## Access

You are not currently logged in.

Access your personal account or get JSTOR access through your library or other institution:

## If You Use a Screen Reader

This content is available through Read Online (Free) program, which relies on page scans. Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.

# An Algebraic Study of Gauss-Kronrod Quadrature Formulae for Jacobi Weight Functions

Walter Gautschi and Sotirios E. Notaris
Mathematics of Computation
Vol. 51, No. 183 (Jul., 1988), pp. 231-248
DOI: 10.2307/2008588
Stable URL: http://www.jstor.org/stable/2008588
Page Count: 18
Preview not available

## 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 $w^{(\alpha, (1 + \gamma)/2)}$.

• 231
• 232
• 233
• 234
• 235
• 236
• 237
• 238
• 239
• 240
• 241
• 242
• 243
• 244
• 245
• 246
• 247
• 248