Skip to main content
Top
Published in: Numerical Algorithms 3/2020

07-12-2019 | Original Paper

A note on generalized averaged Gaussian formulas for a class of weight functions

Author: Miodrag M. Spalević

Published in: Numerical Algorithms | Issue 3/2020

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

In the recent paper Notaris (Numer. Math., 142:129–147, 2019) 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 Spalević (Math. Comp., 76:1483–1492, 2007). Moreover, as almost immediate consequence of our results from Spalević (Math. Comp.,76:1483–1492, 2007) and that theory, we prove the main statements in Notaris (Numer. Math.,142:129–147, 2019) in a different manner, by means of the Jacobi tridiagonal matrix approach.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Literature
1.
go back to reference Calvetti, D., Golub, G.H., Gragg, W.B., Reichel, L.: Computation of Gauss-Kronrod quadrature rules. Math. Comp. 69, 1035–1052 (2000)MathSciNetCrossRef Calvetti, D., Golub, G.H., Gragg, W.B., Reichel, L.: Computation of Gauss-Kronrod quadrature rules. Math. Comp. 69, 1035–1052 (2000)MathSciNetCrossRef
2.
go back to reference Djukić, D.Lj., Reichel, L., Spalević, M.M.: Truncated generalized averaged Gauss quadrature rules. J. Comput. Appl. Math. 308, 408–418 (2016)MathSciNetCrossRef Djukić, D.Lj., Reichel, L., Spalević, M.M.: Truncated generalized averaged Gauss quadrature rules. J. Comput. Appl. Math. 308, 408–418 (2016)MathSciNetCrossRef
3.
go back to reference Djukić, D.Lj., Reichel, L., Spalević, M.M., Tomanović, J.D.: Internality of the averaged Gaussian quadratures and their truncated variants with Bernstein-Szegö weight functions. Electron. Trans. Numer. Anal. 45, 405–419 (2016)MathSciNetMATH Djukić, D.Lj., Reichel, L., Spalević, M.M., Tomanović, J.D.: Internality of the averaged Gaussian quadratures and their truncated variants with Bernstein-Szegö weight functions. Electron. Trans. Numer. Anal. 45, 405–419 (2016)MathSciNetMATH
4.
go back to reference Djukić, D.Lj., Reichel, L., Spalević, M.M., Tomanović, J.D.: 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)MathSciNetCrossRef Djukić, D.Lj., Reichel, L., Spalević, M.M., Tomanović, J.D.: 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)MathSciNetCrossRef
5.
go back to reference Djukić, D.Lj., Reichel, L., Spalević, M.M.: Internality of generalized averaged Gaussian quadratures and their truncated variants for measures induced by Chebyshev polynomials. Appl. Numer. Math. 142, 190–205 (2019)MathSciNetCrossRef Djukić, D.Lj., Reichel, L., Spalević, M.M.: Internality of generalized averaged Gaussian quadratures and their truncated variants for measures induced by Chebyshev polynomials. Appl. Numer. Math. 142, 190–205 (2019)MathSciNetCrossRef
6.
go back to reference Gauss, C.F.: Methodus nova integralium valores per approximationem inveniendi. Commentationes Societatis Regiae Scientiarum Göttingensis Recentiores, 3. Also in Werke III, 163–196 (1814) Gauss, C.F.: Methodus nova integralium valores per approximationem inveniendi. Commentationes Societatis Regiae Scientiarum Göttingensis Recentiores, 3. Also in Werke III, 163–196 (1814)
8.
go back to reference Gautschi, W.: Orthogonal Polynomials: Computation and Approximation. Oxford University Press, Oxford (2004)MATH Gautschi, W.: Orthogonal Polynomials: Computation and Approximation. Oxford University Press, Oxford (2004)MATH
10.
go back to reference Gautschi, W., Notaris, S.E.: Stieltjes polynomials and related quadrature formulae for a class of weight functions. Math. Comp. 65, 1257–1268 (1996)MathSciNetCrossRef Gautschi, W., Notaris, S.E.: Stieltjes polynomials and related quadrature formulae for a class of weight functions. Math. Comp. 65, 1257–1268 (1996)MathSciNetCrossRef
12.
go back to reference Jagels, C., Reichel, L., Tang, T.: Generalized averaged Szegő quadrature rules. J. Comput. Appl. Math. 311, 645–654 (2017)MathSciNetCrossRef Jagels, C., Reichel, L., Tang, T.: Generalized averaged Szegő quadrature rules. J. Comput. Appl. Math. 311, 645–654 (2017)MathSciNetCrossRef
13.
go back to reference Kahaner, D.K., Monegato, G.: Nonexistence of extended Gauss-Laguerre and Gauss-Hermite quadrature rules with positive weights. Z. Angew. Math. Phys. 29, 983–986 (1978)MathSciNetCrossRef Kahaner, D.K., Monegato, G.: Nonexistence of extended Gauss-Laguerre and Gauss-Hermite quadrature rules with positive weights. Z. Angew. Math. Phys. 29, 983–986 (1978)MathSciNetCrossRef
14.
17.
go back to reference 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)MathSciNetCrossRef 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)MathSciNetCrossRef
18.
go back to reference Notaris, S.E.: Anti-Gaussian quadrature dormulae based on the zeros of Stieltjes polynomials. BIT 58, 179–198 (2018)MathSciNetCrossRef Notaris, S.E.: Anti-Gaussian quadrature dormulae based on the zeros of Stieltjes polynomials. BIT 58, 179–198 (2018)MathSciNetCrossRef
19.
go back to reference Notaris, S.E.: Stieltjes polynomials and related quadrature formulae for a class of weight functions, II. Numer. Math. 142, 129–147 (2019)MathSciNetCrossRef Notaris, S.E.: Stieltjes polynomials and related quadrature formulae for a class of weight functions, II. Numer. Math. 142, 129–147 (2019)MathSciNetCrossRef
20.
go back to reference 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) 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)
21.
22.
go back to reference Peherstorfer, F., Petras, K.: Ultraspherical Gauss-Kronrod quadrature is not possible for λ > 3. SIAM J. Numer. Anal. 37, 927–948 (2000)MathSciNetCrossRef Peherstorfer, F., Petras, K.: Ultraspherical Gauss-Kronrod quadrature is not possible for λ > 3. SIAM J. Numer. Anal. 37, 927–948 (2000)MathSciNetCrossRef
23.
go back to reference Peherstorfer, F., Petras, K.: Stieltjes polynomials and Gauss-Kronrod quadrature for Jacobi weight functions. Numer. Math. 95, 689–706 (2003)MathSciNetCrossRef Peherstorfer, F., Petras, K.: Stieltjes polynomials and Gauss-Kronrod quadrature for Jacobi weight functions. Numer. Math. 95, 689–706 (2003)MathSciNetCrossRef
24.
go back to reference Reichel, L., Rodriguez, G., Tang, T.: New block quadrature rules for the approximation of matrix functions. Linear Algebra Appl. 502, 299–326 (2016)MathSciNetCrossRef Reichel, L., Rodriguez, G., Tang, T.: New block quadrature rules for the approximation of matrix functions. Linear Algebra Appl. 502, 299–326 (2016)MathSciNetCrossRef
25.
go back to reference Reichel, L., Spalević, M.M., Tang, T.: Generalized averaged Gauss quadrature rules for the approximation of matrix functionals. BIT 56, 1045–1067 (2016)MathSciNetCrossRef Reichel, L., Spalević, M.M., Tang, T.: Generalized averaged Gauss quadrature rules for the approximation of matrix functionals. BIT 56, 1045–1067 (2016)MathSciNetCrossRef
29.
go back to reference Wilf, H.S.: Mathematics for the Physical Sciences. Wiley, New York (1962)MATH Wilf, H.S.: Mathematics for the Physical Sciences. Wiley, New York (1962)MATH
Metadata
Title
A note on generalized averaged Gaussian formulas for a class of weight functions
Author
Miodrag M. Spalević
Publication date
07-12-2019
Publisher
Springer US
Published in
Numerical Algorithms / Issue 3/2020
Print ISSN: 1017-1398
Electronic ISSN: 1572-9265
DOI
https://doi.org/10.1007/s11075-019-00848-x

Other articles of this Issue 3/2020

Numerical Algorithms 3/2020 Go to the issue

Premium Partner