Lagrange polynomial interpolation

Naive idea: Interpolate all points at once

• \(\forall t\in[t_0,t_{N-1}]\;,\;\;\displaystyle p(t)=\sum_{i=0}^{N-1} \alpha_i(t)\;p_i\)
  \(\forall i\in[0,N-1]\;\; p(t_i)=p_i\)

• Degree of polynomial : \(N-1\)

• - Known solution: Lagrange polynomial

• • \(\displaystyle p(t)=\sum_{i=0}^{N-1} \alpha_i(t)\;p_i\)
    \(\displaystyle \alpha_i(t) = \prod_{k=0, k\neq i}^{N-1} \frac{t-t_k}{t_i-t_k}\)

• - Explanation: By construction \(\alpha_i(t_i)=1\) and \(\alpha_i(t_k)=0\)