构造插值函数$f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_0$经过$\{(x_i,y_i)\}_{i=0}^{n}$，即方程组
$$ \left\{\begin{matrix} a_nx_0^n+a_{n-1}x_0^{n-1}+\cdots+a_0=y_0\\ a_nx_1^n+a_{n-1}x_1^{n-1}+\cdots+a_0=y_1\\ \cdots\\a_nx_n^n+a_{n-1}x_n^{n-1}+\cdots+a_0=y_n\end{matrix}\right. $$
有解。取$a=[a_0,a_1,\cdots,a_n]^T$, $y=[y_0,y_1,\cdots,y_n]^T$,方程组表示为$V(x_0,x_1,\cdots,x_n)a=y$
$$ V=V(x_0,x_1,\cdots,x_m)=\begin{bmatrix}1&x_0&x_0^2&\ldots&x_0^n\\1&x_1&x_1^2&\ldots&x_1^n\\1&x_2&x_2^2&\ldots&x_2^n\\\vdots&\vdots&\vdots&\ddots&\vdots\\1&x_m&x_m^2&\ldots&x_m^n\end{bmatrix} $$
当每个元素各不相同时，范德蒙行列式不等于零，
Lagrange Interpolating Polynomial| 多项式插值