 Collocation order theorem

For $s$ distinct collocation nodes, let ${\pi }_s(t)={\prod }_{i=1}^s(t-c_i)$. If ${\int }_0^1{t}^j{\pi }_s(t)dt=0$ for $0\le j<k$, and the next moment is nonzero, the associated interpolatory quadrature has order $s+k$, and the collocation Runge-Kutta method has the same classical order for sufficiently smooth differential equations. Here $0\le k\le s$. Equivalently, if the quadrature is exact through degree $p-1$ but not degree $p$, the method has order $p$. The generic order is at least $s$, Gauss collocation methods reach $2s$, and Radau IIA methods reach $2s-1$.