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We only consider $\Pi_{c}^{+}$. The procedure to approximate $\Psi_{c}^{+}$ is analogous.

According to Theorem \[th:main\] and Section \[subsubsec:specialcase\], to approximate $\Pi_{c}^{+}$, we need to compute $
\widehat{\Pi}_{c}^{+}(i,j;q_1,\ldots,q_n)
$ for any $q_{1},\ldots,q_{n}>0$, and then to use the Gaver-Stehfest algorithm. Note that $\widehat{\Pi}_{c}^{+}(i,j;q_1,\ldots,q_n)$ can be computed by solving directly using the diagonalization method of [@rogers_shi_1994]. However, because of the special structure of ${\widetilde}{\mG}$, we can simplify the calculation by working on matrices of smaller dimensions. Towards this end we observe that matrices in can be written the block form as follows, $$\begin{aligned}
  {\widetilde}{\mG}=
  \kbordermatrix{
    & (0,\mathbf{E}^{+}) & (1,\mathbf{E}^{+}) & \cdots &  (n,\mathbf{E}^{+}) & (0,\mathbf{E}^{-})  & (1,\mathbf{E}^{-}) & \cdots & (n,\mathbf{E}^{-}) \\
    (0,\mathbf{E}^{+}) & \mA_{1}-q_{1}\mI^{+} & q_{1}\mI^{+} & \cdots & 0 & \mB_{1} & 0 & \cdots & 0\\
    (1,\mathbf{E}^{+}) & 0 & \mA_{2}-q_{2}\mI^{+} & \cdots & 0 & 0 & \mB_{2} & \cdots & 0\\
    \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots \\
    (n-1,\mathbf{E}^{+}) & 0 & 0 & \cdots  & q_{n}\mI^{+} & 0 & 0 & \cdots & 0\\
    (n,\mathbf{E}^{+}) & 0 & 0 & \cdots &  \mA_{n+1} & 0 & 0 & \cdots & \mB_{n+1}\\
    (0,\mathbf{E}^{-}) & \mC_{1} & 