Lagrange interpolation

The Lagrange interpolation defines a polynomial at each point. The Polynomial interpolating the entire data set is the sum of the product of value of the function $y_i$ at that point and individual polynomial calculated at that point.

• Lagrangian Basis polynomials, $L_i(x)$:

$$L_n^N\left(x_i\right)=\prod _{\begin{array}{c}j=0\\ j\ne n\end{array}}^N\frac{x_i-x_j}{x_n-x_j}=\{\begin{array}{ll}0,& \text{ for }i\ne n\\ 1,& \text{ for }i=n\end{array}$$

• Lagrange interpolation Polynomial, $P(x)$:

$$p(x)=\sum _{n=0}^N{y}_n{L}_n^N(x)$$

Note:

• $n$ → is the Order of the Polynomial.
• $(n+1)$ → Interpolation points.