From the 2-D nonlinear Schrödinger equation, a complex algebraic equation system is obtained. This paper uses Gauss-Seidel, SOR, Complex BI-CG and complex BI-PCG to solve the system and compares the total costs of iterations of these iterative methods. Meanwhile, the complex equation system is also transformed into a real system whose coefficient matrix is hepta-diagonal. Gauss-Seidel, SOR and PCG methods are then used to solve it and the total costs of iterations are also compared. The result shows that the PCG method is most effective comparing with the others. It is discussed as well that how to select the optimal relaxation factor of SOR method for the systems considered.

It was shown in literatures of the preconditioned conjugate gradient (PCG) that the initial guess gives little influence upon the number of PCG iterations when the stopping criterion requests to reduce the residual norm by a factor. However, the examples reported here show that using a zero initial gress or different random initial guess causes the number of PCG iterations varies seriously. Moreover, the number of iterations still varies seriously for the different parameter of the modeb when the same random initial guess is used. This variation should be avoided since it may cause confusions when different methods are compared. This paper shows that if the preconditioner is given, the series {r_k} determined uniquely by the linear systems provided a zero initial guess is used. To avoid the serious variation in the number of PCG iterations, using the zero initial guess is a good choice.