## 2 Materials and Methods
### Method of Successive Approximation  
To extend the scope of the MSA application, a hypothesis was formulated, as follows. Let there exist a scalar function \(F(X)=F(x_{1},x_{2},x_{3},\ldots x_{n})\), that is bounded, defined and continuous in the closed region \(\overline{D}\) of the scalar field \(P\). Then for any point \(M\subset\overline{D}\forall M\in D;\forall\varepsilon\geq 0\), the scalar field \(P\) is defined as:  
\[P=\left( \begin{array}{ccc}1&0&0\\ 0&0&1\end{array} \right) \tag{12}\]  
where \(\varepsilon\) is the scalar field, \(P\) is the scalar field, and \(P\) is the scalar field. The following proposition is a theorem of the present paper. **Proposition 1**: _The mean value of the  
\(0\exists U_{\ell}(M)\subset\overline{D}\) in the vicinity of the point \(M_{0}\big{(}x_{1}^{0},x_{2}^{0},x_{3}^{0},\ldots x_{n}^{0}\big{)}\) (Figure 2) the function \(F(X)\) can be represented in the form:  
\[|F(X)-\phi(X)|\leq\varepsilon\forall M_{0}\in\mathcal{U}_{\varepsilon}(M_{0}) \tag{11}\]  
where  
\(U_{\varepsilon}(M_{0})\)--vicinity of the point \(M_{0}\big{(}x_{1}^{0},x_{2}^{0},x_{3}^{0},\ldots x_{n}^{0}\big{)}\);  
\(\phi(X)=\alpha\prod_{i=1}^{n}g_{i}(x_{i})\);  
\(g_{i}(x_{i})\)--approximation functions for \(f_{1},f_{2},f_{3},\ldots,f_{n}\), which are given as follows:  
\[f_{1}(x_{1})=F\big{(}x_{1}^{0},x_{2}^{0},x_{3}^{0},\ldots,x_{n}^{0}\big{)},\ f_{2}(x_{2})=F\big{(}x_{1}^{0},x_{2}^{0},x_{3}^{0},\ldots,x_{n}^{0} \big{)},\ f_{3}(x_{3})=F\big{(}x_{1}^{0},x_{2}^{0},x_{3}^{0},\ldots,x_{n}^{0} \big{)},\ldots,f_{n}(x_{n})=F\big{(}x_{1}^{0},x_{2}^{0},x_{3}^{0},\ldots,x_{n }^{0}\big{)} \tag{12}\]  
\(\alpha\)--the approximation coefficient is determined in accordance with the formula:  
\[\alpha=\frac{F(M_{0})}{g_{1}\big{(}x_{1}^{0}\big{)}\cdot g_{2}\big{(}x_{2}^{0 }\big{)}\cdot g_{3}\big{(}x_{3}^{0}\big{)}\cdot\ldots\cdot g_{n}\big{(}x_{n}^ {0}\big{)}} \tag{13}\]  
Using the specified approach, the representation of the function \(F(X)=F(x_{1},x_{2},x_{3},\ldots x_{n})\) around the point \(M_{0}\big{(}x_{1}^{0},x_{2}^{0},x_{3}^{0},\ldots x_{n}^{0}\big{)}\) has sufficient accuracy for engineering calculations on the entire definition area \(\overline{D}\). The MSA application algorithm can be represented by a sequence of the following steps:  
1. Selecting a point from the function definition area \[M=M_{0}\Big{(}x_{1}^{0},x_{2}^{0},x_{3}^{0},\ldots x_{n}^{0}\big{)},\ M\in \overline{D}\] (14)
2. Creating the function \(f_{1}(x_{1})=F\big{(}x_{1}^{0},x_{2}^{0},x_{3}^{0},\ldots,x_{n}^{0}\big{)}\). 3. Finding the type of function \(g_{1}(x_{1})\), that is an approximation for the function \(f_{1}(x_{1})\);
4. Finding \(\varphi_{1}(x_{1})\) in accordance with step 1: \(\varphi_{1}(x_{1})=\alpha_{1}g_{1}(x_{1})\), where \(\alpha_{1}\) is the approximation coefficient. 5. Defining the function around the point \(M\) from equality \(F(x_{1})\approx\varphi_{1}(x_{1})\). 6. Repeating the steps 2-5 consecutively for the variables \(x_{j}(j=\overline{2,n})\) and obtaining the representation as follows:  
\[F(x_{1},x_{2},x_{3},\ldots x_{n})\approx\varphi_{1}(x_{1},x_{2},x_{3},\ldots x _{n})=\alpha\cdot g_{1}(x_{1})\cdot g_{2}(x_{2})\cdot g_{3}(x_{3})\cdot \ldots\cdot g_{n}(x_{n}) \tag{15}\]  
The location of a point \(M=M\big{(}x_{1}^{0},x_{2}^{0},\ldots,x_{n}^{0}\big{)},M\in\overline{D}\) in the definition area significantly depends on its topology, and therefore affects the way it is represented. The choice of a point on the definition area is determined by prior knowledge of its features and is determined by the researcher's experience. In the case of complex functions and lack of prior knowledge about behavior of the resulting function, it is convenient to choose it in the center of the definition area, i.e., to determine the coordinates according to the formula:  
\[x_{j}=\frac{b_{j}-a_{j}}{2} \tag{16}\]  
where \(a_{j},b_{j}\) represent the beginning and end proves (respectively) of the interval of changes of parameter \(x_{j}\).