How to Handle Fourth Order Implicit Runge-Kutta Methods?

[Deadstar]

Hey folks I'm experimenting with symplectic integrators and I'm trying to figure out how to deal with fourth order implicit Runge-Kutta methods as shown here (symplectic version second link).

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

where the formula is given as...

$$y_{n+1} = y_n + h \sum_{i=1}^s b_i k_i$$

$$k_i = f\left(t_n + c_i h, y_n + h \sum_{j = 1}^s a_{ij} k_j\right)$$

But I have no idea how to use this for s>1 as each $$k_i$$ is dependent on $$k_j$$. Is it a case of substituting the expression into our system f then solving for k? Seem like it could get very messy for any mildly complicated system. Perhaps we just do a numerical approximation?

The exact formula I'm using (which includes coefficient neccesary for it to be symplectic) is found at the top of page 6 here.

http://math.iisc.ernet.in/~rangaraj/docs/pramana_symplectic.pdf

But the problem with calculating k (which is called Z) here remains...

Any help appreciated.

 

[Deadstar]

This is now solved btw. Too late to edit OP.