Comparison theorems for splittings of M-matrices in (block) Hessenberg form

Some variants of the (block) Gauss–Seidel iteration for solution linear systems with M-matrices in Hessenberg form are discussed. Comparison results asymptotic convergence rate some regular splittings derived: particular, we prove that a lower-Hessenberg M-matrix $$\rho (P_{GS})\ge \rho (P_S)\ge (P_{AGS})$$ , where $$P_{GS}, P_S, P_{AGS}$$ matrices Gauss–Seidel, staircase, and anti-Gauss–Seidel method. This is result does not seem to follow from classical comparison results, as these directly comparable. It shown concept stair partitioning provides powerful tool design new suited parallel computation.