y^0 &=& (x_1^0, ..., \dot{x_1}^0, ...)^T \\
q_1^k &=& A y^k + b \\
q_2^k &=& A (y^k + \frac{1}{2} \Delta t q_1^k) + b \\
q_3^k &=& A (y^k + \frac{1}{2} \Delta t q_2^k) + b \\
q_4^k &=& A (y^k + \Delta t q_3^k) + b \\
y^{k+1} &=& y^k + \frac{1}{6} \Delta t (q_1^k + 2 q_2^k + 2 q_3^k + q_4^k)