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Gaussian quadrature; orthogonal polynomials; Kronrod extensions; Patterson sequences; imbedded quadratures; Laguerre weight function; Hermite weight function; quadrature formulae; maximal polynomial order of precision
We present algorithms for the determination of polynomials orthogonal with respect to a positive weight function multiplied by a polynomial with simple roots inside the interval of integration. We apply these algorithms to search for and calculate all possible sequences of imbedded quadratures of maximal polynomials order of precision for the generalized Laguerre and Hermite weight functions.
[1] S. Elhay, J. Kautsky: A method for computing quadratures of the Kronrod-Patterson type. Australian Computer Science Communications 6 (1) (1984), 15.1-15.8.
[2] G. H. Golub, J. Kautsky: Calculation of Gauss quadratures with multiple free and fixed knots. Numer. Math. 41 (1983), 147-163. DOI 10.1007/BF01390210 | MR 0703119 | Zbl 0525.65010
[3] D. K. Kahaner J. Waldvogel, and L. W. Fullerton: Addition of points to Gauss-Laguerre quadrature formulae. SIAM J. Sci. Stat. Comput. 5 (1984), 42-55. DOI 10.1137/0905003 | MR 0731880
[4] J. Kautsky, S. Elhay: Gauss quadratures and Jacobi matrices for weight functions not of one sign. Math. Соmр. 43 (168) (1984), 543-550. MR 0758201 | Zbl 0556.65012
[5] A. S. Kronrod: Nodes and weights for quadrature formulae. Sixteen place tables. Nauka, Moscow; English translation, Consultants Bureau, New York. MR 0183116
[6] G. Monegato: Stieltjes polynomials, and related quadrature rules. SIAM Review 24(2) (1982), 137-158. DOI 10.1137/1024039 | MR 0652464 | Zbl 0494.33010
[7] T. N. L. Patterson: The optimal addition of points to quadrature formulae. Math. Соmр. 22 (1968), 847-856. MR 0242370
[8] R. Piessens, M. Branders: A note on the optimal addition of abscissae to quadrature formulae of the Gauss-Lobatto type. Math. Соmр. 28 (1974), 135-139. MR 0343552
[9] P. Turan: On the theory of mechanical quadrature. Acta. Sci. Math. (Szeged) 12 (1950), 30-37. MR 0036797
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