Cones and error bounds for linear iterations

Consider an ordered Banach space with a cone of positive elementsK and a norm ∥·∥. Let [a,b] denote an order-interval; under mild conditions, ifx*∈[a,b] then $$||x * - tfrac{1}{2}(a + b)|| leqslant tfrac{1}{2}||b - a||.$$ This inequality is used to generate error bounds in norm, which provide on-line exit criteria, for iterations of the type $$x_r = Ax_{r - 1} + a,A = A^ + + A^ - ,$$ whereA ⁺ andA ^? are bounded linear operators, withA ⁺ K ?K andA ^? K ? ?K. Under certain conditions, the error bounds have the form $$begin{gathered} ||x * - x_r || leqslant ||y_r ||,y_r = (A^ + - A^ - )y_{r - 1} , hfill ||x * - x_r || leqslant alpha ||nabla x_r ||, hfill ||x * - tfrac{1}{2}(x_r + x_{r - 1} )|| leqslant tfrac{1}{2}||nabla x_r ||. hfill end{gathered} $$ These bounds can be used on iterative methods which result from proper splittings of rectangular matrices. Specific applications with respect to certain polyhedral cones are given to the classical Jacobi and Gauss-Seidel splittings.