Skip to main content
SpringerLink
Log in

Estimates of the trace of the inverse of a symmetric matrix using the modified Chebyshev algorithm

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

Abstract

In this paper we study how to compute an estimate of the trace of the inverse of a symmetric matrix by using Gauss quadrature and the modified Chebyshev algorithm. As auxiliary polynomials we use the shifted Chebyshev polynomials. Since this can be too costly in computer storage for large matrices we also propose to compute the modified moments with a stochastic approach due to Hutchinson (Commun Stat Simul 18:1059–1076, 1989).

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

Access this article

Log in via an institution

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. Bai, Z., Fahey, M., Golub, G.H.: Some large scale matrix computation problems. Comp. J. Appl. Math. 74, 71–89 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bai, Z., Fahey, M., Golub, G.H., Menon, M., Richter, E.: Computing partial eigenvalue sum in electronic structure calculation. Report SCCM 98-03, Stanford University (1998)

  3. Bai, Z., Golub, G.H.: Bounds for the trace of the inverse and the determinant of symmetric positive definite matrices. Ann. Numer. Math. 4, 29–38 (1997)

    MATH  MathSciNet  Google Scholar 

  4. Bai, Z., Golub, G.H.: Some unusual matrix eigenvalue problems. In: Palma, J., Dongarra, J., Hernandez, V. (eds.) Proceedings of Vecpar’98—Third International Conference for Vector and Parallel Processing, pp. 4–19. Springer, New York (1999)

    Google Scholar 

  5. Gautschi, W.: Orthogonal Polynomials: Computation and Approximation. Oxford University Press, Oxford (2004)

    MATH  Google Scholar 

  6. Golub, G.H., Meurant, G.: Matrices, moments and quadrature. In: Griffiths, D.F., Watson, G.A. (eds.) Numerical Analysis 1993. Pitman Research Notes in Mathematics, vol. 303, pp. 105–156. Chapman & Hall, London (1994)

    Google Scholar 

  7. Golub, G.H., Meurant, G.: Matrices, Moments and Quadrature with Applications. Princeton University Press, Princeton (2009)

    Google Scholar 

  8. Golub, G.H., Welsch, J.H.: Calculation of Gauss quadrature rules. Math. Comp. 23, 221–230 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  9. Hutchinson, M.: A stochastic estimator of the trace of the influence matrix for Laplacian smoothing splines. Commun. Stat. Simul. 18, 1059–1076 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  10. Sack, R.A., Donovan, A.: An algorithm for Gaussian quadrature given modified moments. Numer. Math. 18(5), 465–478 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  11. Wheeler, J.C.: Modified moments and Gaussian quadrature. In: Proceedings of the international conference on Padé approximants, continued fractions and related topics, Univ. Colorado, Boulder. Rocky Mt. J. Math. 4(2), 287–296 (1974)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. 30, rue du Sergent Bauchat, 75012, Paris, France

    Gérard Meurant

Authors
  1. Gérard Meurant
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gérard Meurant.

Additional information

In memory of Gene H. Golub.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Meurant, G. Estimates of the trace of the inverse of a symmetric matrix using the modified Chebyshev algorithm. Numer Algor 51, 309–318 (2009). https://doi.org/10.1007/s11075-008-9246-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-008-9246-z

Keywords

Search

Navigation