Skip to main content
Springer Nature Link
Log in

Rational gauss quadrature rules for the approximation of matrix functionals involving stieltjes functions

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

This paper is concerned with computing approximations of matrix functionals of the form \(F(A):={{{\varvec{v}}}}^Tf(A){{{\varvec{v}}}}\), where A is a large symmetric positive definite matrix, \({{{\varvec{v}}}}\) is a vector, and f is a Stieltjes function. We approximate F(A) with the aid of rational Gauss quadrature rules. Associated rational Gauss–Radau and rational anti-Gauss rules are developed. Pairs of rational Gauss and rational Gauss–Radau quadrature rules, or pairs of rational Gauss and rational anti-Gauss quadrature rules, can be used to determine upper and lower bounds, or approximate upper and lower bounds, for F(A). The application of rational Gauss rules, instead of standard Gauss rules, is beneficial when the function f has singularities close to the spectrum of A.

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
$34.99 /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 excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alqahtani, H., Reichel, L.: Simplified Anti-Gauss Quadrature Rules with Applications in Linear Algebra. Numer. Algorithms. 77, 577–602 (2018)

    Article  MathSciNet  Google Scholar 

  2. Alqahtani, H., Reichel, L.: Generalized Block Anti-Gauss Quadrature Rules. Numer. Math. 143, 605–648 (2019)

    Article  MathSciNet  Google Scholar 

  3. Ammar, G.S., Gragg, W.B.: Superfast Solution of Real Positive Definite Toeplitz Systems. SIAM J. Matrix Anal. Appl. 9, 61–76 (1988)

    Article  MathSciNet  Google Scholar 

  4. Beckermann, B., Reichel, L.: Error Estimation and Evaluation of Matrix Functions Via the Faber Transform. SIAM J. Numer. Anal. 47, 3849–3883 (2009)

    Article  MathSciNet  Google Scholar 

  5. Bentbib, A.H., El Ghomari, M., Jagels, C., Jbilou, K., Reichel, L.: The Extended Global Lanczos Method for Matrix Function Approximation. Electron. Trans. Numer. Anal. 50, 144–163 (2018)

    Article  MathSciNet  Google Scholar 

  6. Berg, C.: Stieltjes-Pick-Bernstein-Schoenberg and their Connection to Complete Monotonicity, in Positive Definite Functions. From Schoenberg to Space-Time Challenges, Mateu, S., Porcu, E., eds., Dept. of Math., University Jaume I, Castellón de la Plana, Spain, (2008)

  7. Calvetti, D., Reichel, L., Sgallari, F.: Application of Anti-Gauss Quadrature Rules in Linear Algebra. In: Gautschi, W., Golub, G.H., Opfer, G. (eds.) Applications and Computation of Orthogonal Polynomials, pp. 41–56. Basel, Birkhäuser (1999)

    Chapter  Google Scholar 

  8. Deckers, K., Bultheel, A.: Recurrence and Asymptotics for Orthonormal Rational Functions on an Interval. IMA J. Numer. Anal. 29, 1–23 (2009)

    Article  MathSciNet  Google Scholar 

  9. Deckers, K., Bultheel, A.: The Existence and Construction of Rational Gauss-type Rules. Appl. Math. Comput. 218, 10299–10320 (2012)

    MathSciNet  MATH  Google Scholar 

  10. Fenu, C., Martin, D., Reichel, L., Rodriguez, G.: Block Gauss and Anti-Gauss Quadrature with Application to Networks. SIAM J. Matrix Anal. Appl. 34, 1655–1684 (2013)

    Article  MathSciNet  Google Scholar 

  11. Frommer, A.: Monotone Convergence of the Lanczos Approximations to Matrix Functions of Hermitian Matrices. Electron. Trans. Numer. Anal. 35, 118–128 (2009)

    MathSciNet  MATH  Google Scholar 

  12. Frommer, A., Schweitzer, M.: Error Bounds and Estimates for Krylov Subspace Approximations of Stieltjes Matrix Functions. BIT Numer. Math. 56, 865–892 (2016)

    Article  MathSciNet  Google Scholar 

  13. Gautschi, W.: Orthogonal Polynomials: Computation And Approximation. Oxford University Press, Oxford (2004)

    Book  Google Scholar 

  14. Golub, G.H., Meurant, G.: Matrices, Moments And Quadrature with Applications. Princeton University Press, Princeton (2010)

    MATH  Google Scholar 

  15. Gonchar, A.A., Lagomasino, G. López.: On Markov’s Theorem for Multipoint Padé Approximants. Math. USSR Sb. 34, 449–459 (1978)

  16. Henrici, P.: Applied And Computational Complex Analysis, vol. 2. Wiley, New York (1977)

    MATH  Google Scholar 

  17. Jagels, C., Jbilou, K., Reichel, L.: The Extended Global Lanczos Method, Gauss-Radau Quadrature, and Matrix Function Approximation. J. Comput. Appl. Math. 381, 113027 (2021)

    Article  MathSciNet  Google Scholar 

  18. Jagels, C., Reichel, L.: Recursion Relations for the Extended Krylov Subspace Method. Linear Algebra Appl. 434, 1716–1732 (2011)

    Article  MathSciNet  Google Scholar 

  19. Jagels, C., Reichel, L.: The Structure of Matrices in Rational Gauss Quadrature. Math. Comp. 82, 2035–2060 (2013)

    Article  MathSciNet  Google Scholar 

  20. Keady, G., Khajohnsaksumeth, N., Wiwatanapataphee, B.: On Functions and Inverses, Both Positive, Decreasing and Convex: and Stieltjes Functions. Cog. Math. Stat. 5, 1477543 (2018)

    Article  MathSciNet  Google Scholar 

  21. Laurie, D.P.: Anti-Gaussian Quadrature Formulas. Math. Comp. 65, 735–747 (1996)

    Article  MathSciNet  Google Scholar 

  22. Massei, S., Robol, L.: Rational Krylov for Stieltjes Matrix Functions: Convergence and Pole Selection. BIT Numer. Math. 61, 237–273 (2021)

    Article  MathSciNet  Google Scholar 

  23. Pranić, M.S., Reichel, L.: Generalized Anti-Gauss Quadrature Rules. J. Comput. Appl. Math. 284, 235–243 (2015)

    Article  MathSciNet  Google Scholar 

  24. Pranić, M.S., Reichel, L.: Recurrence Relations for Orthogonal Rational Functions. Numer. Math. 123, 629–642 (2013)

    Article  MathSciNet  Google Scholar 

  25. Pranić, M.S., Reichel, L.: Rational Gauss Quadrature. SIAM J. Numer. Anal. 52, 832–851 (2014)

    Article  MathSciNet  Google Scholar 

  26. Sokol, A.D.: Real-variables Characterization of Generalized Stieltjes Functions. Expo. Math. 28, 179–185 (2010)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the referees for comments. Research by LR was supported in part by NSF grant DMS-1720259.

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Kent State University, Kent, OH 44242, USA

    J. Alahmadi & L. Reichel

  2. Department of Mathematics and Informatics, Faculty of Science, University of Banja Luka, M. Stojanovića 2, 51000, Banja Luka, Bosnia and Herzegovina

    M. Pranić

Authors
  1. J. Alahmadi
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. M. Pranić
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. L. Reichel
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to L. Reichel.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Alahmadi, J., Pranić, M. & Reichel, L. Rational gauss quadrature rules for the approximation of matrix functionals involving stieltjes functions. Numer. Math. 151, 443–473 (2022). https://doi.org/10.1007/s00211-022-01293-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-022-01293-0

Mathematics Subject Classification