= Collocation order theorem
{title2=$p=s+k$}
For $s$ distinct collocation nodes, let $\pi_s(t)=\prod_{i=1}^s(t-c_i)$. If $\int_0^1t^j\pi_s(t)dt=0$ 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 $2s-1$.
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