18 June 2009

Trefethen's Gauss Quadrature vs Clenshaw-Curtis Paper

From a quick look into B-spline Galerkin methods, I need to integrate B-spline basis functions against each other. Though I could extract piecewise polynomial representations and integrate them symbolically, it was more fun to see what fixed order Gauss quadrature FOSS existed. I stumbled across Pavel Holoborodko's excellent routines. I'm in the process of preparing a patch for the GNU Scientific Library that ports Pavel's code to GSL conventions (update: patch submitted). Hopefully they'll accept it (update: patch accepted).

While digging around, I ran across a beautiful paper by Trefethen which I just finished on the morning bus ride into campus. Absolutely fantastic.

Is Gauss Quadrature Better than Clenshaw–Curtis?
SIAM Rev. 50, 67 (2008)

The paper is so good that it makes me also want to try implementing Clenshaw-Curtis rules for GSL. Before coding, I need to read this follow up work which it seems may have some numerical details buried within it.

New Quadrature Formulas from Conformal Maps
SIAM J. Numer. Anal. 46, 930 (2008)

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