Error bound for the trapezoidal rule

Remarks

On one subinterval $[x_{i-1}, x_i]$, the interpolation error formula gives $f(x) - L(x) = \tfrac{1}{2} f''(\eta_x)(x - x_{i-1})(x - x_i)$ for the linear interpolant $L$. Since $(x - x_{i-1})(x - x_i) \le 0$ there, the mean value theorem for integrals gives the local error $-\tfrac{h^3}{12} f''(\xi_i)$. Summing over the $N$ subintervals and using the intermediate value theorem for $f''$ gives the global formula with $N h^3 = (b - a) h^2$. Example: $f(x) = x^2$ on $[0, 1]$ with $N = 2$: the error is $\tfrac{1}{3} - \tfrac{3}{8} = -\tfrac{1}{24}$, and the formula gives $-\tfrac{1 \cdot 1/4}{12} \cdot 2 = -\tfrac{1}{24}$. Without the smoothness hypothesis the rate $h^2$ can fail: for $f(x) = \sqrt{x}$ on $[0, 1]$, whose second derivative is unbounded near $0$, the error is only of order $h^{3/2}$. Mathlib proves the bound in the form trapezoidal_error_le_of_c2.

Going deeper

For a student

This bound is the model for error analysis of composite Newton–Cotes formulas: a local error of order $h^{d+2} f^{(d+1)}$ summed over $N$ pieces gives a global error of order $h^{d+1}$. It also tells how many subintervals are needed for a given tolerance, once a bound on $\abs{f''}$ is known. The Euler–Maclaurin formula refines it into an expansion of the error in even powers of $h$, which is the basis of Richardson extrapolation and Romberg's method. For smooth periodic functions all the correction terms vanish, which explains the very fast convergence of the trapezoidal rule in that case.