In case your system of equations leads to a tridiagonal matrix, it is clearly an overkill to employ Gaussian elimination or the standard LU decomposition.
Our algorithm starts with forward substitution with a loop over of the elements i and gives an update of the diagonal elements b_i given by the new diagonals \tilde{b}_i \tilde{b}_i= b_i - \frac{a_ic_{i-1}}{\tilde{b}_{i-1}}, and the new righthand side \tilde{f}_i given by \tilde{f}_i= f_i - \frac{a_i\tilde{f}_{i-1}}{\tilde{b}_{i-1}}. Recall that \tilde{b}_1=b_1 and \tilde{f}_1=f_1 always.