数值分析-追赶法

对于爪形矩阵的LU分解

$(\begin{matrix} b_{1} & c_{1} & 0 & \dots & 0 \\ a_{2} & b_{2} & c_{2} & \dots & 0 \\ 0 & a_{3} & b_{3} & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & c_{n - 1} \\ 0 & 0 & 0 & a_{n} & b_{n} \end{matrix}) (\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \\ ⋮ \\ x_{n} \end{matrix}) = (\begin{matrix} d_{1} \\ d_{2} \\ d_{3} \\ ⋮ \\ d_{n} \end{matrix})$

$前向消去: α_{1} = \frac{c_{1}}{b_{1}}, β_{1} = \frac{d_{1}}{b_{1}}$ $α_{i} = \frac{c_{i}}{b_{i} - a_{i} α_{i - 1}}, β_{i} = \frac{d_{i} - a_{i} β_{i - 1}}{b_{i} - a_{i} α_{i - 1}}, (i = 2, 3, \dots, n)$ $回代: x_{n} = β_{n}, x_{i} = β_{i} - α_{i} x_{i + 1}, (i = n - 1, n - 2, \dots, 1)$ 求解顺序为： $α_{1} - > β_{1} - > α_{2} - > β_{2} - > α_{3} \dots \dots$

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