Skip to main content
SpringerLink
Log in

Estimating the condition number of f(A)b

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

Abstract

New algorithms are developed for estimating the condition number of f(A)b, where A is a matrix and b is a vector. The condition number estimation algorithms for f(A) already available in the literature require the explicit computation of matrix functions and their Fr´echet derivatives and are therefore unsuitable for the large, sparse A typically encountered in f(A)b problems. The algorithms we propose here use only matrix-vector multiplications. They are based on a modified version of the power iteration for estimating the norm of the Fr´echet derivative of a matrix function, and work in conjunction with any existing algorithm for computing f(A)b. The number of matrix-vector multiplications required to estimate the condition number is proportional to the square of the number of matrix-vector multiplications required by the underlying f(A)b algorithm. We develop a specific version of our algorithm for estimating the condition number of e A b, based on the algorithm of Al-Mohy and Higham (SIAM J. Matrix Anal. Appl. 30(4), 1639–1657, 2009). Numerical experiments demonstrate that our condition estimates are reliable and of reasonable cost.

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. Al-Mohy, A.H., Higham, N.J.: Computing the Frechet́ derivative of the matrix exponential, with an application to condition number estimation. SIAM J. Matrix Anal. Appl. 30(4), 1639–1657 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Al-Mohy, A.H., Higham, N.J.: Computing the action of the matrix exponential, with an application to exponential integrators. SIAM J. Sci. Comput. 33(2), 488–511 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Al-Mohy, A.H., Higham, N.J.: Improved inverse scaling and squaring algorithms for the matrix logarithm. SIAM J. Sci. Comput. 34(4), C153–C169 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Al-Mohy, A.H., Higham, N.J., Relton, S.D.: Computing the Frechet́ derivative of the matrix logarithm and estimating the condition number. SIAM J. Sci. Comput. 35(4), C394–C410 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. Björck, Å., Hammarling, S.: A Schur method for the square root of a matrix. Linear Algebra Appl. 52/53, 127–140 (1983)

    Article  Google Scholar 

  6. Burrage, K., Hale, N., Kay, D.: An efficient implicit FEM scheme for fractional-in-space reaction-diffusion equations. SIAM J. Sci. Comput. 34(4), A2145–A2172 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen, T.-Y., Demmel, J.W.: Balancing sparse matrices for computing eigenvalues. Linear Algebra Appl. 309, 261–287 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  8. Davies, P.I., Higham, N.J.: A Schur–Parlett algorithm for computing matrix functions. SIAM J. Matrix Anal. Appl. 25(2), 464–485 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  9. Davies, P.I., Higham, N.J.: Computing f(A)b for matrix functions f. In: Boriçi, A., Frommer, A., Joó, B., Kennedy, A., Pendleton, B. (eds.) QCD and Numerical Analysis III, volume 47 of Lecture Notes in Computational Science and Engineering, pp 15–24. Springer, Berlin (2005)

    Google Scholar 

  10. Davis, T.A., Hu, Y.: The University of Florida sparse matrix collection. ACM Trans. Math. Software 38(1), 1 (2011)

    MathSciNet  Google Scholar 

  11. Deadman, E., Higham, N.J.: Testing matrix function algorithms using identities. MIMS EPrint 2014.13, Manchester Institute for Mathematical Sciences, The University of Manchester, UK (2014)

  12. Deadman, E., Higham, N.J., Ralha, R.: Blocked Schur algorithms for computing the matrix square root. In: Manninen, P., Öster, P. (eds.) Applied Parallel and Scientific Computing: 11th International Conference, PARA 2012, Helsinki, Finland, volume 7782 of Lecture Notes in Computer Science, pp 171–182. Springer, Berlin (2013)

    Chapter  Google Scholar 

  13. Fischer, T.M.: On the stability of some algorithms for computing the action of the matrix exponential. Linear Algebra Appl. 443, 1–20 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. Frommer, A., Simoncini, V.: Matrix functions. In: Schilders, W.H.A., Van der Vorst, H.A., Rommes, J. (eds.) Model Order Reduction: Theory, Research Aspects and Applications, volume 13 of Mathematics in Industry, pp 275–303, Berlin (2008)

  15. Gallopoulos, E., Saad, Y.: Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  16. Güttel, S.: Rational Krylov approximation of matrix functions: Numerical methods and optimal pole selection. GAMM-Mitteilungen 36(1), 8–31 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hale, N., Higham, N.J., Trefethen, L.N.: Computing A α, \(\log (A)\), and related matrix functions by contour integrals. SIAM J. Numer. Anal. 46(5), 2505–2523 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Higham, N.J.: Accuracy and Stability of Numerical Algorithms. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, second edition (2002)

  19. Higham, N.J.: Functions of Matrices: Theory and Computation. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA (2008)

  20. Higham, N.J., Edvin, D.: A catalogue of software for matrix functions. Version 1.0 MIMS EPrint 2014.8, Manchester Institute for Mathematical Sciences, The University of Manchester, UK (2014)

  21. Higham, N.J., Lin, L.: An improved Schur–Padé algorithm for fractional powers of a matrix and their Frechet́ derivatives. SIAM J. Matrix Anal. Appl. 34(3), 1341–1360 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  22. Higham, N.J., Tisseur, F.: A block algorithm for matrix 1-norm estimation, with an application to 1-norm pseudospectra. SIAM J. Matrix Anal. Appl. 21(4), 1185–1201 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  23. Hochbruck, M., Ostermann, A.: Exponential integrators. Acta Numerica 19, 209–286, 5 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  24. Jones, E., Oliphant, T., Peterson, P., et al.: SciPy: Open source scientific tools for Python (2001)

  25. Kenney, C.S., Laub, A.J.: Condition estimates for matrix functions. SIAM J. Matrix Anal. Appl. 10(2), 191–209 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  26. Kenney, C.S., Laub, A.J.: Small-sample statistical condition estimates for general matrix functions. J. Sci. Comput. 15(1), 36–61 (1994)

    MathSciNet  MATH  Google Scholar 

  27. Knizhnerman, L.A.: Calculation of functions of unsymmetric matrices using Arnoldi’s method. U.S.S.R. Comput. Maths. Math. Phys. 31(1), 1–9 (1991)

    MathSciNet  MATH  Google Scholar 

  28. Moler, C.B., Loan, C.V.: Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Rev. 45(1), 3–49 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  29. Rice, J.: A theory of condition. SIAM J. Numer. Anal. 3(2), 287–310 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  30. Saad, Y.: Analysis of some Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 29(1), 209–228 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  31. Sheehan, B.N., Saad, Y., Sidje, R.B.: Computing exp(- τ A)b with Laguerre polynomials. Electron. Trans. Numer. Anal. 37, 147–165 (2010)

    MathSciNet  MATH  Google Scholar 

  32. Sidje, R.B.: Expokit: A software package for computing matrix exponentials. ACM Trans. Math. Softw. 24(1), 130–156 (1998)

    Article  MATH  Google Scholar 

  33. Trefethen, L.N., Weideman, J.A.C., Schmelzer, T.: Talbot quadratures and rational approximations. BIT 46(3), 653–670 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  34. Van der Vorst, H.A.: 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(2), 249–263 (1987)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, The University of Manchester, Manchester, M13 9PL, UK

    Edvin Deadman

Authors
  1. Edvin Deadman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Edvin Deadman.

Additional information

This work was supported by European Research Council Advanced Grant MATFUN (267526).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Deadman, E. Estimating the condition number of f(A)b . Numer Algor 70, 287–308 (2015). https://doi.org/10.1007/s11075-014-9947-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-014-9947-4

Keywords

Mathematics Subject Classifications (2010)

Search

Navigation