The conjugate gradient method (CG), an iterative method of solving linear systems, has been optimized in conjunction with two preconditioners, the incomplete Cholesky decomposition (IC) and a truncated Neumann series. In both cases, the preconditioning technique generally used has been adapted and fine-tuned to the application described. The first (IC) involves a generalization of the conventional technique, that is it covers all spectral patterns across the lower triangular matrix L. Secondly, the truncated Neumann series takes account of an innovation borrowed from IC. The implementation of the above procedures in finite element method (FEM) analysis is described, with the emphasis on how the innovations lead to results relevant to the application of these models in the optimization of system solving. Finally, there follow some remarks on the implementation of these techniques in parallel computing environments.

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