# Accounting for Interelement Interferences in Atomic Emission Spectroscopy: A Nonlinear Theory
## Appendix A
We will start with \(s=0\) and find the coefficients \(b_{m},\ d_{m}\) by minimizing the functional  
\[S_{i0}=\sum\limits_{j=1}^{M}\Bigl{[}\varphi_{ij}-\phi_{i}^{0}(I_{m})\Bigr{]}^{ 2}\cdot\frac{1}{W_{ij}}=\sum\limits_{j=1}^{M}\Bigl{[}\varphi_{ij}-I_{m}{}^{(j)} \Bigl{(}b_{m}+d_{m}I_{i}{}^{(j)}\Bigr{)}\Bigr{]}^{2}\cdot\frac{1}{W_{ij}}. \tag{10}\]  
Thus, at step 0, the function approximating the array \(C_{i}\) that takes into account the effect produced by element number \(m\) on determining the concentration of element number \(i\) will be equal to  
\[F_{i}^{0}=F_{i}(I_{i},a_{1},\ldots,a_{K})+\phi_{i}^{0}(I_{m},I_{i}). \tag{11}\]  
Then, using the retrieved values of the coefficients, \(b_{m},\ d_{m}\) we assume that \(s=1\) and introduce the value  
\[\varphi_{i}^{1}=C_{i}-F_{i}(I_{i})-\phi_{i}^{0}(I_{m},I_{i})=C_{i}-F_{i}(I_{i} )-I_{m}(b_{m}+d_{m}I_{i}), \tag{12}\]  
which we approximate by the function  
\[\phi_{i}^{1}(I_{m+1},I_{i})=I_{m+1}(b_{m+1}+d_{m+1}I_{i}). \tag{13}\]  
Next, we minimize the functional  
\[S_{i1}=\sum\limits_{j=1}^{M}\Bigl{[}\varphi_{ij}^{1}-\phi_{i}^{1}(I_{m},I_{i} )\Bigr{]}^{2}\cdot\frac{1}{W_{ij}}=\sum\limits_{j=1}^{M}\Bigl{[}\varphi_{ij}^{ 1}-I_{m+1}{}^{(j)}\Bigl{(}b_{m+1}+d_{m+1}I_{i}{}^{(j)}\Bigr{)}\Bigr{]}^{2}\cdot \frac{1}{W_{ij}} \tag{14}\]  
and find the coefficients \(b_{m+1},\ d_{m+1}\), etc. Thus, at \(s\)-step, the function that approximates the array \(C_{i}\) and takes into account the effects caused by elements numbered from \(m\) to \(m+s\) will have the following form:  
\[F_{i}^{s}=F_{i}(I_{i},a_{1},\ldots,a_{K})+\sum\limits_{q=0}^{s}\phi_{i}^{q} \bigl{(}I_{m+q},I_{i}\bigr{)}=F_{i}(I_{i},a_{1},\ldots,a_{K})+\sum\limits_{q=0 }^{s}I_{m+q}\bigl{(}b_{m+q}+d_{m+q}I_{i}\bigr{)}. \tag{15}\]  
Minimizing the functional \(S_{is}\) at the \(s\)-th step results in:  
\[ \begin{array}{c}b_{m+s}=\frac{\bigl{(}A_{22}^{s}f_{1}^{s}-A_{12}^{s}f_{2}^{ s}\bigr{)}}{A_{11}^{s}A_{22}^{s}-A_{12}^{s}A_{12}^{s}};d_{m+s}=\frac{\bigl{(}A_{11}^{s}f_{2}^ {s}-A_{21}^{s}f_{1}^{s}\bigr{)}}{A_{11}^{s}A_{22}^{s}-A_{12}^{s}A_{22}^{s}};\\ A_{11}^{s}=\sum\limits_{j=1}^{M}\frac{I_{m+s}{}^{(j)}{}^{2}}{W_{ij}};A_{12}^{s}= A_{21}^{s}=\sum\limits_{j=1}^{M}\frac{I_{m+s}{}^{(j)}{}^{2}I_{i}{}^{(j)}}{W_{ij}} ;A_{22}^{s}=\sum\limits_{j=1}^{M}\frac{I_{m+s}{}^{(j)}{}^{2}I_{i}{}^{(j)}}{W_{ ij}};\\ f_{1}^{s}=\sum\limits_{j=1}^{M}\frac{\varphi_{ij}^{s}I_{m+s}{}^{n+s}{}^{(j)}}{W_{ ij}};f_{2}^{s}=\sum\limits_{j=1}^{M}\frac{\varphi_{ij}^{s}I_{m+s}{}^{(j)}I_{i}{}^{(j)}}{W_{ ij}};\\ \varphi_{ij}^{s}=C_{ij}-F_{i}\bigl{(}I_{ij}\bigr{)}-\sum\limits_{q=0}^{s}I_{m+q}{}^{(j)}\Bigl{(}b_{m+q}+d_{m+q}I_{i}{}^{(j)} \Bigr{)}.\end{array}  \tag{16}\]  
It is formula (16) that gives the solution to the problem. * [PERSON] et al. (1959) [PERSON], [PERSON]; [PERSON] [PERSON]; [PERSON] Study of the influence of boron on the results of spectral analysis of structural steels. In _Bulletin of Scientific and Technical Information_; Ural Institute of Ferrrous Metals: Sverdlovsk, Russia, 1959; pp. 120-125. (In Russian)
* [PERSON] et al. (1958) [PERSON]; [PERSON]; [PERSON] [PERSON]; [PERSON] [PERSON] Study of the influence of structure on the results of spectral analysis. In Proceedings of the Materials of the 2 nd Urals Meeting on Spectroscopy, Sverdlovsk, Russia, 14-16 May 1958; pp. 62-66. (In Russian)
* [PERSON] et al. (1958) [PERSON] [PERSON]; [PERSON]; [PERSON] [PERSON]; [PERSON] [PERSON]; [PERSON] [PERSON]; [PERSON]; [PERSON] [PERSON]; [PERSON] [PERSON]; [PERSON] [PERSON]; [PERSON]; et al. _10 th All-Union Meeting on Spectroscopy, 4-14 July 1956, Lvov, Russia; Volume 2: Atomic Spectroscopy_; [PERSON], [PERSON], [PERSON]; Publishing House of Lvov University: Lvov, Russia, 1958; 586p. (In Russian)
* [PERSON] et al. (1962) [PERSON] [PERSON]; [PERSON] [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] [PERSON]; [PERSON]; [PERSON] [PERSON]; [PERSON]; [PERSON]; et al. _New Methods of Metal Tests_; [PERSON]; I. P. Bardin Central Research Institute for Ferrous Metallurgy: Moscow, Russia, 1962; Volume 24, 455p.