I have to find a quadrature rule that approximates the integral of $f(x)$ such that
$$\int_{-h}^{h} f(x) = w_1f(x_1) + w_2f(x_2) + w_3f(x_3) + E$$ where the weights $w_i$ and points $x_i$ are to be found. I also have the extra condition:
$$\int_{-h}^{h} f(x) = \int_{-h}^{h} f(-x)$$
I'm not sure how this extra condition factors into finding the required constants; usually I would set $f(x)$ to be a linear function, then quadratic, then cubic etc. up until the last order of polynomial that we know it will be exact to. I don't know how accurate it has to be, but I'm assuming this particular rule will be exact up to cubics as it resembles Simpson's rule, so since I have six constants to find, the symmetric condition will need to provide 2 more (I estimate).
However, running through my usual method of setting $f(x) = 1$, I would get $2h = w_1 + w_2 + w_3$ as expected. However, since $f$ is constant, I derive the exact same condition. Setting $f(x) = x$ then gives me $0 = w_1x_1 + w_2x_2 + w_2x_2$, and again the extra condition won't help since $\int_{-h}^{h} -x = 0$ aswell. This pattern seems to continue as I use higher order polynomials, so I think that I must be using this symmetric condition wrong. Any help with where to go from here would be very much appreciated.
Best Answer
probably what is meant by the symmetry condition is this (though the phrasing is not completely clear):
Exact integrals always have the symmetry $\int f(x) = \int f(-x)$, so it would be nice if your quadrature-rule had this symmetry too. I.e. $$w_1f(x_1) + w_2f(x_2) + w_3f(x_3) \stackrel{!}{=} w_1f(-x_1) + w_2f(-x_2) + w_3f(-x_3)$$ for all functions $f$ (not only for low-order polynomials). You should convince yourself that the only way to achieve this symmetry is to choose $x_2=0$ and $x_1=-x_3$. This should simplify the rest of your calculations a bit as well.