## 3 Criteria of Response Function Selection
### Step One--Accuracy of the Coefficients  
In the first part of the experiment, the influence of the discrete data accuracy on the coefficient values \(D_{i}^{*}\) of the approximation polynomials \(w^{*}\) of the order \(POa\) was analyzed. For this purpose, cases were considered where discrete data were generated on the basis of a given group of functions as well as a reference base polynomial \(w\) and then approximated by the same type function. Due to the assumption of a different a priori error, by rounding the values of the trial points to a certain number of significant digits, it was then possible to analyze its impact on the a posteriori error assumed as the maximum from the value of the relative error (more precisely, its module--this detail is omitted in the further part of this work to shorten the descriptions). It was calculated for individual coefficients of the approximation polynomial in relation to the reference polynomial. This part of the experiment was shown schematically in Figure 3.  
All the groups of functions from Table 1 as well as all the base reference polynomials from Table 2 were analyzed. The results for the polynomial dependences (the first group of functions) are shown in Figure 4a,b (dashed lines), while for all the groups of functions--in Figure 4b (solid lines). In order to increase the readability of the diagrams, only the maximum values of the relative errors from all the obtained ones for a given polynomial degree are presented, denoted as \(\max\Delta_{\text{rel,c}}\):  
\[\max\Delta_{\text{rel,c}}=\max_{ \begin{array}{c}poly=1,2,\ldots\\ y_{i}=y_{1},\ldots\end{array} }\Delta_{\text{rel,c}} \tag{23}\]  
\begin{table}
\begin{tabular}{c c c} \hline
**Polynomial Order** & **Polynomial Number** & **Polynomial Form** \\ **(POb)** & **(poly)** & **Polynomial Form** \\ \hline
1 & 1 & 10+\(x\) \\
1 & 2 & \(10-10^{-1}x\) \\
2 & 1 & \(10+10^{-1}(-x+x^{2})\) \\
2 & 2 & \(10+10  
+++ ==WARNING: Truncated because of repetitions==
^{-1}(x+x^{2})\) \\  
++  
where  
\[\Delta_{\text{rel},c}=\max_{j=0,\ldots,POb}\left|\frac{D_{j}-D_{j}^{*}}{D_{j}}\right| \tag{24}\]  
In the case of polynomial dependences (dashed lines in Figure 4), the relative error value was initially 1.0 regardless of the degree of the polynomial under consideration. This is due to the fact that the generated discrete data, when rounded to a very small number of significant digits, had identical values. The approximating function was then constant, the values of the coefficients at the nonzero powers of the variable were equal to 0, so their relative error in relation to the respective nonzero coefficients of the reference polynomials was 100%. At a slightly larger number of significant digits of rounding, the individual discrete data generated started to differ from the others, but only in the last place of their decimal notation. An attempt to approximate these values--the diagram of which is clearly a polygonal chain--with smooth polynomials caused \(\max\Delta_{\text{rel},c}\) to start exceeding 100%--except for linear functions. Only when we achieved a sufficiently high accuracy of the values of the test points--which guaranteed their greater differences--it only started to decrease. When considering all the other groups of functions (solid lines in Figure 4b), a greater precision of discrete data was required to obtain the same a posteriori error than for the polynomial dependences (dashed lines). This is equivalent to the fact that with the established accuracy of discrete data, in the case of the polynomial dependences, a smaller relative error of the coefficients \(\max\Delta_{\text{rel},c}\) was obtained than for the other considered functions (see the example in Table 3). This is due to the fact that in the case of the first group of functions, it was possible to directly calculate the coefficients in Equation (10), while in the case of the others, an additional modification of the arguments or the set of values was required in accordance with Equations (11) and (12), correspondingly.