Cfrgtkky

I have read that using Gauss Quadrature integration,
$$int_{-1}^{1}f(x)dx=sum_i f(x_i)w_i$$
for polynomials of degree $leq2n-1$ (and otherwise it is an approximation). Using weight functions $w_i$ from wikipedia, for 2$^{nd}$ order GQ we have
$$int_{-1}^{1}f(x)dx=fbigg(frac{1}{sqrt{3}}bigg) + fbigg(-frac{1}{sqrt{3}}bigg)$$
and for 3$^{rd}$ order,
$$int_{-1}^{1}f(x)dx=frac 59 fbigg(sqrt{frac{3}{5}}bigg)+ frac 59 fbigg(-sqrt{frac{3}{5}}bigg) $$

My problem is that I don't seem to be getting the same answer for second and third order GQ for polynomials of degree $leq 3$, for which they should both yield exact solutions.

For example, for the function $f(x)=a_1 + a_2x + a_3x^2 + a_4x^3$, from first order GQ I get:
$$int_{-1}^{1}f(x)dx=2a_1 + frac{2}{3}a_3$$
which I believe is the correct solution, but for third order GQ I get

$$int_{-1}^{1}f(x)dx=frac{10}{9}a_1 + frac 23 a_3$$

This error seems to occur for any constant term in any polynomial as a result of the fact that the sum of the weight functions aren't equal for second and third order GQ. Why is this? I can't find my mistake!

Thanks in advance for your help.

For 3 point quadrature I believe you are missing $frac{8}{9} f(0)$ in your sum.

It would also introduce a $frac{8}{9}a_{1}$ which would correct your toy example.