Beduino
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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.
[itex]\left\{\begin{matrix}<br /> \overrightarrow{K_{1}}=-\delta t*\mu\frac{\overrightarrow{r}+<br /> a_{11}\overrightarrow{K_{1}}+<br /> a_{12}\overrightarrow{K_{2}}+<br /> a_{13}\overrightarrow{K_{3}}}{\left | \overrightarrow{r}+<br /> a_{11}\overrightarrow{K_{1}}+<br /> a_{12}\overrightarrow{K_{2}}+<br /> a_{13}\overrightarrow{K_{3}} \right |^{3}}\\ <br /> \overrightarrow{K_{2}}=-\delta t*\mu\frac{\overrightarrow{r}+<br /> a_{21}\overrightarrow{K_{1}}+<br /> a_{22}\overrightarrow{K_{2}}+<br /> a_{23}\overrightarrow{K_{3}}}{\left | \overrightarrow{r}+<br /> a_{21}\overrightarrow{K_{1}}+<br /> a_{22}\overrightarrow{K_{2}}+<br /> a_{23}\overrightarrow{K_{3}} \right |^{3}}\\ <br /> \overrightarrow{K_{3}}=-\delta t*\mu\frac{\overrightarrow{r}+<br /> a_{31}\overrightarrow{K_{1}}+<br /> a_{32}\overrightarrow{K_{2}}+<br /> a_{33}\overrightarrow{K_{3}}}{\left | \overrightarrow{r}+<br /> a_{31}\overrightarrow{K_{1}}+<br /> a_{32}\overrightarrow{K_{2}}+<br /> a_{33}\overrightarrow{K_{3}} \right |^{3}}<br /> \end{matrix}\right.[/itex]
What's the most appropriated way to solve this system numerically, all parameters are given, except each K vector.
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}<br /> \overrightarrow{K_{1}}=-\delta t*\mu\frac{\overrightarrow{r}+<br /> a_{11}\overrightarrow{K_{1}}+<br /> a_{12}\overrightarrow{K_{2}}+<br /> a_{13}\overrightarrow{K_{3}}}{\left | \overrightarrow{r}+<br /> a_{11}\overrightarrow{K_{1}}+<br /> a_{12}\overrightarrow{K_{2}}+<br /> a_{13}\overrightarrow{K_{3}} \right |^{3}}\\ <br /> \overrightarrow{K_{2}}=-\delta t*\mu\frac{\overrightarrow{r}+<br /> a_{21}\overrightarrow{K_{1}}+<br /> a_{22}\overrightarrow{K_{2}}+<br /> a_{23}\overrightarrow{K_{3}}}{\left | \overrightarrow{r}+<br /> a_{21}\overrightarrow{K_{1}}+<br /> a_{22}\overrightarrow{K_{2}}+<br /> a_{23}\overrightarrow{K_{3}} \right |^{3}}\\ <br /> \overrightarrow{K_{3}}=-\delta t*\mu\frac{\overrightarrow{r}+<br /> a_{31}\overrightarrow{K_{1}}+<br /> a_{32}\overrightarrow{K_{2}}+<br /> a_{33}\overrightarrow{K_{3}}}{\left | \overrightarrow{r}+<br /> a_{31}\overrightarrow{K_{1}}+<br /> a_{32}\overrightarrow{K_{2}}+<br /> a_{33}\overrightarrow{K_{3}} \right |^{3}}<br /> \end{matrix}\right.[/itex]
What's the most appropriated way to solve this system numerically, all parameters are given, except each K vector.