Question 6.9: Curve fitting with linear combination of nonlinear functions......

In the notation of Eq. (6.91)

F(x)=C_1 f_1(x)+C_2 f_2(x)+C_3 f_3(x)+\ldots+C_m f_m(x)=\sum\limits_{j=1}^m C_j f_j\left(x_i\right)      (6.91)

the approximating function is F(x)=C_{1} f_{1}(x)+C_{2} f_{2}(x) with F(x)=y, C_{1}=A, C_{2}=B, f_{1}(x)=\frac{1}{x}, and f_{2}(x)=\frac{e^{-2 x^{2}}}{x}. The equation has two terms, which means that m=2, and since there are 11 data points, n=11. Substituting this information in Eq. (6.97)

\sum\limits_{i=1}^n \sum\limits_{j=1}^m C_j f_j\left(x_i\right) f_k\left(x_i\right)=\sum\limits_{i=1}^n y_i f_k\left(x_i\right) \text { for } k=1,2, \ldots, m       (6.97)

gives the following system of two linear equations for A and B.

\begin{gathered} \sum\limits_{i=1}^{11} A \frac{1}{x_{i}} \frac{1}{x_{i}}+\sum\limits_{i=1}^{11} B \frac{e^{-2 x_{i}^{2}}}{x_{i}} \frac{1}{x_{i}}=\sum\limits_{i=1}^{11} y_{i} \frac{1}{x_{i}} \quad \text { for } k=1 \\ \sum\limits_{i=1}^{11} A \frac{1}{x_{i}} \frac{e^{-2 x_{i}^{2}}}{x_{i}}+\sum\limits_{i=1}^{11} B \frac{e^{-2 x_{i}^{2}}}{x_{i}} \frac{e^{-2 x_{i}^{2}}}{x_{i}}=\sum\limits_{i=1}^{11} y_{i} \frac{e^{-2 x_{i}^{2}}}{x_{i}} \text { for } k=2 \end{gathered}

These two equations can be rewritten as:

\begin{aligned} A \sum\limits_{i=1}^{11} \frac{1}{x_{i}^{2}}+B \sum\limits_{i=1}^{11} \frac{e^{-2 x_{i}^{2}}}{x_{i}^{2}} & =\sum\limits_{i=1}^{11} y_{i} \frac{1}{x_{i}} \\ A \sum\limits_{i=1}^{11} \frac{e^{-2 x_{i}^{2}}}{x_{i}^{2}}+B \sum\limits_{i=1}^{11} \frac{e^{-4 x_{i}^{2}}}{x_{i}^{2}} & =\sum\limits_{i=1}^{11} y_{i} \frac{e^{-2 x_{i}^{2}}}{x_{i}} \end{aligned}

The system can be written in a matrix form:

\left[\begin{array}{cc} \sum\limits_{i=1}^{11} \frac{1}{x_{i}^{2}} & \sum\limits_{i=1}^{11} \frac{e^{-2 x_{i}^{2}}}{x_{i}^{2}} \\ \sum\limits_{i=1}^{11} \frac{e^{-2 x_{i}^{2}}}{x_{i}^{2}}& \sum\limits_{i=1}^{11} \frac{e^{-4 x_{i}^{2}}}{x_{i}^{2}} \end{array}\right]\left[\begin{array}{l} A \\ B \end{array}\right]=\left[\begin{array}{c} \sum\limits_{i=1}^{11} y_{i} \frac{1}{x_{i}} \\ \sum\limits_{i=1}^{11} y_{i} \frac{e^{-2 x_{i}^{2}}}{x_{i}} \end{array}\right]