Generalized qd algorithm and Markov–Bernstein inequalities for Jacobi weight | Numerical Algorithms
Skip to main content

Advertisement

Springer Nature Link
Log in

Generalized qd algorithm and Markov–Bernstein inequalities for Jacobi weight

  • Oroginal Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

The Markov–Bernstein inequalities for the Jacobi measure remained to be studied in detail. Indeed the tools used for obtaining lower and upper bounds of the constant which appear in these inequalities, did not work, since it is linked with the smallest eigenvalue of a five diagonal positive definite symmetric matrix. The aim of this paper is to generalize the qd algorithm for positive definite symmetric band matrices and to give the mean to expand the determinant of a five diagonal symmetric matrix. After that these new tools are applied to the problem to produce effective lower and upper bounds of the Markov–Bernstein constant in the Jacobi case. In the last part we com pare, in the particular case of the Gegenbauer measure, the lower and upper bounds which can be deduced from this paper, with those given in Draux and Elhami (Comput J Appl Math 106:203–243, 1999) and Draux (Numer Algor 24:31–58, 2000).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
¥17,985 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Japan)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aptekarev, A.: Asymptotics of orthogonal polynomials in a neighborhood of the endpoints of the interval of orthogonality. Math. Sb. 183(5), 43–62 (1992) (English version in Russian Academy of Sciences. Sbornik. Mathematics 76(1), 35–50 (1993))

    MATH  Google Scholar 

  2. Draux, A.: Improvement of the formal and numerical estimation of the constant in some Markov–Bernstein inequalities. Numer. Algorithms 24, 31–58 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  3. Draux, A., El Hami, C.: Hermite-Sobolev and closely connected orthogonal polynomials. Comput. J. Appl. Math. 81, 165–179 (1997)

    Article  MATH  Google Scholar 

  4. Draux, A., Elhami, C.: On the positivity of some bilinear functionals in Sobolev spaces. Comput. J. Appl. Math. 106, 203–243 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  5. Draux, A., Kaliaguine, V.: Markov–Bernstein inequalities for generalized Hermite polynomials. East J. Approx. 12, 1–23 (2006)

    MathSciNet  Google Scholar 

  6. Milovanović, G.V., Mitrinović, D.S., Rassias, Th.M.: Topics in Polynomials: Extremal Problems, Inequalities, Zeros. World Scientific, Singapore (1994)

    MATH  Google Scholar 

  7. Rutishauser, H.: Der Quotienten-Differenzen-Algorithmus. Z. Angew. Math. Phys. 5, 233–251 (1954)

    Article  MATH  MathSciNet  Google Scholar 

  8. Rutishauser, H.: Solution of eigenvalue problems with the LR-transformation. NBS Appl. Math. Series 49, 47–81 (1958)

    MathSciNet  Google Scholar 

  9. Wilkinson, J.H.: Convergence of the LR, QR, and related algorithms. Comput. J. 8, 77–84 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  10. Wilkinson, J.H.: The Algebraic Eigenvalue Problem. Oxford University Press, Oxford (1965)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Département de Génie Mathématique, INSA de Rouen, Place Emile Blondel, BP 08, 76131, Mont-Saint-Aignan Cedex, France

    André Draux & Mohamed Sadik

  2. Académie Militaire, 8012, Fondouk-Jedid, Tunisie

    Borhane Moalla

Authors
  1. André Draux
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Borhane Moalla
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mohamed Sadik
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to André Draux.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Draux, A., Moalla, B. & Sadik, M. Generalized qd algorithm and Markov–Bernstein inequalities for Jacobi weight. Numer Algor 51, 429–447 (2009). https://doi.org/10.1007/s11075-008-9241-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-008-9241-4

Keywords

Mathematics Subject Classifications (2000)

Search

Navigation