Numerical Methods Qualification Exam Problems and Solutions (University of Maryland)/August 2002

Problem 2

Suppose there is a quadrature formula $\underset{a}{\overset{b}{\int }}f(x)dx\approx w_{a}f(a)+w_{b}f(b)+\sum _{j=1}^n{w}_{j}f(x_j)$ which produces the exact integral whenever $f$ is a polynomial of degree $2n+1$. Here the nodes $\{x_j{\}}_{j=1}^n$ are all distinct. Prove that the nodes lies in the open interval $(a,b)$ and the weights $w_a,w_b$ and $\{w_j{\}}_{j=1}^n$ are positive.

Solution 2

All nodes lies in (a,b)

Let $\{x_i{\}}_{i=1}^l$ be the nodes that lie in the interval $(a,b)$.

Let $q_l(x)=\prod _{i=1}^l(x-x_i)$ which is a polynomial of degree $l$.

Let $p_n(x)=\prod _{i=1}^n(x-x_i)=q_l(x)\prod _{i=1}^{n-l}(x-x_i)$ which is a polynomial of degree $n>l$.

Then

$⟨p_n,q_l⟩=\underset{a}{\overset{b}{\int }}{q}_l^2(x)\underset{r(x)}{\underbrace{\prod _{i=1}^{n-l}(x-x_i)}}\ne 0$

since $r(x)$ is of one sign in the interval $(a,b)$ since for $i=1,2,\dots n-l$, $x_i\in ̸(a,b).$

This implies $q_l$ is of degree $n$ since otherwise

$⟨p_n,q_l⟩=0$

from the orthogonality of $p_n$.