In this paper, we propose an iterative method to compute the positive ground states of saturable nonlinear Schrödinger equations. A discretization of the saturable nonlinear Schrödinger equation leads to a nonlinear algebraic eigenvalue problem (NAEP). For any initial positive vector, we prove that this method converges globally with a locally quadratic convergence rate to a positive solution of NAEP. During the iteration process, the method requires the selection of a positive parameter \(\theta _k\) in the kth iteration, and generates a positive vector sequence approximating the eigenvector of NAEP and a scalar sequence approximating the corresponding eigenvalue. We also present a halving procedure to determine the parameters \(\theta _k\), starting with \(\theta _k=1\) for each iteration, such that the scalar sequence is strictly monotonic increasing. This method can thus be used to illustrate the existence of positive ground states of saturable nonlinear Schrödinger equations. Numerical experiments are provided to support the theoretical results.

在本文中，我们提出了一种迭代方法来计算可饱和非线性Schrödinger方程的正基态。饱和非线性Schrödinger方程的离散化导致非线性代数特征值问题（NAEP）。对于任何初始正矢量，我们证明该方法以局部二次收敛速度全局收敛到NAEP的正解。在迭代过程中，该方法需要在第k次迭代中选择正参数\（\ theta _k \），并生成近似NAEP特征向量的正向量序列和近似对应特征值的标量序列。我们还提供了一个减半过程来确定参数\（\ theta _k \），从每次迭代\（\ theta _k = 1 \），因此标量序列严格单调递增。因此，该方法可用于说明可饱和非线性Schrödinger方程的正基态的存在。提供数值实验以支持理论结果。