  We consider the approximation of the ground state of the one-dimensional
cubic nonlinear Schr{\"o}dinger equation by a normalized gradient algorithm
combined with linearly implicit time integrator, and finite difference space
approximation. We show that this method, also called imaginary time evolution
method in the physics literature, is con-vergent, and we provide error
estimates: the algorithm converges exponentially towards a modified solitons
that is a space discretization of the exact soliton, with error estimates
depending on the discretization parameters.

Numerical Analysis