Theoretical consideration on the convergence of the fixed-point iteration method in the CIE mesopic photometry system MES2

Currently the fixed-point iteration method with initial guess 0=0.5 is officially recommended by the CIE MES2 system [CIE 191:2010] in order to compute the adaptation coefficient and the mesopic luminance . However, recently, Gao et al. [Opt. Express 25, 18365 (2017)] and Shpak et al. [Lighting Res. Technol. 49, 111 (2017)] have numerically found that the fixed-point iteration method could be not convergent for large values of / . Shpak et al. suspected that, to achieve convergence, the / ratio cannot be greater than 17. In this paper, a theoretical consideration for the CIE MES2 system is given. Namely, it is shown that the ratio / be smaller than a constant 2 (≈18.1834) is a sufficient condition for the convergence of the fixed-point iteration method. In addition, a new initial guess strategy, achieving faster convergence, is proposed.