Last Edit: 1/13/25

Polynomial Interpolation

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• 如果想要找一个Polynomial，其穿过两个Point，很简单的做法就是建立$P(x)=ax+b$的多项式
• 但是如果问题变成了三个点甚至更多，问题的复杂度就会上升很多，于是需要考虑另一种做法
• 对于点$(-1,-1),(1,3),(2,-2)$我们需要找到一个Polynomial穿过这三个点，可以先写出三个Quadratic Polynomial $$L_1(x) := \begin{cases} 1, & \text{if } x = -1 \ 0, & \text{if } x = 1 \ 0, & \text{if } x = 2, \end{cases}

L_2(x) := \begin{cases} 0, & \text{if } x = -1 \ 1, & \text{if } x = 1 \ 0, & \text{if } x = 2, \end{cases}

L_3(x) := \begin{cases} 0, & \text{if } x = -1 \ 0, & \text{if } x = 1 \ 1, & \text{if } x = 2. \end{cases} $$

• 之后再令$P_2(x)=−1L_1(x) + 3L_2(x) − 2L_3(x)$
• 分别查看三个点的值可以发现 $$P_2(-1) = -1L_1(-1) + 3L_2(-1) - 2L_3(-1) \ = -1(1) + 3(0) - 2(0) \ = -1$$ $$P_2(1) = -1L_1(1) + 3L_2(1) - 2L_3(1) \ = -1(0) + 3(1) - 2(0) \ = 3$$ $$ P_2(2) = -1L_1(2) + 3L_2(2) - 2L_3(2) \ = -1(0) + 3(0) - 2(1) \ = -2$$
• 可以发现这样构建的Polynomial是符合要求的，那么问题就变成了如何构建这三个Quadratic Polynomial
• 已知$L_1(x)$有两个Roots，分别为$x=1$和$x=2$，并且因为$L_1(-1)=1$所以有 $$L_1(-1)=C(-1-1)(-1-2)=1\Rightarrow C=\frac{1}{(-1-1)(-1-2)}$$
• 同理对于$L_2$和$L_3$来说有 $$P_2(x) = -1 \frac{(x - 1)(x - 2)}{(-1 - 1)(-1 - 2)} + 3 \frac{(x + 1)(x - 2)}{(1 + 1)(1 - 2)} - 2 \frac{(x + 1)(x - 1)}{(2 + 1)(2 - 1)} $$
• 这就是Lagrange Interpolation 拉格朗日插值法

ex.

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• 将Lagrange Interpolation推广到四个点上，有 $$L_1(x) := \begin{cases} 1, & \text{if } x = x_1 \ 0, & \text{if } x = x_2 \ 0, & \text{if } x = x_3 \ 0, & \text{if } x = x_4, \end{cases}

L_2(x) := \begin{cases} 0, & \text{if } x = x_1 \ 1, & \text{if } x = x_2 \ 0, & \text{if } x = x_3 \ 0, & \text{if } x = x_4, \end{cases}

L_3(x) := \begin{cases} 0, & \text{if } x = x_1 \ 0, & \text{if } x = x_2 \ 1, & \text{if } x = x_3 \ 0, & \text{if } x = x_4, \end{cases}

L_4(x) := \begin{cases} 0, & \text{if } x = x_1 \ 0, & \text{if } x = x_2 \ 0, & \text{if } x = x_3 \ 1, & \text{if } x = x_4. \end{cases}$$

• 其最终的Polynomial将会长这样 $$P_3(x) = y_1 \frac{(x-x_2)(x-x_3)(x-x_4)}{(x_1-x_2)(x_1-x_3)(x_1-x_4)} + y_2 \frac{(x-x_1)(x-x_3)(x-x_4)}{(x_2-x_1)(x_2-x_3)(x_2-x_4)} + y_3 \frac{(x-x_1)(x-x_2)(x-x_4)}{(x_3-x_1)(x_3-x_2)(x_3-x_4)} + y_4 \frac{(x-x_1)(x-x_2)(x-x_3)}{(x_4-x_1)(x_4-x_2)(x_4-x_3)}$$