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

Abstract:

We study Gauss-Kronrod quadrature formulae for the Jacobi weight function ${w^{(\alpha ,\beta )}}(t) = {(1 - t)^\alpha }{(1 + t)^\beta }$ and its special case $\alpha = \beta = \lambda - \frac {1}{2}$ of the Gegenbauer weight function. We are interested in delineating regions in the $(\alpha ,\beta )$-plane, resp. intervals in $\lambda$, 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 $( - 1,1)$; (c) all weights positive; (d) only real nodes (not necessarily satisfying (a) and/or (b)). We determine the respective regions numerically for $n = 1(1)20(4)40$ in the Gegenbauer case, and for $n = 1(1)10$ in the Jacobi case, where 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 ${w^{(\alpha ,1/2)}}$ is obtained from the one for the weight ${w^{(\alpha ,\alpha )}}$, and similarly, the Gauss-Kronrod formula with an odd number of Gauss nodes for the weight function $w(t) = |t{|^\gamma }{(1 - {t^2})^\alpha }$ is derived from the Gauss-Kronrod formula for the weight ${w^{(\alpha ,(1 + \gamma )/2)}}$.