Lagrange interpolation formula

Let (x1,y1),(x2,y2),,(xn,yn) be n points in the plane (xixj for ij). Then there exists a unique polynomialPlanetmathPlanetmath p(x) of degree at most n-1 such that yi=p(xi) for i=1,,n.

Such polynomial can be found using Lagrange’s interpolation formula:


where f(x)=(x-x1)(x-x2)(x-xn).

To see this, notice that the above formula is the same as

p(x) =y1(x-x2)(x-x3)(x-xn)(x1-x2)(x1-x3)(x1-xn)+y2(x-x1)(x-x3)(x-xn)(x2-x1)(x2-x3)(x2-xn)

and that for all xi, every numerator except one vanishes, and this numerator will be identical to the denominator, making the overall quotient equal to 1. Therefore, each p(xi) equals yi.

Title Lagrange interpolation formula
Canonical name LagrangeInterpolationFormula
Date of creation 2013-03-22 11:46:21
Last modified on 2013-03-22 11:46:21
Owner drini (3)
Last modified by drini (3)
Numerical id 16
Author drini (3)
Entry type Theorem
Classification msc 65D05
Classification msc 41A05
Synonym Lagrange’s Interpolation formula
Related topic SimpsonsRule
Related topic LectureNotesOnPolynomialInterpolation
Defines Lagrange polynomial