Is Gauss Quadrature Better than Clenshaw-Curtis?

By Lloyd N. Trefethen. SIAM Review, 2008

720 citations recorded by the source corpus.

Subfields: Applied Mathematics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/0139b9b7a5b24b0ec5afca73b3137f044ff38e83

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In Applied Mathematics

In Numerical Analysis

Citations in the corpus

An Anisotropic Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data (2008, background): “See the work [27] for an insightful comparison of quadrature formulas based on Clenshaw–Curtis and Gaussian abscissas.”
A Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data (2008, neutral): “See the work [31] for an insightful comparison of quadrature formulas based on Clenshaw-Curtis and Gaussian abscissas.”
Fast Construction of the Fejér and Clenshaw–Curtis Quadrature Rules (2006, background): “The matrix V in (4), V ·w = b, now becomes (indices ≥ 1, as usual) V = (”
Fast, numerically stable computation of oscillatory integrals with stationary points (2010, methodology): “In addition, the method we develop has the interesting property that it converges even when ω is complex or small in magnitude — it becomes equivalent to Clenshaw–Curtis quadrature [4, 27] at ω = 0 — whereas numerical steepest descent can fall pray to Stokes’ phenomena [18] and the error blows up as”
Arithmetic Operations on Independent Random Variables: A Numerical Approach (2012, methodology): “For finite intervals we use Clenshaw–Curtis and Fejér quadratures as described in [20, 21].”
The spatial distribution in infinite dimensional spaces and related quantiles and depths (2014, methodology): “nomials, which form orthonormal bases of X when νis the uniform and the Beta(1/2,1/2) distributions, respectively, then x ∈ X satisfying limn→∞ √ n||x− x(n)|| = 0 can be obtained using Theorem 4.2 in [32] and Theorem 2.1 in [37], respectively. Next, let X = L2(R,ν), where νis the normal distribution”
Linear barycentric rational quadrature (2012, methodology): “Under direct rational quadrature we shall here mean the result of applying existing quadrature rules such as Gauss–Legendre or Clenshaw–Curtis [26, 29], which are known to behave well, to approximate the integrals in (3.2).”
The Exponentially Convergent Trapezoidal Rule (2014, neutral): “See [134, 169, 183, 187].”
Collect at Once, Use Effectively: Making Non-interactive Locally Private Learning Possible (2017, background): “…; r ) , where c is a constant, we have ˆ f p 1) are absolutely continuous over [ 1 , 1] , and f ( k ) T µ 1 ( k ; r ) µ 2 ( k ; r ) k , according to the results in [35], we have It is easy to see there exists c > 0 , such that the term (13) is less than α with chosen k , hence the conclusion holds.”
ON THE OPTIMAL POLYNOMIAL APPROXIMATION OF STOCHASTIC PDES BY GALERKIN AND COLLOCATION METHODS (2012, neutral): “[28]).”
Simulation-based optimal Bayesian experimental design for nonlinear systems (2011, background): “Although Gauss-Legendre quadrature (which is not nested) has a higher degree of polynomial exactness, [67] notes that “the Clenshaw-Curtis and Gauss formulas have essentially the same accuracy unless f is analytic in a sizable neighborhood of the interval of the integration—in which case both method”