i}\phi_{\varepsilon}(\mu,\nu_{i})=\psi_{+}(\mu)+\psi_{-}(\mu),\qquad\mu\in[-1,1],\textrm{ }\nu_{i}\geq0\label{Eqn: HRFR_Discrete}$$

of the discretized boundary condition Eq. (\[Eqn: BC\]), where $\psi_{+}(\mu)$ is by definition the incident flux $\psi(\mu)$ for $\mu\in[0,1]$ and $0$ if $\mu\in[-1,0]$, while $$\psi_{-}(\mu)=\left\{ \begin{array}{ccl}
{\displaystyle \sum_{i=0}^{N}b_{i}^{-}\phi_{\varepsilon}(\mu,\nu_{i})} &  & \textrm{if }\mu\in[-1,0],\textrm{ }\nu_{i}\geq0\textrm{ }\\
0 &  & \textrm{if }\mu\in[0,1]\end{array}\right.$$ is the the emergent angular distribution out of the medium. Equation (\[Eqn: HRFR\_Discrete\]) corresponds to the full-range $\mu\in[-1,1],\textrm{ }\nu_{i}\geq0$ form $$b(\nu_{0})\phi(\mu,\nu_{0})+\int_{0}^{1}b(\nu)\phi(\mu,\nu)d\nu=\psi_{+}(\mu)+\left(b^{-}(\nu_{0})\phi(\mu,\nu_{0})+\int_{0}^{1}b^{-}(\nu)\phi(\mu,\nu)d\nu\right)\label{Eqn: HRFR}$$ of boundary condition (\[Eqn: BC\_HR\]) with the first and second terms on the right having the same interpretation as for Eq. (\[Eqn: HRFR\_Discrete\]). This full-range simulation merely states that the solution (\[Eqn: CaseSolution\_HR\]) of Eq. (\[Eqn: NeutronTransport\]) holds for all $\mu\in[-1,1]$, $x\geq0$, although it was obtained, unlike in the regular full-range case, from the given radiation $\psi(\mu)$ incident on the boundary at $x=0$ over only half the interval $\mu\in[0,1]$. To obtain the simulated full-range coefficients $\{ b_{i}\}$ and $\{ b_{i}^{-}\}$ of the half-range problem, we observe that there are effectively only half the number of coefficients as compared to a normal full-range problem because $\nu$ is now only over half the full interval. This allows us to generate two sets of equations from (\[Eqn: HRFR\]) by integrating with respect to $\mu\in[-1,1]$ with $\nu$ in the half intervals $[-1,0]$ and $[0,1]$ to obtain the two sets of coefficients $b^{-}$ and $b$ respectively. Accordingly we get from Eq. (\[Eqn: HRFR\_Discrete\]) with $\textrm{ }j=0,1,\cdots,N$ the sets of equations

$$\begin{array}{c}
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