Polynomial interpolation

In numerical analysis, polynomial interpolation is the interpolation of a given data set by the polynomial of lowest possible degree that passes through the points in the dataset.

Given a set of n + 1 data points ${\displaystyle (x_{0},y_{0}),\ldots ,(x_{n},y_{n})}$, with no two ${\displaystyle x_{j}}$ the same, a polynomial function ${\displaystyle p(x)=a_{0}+a_{1}x+\cdots +a_{n}x^{n}}$ is said to interpolate the data if ${\displaystyle p(x_{j})=y_{j}}$ for each ${\displaystyle j\in \{0,1,\dotsc ,n\}}$.

There is always a unique such polynomial, commonly given by two explicit formulas, the Lagrange polynomials and Newton polynomials.