A note on extended Gaussian quadrature rules

Abstract

Extended Gaussian quadrature rules of the type first considered by Kronrod are examined. For a general nonnegative weight function, simple formulas for the computation of the weights are given, together with a condition for the positivity of the weights associated with the new nodes. Examples of nonexistence of these rules are exhibited for the weight functions ( 1 − x 2 ) λ − 1 / 2 (1-x^2)^{\lambda - 1/2} , e − x 2 e^{-x^2} and e − x e^{-x} . Finally, two examples are given of quadrature rules which can be extended repeatedly.