Weighted quadrature formulae of Gauss-Christoffel-Stancu type

Authors

  • Dimitrie D. Stancu “Babes-Bolyai” University, Cluj-Napoca, Romania
  • Ioana Taşcu North University, Baia Mare, Romania
  • Alina Beian-Putura Computer Science Highschool, Bistrita, Romania

DOI:

https://doi.org/10.33993/jnaat322-751

Keywords:

weighted quadrature formulae, multiple fixed nodes and simple Gaussian nodes, Stancu method of parameters, Christoffel--Szegő formula
Abstract views: 252

Abstract

In the present paper we consider weighted integrals and develop explicit quadrature formulae of Gauss-Christoffel-Stancu type using simple Gaussian nodes and multiple fixed nodes. Given the multiple fixed nodes and their multiplicities, we present some algorithms for finding the Gaussian nodes, the coefficients and the remainders of the corresponding quadrature formulae. Several illustrative examples are presented in the case of some classical weight functions.

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References

Christoffel, E. B., Über die Gaussiche Quadratur und eine Veralgemeinerung derselben, J. Reine Angew. Math., 55, pp. 61-82, 1858, https://doi.org/10.1515/crll.1858.55.61. DOI: https://doi.org/10.1515/crll.1858.55.61

Christoffel, E. B., Sur une classe particulière de fonctions entières et de fractions continues, Ann. Mat. Pura Appl., 2, 8, pp. 1-10, 1877, https://doi.org/10.1007/BF02420775. DOI: https://doi.org/10.1007/BF02420775

Deruyts. J., Sur le calcul approché de certaines intégrales définies, Bull. Acad. Roy. Belgique, (3) 11, pp. 307-311, 1986.

Gautschi, W., Construction of Gauss-Christoffel quadrature formulas, Math. Comp., 22, pp. 251-270, 1968, https://doi.org/10.2307/2004654. DOI: https://doi.org/10.1090/S0025-5718-1968-0228171-0

Gautschi, W., A survey of Gauss-Christoffel quadrature formulae, in E. B. Christoffel: The influence of his work on mathematics and the physical sciences, edit. by P. Butzer, F. Fehér, Birkhäuser, Basel, pp. 72-147, 1981. DOI: https://doi.org/10.1007/978-3-0348-5452-8_6

Gautschi, W., Recognition of Christoffel work on quadrature during and after his lifetime, ibid., pp. 724-727, 1981. DOI: https://doi.org/10.1007/978-3-0348-5452-8_58

Jacobi, C. G. J., Über Gauss neue Methode, die Werthe der Integrale näherungsweise zu finden, J. Reine Angew. Math., 1, pp. 301-308, 1826, https://doi.org/10.1515/crll.1826.1.301. DOI: https://doi.org/10.1515/crll.1826.1.301

Krylov, V. I., Approximate Calculation of Integrals, McMillan, New York, 1962.

Markov, A., Sur la méthode de Gauss pour le calcul approché des intégrales, Math. Ann., 25, pp. 427-432, 1885, https://doi.org/10.1007/BF01443287. DOI: https://doi.org/10.1007/BF01443287

Mehler, F. G., Bemerkungen zur Theorie der mecanischen Quadraturen, J. Reine Angew. Math., 63, pp. 152-157, 1864, https://doi.org/10.1515/crll.1864.63.152. DOI: https://doi.org/10.1515/crll.1864.63.152

Possé, C., Sur les quadratures, Nouri Ann. Math., (2) 14, pp. 147-156, 1875.

Stancu, D. D., Generalizarea formulei de cuadratură a lui Gauss-Christoffel, Acad. R. P. Rom., Fil. Iaşi, Stud. Cerc. Sti., 8, pp. 1-18, 1957 (in Romanian).

Stancu, D. D., On a class of orthogonal polynomials and on some general quadrature formulae with minimum number of terms, Bull. Math. Soc. Sci. Math. Phys. R. P. Roumaine (N. 5.), 1, no. 49, pp. 479-498, 1957.

Stancu, D. D., O metodă pentru construirea de formule de cuadratură de grad înalt de exactitate, Comunic. Acad. R. P. Rom., 8, pp. 349-358, 1958 (in Romanian).

Stancu, D. D., Sur quelques formules générales de quadrature du type Gauss-Christoffel, Mathematica (Cluj), 1(24), pp. 167-182, 1959.

Stancu, D. D. and Stroud, A. H., Quadrature formulas with simple Gaussian nodes and multiple fixed nodes, Math. Comp., 17, pp. 384-394, 1963. DOI: https://doi.org/10.1090/S0025-5718-1963-0157485-3

Stieltjes, T. J., Quelques recherches sur la théorie des quadratures dites mécaniques, Ann. Sci. Ec. Norm. Paris, Sér. 3, 1, pp. 409-426, 1884. DOI: https://doi.org/10.24033/asens.245

Ssegö, G., Über die Entwickelungen einer analytischen Funktion nach dem Polynomen eines Orthogonalsystems, Math. Ann., 82, pp. 188-212, 1921. DOI: https://doi.org/10.1007/BF01498664

Turan, P., On the theory of the mechanical quadrature, Acta Sci. Math. Szeged, 12, pp. 30-37, 1950.

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Published

2003-08-01

How to Cite

Stancu, D. D., Taşcu, I., & Beian-Putura, A. (2003). Weighted quadrature formulae of Gauss-Christoffel-Stancu type. Rev. Anal. Numér. Théor. Approx., 32(2), 223–234. https://doi.org/10.33993/jnaat322-751

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