If (n+1) pairs of arguments and entries are given, Legrange’s formula is
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This image shows, for four points ((−9, 5), (−4, 2), (−1, −2), (7, 9)), the (cubic) interpolation polynomial L(x) (dashed, black), which is the sum of the scaled basis polynomials y0ℓ0(x), y1ℓ1(x), y2ℓ2(x) and y3ℓ3(x). The interpolation polynomial passes through all four control points, and each scaled basis polynomial passes through its respective control point and is 0 where x corresponds to the other three control points.
In numerical analysis, Lagrange polynomials are used for polynomial interpolation. For a given set of points {\displaystyle (x_{j},y_{j})} with no two {\displaystyle x_{j}} values equal, the Lagrange polynomial is the polynomial of lowest degree that assumes at each value {\displaystyle x_{j}} the corresponding value {\displaystyle y_{j}}, so that the functions coincide at each point.
Although named after Joseph-Louis Lagrange, who published it in 1795, the method was first discovered in 1779 by Edward Waring[1] It is also an easy consequence of a formula published in 1783 by Leonhard Euler.[2]
Uses of Lagrange polynomials include the Newton–Cotes method of numerical integration and Shamir's secret sharing scheme in cryptography.
Lagrange interpolation is susceptible to Runge's phenomenon of large oscillation. As changing the points {\displaystyle x_{j}} requires recalculating the entire interpolant, it is often easier to use Newton polynomials instead.
