New Quadrature Formulas from Conformal Maps

@article{Hale2008NewQF,
  title={New Quadrature Formulas from Conformal Maps},
  author={Nicholas Hale and Lloyd N. Trefethen},
  journal={SIAM J. Numer. Anal.},
  year={2008},
  volume={46},
  pages={930-948}
}
Gauss and Clenshaw-Curtis quadrature, like Legendre and Chebyshev spectral methods, make use of grids strongly clustered at boundaries. From the viewpoint of polynomial approximation this seems necessary and indeed in certain respects optimal. Nevertheless such methods may “waste” a factor of $\pi/2$ with respect to each space dimension. We propose new nonpolynomial quadrature methods that avoid this effect by conformally mapping the usual ellipse of convergence to an infinite strip or another… 
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Comparisons of the convergence behavior of Gauss quadrature with that of its younger brother, Clenshaw-Curtis are compared, and experiments show that the supposed factor-of-2 advantage of Gaussian quadratures is rarely realized.
Is Gauss Quadrature Better than Clenshaw–Curtis? | SIAM Review | Vol. 50, No. 1 | Society for Industrial and Applied Mathematics
We compare the convergence behavior of Gauss quadrature with that of its younger brother, Clenshaw–Curtis. Seven-line MATLAB codes are presented that implement both methods, and experiments show that
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