, Volume 20, Issue 1, pp 170-183

Cones and error bounds for linear iterations

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Abstract

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 + KK 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.