Algorithm 1. splitHMC for the coalescent model
Initialize ${\mathit{\theta }}^{(1)}$ at current $\mathit{\theta }=(f,\tau )$
Sample a new momentum value $p^{(1)}\sim \mathcal{N}(0,\mathbf{I})$
Calculate $H({\mathit{\theta }}^{(1)},p^{(1)})=U({\mathit{\theta }}^{(1)})+K(p^{(1)})$ according to (13)
for $\ell =1$ to L do  $$\begin{array}{l}{p}^{(\ell +\frac{1}{2})}=p^{\left(\ell \right)}+\epsilon /2\left[\begin{array}{c}{s}^{\left(\ell \right)}\\ \left(\right(D-1)/2+\alpha )-\beta \exp \left({\tau }^{\left(\ell \right)}\right)\end{array}\right]\\ p_D^{(\ell +\frac{1}{2})}=p_D^{\left(\ell \right)}-\epsilon /2f^{*\left(\ell \right)}{}^{\text{T}}{f}^{*\left(\ell \right)/2}\\ {\tau }^{(\ell +\frac{1}{2})}={\tau }^{\left(\ell \right)}+\epsilon /2p_D^{(\ell +\frac{1}{2})}\\ \left[\begin{array}{c}{f}^{*(\ell +1)}\\ p_{-D}^{*(\ell +\frac{1}{2})}\end{array}\right]\leftarrow \left[\begin{array}{cc}\cos \left(\sqrt{\mathbf{\Lambda }}{\text{e}}^{\frac{1}{2}{\tau }^{(\ell +\frac{1}{2})}}\epsilon \right)& \sin \left(\sqrt{\mathbf{\Lambda }}{\text{e}}^{\frac{1}{2}{\tau }^{(\ell +\frac{1}{2})}}\epsilon \right)\\ -\sin \left(\sqrt{\mathbf{\Lambda }}{\text{e}}^{\frac{1}{2}{\tau }^{(\ell +\frac{1}{2})}}\epsilon \right)& \cos \left(\sqrt{\mathbf{\Lambda }}{\text{e}}^{\frac{1}{2}{\tau }^{(\ell +\frac{1}{2})}}\epsilon \right)\end{array}\right]\left[\begin{array}{c}{f}^{*\left(\ell \right)}\\ p_{-D}^{*(\ell +\frac{1}{2})}\end{array}\right]\\ {\tau }^{(\ell +1)}={\tau }^{(\ell +\frac{1}{2})}+\epsilon /2p_D^{(\ell +\frac{1}{2})}\\ p_D^{(\ell +1)}=p_D^{(\ell +\frac{1}{2})}-\epsilon /2f^{*(\ell +1)}{}^{\text{T}}{f}^{*(\ell +1)/2}\\ p^{(\ell +1)}=p^{(\ell +\frac{1}{2})}+\epsilon /2\left[\begin{array}{c}{s}^{(\ell +1)}\\ \left(\right(D-1)/2+\alpha )-\beta \exp \left({\tau }^{(\ell +1)}\right)\end{array}\right]\end{array}$$
end for
Calculate $H({\mathit{\theta }}^{(+1)},p^{(L+1)})=U({\mathit{\theta }}^{(L+1)})+K(p^{(L+1)})$ according to (13)
Calculate the acceptance probability $\alpha =\min \{1,\exp [-H({\mathit{\theta }}^{(+1)},p^{(L+1)})+H({\mathit{\theta }}^{(1)},p^{(1)})]\}$
Accept or reject the proposal according to α for the next state $\mathit{\theta }\prime$