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

Abstract

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