Modified Stieltjes polynomials and Gauss–Kronrod quadrature rules

Modified Stieltjes polynomials are defined and used to construct suboptimal extensions of Gaussian rules with one or two degrees less of polynomial exactness than the corresponding Kronrod extension. We prove that, for wide classes of weight functions and a sufficiently large number of nodes, the extended quadratures have positive weights and simple nodes on the interval [ - 1 , 1 ] . The classes of weight functions considered complement those for which the Gauss–Kronrod rule is known to exist. Also, strong asymptotic representations on the whole interval [ - 1 , 1 ] are given for the modified Stieltjes polynomials, which prove that they behave asymptotically as orthogonal polynomials. Finally, we provide some numerical examples.