An algebraic and numerical study of Gauss-Kronrod quadrature formulae

Sotirios E Notaris, Purdue University

Abstract

We study Gauss-Kronrod quadrature formulae for the Jacobi weight function $\omega\sp{(\alpha,\beta)}(t)$ = (1 $-$ t)$\sp\alpha$ (1 + t)$\sp{\beta}$, $\alpha$ $>$ $-$1, $\beta$ $>$ $-$1 on ($-$1,1) and its special case $\alpha$ = $\beta$ = $\lambda$ $-$ 1/2 of the Gegenbauer weight function. We delineate regions in the ($\alpha$, $\beta$)-plane, respectively intervals in $\lambda$, for which the quadrature formula 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 is suggested by the numerical results. Furthermore, the Gauss-Kronrod formula for the weight function $\omega\sp{(\alpha,1/2)}$ is obtained from the one for the weight function $\omega\sp{(\alpha,\alpha)}$, and similarly, the Gauss-Kronrod formula with an odd number of Gauss nodes for the weight function $\sp{\gamma}\omega\sp{(\alpha)}(t)$ = $\vert t\vert\sp{\gamma}(1 - t\sp2)\sp{\alpha}$, $\alpha$ $>$ $-1$, $\gamma$ $>$ $-1$ on ($-$1,1) is derived from the Gauss-Kronrod formula for the weight function $\omega\sp{(\alpha,(1+\gamma)/2)}$. We also discuss the use of Newton's method to compute the Kronrod extension of Gauss-Radau and Gauss-Lobatto quadrature formulae from modified moments. We give formulae for the weights and prove the equivalence between the interlacing property and the positivity of the weights. The underlying non-linear maps are analyzed from the point of view of numerical condition. Numerical examples are given for the weight functions $\omega$(t) = 1 on ($-$1,1) and $\omega$(t) = ln(1/t), $\omega$(t) = $t\sp{\pm 1/2}$ln(1/t) on (0,1) having logarithmic and algebraic singularities at one endpoint. Finally, we apply Newton's method to compute a sequence of quadrature rules constructed by repeated application of Kronrod's idea. We experiment with the weight function $\omega$(t) = 1 on ($-$1,1) and discover that in this case the method fails to converge. An investigation of this nonconvergence concludes our work.

Degree

Ph.D.

Advisors

Gautschi, Purdue University.

Subject Area

Mathematics

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