Positive cones and convergence conditions for iterative methods based on splittings

The study of convergence conditions to solve large and sparse linear systems Ax = b by iterative methods has been discussed by many authors. In this paper, by using the partial order induced by positivity cone of matrices and conditions on the matrices and splittings, we obtain the convergence of the iterative method. The usual partial orders of nonnegative matrices and nonnegative definite matrices satisfy the conditions presented for this new order, and hence, we unify the theory. Moreover, we also show that the new conditions apply to other partial orders as well, thus extending the usual convergence theory to a new situation.