Math 6610 • Analysis of Numerical Methods I

{Numerical Quadrature} Consider the quadrature formula
\[
  \int_a^b w(x)f(x)\,dx=\sum_{i=1}^n \alpha_i f(\xi_i)+E,
\]
for the integral of \(f\) with a given positive weight function \(w(x)>0\). We would like
to use \(n\) knot values \((f(\xi_i), i=1,\ldots,n)\) to achieve the exact integral if \(f\)
is a polynomial of degree up to \(2n-1\).

\textbf{(a)} Start from the special case \(a=-1\), \(b=1\) and \(w(x)=1\), define the
orthogonal polynomials over \([-1,1]\).

\textbf{(b)} Use those orthogonal polynomials and polynomial division to simplify the
integral. How would you pick the knots \(\xi_i\) and the weights \(\alpha_i\) such that
the formula is exact (i.e., \(E=0\)) for all polynomials of degree up to \(2n-1\)? Verify
that your choice works.

\textbf{(c)} Show that you can extend the approach to more general integrals, with
arbitrary \(a<b\) and positive weight function \(w\).