Consider the interpolation problem: find the polynomial through a given set of points $(x_0,y_0),...,(x_n,y_n)$. Suppose we want the polynomial in newtonian form:

$$N(x)=\sum _{j=0}^k[y_0,\ldots ,y_j]\prod _{{i=0}}^{{j-1}}(x-x_{i})$$

Let's define the following matrix $A$.

$$ |A|= \begin{bmatrix} 1 & x_0 & \dots & x_0^n & y_0\\ 1 & x_1 & \dots & x_1^n & y_1\\ \vdots & \vdots & \ddots & \vdots & \vdots\\ 1 & x_n & \dots & x_n^n & y_n\\ 1& x & \dots & x^n & P_n\\ \end{bmatrix} =0 $$

I know how to derive the Lagrange polynomial from $A$, by developing the determinant to the last column.

(Question) How do I derive the Newton polynomial from this matrix $A$?


For $n=1$ we get $$ \begin{vmatrix}1&x_0&y_0\\ 1&x_1&y_1\\ 1&x&P(x)\end{vmatrix}=0 $$ Subtracing the 1st row from the other rows we arrive at $$ \begin{vmatrix}1&x_0&y_0\\ 0&x_1-x_0&y_1-y_0\\ 0&x-x_0&P(x)-y_0\end{vmatrix}=0 $$ Now use the Laplace expansion wrt. the 1st column.


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