Accurate algorithms to transform geocentric to geodetic coordinates

The problem of the transformation is reduced to solving of the equation $$2 sin (psi - Omega ) = c sin 2 psi ,$$ where Ω = arctg[bz/(ar)], c = (a²?b²)/[(ar)²]^1/2 a andb are the semi-axes of the reference ellisoid, andz andr are the polar and equatorial, respectively, components of the position vector in the Cartesian system of coordinates. Then, the geodetic latitude is found as ?=arctg [(a/b tg ψ)], and the height above the ellipsoid as h = (r?a cos ψ)cos ψ + (z?b sin ψ)sin ψ. Two accurate closed solutions are proposed of which one is approximative in nature and the other is exact. They are shown to be superior to others, found in literature and in practice, in both or either accuracy and/or simplicity.