the output of step (4) as the pupil complex amplitude function]{}

This iterative algorithm is a generalization of the iterative algorithm used by [@2000SPIE.4006..377G; @2002AA...391..379G] and detailed in @guyonPhD to compute optimal apodization for the phase mask coronagraph (for which the mask is fully transmissive), and the iterative algorithm used to compute optimal apodization for the APLC (for which the mask is fully opaque) on centrally obscured circular apertures [@2005ApJ...618L.161S; @2010AA...520A.110M] and on more complex pupil shapes [@2009ApJ...695..695S]. showed that the apodization solutions obtained for rectangular and circular apertures are Prolate functions for which analytical expressions exist. Apodization functions can also be computed for centrally obscured apertures [@2005ApJ...618L.161S], and for arbitrary non circular symmetric pupils [@2009ApJ...695..695S]. A remarkable property of the iterative algorithm described above is that it converges for a wide range of pupil shapes and focal plane mask diameters [@guyonPhD]. For small focal plane mask diameters, convergence is due to the fact that modifying the entrance aperture light distribution predominantly affects the “outside focal plane mask” light component. Exact apodization solutions for the APLC and APCMLC therefore exist for most aperture geometries and focal plane mask diameters. An example APCMLC design on a non-circular aperture, for which the 