Fast and accurate computation of Gauss-Legendre and Gauss-Jacobi quadrature nodes and weights. (English) Zbl 1270.65017

Hale, Nicholas; Townsend, Alex

SIAM J. Sci. Comput. 35, No. 2, A652-A674 (2013).

Summary: An efficient algorithm for the accurate computation of Gauss-Legendre and Gauss-Jacobi quadrature nodes and weights is presented. The algorithm is based on Newton’s root-finding method with initial guesses and function evaluations computed via asymptotic formulae. The \(n\)-point quadrature rule is computed in \(\mathcal{O}(n)\) operations to an accuracy of essentially double precision for any \(n\geq 100\).

MSC:

65D32	Numerical quadrature and cubature formulas
41A55	Approximate quadratures

Keywords:

quadrature; Gauss-Legendre; Gauss-Jacobi; asymptotic expansion; numerical examples; algorithm; Newton’s root-finding method

Software:

DLMF; LEGENDRE_RULE_FAST; gaussq.f; Chebfun; ORTHPOL; FastGaussQuadrature; nag