I need to find a numerical approximation for the derivative $F'(x)$ of an appropriately smooth function $F(x)$ at $x=0$. However I do not know the actual function $F$. I have this formula:
$$F'(0) \approx \sum_{i=1}^{n} F(x_i)w_i$$
Where I have the values of $x_i$ and $w_i$, $i=1....n$. However I do not have the actual function $F$ so am unsure how to evaluate it? I know there is a similar thing for integration (Gaussian Quadrature) but I was wondering if anyone has anything for this problem!
Just an idea
Let's assume a line $F(x) = a x + b$ for some unknown $a$ and $b$ this implies $$ a \approx w_1 F(x_1) + w_2 F(x_2) + \cdots $$ $$ a \approx w_1 (a x_1 + b) + w_2(a x_2 + b) + w_3 (a x_3+b) + w_4(a x_4+b) $$ $$ a(1-w_1 x_1 - w_2x_2 - w_3 x_3 - w_4 x_4) \approx b(w_1+w_2+w_3+w_4) $$ $$ a \approx \frac{b(w_1+w_2+w_3+w_4)}{(1-w_1 x_1 - w_2x_2 - w_3 x_3 - w_4 x_4)} $$ or even $$ \frac{a}{b} \approx \frac{(w_1+w_2+w_3+w_4)}{(1-w_1 x_1 - w_2x_2 - w_3 x_3 - w_4 x_4)} $$ that way you have all the terms on the right, does that sound along the right lines? There should be another equation to try and solve for $b$ as well as that is currently unknown.. Might be able to get something out of the Taylor expansions by subtracting different terms similar to the Verlet integrator? $$ \vec{v}(t) = \frac{\vec{x}(t + \Delta t) - \vec{x}(t - \Delta t)}{2\Delta t} + \mathcal{O}(\Delta t^2). $$