Numerical Analysis | Part IB, 2003

Given (n+1)(n+1) distinct points x0,x1,,xnx_{0}, x_{1}, \ldots, x_{n}, let

i(x)=k=0kinxxkxixk\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 nn, let


and let pp be any polynomial of degree n\leq n.

(a) Prove that i=0np(xi)i(x)p(x)\sum_{i=0}^{n} p\left(x_{i}\right) \ell_{i}(x) \equiv p(x).

(b) Hence or otherwise derive the formula

p(x)ω(x)=i=0nAixxi,Ai=p(xi)ω(xi)\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)/ω(x)p(x) / \omega(x) into partial fractions.

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