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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 M. W. Wilson [Math. Comput. 24, 271–282 (1970; Zbl 0219.65028)]. The second approach uses nonnegative least squares methods of C. L. Lawson and 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.
##### MSC:
 65D32 Quadrature and cubature formulas (numerical methods) 41A55 Approximate quadratures 42C05 General theory of orthogonal functions and polynomials 65F20 Overdetermined systems, pseudoinverses (numerical linear algebra)