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Zbl 1170.65016
Huybrechs, Daan
Stable high-order quadrature rules with equidistant points. (English)
[J] J. Comput. Appl. Math. 231, No. 2, 933-947 (2009). ISSN 0377-0427

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.
[Manfred Tasche (Rostock)]

MSC 2000:: *65D32 Quadrature formulas (numerical methods)
41A55 Approximate quadratures
42C05 General theory of orthogonal functions and polynomials
65F20 Overdetermined systems (numerical linear algebra)

Keywords: numerical examples; stable high-order quadrature rules; discrete least squares approximation; discrete orthogonal polynomials; Newton-Cotes quadrature rules; equidistant quadrature points; stability

Citations: Zbl 0219.65028; Zbl 0860.65029

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