Stanford 50: State of the Art and Future Directions of Computational Mathematics and Numerical Computing


  • 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.

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