What techniques are useful for solving this sparse symmetric matrix?

Question

I have the following matrix to solve.

$$\begin{matrix} G_1+G_2&-G_2&0&0&0&0\\ -G_2 & G_2+G_3+G_4&-G_4 &0&0&0\\ 0&-G_4&G_4+G_5+G_6&-G_6&0&0\\ 0&0&-G_6&G_6+G_7+G_8 &-G_8&0\\ 0&0&0&-G_8&G_8+G_9+G_{10}&-G_{10}\\ 0&0&0&0&-G_{10}&G_{10}+G_{11}\end{matrix}$$

I know I could use Gaussian Elimination or LU Decomposition, but my Numerical Methods professor mentioned that there are more useful techniques in the realm of sparse matrices that could be used, but he didn't specify because he was too busy. I've tried tracking down more information on the topic, but I am getting lost in the a lot of complex articles regarding sparse matrices.

I would like to know some which techniques are more useful than LU decomposition for solving this sparse, symmetric, matrix.

Answer 1

You want the Tridiagonal matrix algorithm, also known as the Thomas algorithm.