An overview of results on the existence or nonexistence and the error term of Gauss-Kronrod quadrature formulae

Kronrod in 1964, trying to estimate economically the error of the n-point Gauss-Legendre quadrature formula, developed a new formula by adding to the n Gauss nodes n + 1 new ones, which are determined, together with all weights, such that the new formula has maximum degree of exactness. It turns out that the new nodes are the zeros of a polynomial orthogonal with respect to a variable-sign weight function. This polynomial was considered for the first time by Stieltjes in 1894. Important for the new formula, now appropriately called the Gauss-Kronrod quadrature formula, are properties such as the interlacing of the Gauss nodes with the new nodes, the inclusion of all nodes in the interior of the interval of integration, and the positivity of all quadrature weights. We review, for classical and nonclassical weight functions, the existence or nonexistence and the error term of Gauss-Kronrod formulae having one or more of the aforementioned properties.

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