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Interpolation polynomial in Lagrange Form

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Values $ f_k$ at $ n+1$ distinct points $ x_0,\ldots,x_n$ can be interpolated with a unique polynomial $ p$ of degree $ \le n$:

$\displaystyle p(x_k) = f_k,\quad k=0,\ldots,n

The Lagrange form of $ p$ is

$\displaystyle p(x) = \sum_{k=0}^n f_k q_k(x),\quad
q_k(x) = \prod_{j\ne k} \frac{x-x_j}{x_k-x_j}


The polynomials $ q_k$ are referred to as Lagrange polynomials. They are equal to $ 1$ at $ x_k$ and vanish at all other points $ x_j$:

$\displaystyle q_{k}(x_{j})=\delta_{k,j}

with $ \delta$ the Kronecker symbol.


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  automatically generated 6/22/2016