Skip to main content
SpringerLink
Log in

Structured eigenvalue condition numbers for parameterized quasiseparable matrices

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

The development of fast algorithms for performing computations with \(n\times n\) low-rank structured matrices has been a very active area of research during the last two decades, as a consequence of the numerous applications where these matrices arise. The key ideas behind these fast algorithms are that low-rank structured matrices can be described in terms of O(n) parameters and that these algorithms operate on the parameters instead on the matrix entries. Therefore, the sensitivity of any computed quantity should be measured with respect to the possible variations that the parameters defining these matrices may suffer, since this determines the maximum accuracy of a given fast computation. In other words, it is necessary to develop condition numbers with respect to parameters for different magnitudes and classes of low-rank structured matrices, but, as far as we know, this has not yet been accomplished in any case. In this paper, we derive structured relative eigenvalue condition numbers for the important class of low-rank structured matrices known as \(\{1;1\}\)-quasiseparable matrices with respect to relative perturbations of the parameters in the quasiseparable and in the Givens-vector representations of these matrices, and we provide fast algorithms for computing them. Comparisons among the new structured condition numbers and the unstructured one are also presented, as well as numerical experiments showing that the structured condition numbers can be small in situations where the unstructured one is huge. In addition, the approach presented in this paper is general and may be extended to other problems and classes of low-rank structured matrices.

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. Asplund, E.: Inverses of matrices \(a_{ij}\) which satisfy \(a_{ij}=0\) for \(j>i+p\). Math. Scand. 7, 57–60 (1959)

    MathSciNet  MATH  Google Scholar 

  2. Asplund, S.O.: Finite boundary value problems solved by Green’s matrix. Math. Scand. 7, 49–56 (1959)

    MathSciNet  MATH  Google Scholar 

  3. Aurentz, J.L., Mach, T., Vandebril, R., Watkins, D.S.: Fast and backward stable computation of roots of polynomials. SIAM J. Matrix Anal. Appl. 36(3), 942–973 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  4. Barlow, J.L., Ipsen, I.C.F.: Scaled Givens rotations for the solution of linear least squares problems on systolic arrays. SIAM J. Sci. Stat. Comput. 8(5), 716–733 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bebendorf, M.: Why finite element discretizations can be factored by triangular hierarchical matrices. SIAM J. Numer. Anal. 45(4), 1472–1494 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bella, T., Olshevsky, V., Stewart, M.: Nested product decomposition of quasiseparable matrices. SIAM J. Matrix Anal. Appl. 34(4), 1520–1555 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Berger, W.J., Saibel, E.: On the inversion of continuant matrices. Frankl. Inst. Eng. Appl. Math. 6, 249–253 (1953)

    Article  MathSciNet  Google Scholar 

  8. Bindel, D., Demmel, J.W., Kahan, W., Marques, O.: On computing Givens rotations reliably and efficiently. ACM Trans. Math. Softw. 28(2), 206–238 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bini, D.A., Gemignani, L., Pan, V.Y.: Fast and stable QR eigenvalue algorithms for generalized companion matrices and secular equations. Numer. Math. 100(3), 373–408 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bini, D.A., Gemignani, L., Tisseur, F.: The Ehrlich–Aberth method for the nonsymmetric tridiagonal eigenvalue problem. SIAM J. Matrix Anal. Appl. 27(1), 153–175 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bini, D.A., Boito, P., Eidelman, Y., Gemignani, L., Gohberg, I.: A fast implicit QR eigenvalue algorithm for companion matrices. Linear Algebra Appl. 432, 2006–2031 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Börm, S., Grasedyck, L.: Hybrid cross approximation of integral operators. Numer. Math. 101, 221–249 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Börm, S., Grasedyck, L., Hackbusch, W.: Introduction to hierarchical matrices with applications. Eng. Anal. Bound. Elem. 27, 405–422 (2003)

    Article  MATH  Google Scholar 

  14. Chandrasekaran, S., Gu, M., Xia, J., Zhu, J.: A fast QR algorithm for companion matrices. In: Recent Advances in Matrix and Operator Theory, pp. 111–143. Operator Theory: Advances and Applications, vol. 179. Birkhäuser, Basel (2008)

  15. Delvaux, S., Van Barel, M.: A QR-based solver for rank structured matrices. SIAM J. Matrix Anal. Appl. 30(2), 464–490 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Dopico, F.M., Olshevsky, V., Zhlobich, P.: Stability of QR-based fast system solvers for a subclass of quasiseparable rank one matrices. Math. Comput. 82, 2007–2034 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Eidelman, Y., Gohberg, I.: On a new class of structured matrices. Integral Equ. Oper. Theory 34(3), 293–324 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  18. Eidelman, Y., Gohberg, I.: On generators of quasiseparable finite block matrices. Calcolo 42, 187–214 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  19. Eidelman, Y., Gohberg, I., Haimovici, I.: Separable type representations of matrices and fast algorithms. Volume 1. Basics. Completion problems. Multiplication and inversion algorithms. In: Operator Theory: Advances and Applications, vol. 234. Birkhäuser/Springer, Basel (2014)

  20. Eidelman, Y., Gohberg, I., Haimovici, I.: Separable type representations of matrices and fast algorithms. Volume 2. Eigenvalue method. In: Operator Theory: Advances and Applications, vol. 235. Birkhäuser/Springer, Basel (2014)

  21. Eidelman, Y., Gohberg, I., Olshevsky, V.: The QR iteration method for Hermitian quasiseparable matrices of an arbitrary order. Linear Algebra Appl. 404, 305–324 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  22. Ferreira, C., Parlett, B., Dopico, F.M.: Sensitivity of eigenvalues of an unsymmetric tridiagonal matrix. Numer. Math. 122(3), 527–555 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  23. Fiedler, M., Markham, T.L.: Generalized totally nonnegative matrices. Linear Algebra Appl. 345, 9–28 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  24. Gantmacher, F.R., Krein, M.G.: Oscillation Matrices and Kernels and Small Vibrations of Mechanical Systems. AMS Chelsea Publishing, Providence (2002) (revised edition from the Russian original edition published in 1941)

  25. Geurts, A.J.: A contribution to the theory of condition. Numer. Math. 39, 85–96 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  26. Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. The Johns Hopkins University Press, Baltimore (1996)

    MATH  Google Scholar 

  27. Grasedyck, L., Kriemann, R., Le Borne, S.: Parallel black box H-LU preconditioning for elliptic boundary value problems. Comput. Vis. Sci. 11, 273–291 (2008)

    Article  MathSciNet  Google Scholar 

  28. Higham, N.J.: Accuracy and Stability of Numerical Algorithms, 2nd edn. Society for Industrial and Applied Mathematics, Philadelphia (2002)

    Book  MATH  Google Scholar 

  29. Higham, D.J., Higham, N.J.: Structured backward error and condition of generalized eigenvalue problems. SIAM J. Matrix Anal. Appl. 20(2), 493–512 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  30. Ipsen, I.C.F.: Relative perturbation results for matrix eigenvalues and singular values. Acta Numerica 7, 151–201 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  31. Karow, M., Kressner, D., Tisseur, F.: Structured eigenvalue condition numbers. SIAM J. Matrix Anal. Appl. 28(4), 1052–1068 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  32. Li, R.C.: Relative perturbation theory. III. More bounds on eigenvalue variations. Linear Algebra Appl. 266, 337–345 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  33. Martinsson, P.G., Rokhlin, V.: A fast direct solver for boundary integral equations in two dimensions. J. Comput. Phys. 205, 1–23 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  34. Martinsson, P.G., Rokhlin, V., Tygert, M.: A fast algorithm for the inversion of general Toeplitz matrices. Comput. Math. Appl. 50, 741–752 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  35. Parlett, B.N., Reinsch, C.: Balancing a matrix for calculation of eigenvalues and eigenvectors. Numer. Math. 13, 292–304 (1969)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  37. Roy, S.N., Sarhan, A.E.: On inverting a class of patterned matrices. Biometrika 43, 227–231 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  38. Stewart, G.W., Sun, J.: Matrix Perturbation Theory. Academic Press, Boston (1990)

    MATH  Google Scholar 

  39. Van Barel, M., Vandebril, R., Van Dooren, P., Frederix, K.: Implicit double shift QR-algorithm for companion matrices. Numer. Math. 116, 177–212 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  40. Vandebril, R., Van Barel, M., Mastronardi, N.: A note on the representation and definition of semiseparable matrices. Numer. Linear Algebra Appl. 12, 839–858 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  41. Vandebril, R., Van Barel, M., Golub, G., Mastronardi, N.: A bibliography on semiseparable matrices. Calcolo 42, 249–270 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  42. Vandebril, R., Van Barel, M., Mastronardi, N.: Matrix Computations and Semiseparable Matrices. Linear Systems, vol. 1. The Johns Hopkins University Press, Baltimore (2008)

  43. Vandebril, R., Van Barel, M., Mastronardi, N.: Matrix Computations and Semiseparable Matrices. Eigenvalue and Singular Value Methods, vol. II. The Johns Hopkins University Press, Baltimore (2008)

  44. Wilkinson, J.H.: The Algebraic Eigenvalue Problem. Oxford University Press, New York (1965)

    MATH  Google Scholar 

Download references

Acknowledgments

The authors sincerely thank two anonymous referees and the Associate Editor handling this manuscript (Prof. Ilse Ipsen) for their extremely detailed readings of this long and technical paper, for correcting many grammatical typos, and for providing constructive and insightful comments that have contributed to a significant improvement of the original manuscript.

Author information

Authors and Affiliations

  1. Departamento de Matemáticas, Universidad Carlos III de Madrid, Avda. Universidad 30, 28911, Leganés, Spain

    Froilán M. Dopico & Kenet Pomés

Authors
  1. Froilán M. Dopico
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kenet Pomés
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Froilán M. Dopico.

Additional information

This research was partially supported by Ministerio de Economía y Competitividad of Spain through Grant MTM2012-32542.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dopico, F.M., Pomés, K. Structured eigenvalue condition numbers for parameterized quasiseparable matrices. Numer. Math. 134, 473–512 (2016). https://doi.org/10.1007/s00211-015-0779-5

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-015-0779-5

Mathematics Subject Classification

Search

Navigation