Skip to main content
SpringerLink
Log in

Bounds for the Entries of Matrix Functions with Applications to Preconditioning

  • Published:
BIT Numerical Mathematics Aims and scope Submit manuscript

Abstract

Let A be a symmetric matrix and let f be a smooth function defined on an interval containing the spectrum of A. Generalizing a well-known result of Demko, Moss and Smith on the decay of the inverse we show that when A is banded, the entries of f(A)are bounded in an exponentially decaying manner away from the main diagonal. Bounds obtained by representing the entries of f(A)in terms of Riemann-Stieltjes integrals and by approximating such integrals by Gaussian quadrature rules are also considered. Applications of these bounds to preconditioning are suggested and illustrated by a few numerical examples.

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. O. Axelsson, Iterative Solution Methods, Cambridge University Press, Cambridge, 1994.

    Google Scholar 

  2. O. Axelsson and B. Polman, On approximate factorization methods for block matrices suitable for vector and parallel processors, Linear Algebra Appl., 77 (1986), pp. 3–26.

    Google Scholar 

  3. Z. Bai, M. Fahey, and G. H. Golub, Some large-scale matrix computation problems, J. Comput. Appl. Math., 74 (1996), pp. 71–89.

    Google Scholar 

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

    Google Scholar 

  5. M. Benzi and G. H. Golub, Bounds for the entries of matrix functions with applications to preconditioning, Stanford University Report SCCM–98–04, February 1998.

  6. M. Benzi, C. D. Meyer, Jr., and M. Tůma, A sparse approximate inverse preconditioner for the conjugate gradient method, SIAM J. Sci. Comput., 17 (1996), pp. 1135–1149.

    Google Scholar 

  7. S. N. Bernstein, Leçons sur les Propriétés Extrémales et la Meilleure Approximation des Fonctions Analytiques d'une Variable Réelle, Gauthier-Villars, Paris, 1926.

    Google Scholar 

  8. P. Castillo and Y. Saad, Preconditioning the matrix exponential operator with applications, University of Minnesota Supercomputing Institute Technical Report UMSI 97/142, Minneapolis, MN, 1997.

  9. R. H. Chan and G. Strang, Toeplitz equations by conjugate gradients with circulant preconditioner, SIAM J. Sci. Stat. Comput., 10 (1989), pp. 104–119.

    Google Scholar 

  10. T. Chan, An optimal circulant preconditioner for Toeplitz systems, SIAM J. Sci. Stat. Comput., 9 (1998), pp. 766–771.

    Google Scholar 

  11. E. Chow and Y. Saad, Approximate inverse preconditioners via sparse-sparse iterations, SIAM J. Sci. Comput., 19 (1998), pp. 995–1023.

    Google Scholar 

  12. P. Concus, G. H. Golub, and G. Meurant, Block preconditioning for the conjugate gradient method, SIAM J. Sci. Stat. Comput., 6 (1985), pp. 220–252.

    Google Scholar 

  13. P. Concus and G. Meurant, On computing INV block preconditionings for the conjugate gradient method, BIT, 26 (1986), pp. 493–504.

    Google Scholar 

  14. P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, 2nd ed., Academic Press, New York, 1984.

    Google Scholar 

  15. G. Dahlquist, S. C. Eisenstat and G. H. Golub, Bounds for the error of linear systems using the theory of moments, J. Math. Anal. Appl., 37 (1972), pp. 151–166.

    Google Scholar 

  16. G. Dahlquist, G. H. Golub, and S. G. Nash, Bounds for the error in linear systems, in Proceedings of the Workshop on Semi-Infinite Programming, R. Hettich, ed., Springer-Verlag, New York, 1978, pp. 154–172.

    Google Scholar 

  17. S. Demko, Inverses of band matrices and local convergence of spline projections, SIAM J. Numer. Anal., 14 (1977), pp. 616–619.

    Google Scholar 

  18. S. Demko, W. F. Moss, and P. W. Smith, Decay rates for inverses of band matrices, Math. Comp., 43 (1984), pp. 491–499.

    Google Scholar 

  19. V. L. Druskin and L. A. Knizhnerman, Two polynomial methods of calculating functions of symmetric matrices, USSR Comput. Maths. Math. Phys., 29 (1989), pp. 112–121.

    Google Scholar 

  20. M. Fiedler and H. Schneider, Analytic functions of M-matrices and generalizations, Linear and Multilinear Algebra, 13 (1983), pp. 185–201.

    Google Scholar 

  21. F. R. Gantmacher, The Theory of Matrices, Vol. I, Chelsea, New York, 1959.

  22. G. H. Golub, Some uses of the Lanczos algorithm in numerical linear algebra, in Topics in Numerical Analysis, J. H. Miller, ed., Academic Press, New York, 1973, pp. 173–184.

    Google Scholar 

  23. G. H. Golub, Bounds for matrix moments, Rocky Mountain J. Math., 4 (1974), pp. 207–211.

    Google Scholar 

  24. G. H. Golub and G. Meurant, Matrices, moments and quadratures, in Numerical Analysis 1993, D. F. Griffiths and G. A. Watson, eds., Pitman Research Notes in Mathematics, Vol. 303, Essex, England, 1994, pp. 105–156.

  25. G. H. Golub and G. Meurant, Matrices, moments and quadratures II; how to compute the norm of the error in iterative methods, BIT, 37 (1997), pp. 687–705.

    Google Scholar 

  26. G. H. Golub and Z. Strakoš, Estimates in quadratic formulas, Numer. Algorithms, 8 (1994), pp. 241–268.

    Google Scholar 

  27. G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed., The Johns Hopkins University Press, Baltimore, 1996.

    Google Scholar 

  28. G. H. Golub and U. von Matt, Quadratically constrained least squares and quadratic problems, Numer. Math., 59 (1991), pp. 561–580.

    Google Scholar 

  29. G. H. Golub and J. H. Welsh, Calculation of Gauss quadrature rules, Math. Comp., 23 (1969), pp. 221–230.

    Google Scholar 

  30. M. Grote and T. Huckle, Parallel preconditioning with sparse approximate inverses, SIAM J. Sci. Comput., 18 (1997), pp. 838–853.

    Google Scholar 

  31. L. A. Knizhnerman, The simple Lanczos procedure: estimates of the error of the Gauss quadrature formula and their applications, Comput. Math. Math. Phys., 36 (1996), pp. 1481–1492.

    Google Scholar 

  32. L. Yu. Kolotilina and A. Yu. Yeremin, Factorized sparse approximate inverse preconditionings I. Theory, SIAM J. Matrix Anal. Appl., 14 (1993), pp. 45–58.

    Google Scholar 

  33. L. Yu. Kolotilina and A. Yu. Yeremin, On a family of two-level preconditionings of the incomplete block factorization type, Sov. J. Numer. Anal. Math. Modelling, 1 (1986), pp. 293–320.

    Google Scholar 

  34. G. Meinardus, Approximation of Functions: Theory and Numerical Methods, Springer-Verlag, New York, 1967.

    Google Scholar 

  35. G. Meurant, A review on the inverse of symmetric tridiagonal and block tridiagonal matrices, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 707–728.

    Google Scholar 

  36. P. D. Robinson and A. Wathen, Variational bounds on the entries of the inverse of a matrix, IMA J. Numer. Anal., 12 (1992), pp. 463–486.

    Google Scholar 

  37. Y. Saad, Iterative Methods for Sparse Linear Systems, PWS Publishing, Boston, 1996.

    Google Scholar 

  38. G. Strang, A proposal for Toeplitz matrix calculations, Stud. Appl.Math., 74 (1986), pp. 171–176.

    Google Scholar 

  39. R. S. Varga, Nonnegatively posed problems and completely monotonic functions, Linear Algebra Appl., 1 (1968), pp. 329–347.

    Google Scholar 

  40. H. A. van der Vorst, An iterative solution method for solving f(A)x = b, using Krylov subspace information obtained for the symmetric positive definite matrix A, J. Comput. Appl. Math., 18 (1987), pp. 249–263.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Scientific Computing Group, CIC-19, Los Alamos National Laboratory, MS B256, Los Alamos, NM, 87545, USA. email: benzi@lanl.gov

    Michele Benzi

  2. Department of Computer Science, Stanford University, Gates 2B MC 9025, Stanford, CA, 94305-9025, USA. email: golub@sccm.stanford.edu

    Gene H. Golub

Authors
  1. Michele Benzi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gene H. Golub
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Benzi, M., Golub, G.H. Bounds for the Entries of Matrix Functions with Applications to Preconditioning. BIT Numerical Mathematics 39, 417–438 (1999). https://doi.org/10.1023/A:1022362401426

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1022362401426

Search

Navigation