I posted a short program which implements the algorithm presented here.
As an example it calculates the following cyclic tridiagonal system \[\begin{bmatrix}
-2 & 1 & 0 & 0 & 2 \\
1 & -2 & 1 & 0 & 0 \\
0 & 1 & -2 & 1 & 0 \\
0 & 0 & 1 & -2 & 1 \\
-1 & 0 & 0 & 1 & -2 \\
\end{bmatrix} x = \begin{bmatrix}
1 \\
2 \\
3 \\
4 \\
5 \\
\end{bmatrix}.\]
As an example it calculates the following cyclic tridiagonal system \[\begin{bmatrix}
-2 & 1 & 0 & 0 & 2 \\
1 & -2 & 1 & 0 & 0 \\
0 & 1 & -2 & 1 & 0 \\
0 & 0 & 1 & -2 & 1 \\
-1 & 0 & 0 & 1 & -2 \\
\end{bmatrix} x = \begin{bmatrix}
1 \\
2 \\
3 \\
4 \\
5 \\
\end{bmatrix}.\]
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