 Solution

Richardson extrapolation cancels a leading discretization error by combining two mesh sizes. If $A(h)=I+c{h}^p+O(h^q)$ with $q>p$, thenUse a uniform grid $x_i=a+ih$, $0\le i\le n$, with even $n$. The composite trapezium rule isThe composite Simpson rule, formed by integrating a quadratic interpolant on each pair of successive panels, isAssume an even-power trapezium error expansion $T_h=I+c_2{h}^2+O(h^4)$; continuous fourth derivative is a standard sufficient smoothness hypothesis. The coarse grid has step $2h$, so one extrapolation step is $(4T_h-T_{2h})/3$. Its endpoint coefficients are $h/3$, its odd-index coefficients are $4h/3$, and its interior even-index coefficients are $2h/3$. ThereforeThe algebraic identity between the two rules holds for any sampled values; the smoothness and error assumptions are used to infer the improved accuracy.