Variable preconditioning in complex Hilbert space and its application to the nonlinear Schrödinger equation

An iterative method is developed for nonlinear equations in complex Hilbert spaces, extending the method of variable preconditioning defined earlier in real spaces. We derive convergence of our method. The motivating example for this extension is the time-dependent nonlinear Schrödinger equation, where we use our iteration for the time discretization of the problem and test it numerically.