Editing Lagrange polynomial (section)

==Definition==
Given a set of <math display=inline>k + 1</math> nodes <math>\{x_0, x_1, \ldots, x_k\}</math>, which must all be distinct, <math>x_j \neq x_m</math> for indices <math>j \neq m</math>, the '''Lagrange basis''' for polynomials of degree <math display=inline>\leq k</math> for those nodes is the set of polynomials <math display=inline>\{\ell_0(x), \ell_1(x), \ldots, \ell_k(x)\}</math> each of degree <math display=inline>k</math> which take values <math display=inline>\ell_j(x_m) = 0</math> if <math display=inline>m \neq j</math> and <math display=inline>\ell_j(x_j) = 1</math>. Using the [[Kronecker delta]] this can be written <math display=inline>\ell_j(x_m) = \delta_{jm}.</math> Each basis polynomial can be explicitly described by the product:

<math display=block>\begin{aligned}
\ell_j(x) &= \frac{(x-x_0)}{(x_j-x_0)} \cdots \frac{(x-x_{j-1})}{(x_j-x_{j - 1})} \frac{(x-x_{j+1})}{(x_j-x_{j+1})} \cdots \frac{(x-x_k)}{(x_j-x_k)} \\[8mu]
&= \prod_{\begin{smallmatrix}0\le m\le k\\ m\neq j\end{smallmatrix}} \frac{x-x_m}{x_j-x_m} \vphantom\Bigg|.
\end{aligned}</math>

Notice that the numerator <math display=inline>\prod_{m \neq j}(x - x_m)</math> has <math display=inline>k</math> roots at the nodes <math display=inline>\{x_m\}_{m \neq j}</math> while the denominator <math display=inline>\prod_{m \neq j}(x_j - x_m)</math> scales the resulting polynomial so that <math display=inline>\ell_j(x_j) = 1.</math>

The Lagrange interpolating polynomial for those nodes through the corresponding ''values'' <math>\{y_0, y_1, \ldots, y_k\}</math> is the linear combination:

<math display=block>L(x) = \sum_{j=0}^{k} y_j \ell_j(x).</math>

Each basis polynomial has degree <math display=inline>k</math>, so the sum <math display=inline>L(x)</math> has degree <math display=inline>\leq k</math>, and it interpolates the data because <math display=inline>L(x_m) = \sum_{j=0}^{k} y_j \ell_j(x_m) = \sum_{j=0}^{k} y_j \delta_{mj} = y_m.</math>

The interpolating polynomial is unique. Proof: assume the polynomial <math display=inline>M(x)</math> of degree <math display=inline>\leq k</math> interpolates the data. Then the difference <math display=inline>M(x) - L(x)</math> is zero at <math display=inline>k + 1</math> distinct nodes <math display=inline>\{x_0, x_1, \ldots, x_k\}.</math> But the only polynomial of degree <math display=inline>\leq k</math> with more than <math display=inline>k</math> roots is the constant zero function, so <math display=inline>M(x) - L(x) = 0,</math> or <math display=inline>M(x) = L(x).</math>