Lagrange Interpolation Formula

For any distinct reals $x_0, \ldots , x_n$ and any reals $y_0, \ldots, y_n$ , there exists a unique polynomial $\displaystyle P(x)$ of degree less than or equal to $\displaystyle n$ such that for all integers $0 \le i \le n$ , $P(x_i) = y_i$ , and this polynomial is

$P(x) = \sum_{i=0}^{n}y_i \frac{(x-x_0) \cdots (x-x_{i-1}) (x-x_{i+1}) \cdots (x-x_n)}{(x_i-x_0) \cdots (x_i-x_{i-1}) (x_i - x_{i+1}) \cdots (x_i - x_n)}$ .

The formula explains itself.

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Lagrange Interpolation Formula