[Math] Degree of Precision Effect on Quadrature Accuracy

Question

For an $n$ point Gaussian quadrature, one can show that it has degree of precision $2n – 1$ meaning it will exactly integrate polynomials of that degree or lower. Is it always true that a quadrature with a higher degree of precision will give better results for the integration of a non-polynomial function?

Answer 1

The answer to "is it always true" is always "no", especially in numerical methods. The issue is in how closely the integrand resembles a polynomial function. If the integrand is analytic in a large neighborhood of the interval of integration, then Gaussian quadrature converges extremely fast. But for integrals like $\int_{-1}^1\sqrt{1-x^4}\,dx$ or worse yet, $\int_{-1}^1 1/\sqrt{1-x^4}\,dx$, where the behavior is markedly non-polynomial, high degree of the method does not pay off. One is better off using a lower degree method on smaller subintervals. This is same story as with Legendre polynomial vs. polynomial splines.

For a technical treatment, see Is Gauss Quadrature Better than Clenshaw-Curtis? by Lloyd N. Trefethen.

We compare the convergence behavior of Gauss quadrature with that of its younger brother, Clenshaw–Curtis. Seven-line MATLAB codes are presented that implement both methods, and experiments show that the supposed factor-of-$2$ advantage of Gauss quadrature is rarely realized.

For an example where Gaussian quadrature loses to the left endpoint rule, see this post.