对于爪形矩阵的LU分解

\[ \begin{pmatrix} b_1 & c_1 & 0 & \cdots & 0 \\ a_2 & b_2 & c_2 & \cdots & 0 \\ 0 & a_3 & b_3 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & c_{n-1} \\ 0 & 0 & 0 & a_n & b_n \\ \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \\ x_3 \\ \vdots \\ x_n \\ \end{pmatrix}= \begin{pmatrix} d_1 \\ d_2 \\ d_3 \\ \vdots \\ d_n \\ \end{pmatrix} \]

\[ \textbf{前向消去:} \alpha_1 = \frac{c_1}{b_1}, \quad \beta_1 = \frac{d_1}{b_1} \] \[ \alpha_i = \frac{c_i}{b_i - a_i \alpha_{i-1}}, \quad \beta_i = \frac{d_i - a_i \beta_{i-1}}{b_i - a_i \alpha_{i-1}}, \quad (i = 2, 3, \dots, n) \] \[ \textbf{回代:} x_n = \beta_n, \quad x_i = \beta_i - \alpha_i x_{i+1}, \quad (i = n-1, n-2, \dots, 1) \] 求解顺序为：\[ \alpha_{1}->\beta_{1}->\alpha_{2}->\beta_{2}->\alpha_{3}…… \]