Lagmental Vicfred

Lagrange Interpolation Uses Cardinal Basis Polynomials by Vicfred

Last updated: Thu 01 January 2015

tags: mathematics, numerical analysis, root finding, interpolation

There is one polynomial of degree at most n through n plus one data points with distinct nodes. The point is to make the formal expression readable enough to audit line by line.

Definitions first

Numerical analysis replaces an exact object \(x^\ast\) by approximations \(x_n\) with controlled error. Stability asks how rounding or data perturbations affect the answer.

$$ \ell_j(x)=\prod_{\substack{0\le m\le n\\m\ne j}}\frac{x-x_m}{x_j-x_m} $$

The definition determines which expressions are legal; only then does the identity become meaningful. An equality in \(\mathcal A\) may change ambient meaning, so I keep \(\mathsf D\) separate from \(\mathsf C\).

$$ p_n(x)=\sum_{j=0}^{n}y_j\ell_j(x),\qquad p_n(x_i)=y_i $$

A small case in full

This is the algebraic core of the note. Once this line is correct, the surrounding interpretation has something solid to refer to.

$$ f(x)-p_n(x)=\frac{f^{(n+1)}(\xi_x)}{(n+1)!}\prod_{j=0}^{n}(x-x_j) $$

The reusable statement

The formula is reusable precisely because it says which pieces are structural and which belong only to the worked example.

$$ \begin{aligned} \mathsf{D}\;&:\quad \ell_j(x)=\prod_{\substack{0\le m\le n\\m\ne j}}\frac{x-x_m}{x_j-x_m},\\[5pt] \mathsf{C}\;&:\quad p_n(x)=\sum_{j=0}^{n}y_j\ell_j(x),\qquad p_n(x_i)=y_i. \end{aligned} $$

A nearby false statement

Fast local convergence is not a global guarantee. Newton's method can diverge, interpolation can oscillate, and a small residual can coexist with a large forward error.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] p_n(x)=\sum_{j=0}^{n}y_j\ell_j(x),\qquad p_n(x_i)=y_i \end{gathered}} $$

A symbolic summary is trustworthy only because the example and limitation remain visible beside it. The box compresses the conclusion without hiding its origin.

This article was posted on Sat 10 September 2011. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.