GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems
SIAM Journal on Scientific and Statistical Computing
Introduction to parallel computing: design and analysis of algorithms
Introduction to parallel computing: design and analysis of algorithms
Using MPI: portable parallel programming with the message-passing interface
Using MPI: portable parallel programming with the message-passing interface
Matrix computations (3rd ed.)
ScaLAPACK user's guide
A parallel Newton-GMRES algorithm for solving large scale nonlinear systems
VECPAR'02 Proceedings of the 5th international conference on High performance computing for computational science
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In this work we describe three 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. Two of the three 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 criteria and used parallel preconditioning techniques. The third algorithm is based on parallelizing the BFGS method. 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.