Lagrange's interpolation formula

名词noun

1. 1(mathematics) A formula which when given a set of n points (xi,yi), 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: ∑inyi∏j≠ix−xjxi−xj. When x=xi then all terms in the sum other than the i th contain a factor x−xi 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 xi−xj both in the numerator and denominator, which simplify to yield 1, thus the polynomial should return yi as the function of xi for any i in the set {1,...,n}.