Mathematics-Online lexicon: Interpolation polynomial in Lagrange Form

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	Mathematics-Online lexicon:
	Interpolation polynomial in Lagrange Form

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

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

The Lagrange form of

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

$\includegraphics[width=.45\linewidth]{interpolation_Bild}$

The polynomials are referred to as Lagrange polynomials. They are equal to at and vanish at all other points :

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

with $\delta$ the Kronecker symbol.

Example:

automatically generated 6/22/2016