Lagrange's interpolation formula

English

Etymology

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

Noun

Lagrange's interpolation formula (uncountable)

1. (mathematics) A formula which when given a set of n points $(x_{i},y_{i})$, 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: $\sum _{i}^{n}y_{i}\prod _{j\neq i}{x-x_{j} \over x_{i}-x_{j}}$. When $x=x_{i}$ then all terms in the sum other than the i ^th contain a factor $x-x_{i}$ 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 $x_{i}-x_{j}$ both in the numerator and denominator, which simplify to yield 1, thus the polynomial should return $y_{i}$ as the function of $x_{i}$ for any i in the set $\{1,...,n\}$.