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Rates of convergence of Gaussian quadrature for singular integrands


Authors: D. S. Lubinsky and P. Rabinowitz
Journal: Math. Comp. 43 (1984), 219-242
MSC: Primary 65D30
DOI: https://doi.org/10.1090/S0025-5718-1984-0744932-2
MathSciNet review: 744932
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Abstract: The authors obtain the rates of convergence (or divergence) of Gaussian quadrature on functions with an algebraic or logarithmic singularity inside, or at an endpoint of, the interval of integration. A typical result is the following: For a bounded smooth weight function on $ [ - 1,1]$, the error in n-point Gaussian quadrature of $ f(x) = \vert x - y{\vert^{ - \delta }}$ is $ O({n^{ - 2 + 2\delta }})$ if $ y = \pm 1$ and $ O({n^{ - 1 + \delta }})$ if $ y \in ( - 1,1)$, provided we avoid the singularity. If we ignore the singularity y, the error is $ O({n^{ - 1 + 2\delta }}{(\log n)^\delta }{(\log \log n)^{\delta (1 + \varepsilon )}})$ for almost all choices of y. These assertions are sharp with respect to order.


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DOI: https://doi.org/10.1090/S0025-5718-1984-0744932-2
Article copyright: © Copyright 1984 American Mathematical Society

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