A new representation of generalized averaged Gauss quadrature rules

Abstract

Gauss quadrature rules associated with a nonnegative measure with support on (part of) the real axis find many applications in Scientific Computing. It is important to be able to estimate the quadrature error when replacing an integral by an l-node Gauss quadrature rule in order to choose a suitable number of nodes. A classical approach to estimate this error is to evaluate the associated (2l + 1)-node Gauss-Kronrod rule. However, Gauss-Kronrod rules with 2l + 1 real nodes might not exist. The (2l + 1)-node generalized averaged Gauss formula associated with the l-node Gauss rule described in Spalevic (2007) [16] is guaranteed to exist and provides an attractive alternative to the (2l + 1)-node Gauss-Kronrod rule. This paper describes a new representation of generalized averaged Gauss formulas that is cheaper to evaluate than the available representation.