# 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}
}
• Published 1 March 2008
• Computer Science, Mathematics
• SIAM J. Numer. Anal.
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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## References

SHOWING 1-10 OF 63 REFERENCES
Bounds on the error of fejer and clenshaw-curtis type quadrature for analytic functions
• Mathematics
• 1993
Abstract We consider the problem of integrating a function f : [-1,1] → R which has an analytic extensionf to an open disk Dr of radius r and center the origin, such that ¦ f (z)¦ ≤1 for any z ∈ D r
Is Gauss Quadrature Better than Clenshaw-Curtis?
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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
Is Gauss quadrature optimal for analytic functions?
• Mathematics
• 1985
SummaryWe consider the problem of optimal quadratures for integrandsf: [−1,1]→ℝ which have an analytic extension $$\bar f$$ to an open diskDr of radiusr about the origin such that \left| {\bar f}
The conformal ‘bratwurst’ mapsand associated Faber polynomials
• Computer Science, Mathematics
Numerische Mathematik
• 2000
The main advantage of the approach is that the conformal maps are derived from elementary transformations, allowing an easy computation of the associated transfinite diameter, asymptotic convergence factor and Faber polynomials.
The kink phenomenon in Fejér and Clenshaw–Curtis quadrature
• Mathematics, Computer Science
Numerische Mathematik
• 2007
This paper derives explicit as well as asymptotic error formulas that provide a complete description of the Fejér and Clenshaw–Curtis rules for numerical integration phenomenon.
Accuracy, Resolution, and Stability Properties of a Modified Chebyshev Method
• Mathematics, Computer Science
SIAM J. Sci. Comput.
• 2002
While $\alpha$ can be chosen such that the mapped method improves the accuracy and resolution of the Chebyshev method, for practical choices of $N, it is not possible to achieve both single precision accuracy and gain the advantage of an$O(N^{-M})\$ time step.
Clenshaw--Curtis and Gauss--Legendre Quadrature for Certain Boundary Element Integrals
• Mathematics, Computer Science
SIAM J. Sci. Comput.
• 2008
The authors have concluded, after considering asymptotic estimates of the truncation errors for certain proto-type functions arising in this context, that Gauss-Legendre quadratures should continue to be the preferred quadrature rule.
High-Order Corrected Trapezoidal Quadrature Rules for Singular Functions
• Mathematics
• 1997
A class of quadrature formulae is presented applicable to both nonsingular and singular functions, generalizing the classical endpoint corrected trapezoidal quadrature rules. While the latter rules
Gaussian Versus Optimal Integration of Analytic Functions
Abstract. We consider error estimates for optimal and Gaussian quadrature formulas if the integrand is analytic and bounded in a certain complex region. First, a simple technique for the derivation