Lagrange's interpolation formula

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

Noun

 * 1)  A formula which when given a set of n points $$(x_i, y_i)$$, gives back the unique polynomial of degree (at most) n &minus; 1 in one variable which describes a function passing through those points. The formula is a sum of products, like so: $$\sum_i^n y_i \prod_{j \ne i} {x - x_j \over x_i - x_j}$$. When $$x = x_i$$ then all terms in the sum other than the ith contain a factor $$x - x_i$$ in the numerator, which becomes equal to zero, thus all terms in the sum other than the ith vanish, and the ith 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 $$\{1, ..., n\}$$.