Enhanced Averaged Quadrature Rules with Application to Error Estimation

Gauss quadrature is a popular approach to approximate the value of an integral determined by a measure with support on the real axis. Laurie proposed an ( n + 1 ) $$(n+1)$$ -point quadrature rule, referred to as an anti-Gauss rule, that gives an error of the same magnitude and of opposite sign as the associated n-point Gauss quadrature rule for all polynomials of degree up to 2 n + 1 $$2n+1$$ . Laurie also described averaged rules that are the average of an n-point Gauss rule and the associated ( n + 1 ) $$(n+1)$$ -point anti-Gauss rule. The difference between an averaged rule and the associated Gauss rule has recently been used to estimate the quadrature error in the Gauss rule. For many integrands and measures, the error estimate so obtained is quite accurate, but not for all integrands and measures. This chapter proposes to use the difference between enhanced averaged rules introduced in Pranić and Reichel (J Comput Appl Math 284:235–243, 2015 [Eq. (1.14)]) and the associated Gauss rule to estimate the quadrature error in the latter. The enhanced averaged rules generalize averaged rules introduced by Laurie. Also enhanced averaged rules associated with Gauss rules determined by measures with support in the complex plane are described. Computed examples illustrate the performance enhanced averaged rules applied to error estimation of Gauss rules.