= Collocation endpoint superconvergence by quadrature orthogonality
{title2=$p=q$}
For $s$ collocation nodes, the residual of the collocation polynomial vanishes at each node and contains the factor $\omega(\tau)=\prod_i(\tau-c_i)$. Quadrature exactness through degree $q-1$ implies $\int_0^1\omega(\tau)p(\tau)d\tau=0$ for every polynomial of degree at most $q-s-1$. Expand the smooth vector-field interpolation error and the derivative of the exact flow in powers of the step. Every endpoint-error term of degree below $q$ has this vanishing weighted moment, so the endpoint local error is $O(h^{q+1})$, although the interior stage accuracy is generally only $s$. A step Lipschitz estimate and <discrete Gronwall inequality> then give global order $q$. Gaussian nodes attain $q=2s$, while the first nonexact quadrature moment bounds the order from above.
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