Lagrange's interpolation formula

English

Etymology

Named after Template:W (1736–1813), an Italian Enlightenment Era mathematician and astronomer.

Noun

1. Template:Lb 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: $\sum _i^n{y}_i\prod _{j\ne i}\frac{x-x_j}{x_i-x_j}$. 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 $\{1,...,n\}$.