In numerical analysis,
Lagrange polynomials are used for polynomial interpolation. For a
given set of distinct points
and numbers
,
the Lagrange polynomial is the polynomial of the least degree that at
each point
assumes the corresponding value
(i.e. the functions coincide at each point). The interpolating
polynomial of the least degree is unique, however, and it is
therefore more appropriate to speak of "the Lagrange form"
of that unique polynomial rather than "the Lagrange
interpolation polynomial," since the same polynomial can be
arrived at through multiple methods. Although named after Joseph
Louis Lagrange, who published it in 1795, it was first discovered in
1779 by Edward Waring and it is also an easy consequence of a formula
published in 1783 by Leonhard Euler.
Lagrange interpolation is susceptible to Runge's phenomenon, and the fact that changing the interpolation points requires recalculating the entire interpolant can make Newton polynomials easier to use. Lagrange polynomials are used in the Newton–Cotes method of numerical integration and in Shamir's secret sharing scheme in cryptography.
Example:
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.
Lagrange interpolation is susceptible to Runge's phenomenon, and the fact that changing the interpolation points requires recalculating the entire interpolant can make Newton polynomials easier to use. Lagrange polynomials are used in the Newton–Cotes method of numerical integration and in Shamir's secret sharing scheme in cryptography.
Example:
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.
From Wikipedia.
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