linear system (Q1856056)

Iterative methods for solving restricted rectangular but consistent linear systems of equations \(Ax = b\), \(x \in T\), where \(A \in\mathbb{C}^{m \times n}\) and \(T\) is a subspace of \(\mathbb{C}^n\), are considered. The methods are based on splittings \(A = M-N\). The authors give a new definition of a subproper splitting of the matrix \(A\) which is more general than that given by \textit{M. Neumann} [Linear Algebra Appl. 14, 41-51 (1976; Zbl 0334.65034)] and by \textit{A. Berman} and \textit{M. Neumann} [SIAM J. Numer. Anal. 13, 877-888 (1976; Zbl 0368.65023)]. A necessary and sufficient condition on the splitting is established such that the iterates converge to a solution of \(Ax = b\) with \(b \in AT\) for every initial guess. In the case of a real matrix \(A\), monotonicity type conditions and regular subproper splittings are applied to obtain a necessary and a sufficient condition for the convergence of subproper splitting methods. Two numerical examples illustrate the convergence of the presented methods.