Lagrange's interpolation formula

Named after Joseph Louis Lagrange (1736–1813), an Italian Enlightenment Era mathematician and astronomer.

Lagrange's interpolation formula (uncountable)

1. (mathematics) A formula which when given a set of n points ${\displaystyle (x_{i},y_{i})}$ , gives back the unique polynomial of degree (at most) n − 1 in one variable which describes a function passing through those points. The formula is a sum of products, like so: ${\displaystyle \sum _{i}^{n}y_{i}\prod _{j\neq i}{x-x_{j} \over x_{i}-x_{j}}}$ . When ${\displaystyle x=x_{i}}$ then all terms in the sum other than the i ^th contain a factor ${\displaystyle x-x_{i}}$ in the numerator, which becomes equal to zero, thus all terms in the sum other than the i ^th vanish, and the i ^th term has factors ${\displaystyle x_{i}-x_{j}}$ both in the numerator and denominator, which simplify to yield 1, thus the polynomial should return ${\displaystyle y_{i}}$ as the function of ${\displaystyle x_{i}}$ for any i in the set ${\displaystyle \{1,...,n\}}$ .