NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS Numer. Linear Algebra Appl., 6, 79–92 (1999) Non-stationary Parallel Multisplitting Algorithms for Almost Linear Systems ∗ Josep Arnal, Violeta Migall´ on and Jos´ e Penad´ es Departamento de Ciencia de la Computaci´ on e Inteligencia Artificial, Universidad de Alicante, E-03071 Alicante, Spain ({arnal, violeta, jpenades}@dtic.ua.es) Non-stationary parallel multisplitting iterative methods are introduced for the solution of almost linear sys- tems. A non-stationary parallel algorithm based on the AOR-type methods and its extension to asynchronous models are considered. Convergence properties of the synchronous and asynchronous versions of these meth- ods are studied for M-matrices and H -matrices. Furthermore, computational results about these methods on a distributed memory multiprocessor, which illustrate the performance of the algorithms studied, are discussed. Copyright © 1999 John Wiley & Sons, Ltd. KEY WORDS Almost linear systems, non-stationary multisplitting methods, asynchronous algorithms, parallel implementation, distributed memory. 1. Introduction We are interested in the parallel solution of almost linear systems of the form Ax + (x) = b, (1.1) where A = (a ij ) is a real n × n matrix, x and b are n-vectors and  :  n →  n is a nonlinear diagonal mapping (i.e., the i th component  i of  is a function only of x i ). These systems appear in practice from the discretization of differential equations, which arise in many fields of applications such as trajectory calculation or the study of oscillatory systems; see e.g., [3], [5] for some examples. Considering that system (1.1) has in fact a unique solution, White [22] introduced the par- allel nonlinear Gauss-Seidel algorithm, based on both the classical nonlinear Gauss-Seidel method (see [17]) and the multisplitting technique (see [16]). Until then, the multisplitting ∗ This research was supported by Spanish DGICYT grant number PB96-1054-C02-01. CCC 1070–5325/99/020079–14 $17.50 Received November 1997 Copyright © 1999 John Wiley & Sons, Ltd. Revised May 1998