Comparison theorems for weak splittings in respect to a proper cone of nonsingular matrices

AbstractThe asymptotic rates of convergence of two iteration matrices induced by two splittings of the same nonsingular matrix have been compared in the works of many authors. In this paper new comparison theorems which extend Song's results [Y. Song, Numer. Math. 65 (1993) 245–252] for weak splittings of the same or different types are presented. These new comparison theorems are derived by using nonnegativity in respect to a proper cone K. New converses which partially solve an open question asked by [V.A. Miller, M. Neumann, Numer. Math. 47 (1985) 427–434] are also presented