Lagrange Interpolating Polynomial

The Lagrange interpolating polynomial is the polynomial of degree that passes through the points , , ..., , and is given by

(1)

where

(2)

Written explicitly,

(3)

The formula was first published by Waring (1779), rediscovered by Euler in 1783, and published by Lagrange in 1795 (Jeffreys and Jeffreys 1988).

Lagrange interpolating polynomials are implemented in the Wolfram Language as InterpolatingPolynomial[data, var]. They are used, for example, in the construction of Newton-Cotes formulas.

When constructing interpolating polynomials, there is a tradeoff between having a better fit and having a smooth well-behaved fitting function. The more data points that are used in the interpolation, the higher the degree of the resulting polynomial, and therefore the greater oscillation it will exhibit between the data points. Therefore, a high-degree interpolation may be a poor predictor of the function between points, although the accuracy at the data points will be "perfect."

For points,

			(4)
			(5)

Note that the function passes through the points , as can be seen for the case ,

			(6)
			(7)
			(8)

Generalizing to arbitrary ,

(9)

The Lagrange interpolating polynomials can also be written using what Szegö (1975) called Lagrange's fundamental interpolating polynomials. Let

			(10)
			(11)
			(12)
			(13)

so that is an th degree polynomial with zeros at , ..., . Then define the fundamental polynomials by

(14)

which satisfy

(15)

where is the Kronecker delta. Now let , ..., , then the expansion

(16)

gives the unique Lagrange interpolating polynomial assuming the values at . More generally, let be an arbitrary distribution on the interval , the associated orthogonal polynomials, and , ..., the fundamental polynomials corresponding to the set of zeros of a polynomial . Then

(17)

for , 2, ..., , where are Christoffel numbers.

Lagrange interpolating polynomials give no error estimate. A more conceptually straightforward method for calculating them is Neville's algorithm.