An algebraic study of Gauss-Kronrod quadrature formulae for Jacobi weight functions

We study Gauss-Kronrod quadrature formulae for the Jacobi weight function $ {w^{(\alpha ,\beta )}}(t) = {(1 - t)^\alpha }{(1 + t)^\beta }$ $ \alpha = \beta = \lambda - \frac{1}{2}$ $ (\alpha ,\beta )$, for which the quadrature rule has (a) the interlacing property, i.e., the Gauss nodes and the Kronrod nodes interlace; (b) all nodes contained in $ ( - 1,1)$ $ n = 1(1)20(4)40$ $ n = 1(1)10$n is the number of Gauss nodes. Algebraic criteria, in particular the vanishing of appropriate resultants and discriminants, are used to determine the boundaries of the regions identifying properties (a) and (d). The regions for properties (b) and (c) are found more directly. A number of conjectures are suggested by the numerical results. Finally, the Gauss-Kronrod formula for the weight $ {w^{(\alpha ,1/2)}}$ $ {w^{(\alpha ,\alpha )}}$ $ w(t) = \vert t{\vert^\gamma }{(1 - {t^2})^\alpha }$ <IMG WIDTH="107" HEIGHT="23" ALIGN="BOTTOM" BORDER="0" SRC="images/img11.gif" ALT="$ {w^{(\alpha ,(1 + \gamma )/2)}}$">.