The Solution of Systems of Linear Equationsusing the Conjugate Gradient

The solution of large sparse systems of linear equations is one of the most compu-tationally intensive parts of nite element simulations. In order to solve these systems of linear equations, we have implemented a parallel conjugate gradient solver on the SPMD-programmable MUSIC-system. We outline the conjugate gradient method, give a formal speciication in Maple, and describe a data-parallel program. We illustrate how the number of processors innuences the speed of convergence due to diierent data distributions and the non-associativity of the oating point addition. We investigate the speed of convergence of the conjugate gradient method for diierent oating point precisions (32, 44, 64, and 128 bit) and various nite element models (linear beams, human spine segments). The results show that it is more important to concentrate on appropriate numerical methods depending on the nite element models considered than on the oating point precision used. Finally, we give the results of our speedup measurements.