### Information

- March 31, 2007
- 2:25 pm - 2:50 pm

# Beating Gauss quadrature

##### Nick Trefethen (University of Oxford)

We all know that Gauss quadrature points are in some sense
optimal, and that they can be computed by the marvelous
algorithm of Golub and Welsch. But as so often happens in
mathematics, the optimality theorem conceals an assumption
that may not always be reasonable---in this case, that
the quality of a quadrature formula is determined by how high a
degree of polynomial it can integrate exactly.
If you drop this assumption, you find that alternative quadrature
formulas can outperform Gauss for many integrands by a factor of
about π*/2*. The new formulas involve nearly uniformly spaced
nodes,
without the usual clustering at endpoints, which can
be a big advantage in PDE simulations by spectral methods.
We show how to derive such formulas by conformal mapping
and point out connections with previous work by Kosloff and
Tal-Ezer, Alpert, and others. Fortunately, the Golub-Welsch
algorithm is still applicable.