I'm trying to solve a implicit runge kutta algorithm numerically in ℝ(adsbygoogle = window.adsbygoogle || []).push({}); ^{3}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.

[itex]\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.[/itex]

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

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# Implicit Runge–Kutta in R^3 space

Can you offer guidance or do you also need help?

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