Runge-Kutta Methods

These algorithms involve multiple evaluations of the right hand side function f at points intermediate to x_i and x_i+1.

   $$\begin{eqnarray} y_{i+1} &=& y_i + \sum_{j=1}^{\nu} w_j k_j \nonumber \\ k_j &=&... ..._i + \sum_{l=1}^{\left[\j-1, \nu \right]} \alpha_{jl} k_l \right)\end{eqnarray}$$

l=1 to j-1 corresponds to an explicit scheme and l=1 to $\nu$ corresponds to am implicit scheme. The IMSL subroutine IVPRK can be used to solve IVP for ODEs using fifth and sixth order Runge Kutta method.