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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.

MSC:
65D32Quadrature and cubature formulas (numerical methods)
41A55Approximate quadratures
42C05General theory of orthogonal functions and polynomials
65F20Overdetermined systems, pseudoinverses (numerical linear algebra)
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References:
[1] Stroud, A. H.: Approximate calculation of multiple integrals, (1971) · Zbl 0379.65013
[2] Engels, H.: Numerical quadrature and cubature, (1980) · Zbl 0435.65013
[3] Davis, P. J.; Rabinowitz, P.: Methods of numerical integration, (1984) · Zbl 0537.65020
[4] Evans, G. A.: Practical numerical integration, (1993) · Zbl 0811.65015
[5] Krommer, A. R.; Ueberhuber, C. W.: Computational integration, (1998) · Zbl 0903.65019
[6] Morton, K. W.; Mayers, D. F.: Numerical solution of partial differential equations, (1994) · Zbl 0811.65063
[7] Boyd, J. P.: Chebyshev and Fourier spectral methods, (2001) · Zbl 0994.65128
[8] Brunner, H.: Collocation methods for Volterra integral and related functional equations, Cambridge monographs on applied and computational mathematics 15 (2004) · Zbl 1059.65122
[9] Tchakaloff, V.: Formules de cubature mécaniques à coefficients non négatifs, Bull. sci. Math. 81, 123-134 (1957) · Zbl 0079.13908
[10] Davis, P. J.: A construction of nonnegative approximate quadrature, Math. comp. 21, 578-582 (1967) · Zbl 0189.16401 · doi:10.2307/2005001
[11] Cools, R.: Constructing cubature formulae: the science behind the art, Acta numer. 6, 1-54 (1997) · Zbl 0887.65028
[12] Steinitz, E.: Über bedingt konvergente reihen und konvexe systeme, J. reine angew. Math. 143, 128-175 (1913) · Zbl 44.0287.01 · doi:10.1515/crll.1913.143.128 · crelle:GDZPPN002167859
[13] Wilson, M. W.: Necessary and sufficient conditions for equidistant quadrature formula, SIAM J numer. Anal. 7, No. 1, 134-141 (1970) · Zbl 0197.04901 · doi:10.1137/0707009
[14] Wilson, M. W.: Discrete least squares and quadrature formulas, Math. comp. 24, No. 110, 271-282 (1970) · Zbl 0219.65028 · doi:10.2307/2004477
[15] Lawson, C. L.; Hanson, R. J.: Solving least squares problems, (1996) · Zbl 0860.65028
[16] Karlin, S.; Studden, W.: Tchebysheff systems with applications in analysis and statistics, (1966) · Zbl 0153.38902
[17] Pólya, G.: Über die konvergenz von quadraturverfahren, Math. Z. 37, 264-286 (1933) · Zbl 0007.00703 · doi:10.1007/BF01474574
[18] Davis, P. J.: On a problem in the theory of mechanical quadratures, Pacific J. Math. 5, 669-674 (1955) · Zbl 0065.29401
[19] Golub, G. H.; Van Loan, C. F.: Matrix computations, (1996) · Zbl 0865.65009
[20] Alpert, B. K.: Hybrid Gauss-trapezoidal quadrature rules, SIAM J. Sci. comput. 20, No. 5, 1551-1584 (1999) · Zbl 0933.41019 · doi:10.1137/S1064827597325141
[21] Forsythe, G. E.: Generation and use of orthogonal polynomials for data-Fitting with a digital computer, J. SIAM 5, No. 2, 74-88 (1956) · Zbl 0083.35503 · doi:10.1137/0105007
[22] Gautschi, W.: Orthogonal polynomials: computation and approximation, (2004) · Zbl 1130.42300
[23] Gautschi, W.: Moments in quadrature problems, Comput. math. Appl. 33, 105-118 (1997) · Zbl 0930.65011 · doi:10.1016/S0898-1221(96)00223-4
[24] Gautschi, W.: Is the recurrence relation for orthogonal polynomials always stable?, Bit 33, No. 2, 277-284 (1993) · Zbl 0783.65009 · doi:10.1007/BF01989750
[25] Gautschi, W.: The computation of special functions by linear difference equations, Advances in difference equations, 213-243 (1997) · Zbl 0892.65008
[26] Trefethen, L. N.: Is Gauss quadrature better than clenshaw-curtis?, SIAM rev. 50, No. 1, 67-87 (2008) · Zbl 1141.65018 · doi:10.1137/060659831
[27] Davis, P. J.: Approximate integration rules with nonnegative weights, Lectures in differential equations 2, 233-256 (1969) · Zbl 0187.40104