Lagrange Interpolation

Algebra Level 3

Let $\displaystyle\ell _i(x)=\prod_{\begin{matrix}0\leq j\leq n\\j\neq i\end{matrix}}{\frac {x-x_{j}}{x_{i}-x_{j}}}$ , where no two $x_j$ are the same.

Simplify $\displaystyle\sum_{i=0}^n\ell_i(0)x_i^{n+1}$ .

Resource: My friend Carwaniwer Qee .

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\prod_{i=0}^nx_i

(-1)^n\prod_{i=0}^nx_i

1 solution

Carwaniwer Qee
Jan 13, 2019

According to the Lagrangian interpolation formula： $f(x) = \sum\limits_{i=0}^nl_i(x)×f(x_i)+\frac{f^{(n+1)} (ξ)}{(n+1)!}ω_{n+1}(x),$ when $f(x)=x^{n+1},$ we have $f^{n+1}(ξ)=(n+1)!,$ $R_n(x)=ω_{n+1}(x),$ so, $\sum\limits_{i=0}^nl_i(x)x_i^{n+1}+ω_{n+1}(x)=x^{n+1},$ let $x=0$ , we have $\sum\limits_{i=0}^nl_i(0)x_i^{n+1}+ω_{n+1}(0)=0,$ so we have $\sum\limits_{i=0}^nl_i(0)x_i^{n+1}=-ω_{n+1}(0)=(-1)^nx_0x_1·····x_n$

What's the meaning of $\omega_{n+1}(x)$ in Lagrange interpolation formula? I forgot. :(

Brian Lie - 2 years, 4 months ago

where $\omega_{n+1}=(x-x_0)(x-x_1).....(x-x_n)$

Carwaniwer Qee - 2 years, 4 months ago

How to proof Lagrange interpolation formula?

Brian Lie - 2 years, 4 months ago

Lagrange Interpolation

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