Butcher tableau

Remarks

The method is explicit when $a_{ij} = 0$ for $j \ge i$, so the tableau is strictly lower triangular and blank entries are read as $0$. Usually $c_i = \sum_j a_{ij}$ for each $i$. An embedded pair adds a second row $\hat b^{T}$ of weights that gives a method of another order from the same stages. Example: the classical fourth-order method has $c = (0, \tfrac12, \tfrac12, 1)^T$, $a_{21} = a_{32} = \tfrac12$, $a_{43} = 1$, all other $a_{ij} = 0$, and $b = (\tfrac16, \tfrac13, \tfrac13, \tfrac16)^T$. Check of two order conditions: $\sum b_i = 1$ and $\sum b_i c_i = \frac16 + \frac16 + \frac16 = \frac12$. Non-example: an array with $c \in \R^2$, $A \in \R^{2 \times 2}$ and $b \in \R^3$ is not a Butcher tableau, since the number of weights must equal the number of stages.