The two-stage iterative methods for symmetric positive definite matrices

Two-stage iterative methods for the solution of the symmetric positive defi nite linear system Ax = b are considered. In this paper, in particular: str ong conditions and feeble conditions on splittings are given so that the tw o-stage methods are convergent for any number of inner iterations and for a ny nonnegative even number of inner iterations, respectively. Some suitable examples of strong P-regular splittings are presented and the convergence of some feasible two-stage methods is discussed, in which the outer splitti ng is weak P-regular and the inner splitting is strong or weak P-regular sp littings. (C) 2000 Elsevier Science Inc. All rights reserved.