On Some Convergence Results of the k-Step Interactive Methods

For the iterative solution of the nonsingular linear system (1) (I T)x = c we consider the class of monoparametric k-step methods (2) x(m) = roTx(m-l) + (I ro) x(m-k) + roc for k = I, 2, 3, .... with CO being a real parameter. The main objectives of this paper are the following: i) To determine the value of k = I, 2, 3, ... for which the method in (2) converges asymptotically as fast as possible under the assumption that aCT) e [ex, J31. ~ < a'; J3 < 1 and ti) For a giveo aCT). not necessari1y on the real axis, and for a given k ~ 3 to make an attempt toward the determination of an "optimal" ro in the sense of (i) above. Finally based on a recent result by Eirmann, Neithammer and Ruttan for the k-cyclic SOR method. we discuss and suggest possible ways of extending and improving the results in (i) and (ti) above. 1980 Mathematics Subject Classifications AMS (MOS): Primary 65F1O. CR. Categories: 5.14. '" Visiling. Department of Computer Science, Purdue University, West Lafayette, m 47907. The work of the second author was supponed in pan by NSF grant CCR.8619817. •• Department of Mathematics, University of Ioannina. GR-451 10. Ioannina. Greece.