ON SOME CONVERGENCE RESULTS OF THE K-STEP ITERATIVE METHODS

For the iterative solution of the nonsingular linear system (I - T)x = c we consider the class of monoparametric k-step methods x(m) = omega-Tx(m-1) +(1 - omega) x(m-k) + omega-c for k = 1,2,3,..., with omega being a real parameter. The main objectives of this paper are the following: (i) to determine the value of k = 1,2,3,... for which the above mentioned k-step method converges asymptotically as fast as possible under the assumption that sigma(T) is-an-element-of [alpha, beta], - infinity < alpha less-than-or-equal-to beta < 1; (ii) for a given sigma(T), not necessarily on the real axis, and for a given k greater-than-or-equal-to 3 to make an attempt toward the determination of an "optimal" omega in the sense of (i) above. Finally based on a recent result by Eiermann, Niethammer and Ruttan for the k-cyclic SOR method we discuss and suggest possible ways of extending and improving the results in (i) and (ii) above.