2009-59.md

course	course_year	question_number	tags	title	year
Numerical Analysis	IB	59	IB 2009 Numerical Analysis	Paper 4, Section I, C	2009

Suppose that $w(x)>0$ for all $x \in(a, b)$. The weights $b_{1}, \ldots, b_{n}$ and nodes $x_{1}, \ldots, x_{n}$ are chosen so that the Gaussian quadrature formula

$$\int_{a}^{b} w(x) f(x) d x \sim \sum_{k=1}^{n} b_{k} f\left(x_{k}\right)$$

is exact for every polynomial of degree $2 n-1$. Show that the $b_{i}, i=1, \ldots, n$ are all positive.

When $w(x)=1+x^{2}, a=-1$ and $b=1$, the first three underlying orthogonal polynomials are $p_{0}(x)=1, p_{1}(x)=x$, and $p_{2}(x)=x^{2}-2 / 5$. Find $x_{1}, x_{2}$ and $b_{1}, b_{2}$ when $n=2$.