\chapter{The integration weights for the SSDD case}
\label{app:ssdd}  
In this appendix we will write up the equations for the tetrahedral
integration weights of the SSDD case (see \article\ref{sec:comp-phi}).  These
equations can be obtained very simply from the equations of Lambin and
Vigneron in appendix \ref{app:atm}.  Namely we have:
\begin{eqnarray}
w_i(z) &\equiv& 6\int_{0}^{1}dc\int_{0}^{1-c}db\int_{0}^{1-b-c}da
\frac{(1-a-b-c)\delta_{i1}+a\delta_{i2}+b\delta_{i3}+c\delta_{i4}}
{\left(z_1-aE_{21}-bE_{31}-cE_{41}\right)^2} 
\nonumber \\ &=&
 -\frac{d}{dz}\int_{0}^{1}dc\int_{0}^{1-c}db\int_{0}^{1-b-c}da
\frac{(1-a-b-c)\delta_{i1}+a\delta_{i2}+b\delta_{i3}+c\delta_{i4}}
{z_1-aE_{21}-bE_{31}-cE_{41}}
 \nonumber \\ &=&
-\frac{d}{dz}r_i(z).
\end{eqnarray}
Thus the only thing we need to do is to differentiate the equation from
appendix \ref{app:atm} to obtain our integration weights.  We will write the
results in terms of the Lambin-Vigneron function from appendix \ref{app:atm},
its derivative $\dlv$ and the associated function $\lvd$ which we define as:
\begin{equation}
  \label{eq:lv2function}
  \lvd(x) = -\lv(x)-x\dlv(x).
\end{equation}
All of these functions are quite simple and have nice asymptotic limits.

The weight factors are now as follows:
\subsection*{Case 1: All corners different}
\begin{eqnarray}
w_1 &=& 
 \frac{1}{E_{32}E_{42}}\lvd\left(\frac{z_2}{E_{21}}\right)
+\frac{1}{E_{23}E_{43}}\lvd\left(\frac{z_3}{E_{31}}\right)
+\frac{1}{E_{24}E_{34}}\lvd\left(\frac{z_4}{E_{41}}\right)
\nonumber \\
w_2 &=& 
 \frac{1}{E_{31}E_{41}}\lvd\left(\frac{z_1}{E_{12}}\right)
+\frac{1}{E_{13}E_{43}}\lvd\left(\frac{z_3}{E_{32}}\right)
+\frac{1}{E_{14}E_{34}}\lvd\left(\frac{z_4}{E_{42}}\right)
\nonumber \\
w_3 &=& 
 \frac{1}{E_{21}E_{41}}\lvd\left(\frac{z_1}{E_{13}}\right)
+\frac{1}{E_{12}E_{42}}\lvd\left(\frac{z_2}{E_{23}}\right)
+\frac{1}{E_{14}E_{24}}\lvd\left(\frac{z_4}{E_{43}}\right)
\nonumber \\
w_4 &=& 
 \frac{1}{E_{21}E_{31}}\lvd\left(\frac{z_1}{E_{14}}\right)
+\frac{1}{E_{12}E_{32}}\lvd\left(\frac{z_2}{E_{24}}\right)
+\frac{1}{E_{13}E_{23}}\lvd\left(\frac{z_3}{E_{34}}\right)
\end{eqnarray}

\subsection*{Case 2: Two corners equal}
\begin{eqnarray}
w_2 &=&    
 \frac{-1}{2E_{32}E_{42}}
+\frac{1}{E_{23}E_{43}}\lvd\left(\frac{z_3}{E_{32}}\right)
+\frac{1}{E_{24}E_{34}}\lvd\left(\frac{z_4}{E_{42}}\right)
\nonumber \\
w_3 &=& 
 \frac{-1}{E_{23}E_{24}}
 +\frac{1}{E_{24}^2}\lvd\left(\frac{z_4}{E_{43}}\right)
 \nonumber \\ &+&
 \left[\frac{2}{E_{23}E_{24}}+\frac{1}{E_{24}^2}\right]
 \lv\left(\frac{z_2}{E_{23}}\right)
 +\left[\frac{2z_3}{E_{23}^2E_{24}}+\frac{z_4}{E_{23}E_{24}^2}\right]
 \dlv\left(\frac{z_2}{E_{23}}\right)
\nonumber \\
w_4 &=& 
 \frac{-1}{E_{23}E_{24}}
 +\frac{1}{E_{23}^2}\lvd\left(\frac{z_3}{E_{34}}\right)
 \nonumber \\ &+&
 \left[\frac{2}{E_{23}E_{24}}+\frac{1}{E_{23}^2}\right]
 \lv\left(\frac{z_2}{E_{24}}\right)
 +\left[\frac{2z_4}{E_{23}E_{24}^2}+\frac{z_3}{E_{23}^2E_{24}}\right]
 \dlv\left(\frac{z_2}{E_{24}}\right)
\end{eqnarray}

\subsection*{Case 3: Two pairs of equal corners}
\begin{eqnarray}
w_2 &=& 
 \frac{z_3}{E_{32}^2}+\frac{1}{2}\frac{z_2}{E_{32}^2}
 -3\frac{z_2}{E_{32}^2}\lv\left(\frac{z_3}{E_{32}}\right)
\nonumber \\
w_3 &=& 
 \frac{-3}{2E_{23}^2}
 +\frac{3}{E_{23}^2}\lv\left(\frac{z_2}{E_{23}}\right)
 +3\frac{z_3}{E_{23}^3}\dlv\left(\frac{z_2}{E_{23}}\right)
\end{eqnarray}

\subsection*{Case 4: Three equal corners}
\begin{eqnarray}
w_3 &=& 
 \frac{1}{2E_{34}^2}+
 \frac{1}{E_{34}^2}\lvd\left(\frac{z_4}{E_{43}}\right)
\nonumber \\
w_4 &=& 
 \frac{-3}{2E_{43}^2}
 +\frac{3}{E_{43}^2}\lv\left(\frac{z_4}{E_{43}}\right)
 +3\frac{z_3}{E_{43}^3}\dlv\left(\frac{z_4}{E_{43}}\right)
\end{eqnarray}

\subsection*{Case 5: All corners equal}
\begin{equation}
w_4 = \frac{-1}{4}\frac{1}{z_4^2}
\end{equation}








