| course |
course_year |
question_number |
tags |
title |
year |
Numerical Analysis |
IB |
48 |
IB |
2003 |
Numerical Analysis |
|
3.I.6B |
2003 |
Given $(n+1)$ distinct points $x_{0}, x_{1}, \ldots, x_{n}$, let
$$\ell_{i}(x)=\prod_{\substack{k=0 \ k \neq i}}^{n} \frac{x-x_{k}}{x_{i}-x_{k}}$$
be the fundamental Lagrange polynomials of degree $n$, let
$$\omega(x)=\prod_{i=0}^{n}\left(x-x_{i}\right)$$
and let $p$ be any polynomial of degree $\leq n$.
(a) Prove that $\sum_{i=0}^{n} p\left(x_{i}\right) \ell_{i}(x) \equiv p(x)$.
(b) Hence or otherwise derive the formula
$$\frac{p(x)}{\omega(x)}=\sum_{i=0}^{n} \frac{A_{i}}{x-x_{i}}, \quad A_{i}=\frac{p\left(x_{i}\right)}{\omega^{\prime}\left(x_{i}\right)}$$
which is the decomposition of $p(x) / \omega(x)$ into partial fractions.