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

Currently the fixed-point iteration method with initial guess m₀=0.5 is officially recommended by the CIE MES2 system [CIE 191:2010] in order to compute the adaptation coefficient m and the mesopic luminance L _{m e s} . However, recently, Gao et al. [Opt. Express25, 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 S/P. Shpak et al. suspected that, to achieve convergence, the S/P 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 S/P be smaller than a constant C₂ (≈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.