= Butcher tableau
{c}
A Butcher tableau records a <Runge-Kutta method>'s stage <matrix> $A$, time nodes $c$, and output weights $b$ in the block array
$$
\begin{array}{c|c}c&A\\\hline&b^T\end{array}.
$$
Read row $i$ above the line as the recipe for <Runge-Kutta stage> $i$: the left entry gives $t_n+c_i h$, and the right entries give the coefficients in $Y_i=y_n+h\sum_ja_{ij}k_j$. Below the line, $b_i$ weights slope $k_i$ in the final update. Zeros on and above the diagonal identify an <Explicit Runge-Kutta method>. For the internally consistent convention, row sums of $A$ equal $c$; first-order <consistency of a numerical method> also requires $\sum_i b_i=1$. An <embedded Runge-Kutta pair> adds another weight row while retaining $A,c$.
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