Several Parallel Algorithms for Solving Nonlinear Systems with Symmetric and Positive Definite Jacobians

In this work we describe two sequential algorithms and their parallel counterparts for solving nonlinear systems, when the Jacobian matrix is symmetric and positive definite. This case appears frequently in unconstrained optimization problems. Both algorithms are based on Newton’s method. The first solves the inner iteration with Cholesky decomposition while the second is based on the inexact Newton methods family, where a preconditioned CG method has been used for solving the linear inner iteration. In this latter case and to control the inner iteration as far as possible and avoid the oversolving problem, we also parallelized several forcing term criterions and used parallel preconditioning techniques. We implemented the parallel algorithms using the SCALAPACK library. Experimental results have been obtained using a cluster of Pentium II PC's connected through a Myrinet network. To test our algorithms we used four different problems. The algorithms show good scalability in most cases.