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A Butcher tableau records a Runge-Kutta method's stage matrix $A$, time nodes $c$, and output weights $b$ in the block arrayRead 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 $Y_i=y_n+h{\sum }_j{a}_{ij}{k}_j$. 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 $A,c$.

An implicit Runge-Kutta method defines at least one Runge-Kutta stage using an equation that involves that stage or a later unresolved stage. In the slope convention,An Explicit Runge-Kutta method has strictly lower-triangular $A$, so the slopes are computed sequentially. An implicit tableau instead requires one or more algebraic solves, usually nonlinear when the vector field is nonlinear. Existence, uniqueness and numerical accuracy of those solves are additional requirements; A-stability of the scalar test equation does not establish them.

For the implicit midpoint rule, $Y=y_n+(h/2)f(Y)$ and $y_{n+1}=y_n+hf(Y)$. On $y^{\prime }=-\lambda y$ with $\lambda >0$, the stage is $Y=y_n/(1+h\lambda /2)$ and the update factor is $(1-h\lambda /2)/(1+h\lambda /2)$. This linear example needs only one division, whereas a nonlinear $f$ generally requires an iterative solve. The stage solvability of an implicit Runge-Kutta method gives a sufficient small-step contraction condition.