Gauss-type quadrature rules for variable-sign weight functions

Abstract

When the Gauss quadrature formula Gn is applied, it is often assumed that the weight function (or the measure) is non-negative on the integration interval [a,b]. In the present paper, we introduce a Gauss-type quadrature formula Qn for weight functions that change the sign in the interior of [a,b]. Construction of Qn is based on the idea to transform the given integral into a sum of one integral that does not cause a quadrature error and the other integral with a property that the points from the interior of [a,b] at which the weight function changes sign are the zeros of its integrand. It proves that all nodes of Qn are pairwise distinct and contained in the interior of [a,b]. Moreover, Gn (with a non-negative weight function) turns out to be a special case of Qn. Obtained results on the remainder term of Qn suggest that the application of Qn makes sense both when the points from the interior of [a,b] at which the weight function changes sign are known exactly, as well as when those points are known approximately. The accuracy of Qn is confirmed by numerical examples.