A parallel Newton-GMRES algorithm for solving large scale nonlinear systems

In this work we describe a portable sequential and parallel algorithm based on Newton's method, for solving nonlinear systems. We used the GMRES iterative method to solve the inner iteration. To control the inner iteration as much as possible and avoid the oversolving problem, we also parallelized several forcing term criterions. We implemented the parallel algorithms using the parallel numerical linear algebra library SCALAPACK based on the MPI environment. Experimental results have been obtained using a cluster of Pentium II PC's connected through a Myrinet network. To test our algorithms we used three different test problems, the H-Chandrasekhar problem, computing the intersection point of several hyper-surfaces, and the Extended Rosenbrock Problem. The latter requires some improvements for the method to work with structured sparse matrices and chaotic techniques. The algorithm obtained shows a good scalability in most cases. This work is included in a framework tool we are developing where, given a problem that implies solving a nonlinear system, the best nonlinear method must be chosen to solve the problem. The method we present here is one of the methods we implemented.