A note on generalized averaged Gaussian formulas for a class of weight functions. (English) Zbl 1453.65056
Summary: In the recent paper [S. E. Notaris, Numer. Math. 142, No. 1, 129–147 (2019; Zbl 1411.41022)] it has been introduced a new and useful class of nonnegative measures for which the well-known Gauss-Kronrod quadrature formulae coincide with the generalized averaged Gaussian quadrature formulas. In such a case, the given generalized averaged Gaussian quadrature formulas are of the higher degree of precision, and can be numerically constructed by an effective and simple method; see [M. M. Spalević, Math. Comput. 76, No. 259, 1483–1492 (2007; Zbl 1113.65025)]. Moreover, as almost immediate consequence of our results from [Spalević, loc. cit.] and that theory, we prove the main statements in [Notaris, loc. cit.] in a different manner, by means of the Jacobi tridiagonal matrix approach.
MSC:
| 65D32 | Numerical quadrature and cubature formulas |
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
Software:
OPQReferences:
| [1] | Calvetti, D.; Golub, GH; Gragg, WB; Reichel, L., Computation of Gauss-Kronrod quadrature rules, Math. Comp., 69, 1035-1052 (2000) · Zbl 0947.65022 · doi:10.1090/S0025-5718-00-01174-1 |
| [2] | Djukić, DLj; Reichel, L.; Spalević, MM, Truncated generalized averaged Gauss quadrature rules, J. Comput. Appl. Math., 308, 408-418 (2016) · Zbl 1346.65009 · doi:10.1016/j.cam.2016.06.016 |
| [3] | Djukić, DLj; Reichel, L.; Spalević, MM; Tomanović, JD, Internality of the averaged Gaussian quadratures and their truncated variants with Bernstein-Szegö weight functions, Electron. Trans. Numer. Anal., 45, 405-419 (2016) · Zbl 1355.65043 |
| [4] | Djukić, DLj; Reichel, L.; Spalević, MM; Tomanović, JD, Internality of generalized averaged Gaussian quadrature rules and their truncated variants for modified Chebyshev measures of the second kind, J. Comput. Appl. Math., 345, 70-85 (2019) · Zbl 1401.65027 · doi:10.1016/j.cam.2018.06.017 |
| [5] | Djukić, DLj; Reichel, L.; Spalević, MM, Internality of generalized averaged Gaussian quadratures and their truncated variants for measures induced by Chebyshev polynomials, Appl. Numer. Math., 142, 190-205 (2019) · Zbl 1416.65074 · doi:10.1016/j.apnum.2019.03.008 |
| [6] | Gauss, C.F.: Methodus nova integralium valores per approximationem inveniendi. Commentationes Societatis Regiae Scientiarum Göttingensis Recentiores, 3. Also in Werke III, 163-196 (1814) |
| [7] | Gautschi, W., On generating orthogonal polynomials, SIAM J. Sci. Stat. Comput., 3, 289-317 (1982) · Zbl 0482.65011 · doi:10.1137/0903018 |
| [8] | Gautschi, W., Orthogonal Polynomials: Computation and Approximation (2004), Oxford: Oxford University Press, Oxford · Zbl 1130.42300 |
| [9] | Gautschi, W.: OPQ: A MATLAB suite of programs for generating orthogonal polynomials and related quadrature rules. http://www.cs.purdue.edu/archives/2001/wxg/codes |
| [10] | Gautschi, W.; Notaris, SE, Stieltjes polynomials and related quadrature formulae for a class of weight functions, Math. Comp., 65, 1257-1268 (1996) · Zbl 0853.41019 · doi:10.1090/S0025-5718-96-00732-6 |
| [11] | Golub, GH; Welsch, JH, Calculation of Gauss quadrature rules, Math. Comp., 23, 221-230 (1969) · Zbl 0179.21901 · doi:10.1090/S0025-5718-69-99647-1 |
| [12] | Jagels, C.; Reichel, L.; Tang, T., Generalized averaged Szegő quadrature rules, J. Comput. Appl. Math., 311, 645-654 (2017) · Zbl 1382.65070 · doi:10.1016/j.cam.2016.08.038 |
| [13] | Kahaner, DK; Monegato, G., Nonexistence of extended Gauss-Laguerre and Gauss-Hermite quadrature rules with positive weights, Z. Angew. Math. Phys., 29, 983-986 (1978) · Zbl 0399.41027 · doi:10.1007/BF01590820 |
| [14] | Kronrod, AS, Integration with control of accuracy, Soviet Phys. Dokl., 9, 17-19 (1964) · Zbl 0131.15003 |
| [15] | Laurie, DP, Anti-Gaussian quadrature formulas, Math. Comp., 65, 739-747 (1996) · Zbl 0843.41020 · doi:10.1090/S0025-5718-96-00713-2 |
| [16] | Laurie, DP, Calculation of Gauss-Kronrod quadrature rules, Math. Comp., 66, 1133-1145 (1997) · Zbl 0870.65018 · doi:10.1090/S0025-5718-97-00861-2 |
| [17] | Máté, A.; Nevai, P.; Van Assche, W., The supports of measures associated with orthogonal polynomials and the spectra of the related self-adjoint operators, Rocky Mountain J. Math., 21, 501-527 (1991) · Zbl 0731.42021 · doi:10.1216/rmjm/1181073020 |
| [18] | Notaris, SE, Anti-Gaussian quadrature dormulae based on the zeros of Stieltjes polynomials, BIT, 58, 179-198 (2018) · Zbl 06858774 · doi:10.1007/s10543-017-0672-y |
| [19] | Notaris, SE, Stieltjes polynomials and related quadrature formulae for a class of weight functions, II, Numer. Math., 142, 129-147 (2019) · Zbl 1411.41022 · doi:10.1007/s00211-018-1009-8 |
| [20] | Peherstorfer, F.: On positive quadrature formulas. In: Brass, H., Hämmerlin, G. (eds.) Numerical Integration IV, Intern. Ser. Numer. Math. # 112, pp 297-313. Basel, Birkhäuser (1993) · Zbl 0791.41025 |
| [21] | Peherstorfer, F., Positive quadrature formulas III: Asymptotics of weights, Math. Comp., 77, 2241-2259 (2008) · Zbl 1198.65051 · doi:10.1090/S0025-5718-08-02119-4 |
| [22] | Peherstorfer, F.; Petras, K., Ultraspherical Gauss-Kronrod quadrature is not possible for λ > 3, SIAM J. Numer. Anal., 37, 927-948 (2000) · Zbl 0947.33003 · doi:10.1137/S0036142998327744 |
| [23] | Peherstorfer, F.; Petras, K., Stieltjes polynomials and Gauss-Kronrod quadrature for Jacobi weight functions, Numer. Math., 95, 689-706 (2003) · Zbl 1035.41020 · doi:10.1007/s00211-002-0412-2 |
| [24] | Reichel, L.; Rodriguez, G.; Tang, T., New block quadrature rules for the approximation of matrix functions, Linear Algebra Appl., 502, 299-326 (2016) · Zbl 1338.65122 · doi:10.1016/j.laa.2015.07.007 |
| [25] | Reichel, L.; Spalević, MM; Tang, T., Generalized averaged Gauss quadrature rules for the approximation of matrix functionals, BIT, 56, 1045-1067 (2016) · Zbl 1352.65089 · doi:10.1007/s10543-015-0592-7 |
| [26] | Spalević, MM, On generalized averaged Gaussian formulas, Math. Comp., 76, 1483-1492 (2007) · Zbl 1113.65025 · doi:10.1090/S0025-5718-07-01975-8 |
| [27] | Spalević, MM, A note on generalized averaged Gaussian formulas, Numer. Algor., 46, 253-264 (2007) · Zbl 1131.65029 · doi:10.1007/s11075-007-9137-8 |
| [28] | Spalević, MM, On generalized averaged Gaussian formulas. II, Math. Comp., 86, 1877-1885 (2017) · Zbl 1361.65012 · doi:10.1090/mcom/3225 |
| [29] | Wilf, HS, Mathematics for the Physical Sciences (1962), New York: Wiley, New York · Zbl 0105.26803 |
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