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 recommended by the CIE MES2 system [CIE 191:2010] in order to compute the adaptation coefficient m 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 / 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 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.

Abstract: Currently the fixed-point iteration method with initial guess 0 0.5 m = is officially recommended by the CIE MES2 system [CIE 191:2010] in order to compute the adaptation coefficient m and the mesopic luminance . 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 2 ( 18.1834) C ≈ 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.

Introduction
The CIE MES2 system [1] was proposed in 2010 as an intermediate between the USP-system developed by Rea et al. [2] in 2004, and the Move-system developed by Goodman et al. [3] in 2007. In the MES2 system, the spectral luminance efficiency function in the mesopic range from where the values for the parameters a and b adopted by CIE [1] are then the coefficient of adaptation m , defined by (4) and (5), should be also the solution of the Gao et al. [4] have shown that, with the values for a and b given by (6), the equation ( ) 0 F m = may have either no solution or more than one, and, in agreement with Shpak et al. [5], they have recommended that the values for the parameters a and b should be better replaced by 10 5 1 1 , . 3 3 Gao et al. [4] have shown that, with the new values for a and b given by (8), the equation ( ) 0 F m = has a unique solution between 0 and 1 , when the following condition is satisfied: 2 2 0.005 and .0 . 5 Thus, in this paper we will use the values for a and b given by (8), together with the remaining equations of the CIE MES2 system. Note first that a and b given by (8), also satisfy ( ) with 0 0.5, m = until 'convergence' is achieved. The term 'convergence' in this algorithm (14) means that the iteration process is stopped when two consecutive values n m and 1 n m + are close enough, i.e., the difference among them in absolute value is smaller than a prefixed small tolerance .
ε Therefore, when we have  since g is a continuous function. Hence, * m is a fixed point of the function , g and this is the reason this iteration method is also named in the literature [6] as fixed-point iteration.
It is clear that the function , g or the fixed-point iteration method, is dependent on both, p L and the ratio / .
Recently, Gao et al. [4] and Shpak et al. [5] have reported that the convergence of the fixed-point iteration method depends on the ratio / , S P and for large values of / S P the method does not converge. Shpak et al. [5] suspected that, to achieve convergence, the ratio / S P cannot be larger than 17. Since currently the fixed-point iteration method is officially recommended by the CIE MES2 system [1], and it may be also implemented in automatic devices, it is appropriate to provide a full theoretical consideration on the convergence of such method. This is the main goal of this paper.

Convergence analysis for fixed-point iteration method
We start quoting a result about a sufficient condition for the convergence of the fixed-point iteration method.
and * x is a fixed point of .
n n x ≥ is monotonically decreasing, and we get the same conclusion. Now, from Theorem 1, we have: then the fixed-point iteration method given by (14) is convergent.
Proof: First, we note that when / 1, S P = the fixed-point iteration method (14) is convergent. In fact, when / 1, S P = we have , It is easy to check that ( ) g m ′ is given by And by Theorem 1, the fixed-point iteration method given by (14) is convergent.
We note that Lemma 1 cannot be applied to prove Theorem 2, since ( ) g m ′ is greater than 1, when / S P and m are sufficiently small. Now, in order to investigate the convergence of the fixed-point iteration method when / 1, S P > we need the second derivative of the function , g given by With the expression of ( ) g m ′′ given by (23), we have: Then, from (24), we have 0 A < and 0 B < . And from (23), it is obvious that , it is clear from (24) that 0 A = and 0 B < . Therefore, from (23), Then, from (24), we have 0 A > and 0 B < . And from (23), it is obvious that is a decreasing function of / S P , as it can be easily shown, we have and, again . Then, from (24), we have 0 A > and 0 B < , and now, since and the proof is concluded. h(S/P)
The left-hand side of the above inequality is a constant, while the right-hand side depends on p L . If we let We are now ready to state another convergence theorem for the fixed-point iteration method when / 1 S P > .  The above information can help to choose a "better" initial guess 0 m , and will be discussed in the next section.

Performance of fixed-point iteration method with new initial strategy
We have shown that the fixed-point iteration method (14)  which make a total number of 37 values for the ratio / . S P Therefore, we have considered 925 25 37 = × cases for testing the performance of the fixedpoint iteration method with the original initial guess 0 0.5 m = (the current CIE MES2 method [1]), and also with a new initial strategy. Regarding the selected range of values for the ratio / S P from 0.1 to 18.1, it must be remarked that, for most current conventional light sources, these values are low, in a range around 0.0-3.0 [7]. However, higher values, up to a maximum of around 73.0, which are associated to blue monochromatic lights, are also possible [5]. For example, Nizamoglu et al. [8] have reported / S P values of 5.15 for nanocrystal hybridized LEDs, and previous researchers [4] have considered / S P values up to 50, for theoretical light sources based on Hung et al. method [9]. The initial value 0 m in the new strategy, that we propose, is given by where 0 g and 1 g are given by (34). We have fixed tolerance 5 10 ε − = for the convergence, and have limited the number of iterations to 200, in order to avoid the program running during a very long time. Figure 4 shows the contours with the number of iterations needed for the convergence of the fixed-point iteration method, using the initial value 0 0.5, m = as recommended by CIE [1]. In 921 cases, from the total of 925, the convergence is obtained when computing less than 100 iterations. In two cases, namely / 18.1 S P =  Figure 5 shows the contours with the number of iterations needed for the convergence of the fixed-point iteration method, using as initial value 0 , m in each case, the value provided by the new strategy proposed in (47). Now, in all 925 cases under study, the convergence is obtained when computing less than 70 iterations. This fact proves the validity of our convergence analysis. In addition, let 1 N and 2 N be the number of iterations needed for the convergence, when using 0 0.5 m = and the initial value provided by (47), respectively. It has been found that 2 cases. Thus, with the proposed new initial strategy (47), the fixed-point iteration method converges faster than with the CIE recommended initial value [1] in the 80% of the cases.

Conclusions
The MES2 system was recommended by CIE [1] to compute the mesopic luminance, using a fixed-point iteration method (see (14)). In this process of computation of the mesopic luminance, a numerical solution of a nonlinear equation ( ) 0 F m = is searched (see (7)). Shpak et al. [5] have proposed new values for the parameters a and b involved in that equation (see (8)). With these new values for a and b , Gao et al. [4] have shown that the nonlinear equation ( ) 0 F m = has a unique solution in (0,1 ) , whenever a condition for s L and p L , given by (3), is satisfied (see (9)). However, Gao et al. [4] and Shpak et al. [5] have found that the fixed-point iteration method may be not convergent for large values of / / s p S P L L = . Shpak et al. [5] pointed out that this ratio should not be larger than 17 in order to have convergence. In this paper a theoretical consideration on the convergence has been given, and it has been found that the fixed-point iteration method converges when using appropriate initial values, and the ratio / S P is smaller than 2 18.1834 C ≈ . For values of / S P larger than 2 C , there is no guarantee for the convergence of the fixed-point iteration method. Values of the ratio / S P for current light sources are usually lower than 3.0 [7], but Nizamoglu et al. [8] have reported higher / S P values of 5.15 for nanocrystal hybridized LEDs. The theoretical upper limit for the ratio / S P is around 73.0 [5]. Therefore, our current analyses considering sources with high / S P values make sense, because we are proposing a valid CIE method for both current and future light sources. Moreover, for values of / S P smaller than 2 C , a new strategy (47) for the choice of the initial value 0 m has been proposed. In many cases, this new strategy produces a remarkable reduction of the number of iterations needed to achieve convergence, compared when using 0 0.5 m = as initial value, as currently recommended in CIE MES2 method [1].