Asynchronous Two-Stage Iterative Methods

this paper; see [12], [16], and the references given therein. We point out that, since the number of inner iterations may vary from block to block, the convergence results in Baudet [1] or Chazan and Miranker [5] cannot be applied to our situation. In Sect. 2, we derive two new convergence results for block two-stage iterative methods. We then investigate asynchronous variations of our two-stage methods. These asynchronous methods arise naturally in parallel computations if one tries to reduce idle times of the processors. In Sects. 3 and 4, we introduce two different asynchronous models for block two-stage methods and investigate their convergence. As our major result, we establish convergence for both asynchronous models under the same conditions as those for the synchronous method. In the rest of this section we present some notation, definitions and preliminary results which we refer to later. We say that a vector x is nonnegative (positive), denoted x 0 (x ? 0), if all its entries are nonnegative (positive). Similarly, a matrix B is said to be nonnegative, denoted B O, if all its entries are nonnegative or, equivalently, if it leaves invariant the set of all nonnegative vectors. We compare two matrice