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Stable high-order quadrature rules with equidistant points. (English) Zbl 1170.65016
Newton-Cotes quadrature rules become unstable for high orders. In this paper, the author reviews two techniques to construct stable high-order quadrature rules using $N$ equidistant quadrature points. The first method is based on results of {\it M. W. Wilson} [Math. Comput. 24, 271--282 (1970; Zbl 0219.65028)]. The second approach uses nonnegative least squares methods of {\it C. L. Lawson} and {\it R. J. Hanson} [Solving least squares problems, SIAM Philadelphia (1995; Zbl 0860.65029)]. The stability follows from the fact that all weights are positive. These results can be achieved in the case $N\sim d^2$, where $d$ is the polynomial order of accuracy. Then the computed approximation corresponds implicitly to the integral of a (discrete) least squares approximation of the (sampled) integrand. The author shows how the underlying discrete least squares approximation can be optimized for the numerical integration. Numerical tests are presented.

65D32Quadrature and cubature formulas (numerical methods)
41A55Approximate quadratures
42C05General theory of orthogonal functions and polynomials
65F20Overdetermined systems, pseudoinverses (numerical linear algebra)
Full Text: DOI
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