  The discretization of Gross-Pitaevskii equations (GPE) leads to a nonlinear
eigenvalue problem with eigenvector nonlinearity (NEPv). In this paper, we use
two Newton-based methods to compute the positive ground state of GPE. The first
method comes from the Newton-Noda iteration for saturable nonlinear
Schr\"odinger equations proposed by Liu, which can be transferred to GPE
naturally. The second method combines the idea of the Bisection method and the
idea of Newton method, in which, each subproblem involving block tridiagonal
linear systems can be solved easily. We give an explicit convergence and
computational complexity analysis for it. Numerical experiments are provided to
support the theoretical results.
