Suboptimal Kronrod extension formulae for numerical quadrature

Summary We consider cases where the Stieltjes polynomial associated with a Gaussian quadrature formula has complex zeros. In such cases a Kronrod extension of the Gaussian rule does not exist. A method is described for modifying the Stieltjes polynomial so that the resulting polynomial has no complex zeros. The modification is performed in such a way that the Kronrod-type extension rule resulting from the addition of then+1 zeros of the modified Stieltjes polynomial to the original knots of the Gaussian rule has only slightly lower degree of precision than normally achieved when the Kronrod extension rule exists. As examples of the use of the method, we present some new formulae extending the classical Gauss-Hermite quadrature rules. We comment on the limited success of the method in extending Gauss-Laguerre rules and show that several modified extensions of the Gauss Gegenbauer formulae exist in cases where the standard Kronrod extension does not.