Preconditioning techniques and numbering methods for the preconditioned conjugate gradient method are investigated to solve simultaneous linear equation iteratively and efficiently. One level fill-in technique with incomplete Cholesky decomposition and the reverse Cuthill-McKee numbering algorithm result in the best results in view of robustness and efficiency. For the numbering methods, the minimal degree ordering method, the nested dissection method, the frontal profile reduction algorithm, the Cuthill-Mckee algorithm, and the spiral ordering method are compared with numerically. It is well known that iterative methods are less efficient than direct methods for solving simultaneous linear equations. But for large problems of several tens of thousands des, the iterative method can be more efficient than the direct method. And it is ascertained through many test problems in this study. For several examples, the spiral ordering method results in better results but the method shows non-robustness. It would be desirable that the correlation between the bandwidth minimization and the efficiency of the PCG is further examined and the PCG is extended to solution of linearly constrained problems
