On Gauss-Type Quadrature Rules

Abstract

Some Gauss-type Quadrature rules over [0, 1], which involve values and/or the derivative of the integrand at 0 and/or 1, are investigated. Our work is based on the orthogonal polynomials with respect to linear weight function ω(t): = 1 − t over [0, 1]. These polynomials are also linked with a class of recently developed “identity-type functions”. Along the lines of Golub's work, the nodes and weights of the quadrature rules are computed from Jacobi-type matrices with simple rational entries. Computational procedures for the derived rules are tested on different integrands. The proposed methods have some advantage over the respective Gauss-type rules with respect to the Gauss weight function ω(t): = 1 over [0, 1].

Keywords:

• Gauss/Gauss–Radau/Gauss–Lobatto quadrature rules
• Hypergeometric functions
• Jacobi-matrix
• Orthogonal polynomials
• Three-term recurrence relation

AMS Subject Classification:

• 65D30
• 65F15
• 33C45

ACKNOWLEDGMENTS

The authors are grateful to KFUPM for the excellent research facilities availed during the preparation of this article. The authors are also thankful to the anonymous referee for the comments which helped improve the quality of this manuscript.