A Set of GMRES

In this report we describe the 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 simplicity, exibility and eeciency the GMRES solvers have been implemented 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, that is a well-known bottleneck of eeciency for the Krylov methods. Finally the implemented stopping criterion is based on a normwise backward error. After a short presentation of the GMRES methods and of the solution of the least-squares problems in real and complex arithmetic, we give a detailed description of the user interface. 1 The GMRES algorithm 1.1 General description The Generalized Minimum RESidual (GMRES) method was proposed by Saad and Schultz in 1986 6] in order to solver large, sparse and nonsymmetric (or non Hermitian) linear systems. GMRES belongs to the class of Krylov based iterative methods. Let A be a square nonsingular n n complex matrix, and b be a complex vector of length n, deening the linear system Ax = b (1) to be solved. Let x 0 2 C n be an initial guess for this linear system and r 0 = b ? Ax 0 be its corresponding residual. The GMRES algorithm builds an approximation of the solution of (1) under the form x m = x 0 + V m y (2)