We present two complementary algorithms suitable for using focal-plane measurements to control a wavefront corrector with an extremely high-spatial resolution. The algorithms use linear approximations to iteratively minimize the aberrations seen by the focal-plane camera. The first algorithm, Fast & Furious (FF), uses a weak-aberration assumption and pupil symmetries to achieve fast wavefront reconstruction. The second algorithm, an extension to FF, can deal with an arbitrary pupil shape; it uses a Gerchberg-Saxton (GS)-style error reduction to determine the pupil amplitudes. Simulations and experimental results are shown for a spatial-light modulator controlling the wavefront with a resolution of 170 × 170 pixels. The algorithms increase the Strehl ratio from ～0.75 to 0.98-0.99, and the intensity of the scattered light is reduced throughout the whole recorded image of 320 × 320 pixels. The remaining wavefront rms error is estimated to be ～0.15 rad with FF and ～0.10 rad with FF-GS.
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