How is this matrix called (two diagonals)?

Question

I need to write an algorithm for solving this matrix but I wanted to first make a search online and that's why I need its name.

Answer 1

You can rearrange terms, to get the following system $$ \left [ \begin{array}{ccccccc} d_1 & a_{2n+1} \\ a_1 & d_{2n+1} \\ & & d_2 & a_{2n} \\ & & a_2 & d_{2n} \\ & & & & \ddots \\ & & & & & d_n & a_{n+2} \\ & & & & & a_n & d_{n+2} \\ & & & & & & & d_{n+1} \end{array}\right ] \left [ \begin{array}{c} x_1 \\ x_{2n+1} \\ x_2 \\ x_{2n} \\ \vdots \\ x_n \\ x_{n+2} \\ x_{n+1} \end{array}\right ] = \left [ \begin{array}{c} b_1 \\ b_{2n+1} \\ b_2 \\ b_{2n} \\ \vdots \\ b_n \\ b_{n+2} \\ b_{n+1} \end{array}\right ] $$ which is equivalent to solving $n$ $2\times 2$ linear systems and $1$ algebraic equation of $1^{st}$ order.