In numerical analysis, a Lagrange polynomial, named after Joseph Louis Lagrange, is the interpolating polynomial for a given set of data points in the Lagrange form. It was first discovered by Edward Waring in 1779 and later rediscovered by Leonhard Euler in 1783.
Notice that, for any given set of data points, there is only one polynomial (of least possible degree) that interpolates these points. Thus, it is more appropriate to call it "the Lagrange form of the interpolation polynomial" rather than "the Lagrange interpolation polynomial".
This image shows, for four points ((−9, 5), (−4, 2), (−1, −2), (7, 9)), the (cubic) interpolation polynomial L(x), 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.
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