Interpolation

In what follows are various notes dealing with interpolation.

Lagrange Interpolation

\(\begin{align} \ell_j(x) &= \prod_{m=0, m \ne j}^k\dfrac{x - x_m}{x_j - x_m}\\ &= \dfrac{x - x_0}{x_j - x_0} \ldots \dfrac{x - x_{j-1}}{x_j - x_{j-1}}\dfrac{x - x_{j+1}}{x_j - x_{j+1}} \ldots \dfrac{x - x_k}{x_j - x_k}\\ \end{align}\)

\(L(x) = \sum_{j=0}^ky_j\ell_j(x)\)