Anti-Gaussian quadrature formulas

Abstract

An anti-Gaussian quadrature formula is an ( n + 1 ) (n+1) -point formula of degree 2 n − 1 2n-1 which integrates polynomials of degree up to 2 n + 1 2n+1 with an error equal in magnitude but of opposite sign to that of the n n -point Gaussian formula. Its intended application is to estimate the error incurred in Gaussian integration by halving the difference between the results obtained from the two formulas. We show that an anti-Gaussian formula has positive weights, and that its nodes are in the integration interval and are interlaced by those of the corresponding Gaussian formula. Similar results for Gaussian formulas with respect to a positive weight are given, except that for some weight functions, at most two of the nodes may be outside the integration interval. The anti-Gaussian formula has only interior nodes in many cases when the Kronrod extension does not, and is as easy to compute as the ( n + 1 ) (n+1) -point Gaussian formula.