New Quadrature Formulas from Conformal Maps

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 π/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 approximately straight-sided domain. The new methods are compared with related ideas of Bakhvalov, Kosloff and Tal-Ezer, Rokhlin and Alpert, and others. An advantage of the conformal mapping approach is that it leads to theorems guaranteeing geometric rates of convergence for analytic integrands. For example, one of the formulas presented is proved to converge 50% faster than Gauss quadrature for functions analytic in an ε-neighborhood of [−1, 1].