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A family of Gauss-Kronrod quadrature formulae

Authors: Walter Gautschi and Theodore J. Rivlin
Journal: Math. Comp. 51 (1988), 749-754
MSC: Primary 65D32
MathSciNet review: 958640
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Abstract: We show, for each $ n \geq 1$, that the $ (2n + 1)$-point Kronrod extension of the n-point Gaussian quadrature formula for the measure

$\displaystyle d{\sigma _\gamma }(t) = {(1 + \gamma )^2}{(1 - {t^2})^{1/2}}dt/({(1 + \gamma )^2} - 4\gamma {t^2}),\quad - 1 < \gamma \leq 1,$

has the properties that its $ n + 1$ Kronrod nodes interlace with the n Gauss nodes and all its $ 2n + 1$ weights are positive. We also produce explicit formulae for the weights.

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Keywords: Gauss-Kronrod quadrature, Geronimus measure, indefinite inner product, orthogonal polynomials
Article copyright: © Copyright 1988 American Mathematical Society

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