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**arbolis** I need help for today on this exercise :

Approximate the data in the following table $\displaystyle \begin{array}{c|cccc} x & -1 & 0 & 1 & 2\\ \hline y & -1.1 & -0.4 & -0.9 & -2.7\end{array}$ with a function of the form $\displaystyle f(x)~-e^{ax^2+bx+c}$ in the sens of least squares for the function $\displaystyle ln(-f(x))$.

I understand the problem and I will simplify it for you, so you can help me. They want me to approximate the data given in the table, via a function of the type $\displaystyle f(x)~-e^{ax^2+bx+c}$, which is a too complicated problem. Therefore, in order to do it, we take the natural log of $\displaystyle f(x)$, to approximate a new set of data, which will be $\displaystyle ln(-f(x))$. This is much less complicated to determine a, b and c. After this we just replace a, b and c in f(x) and we're done. But the problem is that I don't know how to get these values.

The problem is simplificated to approximate $\displaystyle \begin{array}{c|cccc} x & -1 & 0 & 1 & 2\\ \hline y & 0.0953 & -0.9163 & -0.105 & 0.993\end{array}$ with a function of the form $\displaystyle ax^2+bx+c$. To get a linear system, I must use the inner product theory to solve the problem.

Let V be the set of $\displaystyle f(-1)$, $\displaystyle f(0)$, $\displaystyle f(1)$, $\displaystyle f(2)$ such that f is a function. Let W be a subset of V, so W={...}. Then I define the inner product between x and y as $\displaystyle \sum_{i=1}^{n}x_iy_i$. From it I must use the normal equations to solve the problem and get a, b and c. But as I don't know how to define W, nor how to construct the normal equations, I'm totally stuck on this.