\begin{Definition*}{one-step}{Explicit one-step method}
  \defindex{One-step method!explicit} An \textbf{explicit one-step
    method} \defindex{Explicit one-step method} is a method which,
  given $u_0$ at $t_0 = 0$ computes a sequence of approximations
  $y_1\,\dots,y_n$ to the solution of an IVP in the time steps
  $t_1,\dots,t_n$ using an update formula of the form\footnote{The adjective
  `explicit' is here in contrast to `implicit'
  one-step methods, where the increment function depends 
	on $y_{k}$ and equation~\eqref{eq:explicit:8} must be solved
        for $y_{k}$.}
  \begin{gather}
    \label{eq:explicit:8}
    y_{k} = y_{k-1} + h_{k} \verfahren_{h_k}(t_{k-1},y_{k-1}).
  \end{gather}
  The function $\verfahren()_{h_k}$ is called \define{increment
    function}.  We will often omit the index $h_k$ on
  $\verfahren_{h_k}()$ because it is clear that the method is always
  applied to time intervals.

  The method is called \textbf{one-step method}
  because the value $y_{k}$ explicitly depends only on the values
  $y_{k-1}$ and $f(t_{k-1}, y_{k-1})$, not on previous values.
\end{Definition*}

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