Disparities in convergence rate of a composite Gaussian quadrature rule

Question

I have two functions, $f(x) = \frac{1}{1+x^{2}}$ and $g(x) = \sqrt{x}$. Consider the definite integrals $\int_{0}^{1}f(x)dx$ and $\int_{0}^{1}g(x)dx$.

For a composite Gaussian quadrature rule, I have that $\int f$ converges at $\mathcal{O}(h^{6})$ and $\int g$ converges at $\mathcal{O}(h^{1.5})$.

How do I articulate the disparity here? I suspect it has something to do with the fact that $\frac{d}{dx}\sqrt{x} \to \infty$ as $x\to 0^{+}$ but I would like to understand the reasons more sufficiently.

Answer 1

Check the error estimates, e.g., here:

https://en.wikipedia.org/wiki/Gaussian_quadrature#Error_estimates

As one can notice, the error estimates depend on the norms of derivatives of the function to be integrated. Thus, for smooth $g$ you have the expected order of convergence, while $f$ is not smooth enough which implies the convergence deterioration.