Gaussian elimination $$ \begin{bmatrix} u_1 & h_1 &0 &\dots & & & & \\ h_1 & u_2 & h_2 &0 &\dots & & & \\ 0 & h_2 & u_3 & h_3 &0 &\dots & & \\ \dots& & \dots &\dots &\dots &\dots &\dots & \\ &\dots & & &0 &h_{n-3} &u_{n-2} &h_{n-2} \\ & && & &0 &h_{n-2} &u_{n-1} \end{bmatrix} \begin{bmatrix} f_1 \\ f_2 \\ f_3\\ \dots \\ f_{n-2} \\ f_{n-1} \end{bmatrix} = \begin{bmatrix} v_1 \\ v_2 \\ v_3\\ \dots \\ v_{n-2}\\ v_{n-1} \end{bmatrix}. $$ Note that this is a set of tridiagonal equations and can be solved through only \( O(n) \) operations.