Accurate computation of weights in classical Gauss-Christoffel quadrature rules

Journal Article · Sun Dec 01 04:00:00 EST 1996 · Journal of Computational Physics

DOI:https://doi.org/10.1006/jcph.1996.0258· OSTI ID:530649

Yakimiw, E ^[1]

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1. Recherche en Prevision Numerique, Dorval, Quebec (Canada)

For many classical Gauss-Christoffel quadrature rules there does not exist a method which guarantees a uniform level of accuracy for the Gaussian quadrature weights at all quadrature nodes unless the nodes are known exactly. More disturbing, some algebraic expressions for these weights exhibit an excessive sensitivity to even the smallest perturbations in the node location. This sensitivity rapidly increases with high order quadrature rules. Current uses of very high order quadratures are common with the advent of more powerful computers, and a loss of accuracy in the weights has become a problem and must be addressed. A simple but efficient and general method for improving the accuracy of the computation of the quadrature weights even though the nodes may carry a significant large error. In addition, a highly efficient root-finding iterative technique with superlinear converging rates for computing the nodes is developed. It uses solely the quadrature polynomials and their first derivatives. A comparison of this method with the eigenvalue method of Golub and Welsh implemented in most standard software libraries is made. The proposed method outperforms the latter from the point of view of both accuracy and efficiency. The Legendre, Lobatto, Radau, Hermite, and Laguerre quadrature rules are examined. 22 refs., 7 figs., 5 tabs.

• OSTI ID: 530649
• Journal Information: Journal of Computational Physics, Journal Name: Journal of Computational Physics Journal Issue: 2 Vol. 129; ISSN JCTPAH; ISSN 0021-9991
• Country of Publication: United States
• Language: English

Related Subjects

99 GENERAL AND MISCELLANEOUS
CONVERGENCE
ITERATIVE METHODS
POLYNOMIALS
QUADRATURES