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