Gaussian Quadrature and Error $O(h^4)$

Question

I'm having a bit of confusion with a problem I'm trying to solve regarding the error. Given that the error of the second order composite Gaussian quadrature method should be $O(h^4)$, how am I able to use that information and the actual error at a given number of subintervals, $N$, to predict the error if $N$ was increased or decreased by a factor of $5$?

Answer 1

For the $2$-points Gauss-Legendre quadrature the error over a single interval of width $b-a$ is $$e_1=\frac{(b-a)^5}{4320}f^{(4)}(\xi)$$ For some $a<\xi<b$. If you increase the number of subintervals by a factor of $N$ the maximum error per subinterval goes down by a factor of $N^5$ but the number of subintervals goes up by a factor of $N$, so the maximum error goes down by a factor of $N^4=5^4=625$. It should be clear that if the number of subintervals were decreased by a factor of $5$ the maximum error would go up by that same factor of $625$.