A family of Gauss-Kronrod quadrature formulae

Abstract:

We show, for each $n \geq 1$, that the $(2n + 1)$-point Kronrod extension of the n-point Gaussian quadrature formula for the measure \[ d{\sigma _\gamma }(t) = {(1 + \gamma )^2}{(1 - {t^2})^{1/2}}dt/({(1 + \gamma )^2} - 4\gamma {t^2}),\quad - 1 < \gamma \leq 1,\] has the properties that its $n + 1$ Kronrod nodes interlace with the n Gauss nodes and all its $2n + 1$ weights are positive. We also produce explicit formulae for the weights.