Abstract: 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.

Key words: nonlinear Schrödinger equation, iterative method, BI-conjugate, preconditioned conjugate gradient