Algorithm 842 : A set of GMRES routines for real and complex arithmetics on high performance computers

• Published in: ACM transactions on mathematical software 2005-06, Vol.31 (2), p.228-238
• Main Authors: FRAYSSE, Valérie, GIRAUD, Luc, CERFACS, Serge Gratton, LANGOU, Julien
• Format: Article
• Language: English
• Subjects: Algorithms; Applied sciences; Computer science; control theory; systems; Distributed shared memory; Exact sciences and technology; Mathematics; Matrix; Numerical analysis; Numerical analysis. Scientific computation; Numerical linear algebra; Sciences and techniques of general use; Software; Software engineering; Studies
• ISSN: 0098-3500; 1557-7295
• DOI: 10.1145/1067967.1067970

In this article we describe our implementations of the GMRES algorithm for both real and complex, single and double precision arithmetics suitable for serial, shared memory and distributed memory computers. For the sake of portability, simplicity, flexibility and efficiency the GMRES solvers have been implemented in Fortran 77 using the reverse communication mechanism for the matrix-vector product, the preconditioning and the dot product computations. For distributed memory computation, several orthogonalization procedures have been implemented to reduce the cost of the dot product calculation, which is a well-known bottleneck of efficiency for the Krylov methods. Either implicit or explicit calculation of the residual at restart are possible depending on the actual cost of the matrix-vector product. Finally the implemented stopping criterion is based on a normwise backward error.