  In this paper, we propose an iterative method to compute the positive ground
states of saturable nonlinear Schr\"odinger equations. A discretization of the
saturable nonlinear Schr\"odinger 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 $k$th 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 Schr\"odinger equations.
Numerical experiments are provided to support the theoretical results.
