Barycentric interpolation formula

Builds on Lagrange basis polynomials Lagrange polynomial Polynomial interpolation Field

Statement

Let $K$ be a field, let $n \ge 0$ , let $x_0, \dots, x_n \in K$ be pairwise distinct, and let $y_0, \dots, y_n \in K$ . Let $p$ be the interpolation polynomial of degree at most $n$ with $p(x_j) = y_j$ for all $j$ . Define the barycentric weights $w_j = \prod_{k \ne j} (x_j - x_k)^{-1}$ and let $\ell(x) = \prod_{k=0}^{n} (x - x_k)$ . Then for every $x \in K$ that is not a node:

(first form) $\displaystyle p(x) = \ell(x) \sum_{j=0}^{n} \frac{w_j}{x - x_j} \, y_j$ ;
(second form) $\displaystyle p(x) = \frac{\sum_{j=0}^{n} \frac{w_j}{x - x_j} \, y_j}{\sum_{j=0}^{n} \frac{w_j}{x - x_j}}$ .

Remarks

The Lagrange basis polynomials satisfy $\ell_j(x) = \ell(x) \, w_j / (x - x_j)$ , so the first form is the Lagrange formula $p = \sum_j y_j \ell_j$ rewritten. Applying the first form to the constant data $y_j = 1$ , whose interpolant is $1$ , gives $1 = \ell(x) \sum_j w_j / (x - x_j)$ . Dividing the first form by this identity gives the second form.

Example: nodes $0, 1$ have weights $w_0 = -1$ and $w_1 = 1$ . The second form gives $p(x) = \dfrac{-y_0/x + y_1/(x - 1)}{-1/x + 1/(x - 1)} = y_0 (1 - x) + y_1 x$ , the line through the data.

Once the weights are computed, in $O(n^2)$ operations, each evaluation costs $O(n)$ operations, and adding a new node updates the weights in $O(n)$ operations. The weights do not depend on the data, so the same weights serve for many data sets.

Going deeper

For a student

The second form is the standard way to evaluate an interpolation polynomial in floating point arithmetic. For good node families, such as Chebyshev nodes, the weights have simple closed forms and the evaluation is numerically stable. Unlike the Newton form, it needs no ordering of the nodes and no table of divided differences.

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Field concept
Lagrange basis polynomials concept
Lagrange polynomial concept
Polynomial interpolation concept
Barycentric interpolation formula statement

Tag	`TPRFJNZ7S2`
URI	`m2://TPRFJNZ7S2`
Kind	statement, proposition
Writing	W0 stub
MSC	65D05, 41A05
Status	This copy of the site has no build data. Run `m2 status TPRFJNZ7S2`.
About	Lagrange polynomial
Uses	Lagrange basis polynomials, Lagrange polynomial, Polynomial interpolation, Field