Improved Lagrange Formula

The improved Lagrange polynomial formula is a modification of the Lagrange polynomial formula, and it also allows us to derive the barycentric formula.

P(x) = \sum_{i=0}^n{y_i l_i(x)}

where $l_i(x) = \prod_{j \ne i}{\frac{x-x_j}{x_i-x_j}}$ is the Lagrange basis polynomial.

We introduce

l(x) = (x-x_0)(x-x_1)\cdots(x-x_n)

and define the barycentric weights as

w_i = \frac{1}{\prod_{j \ne i}{(x_i-x_j)}}

Note that $w_i = 1/l'(x_i)$ .

Thus, the Lagrange basis polynomial can be expressed as:

l_i(x) = l(x)\frac{w_i}{x-x_i}

Substituting this into the original formula, we can factor out $l(x)$ :

\boxed{P(x) = l(x) \sum_{i=0}^n{\frac{w_i}{x-x_i} y_i}}

This is the improved Lagrange polynomial formula.

References #

Jean-Paul Berrut and Lloyd N. Trefethen, Barycentric Lagrange Interpolation, SIAM Review, Vol. 46, No. 3, pp. 501–517, 2004.