We investigate methods to calibrate the non-common path aberrations at an adaptive optics system having a wavefront-correcting device working with an extremely high resolution (larger than 150x150 correcting elements). We use focal-plane images collected successively, the corresponding phase-diversity information and numerically efficient algorithms to calculate the required wavefront updates. Different approaches are considered in numerical simulations, and laboratory experiments are shown to confirm the results. We compare the performances of the standard Gerchberg-Saxton algorithm, Fast and Furious (use of small-phase assumption to take advantage of linearisation) and recently proposed phase-retrieval methods based on convex optimisation. The results indicate that the calibration task is easiest with algorithms similar to Fast and Furious, at least in the framework we considered.