函数逼近——两点三次Hermite(埃尔米特)、分段Hermite插值法 | 北太天元 or Matlab

一、两点三次Hermite插值

已知: 两个插值节点机器对应的导数值 $(x_0,y_0),(x_1,y_1),y_0^{\prime },y_1^{\prime }.$

$$H(x)={\alpha }_0(x)y_0+{\alpha }_1(x)y_1+{\beta }_0(x)y_0^{\prime }+{\beta }_1(x)y_1^{\prime }$$

其中
{ α0(x)=(1+2x−x0x1−x0)(x−x1x0−x1)2α1(x)=(1+2x−x1x0−x1)(x−x0x1−x0)2{ β0(x)=(x−x0)(x−x1x0−x1)2β1(x)=(x−x1)(x−x0x1−x0)2\begin{cases}\alpha_0(x)=\left(1+2\dfrac{x-x_0}{x_1-x_0}\right)\left(\dfrac{x-x_1}{x_0-x_1}\right)^2\\\alpha_1(x)=\left(1+2\dfrac{x-x_1}{x_0-x_1}\right)\left(\dfrac{x-x_0}{x_1-x_0}\right)^2\end{cases}\begin{cases}\beta_0(x)=(x-x_0)\left(\dfrac{x-x_1}{x_0-x_1}\right)^2\\\beta_1(x)=(x-x_1)\left(\dfrac{x-x_0}{x_1-x_0}\right)^2\end{cases}⎩ ⎨ ⎧​α<