Find an equation of the form $y= a e^ {x ^2} + bx^3$ which best fits the points $(−1, 0), (0, 1), (1, 2)$
I am given the hint don’t form a summation – explicitly form S with three terms, and find the derivatives.
I have tried using the formula $$S=\sum_{i=1}^{n}\sum_{j=0}^{m}(C_jx_i^j-y_i )^2$$ and expanding it out with the points that are given, then simplifying some, and then taking the partial derivatives with respect to the $C's$, setting them equal to zero and then trying to solve.
However, this process is really long and I don't think that I am able to solve it. I am wondering if I am missing something or if there is a better way that I am overlooking.
This is not a statistics class, this is a numerical analysis class.
Write $S= \sum_{k=1}^3 (a e^{x_k^2} + b x_k^3-y_k)^2$.
Differentiate with respect to $a,b$ to get the least squares approximation.