The exact degree of precision of generalized Gauss-Kronrod integration rules

Abstract:

It is shown that the Kronrod extension to the $n$-point Gauss integration rule, with respect to the weight function $(1 - x^2)^{\mu - 1/2}$, $0 < \mu \leqslant 2$, $\mu \ne 1$, is of exact precision $3n + 1$ for n even and $3n + 2$ for n odd. Similarly, for the $(n + 1)$-point Lobatto rule, with $-1/2< \mu \leqslant 1$, $\mu \ne 0$, the exact precision is $3n$ for $n$ odd and $3n + 1$ for n even.