Numerical Algorithms

, Volume 70, Issue 2, pp 287–308 | Cite as

Estimating the condition number of f(A)b

Original Paper

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.

Keywords

Matrix function Matrix exponential Condition number estimation Fréchet derivative Power iteration Block 1-norm estimator Python 

Mathematics Subject Classifications (2010)

15A60 65F35 65F60 

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References

  1. 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)MathSciNetCrossRefMATHGoogle Scholar
  2. 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)MathSciNetCrossRefMATHGoogle Scholar
  3. 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)MathSciNetCrossRefMATHGoogle Scholar
  4. 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)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Björck, Å., Hammarling, S.: A Schur method for the square root of a matrix. Linear Algebra Appl. 52/53, 127–140 (1983)CrossRefGoogle Scholar
  6. 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)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Chen, T.-Y., Demmel, J.W.: Balancing sparse matrices for computing eigenvalues. Linear Algebra Appl. 309, 261–287 (2000)MathSciNetCrossRefMATHGoogle Scholar
  8. 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)MathSciNetCrossRefMATHGoogle Scholar
  9. 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. 10.
    Davis, T.A., Hu, Y.: The University of Florida sparse matrix collection. ACM Trans. Math. Software 38(1), 1 (2011)MathSciNetGoogle Scholar
  11. 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)Google Scholar
  12. 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)CrossRefGoogle Scholar
  13. 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)MathSciNetCrossRefMATHGoogle Scholar
  14. 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)Google Scholar
  15. 15.
    Gallopoulos, E., Saad, Y.: Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13(5), 1236–1264 (1992)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Güttel, S.: Rational Krylov approximation of matrix functions: Numerical methods and optimal pole selection. GAMM-Mitteilungen 36(1), 8–31 (2013)MathSciNetCrossRefMATHGoogle Scholar
  17. 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)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Higham, N.J.: Accuracy and Stability of Numerical Algorithms. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, second edition (2002)Google Scholar
  19. 19.
    Higham, N.J.: Functions of Matrices: Theory and Computation. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA (2008)Google Scholar
  20. 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)Google Scholar
  21. 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)MathSciNetCrossRefMATHGoogle Scholar
  22. 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)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Hochbruck, M., Ostermann, A.: Exponential integrators. Acta Numerica 19, 209–286, 5 (2010)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Jones, E., Oliphant, T., Peterson, P., et al.: SciPy: Open source scientific tools for Python (2001)Google Scholar
  25. 25.
    Kenney, C.S., Laub, A.J.: Condition estimates for matrix functions. SIAM J. Matrix Anal. Appl. 10(2), 191–209 (1989)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Kenney, C.S., Laub, A.J.: Small-sample statistical condition estimates for general matrix functions. J. Sci. Comput. 15(1), 36–61 (1994)MathSciNetMATHGoogle Scholar
  27. 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)MathSciNetMATHGoogle Scholar
  28. 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)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Rice, J.: A theory of condition. SIAM J. Numer. Anal. 3(2), 287–310 (1966)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Saad, Y.: Analysis of some Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 29(1), 209–228 (1992)MathSciNetCrossRefMATHGoogle Scholar
  31. 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)MathSciNetMATHGoogle Scholar
  32. 32.
    Sidje, R.B.: Expokit: A software package for computing matrix exponentials. ACM Trans. Math. Softw. 24(1), 130–156 (1998)CrossRefMATHGoogle Scholar
  33. 33.
    Trefethen, L.N., Weideman, J.A.C., Schmelzer, T.: Talbot quadratures and rational approximations. BIT 46(3), 653–670 (2006)MathSciNetCrossRefMATHGoogle Scholar
  34. 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)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.School of MathematicsThe University of ManchesterManchesterUK

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