Error bounds for Gaussian quadrature rules using linear kernels

Abstract

It is well-known that the remaining term of a n-point Gaussian quadrature depends on the 2n-order derivative of the integrand function. Discounting the fact that calculating a 2n-order derivative requires a lot of differentiation, the main problem is that an error bound for a n-point Gaussian quadrature is only relevant for a function that is 2n times differentiable, a rather stringent condition. In this paper, by defining some specific linear kernels, we resolve this problem and obtain new error bounds (involving only the first derivative of the weighted integrand function) for all Gaussian weighted quadrature rules whose nodes and weights are pre-assigned over a finite interval. The advantage of using linear kernels is that their L 1 -norm, L ∞ -norm, maximum and minimum can easily be computed. Three illustrative examples are given in this direction.

A note on weighted quadrature rules

“…We observe that the left-hand side of (9) has appeared in the left-hand side of (12) for w i D a i and x i D b i . Therefore, changing the variable u.x/ D t in the integral (12)…”

Revisited Optimal Error Bounds for Interpolatory Integration Rules

“…These bounds depend on the regularity of the integrand. Other similar approaches based on kernels have been recently presented in [11] for Newton-Cotes quadrature rules and in [12] for Gaussian weighted quadrature rules.…”

A Modification of Newton's Method Using Gauss-Legendre Quadrature Rules

“…For some results on numerical improvement of Gauss-Legendre quadrature rules see e.g. [3] and for more details on error bounds for Gaussian quadrature rules , we refer [14].…”