Abstract. We present an iterative method for solving linear systems, which has the property of minimizing at every step the norm of the residual vector over a Krylov subspace. The algorithm is derived from the Arnoldi process for constructing an /2-orthogonal basis of Krylov subspaces. It can be considered as a generalization of Paige and Saunders' MINRES algorithm and is theoretically equivalent to the Generalized Conjugate Residual (GCR) method and to ORTHODIR. The new algorithm presents several advantages over GCR and ORTHODIR.

We consider a class of iterative algorithms for solving systems of linear equations where the coefficient matrix is nonsymmetric with positive-definite symmetric part. The algorithms are modelled after the conjugate gradient method, and are well suited for large sparse systems. They do not make use of any associated symmetric problems. Convergence results and error bounds are presented.

AThis work was supported in part by ONR grant N00014-82-K-0184 and in part by a joint study with IBM/Kingston i 'f /-----.-~ --1. introduction * AThe hypercube is a loosely coupled parallel multiprocessor based on a binary n-cube network and introduced under different names (for example, cosmic cube, n-cube, binary n-cube, and boolean n-cube). A few machines based on the hypercube topology have been implemented by several groups, see (7 for references, and others are now being built. An n-cube parallel multiprocessor consists of 20 identical processors, each provided with its own sizable memory, and connected to n neighbors in the form of of a binary n-cube network.There are essentially two broad classes of MIMD parallel multiprocessor processor designs with large number of processors. The first type of architecture consists of a large number of identical processors conneted to one another ,ccording to some convenient pattern. In these machines there is no shared memory and no global synchronization.,, Moreover, communication is achieved by message passing and computation is data driven (some designs incoporate a global bus which does not constitute the main way of intercomunication). By message passing we mean that data or possibly code are transferred from processor Ao processor B by traveling accross a sequence of nearest neighbor nodes starting with node A and ending with B. Synchronization is driven by data in the sense that computation in somelnode is performed only when its necessary data is available. Examples include grid networks such as the finite element machine 1I), tree machines 13], and the cosmic cube 17]. At the border line of this class we might also include the datafiow machines which utilize the same concept of data driven synchronization but adopt a more dynamic way of circulating data. The main advantage of such architectures, is their simplicity, of design. The nodes are identical, or are of a few different kinds, and can therefore be fabricated at relatively low cost.T'e second important class of parallel multiprocessors consists of those systems with N identical processors connected via a large switching network to N memorieJ!Lus the memory can be viewed as split into N 'bank ", and shared between the N processors. V i~ons on this scheme are numerous, but the essential feature here is the switching network. Examples include the Ultra computer developed at NYU [5] which uses an Omega network. The main advantage of this second configuration is that it makes data access transparent to the user who may regard data as being held in a large memory which is readily accessible to any processor. This greatly facilitates the programing of the machine but memory conflicts can lead to degraded preformance. Also, the network can simulate any of the communication patterns of the first type of architectures. However, switched network models cannot easily take advantage of proximity of data in problems where communication is local. Moreover, the switching network becomes exceedingly complex to build as ...

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