Lagrange Polynomial

Topics: Polynomial - Numerical Analysis

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Let us have $n+1$ points $(x_0,y_0),\dots ,(x_n,y_n)$, with $x_i\ne x_j$ when $i\ne j$ (i.e. all abscissas distinct).

We will build a polynomial of degree $n$, called the Lagrange (interpolation) polynomial, that will allow us to interpolate this sequence of points.

Lagrange Basis §

(definition)

We define the Lagrange basis as the set of polynomials $\{l_0(X),l_1(X),\dots ,l_n(X)\}$, where:

$$\begin{array}{rl}{l}_i(X)& =\prod _{\begin{array}{c}j=0\\ j\ne i\end{array}}^n\frac{X-x_j}{x_i-x_j}\\ & =\left(\frac{X-x_0}{x_i-x_0}\right)\dots \left(\frac{X-x_{i-1}}{x_i-x_{i-1}}\right)\left(\frac{X-x_{i+1}}{x_i-x_{i+1}}\right)\dots \left(\frac{X-x_n}{x_i-x_n}\right)\end{array}$$

In particular, notice that:

1. $l_i(X)$ is of degree $n$ for all $i$
2. $l_i=\{\begin{array}{l}0\text{ if }X\ne x_i\\ 1\text{ if }X=x_i\end{array}$

(see Kronecker delta)

Lagrange Polynomial §

(definition)

We define the Lagrange (interpolation) polynomial as the linear combination:

$$\begin{array}{rl}L(X)& =\sum _{j=0}^n{l}_j(X)y_j\\ & =l_0(X)y_0+l_1(X)y_1+\cdots +l_n(X)y_n\end{array}$$

This polynomial is special in that for all $i$, $L(x_i)=y_i$. This happens thanks to the Kronecker delta-like behaviour of the polynomials in the basis.

This allows us to use the polynomial to interpolate the initial sequence of points.