# Implicit Runge–Kutta in R^3 space

## Main Question or Discussion Point

I'm trying to solve a implicit runge kutta algorithm numerically in ℝ3 space as a integrator for orbital simulation.

http://en.wikipedia.org/wiki/Runge–Kutta_methods#Implicit_Runge.E2.80.93Kutta_methods

More specifically a 6th order Gauss–Legendre method
http://en.wikipedia.org/wiki/Gauss–Legendre_method

I have worked out the three K parameters needed in the method, which forms a system of non linear vector equations below.

$\left\{\begin{matrix} \overrightarrow{K_{1}}=-\delta t*\mu\frac{\overrightarrow{r}+ a_{11}\overrightarrow{K_{1}}+ a_{12}\overrightarrow{K_{2}}+ a_{13}\overrightarrow{K_{3}}}{\left | \overrightarrow{r}+ a_{11}\overrightarrow{K_{1}}+ a_{12}\overrightarrow{K_{2}}+ a_{13}\overrightarrow{K_{3}} \right |^{3}}\\ \overrightarrow{K_{2}}=-\delta t*\mu\frac{\overrightarrow{r}+ a_{21}\overrightarrow{K_{1}}+ a_{22}\overrightarrow{K_{2}}+ a_{23}\overrightarrow{K_{3}}}{\left | \overrightarrow{r}+ a_{21}\overrightarrow{K_{1}}+ a_{22}\overrightarrow{K_{2}}+ a_{23}\overrightarrow{K_{3}} \right |^{3}}\\ \overrightarrow{K_{3}}=-\delta t*\mu\frac{\overrightarrow{r}+ a_{31}\overrightarrow{K_{1}}+ a_{32}\overrightarrow{K_{2}}+ a_{33}\overrightarrow{K_{3}}}{\left | \overrightarrow{r}+ a_{31}\overrightarrow{K_{1}}+ a_{32}\overrightarrow{K_{2}}+ a_{33}\overrightarrow{K_{3}} \right |^{3}} \end{matrix}\right.$

What's the most appropriated way to solve this system numerically, all parameters are given, except each K vector.