用GSL解决大型ODE系统

Question

I have general question that I cannot wrap my head around and I am not certain if this is a right place to ask. At the moment I am clueless and any literature and advice is welcome. What I personally hate about GSL is that their examples are completely unhelpful, while the explanation on different solvers is too general and lacks an example. The other slightly annoying bit is that every GSL function needs a pointer to its parameters, which forces a lot of users to make messy structures, however the neatest approach (in my opinion) is to make a class and simply send this pointer to it and use reinterpet_cast(params) - I learned this after hearing it from a collegue, it is definitely not in any of GSL examples, and it doesn't really naturally comes to mind to the beginners.

In any case, my question is how do you solve a system of first order ODE that is given in matrix form? I have a system of 900 first order ODEs given in matrix form as

y'=M*y

where M is 900x900 matrix of the system. System consists of complex numbers and I am using armadillo library for matrix M. I assume complex numbers in my problem may be tricky as well, however I assume GSL has something or this can be done by separating real and imaginary part.

My entire code is in C++, I am solving this in steady state normally (there is a normalization trick that simply inverts M and avoids det(M)=0), however I have no idea how to do it with GSL. The help page there is not helpful at all (since it is for 2nd order ODE). I very roughly know how ODE solvers work in Matlab, and I see the logic behind GSL version, but my issue is that I need to pass the entire system as a matrix, and I do not know how to obtain Jacobian (if I even need it). All the examples I found so far, take very small systems where it is quite straightforward to write down the system. Can someone please explain how does GSL's ode funciton generalize for matrix approach, I assume it may be as simple as putting:

int odefunc (double t, const double y[], double f[], void *params)
{
    p = reinterpert_cast<Whatever_class *>(params)
    cx_mat M = p->get_matrix_M();
    f=M*y; %Probably does not work since M is armadillo matrix and f needs 
            to be accessed as f[i]
    return GSL_SUCCESS;
}

Fortunate thing in all this is that time t does not show in the equations, so the solution to this probably can be done much simpler through eigenvalues and eigenvectors, but I really wish to know how to do it with GSL.

Further to my question, what I must actually do is solve a coupled system as:

y'=M(x)*y
x'=G(y,t);

Matrix of the first system M depends on another variable x that also depends on time through, fortunately, just one differential equation. The issue is that I really want to avoid outcopuling the system. G(y,t) is very complicated and requires to reshape vector y into a matrix, and seek its trace with another (constant) matrix, so I can calculate G(y,t) only when I know what y is. Similarly M(x) is generated by very complicated matrix transformations (Khatri-Rao and Kronecker tensor products). It is clear to me that solution vector should be in form [y; x], but I do not understand how to fill up the system in a way GSL needs.

Is there a way to do this with GSL or do I need to develop a custom numerical method for this?