function pointer with Eigen - c++

Question

I am pretty competent with Python, but I'm pretty new to C++ and things like pointers. I try to write some codes for solving ODE with the Eigen package for linear algebra (I will need to deal with lots of matrices later, so I plan to start with it). I have the following code for RK4 and they work:
#include "../eigen-eigen-b3f3d4950030/Eigen/Dense"
using namespace Eigen;
VectorXd Func(const VectorXd& a)
{ // equations for solving simple harmonic oscillator
Vector2d ans;
ans(0) = a(1); // dy/dt
ans(1) = -a(0); // d2y/dt2
return ans;
}
MatrixXd RK4(VectorXd Func(const VectorXd& y), const Ref<const VectorXd>& y0, double h, int step_num)
{
MatrixXd y(step_num, y0.rows());
y.row(0) = y0;
for (int i=1; i<step_num; i++){
VectorXd y_old = y.row(i-1).transpose();
VectorXd k1 = h*Func(y_old);
VectorXd k2 = h*Func(y_old+k1/2);
VectorXd k3 = h*Func(y_old+k2/2);
VectorXd k4 = h*Func(y_old+k3);
VectorXd dy = (k1 + 2*k2 + 2*k3 + k4)/6;
y.row(i) = y.row(i-1) + dy.transpose();
}
return y;
}
int main()
{
Vector2d v1;
v1(0) = 1.4; v1(1) = -0.1;
double h = 0.1;
int step_num = 50;
MatrixXd sol = RK4(Func,v1,h,step_num);
return 0;
}
I have the following questions:
What's the meaning of & in the function argument? Pass by reference? I just copied the code from the official documentation, but I'm not too sure if I understand every bit in the function arguments of RK4 such as VectorXd Func(const VectorXd& y). Are there alternative ways of accepting Eigen::MatrixXd and functions which accept Eigen::MatrixXd as function arguments?
From what I understand, we cannot return a whole 2D array from a function, and what we are returning is just the first element of the array (correct me if I'm wrong). What about the Eigen::MatrixX? What are we actually passing/returning? The first element of the matrix, or a completely new object defined by the Eigen library?
I'm not sure if these codes are written efficiently. Are there anything I can do to optimize this part? (Just wondering if I have done anything that may significantly slow down the speed).
Thanks

Answer 1

Yes, & is pass-by-reference; The latter one is syntax for passing a function, that takes a vector by reference and returns a vector. An Eigen::Matrix should always be passed by reference. There are tons of ways to pass one function to another, the most idiomatic ones in C++ are probably template arguments and std::function.
You can't have multiple return arguments, but you can return a pair or a tuple or a Matrix object. RK4 returns a whole matrix.
The code is fairly efficient. If it was really performance-critical there might be a few things that could be optimized, but I would not worry for now.
The biggest point is that RK4 is very general and works with dynamically sized types, which are a lot more expensive than their statically sized counter parts (VectorXf vs Vector2d). But this would require you to create a specialized version for all dimensions you are interested in or to get the compiler to do it for you by using templates.
Generally: Read a good book to get you started.