Open AccessTheoretical consideration on convergence of the fixed-point iteration method in CIE mesopic photometry system MES2
Author(s) -
Cheng Gao,
Xiaohui Zhang,
Xu Yang,
Zhifeng Wang,
Manuel Melgosa,
José Juan Quesada-Molina,
Kaida Xiao,
Changjun Li
Publication year - 2018
Publication title -
optics express
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.394
H-Index - 271
ISSN - 1094-4087
DOI - 10.1364/oe.26.031351
Subject(s) - mesopic vision , fixed point , photometry (optics) , convergence (economics) , optics , fixed point iteration , luminance , mathematics , physics , mathematical analysis , retina , stars , astronomy , economics , photopic vision , economic growth
Currently the fixed-point iteration method with initial guess m 0 =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 C 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.