Gauss-Legendre vs Trapezoid Quadrature

What you are seeing: two ways of approximating $\int_{-1}^{1} f(x)\, dx$ . The trapezoidal rule uses $n + 1$ equispaced points and converges as $O(h^2)$ for smooth $f$ . Gauss-Legendre uses $n$ optimized nodes (roots of $P_n$ ) with matching weights, exact for polynomials up to degree $2n - 1$ . For analytic $f$ the GL error decays exponentially.

Top panel: $f(x)$ with nodes overlaid (orange = trapezoid, cyan = GL). Bottom: log-error vs $n$ . For smooth $\cos(2x)$ , GL reaches machine precision by $n = 16$ while trapezoid still loses accuracy as $O(h^2)$ .

Figure 1. Quadrature shootout. Method: Gauss-Legendre via Golub-Welsch eigenproblem and trapezoid via equispaced sum.

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WHAT TO TRY

Raise the node count n: the trapezoid error falls slowly as 1/n-squared, while Gauss-Legendre plunges to machine precision within a handful of nodes. The error-vs-n panel shows the exponential gap.
Note where Gauss puts its nodes: clustered, unequally weighted, and never at the endpoints, which is how n points integrate polynomials up to degree 2n-1 exactly.
Change the integrand: smooth functions let Gauss converge spectrally, but a kink or singularity drags both schemes back to algebraic convergence.