Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Restarted Q-Arnoldi-type methods exploiting symmetry in quadratic eigenvalue problems


We investigate how to adapt the Q-Arnoldi method for the case of symmetric quadratic eigenvalue problems, that is, we are interested in computing a few eigenpairs \((\lambda ,x)\) of \((\lambda ^2M+\lambda C+K)x=0\) with MCK symmetric \(n\times n\) matrices. This problem has no particular structure, in the sense that eigenvalues can be complex or even defective. Still, symmetry of the matrices can be exploited to some extent. For this, we perform a symmetric linearization \(Ay=\lambda By\), where AB are symmetric \(2n\times 2n\) matrices but the pair (AB) is indefinite and hence standard Lanczos methods are not applicable. We implement a symmetric-indefinite Lanczos method and enrich it with a thick-restart technique. This method uses pseudo inner products induced by matrix B for the orthogonalization of vectors (indefinite Gram-Schmidt). The projected problem is also an indefinite matrix pair. The next step is to write a specialized, memory-efficient version that exploits the block structure of A and B, referring only to the original problem matrices MCK as in the Q-Arnoldi method. This results in what we have called the Q-Lanczos method. Furthermore, we define a stabilized variant analog of the TOAR method. We show results obtained with parallel implementations in SLEPc.

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

Fig. 1


  1. 1.

    Bai, Z., Su, Y.: SOAR: a second-order Arnoldi method for the solution of the quadratic eigenvalue problem. SIAM J. Matrix Anal. Appl. 26(3), 640–659 (2005)

  2. 2.

    Bai, Z., Day, D., Ye, Q.: ABLE: an adaptive block Lanczos method for non-Hermitian eigenvalue problems. SIAM J. Matrix Anal. Appl. 20(4), 1060–1082 (1999)

  3. 3.

    Bai, Z., Ericsson, T., Kowalski, T.: Symmetric indefinite Lanczos method. In: Bai, Z., Demmel, J., Dongarra, J., Ruhe, A., van der Vorst, H. (eds.) Templates for the solution of algebraic eigenvalue problems: a practical guide, pp. 249–260. Society for Industrial and Applied Mathematics, Philadelphia (2000)

  4. 4.

    Balay, S., Abhyankar, S., Adams, M., Brown, J., Brune, P., Buschelman, K., Dalcin, L., Eijkhout, V., Gropp, W., Kaushik, D., Knepley, M., McInnes, L.C., Rupp, K., Smith, B., Zampini, S., Zhang, H.: PETSc users manual. Tech. Rep. ANL-95/11 - Revision 3.6, Argonne National Laboratory (2015)

  5. 5.

    Benner, P., Faßbender, H., Stoll, M.: Solving large-scale quadratic eigenvalue problems with Hamiltonian eigenstructure using a structure-preserving Krylov subspace method. Electron. Trans. Numer. Anal. 29, 212–229 (2008)

  6. 6.

    Betcke, T., Higham, N.J., Mehrmann, V., Schröder, C., Tisseur, F.: NLEVP: a collection of nonlinear eigenvalue problems. ACM Trans. Math. Softw. 39(2), 7:1–7:28 (2013)

  7. 7.

    Campos, C., Roman, J.E.: Parallel Krylov solvers for the polynomial eigenvalue problem in SLEPc (2015, submitted)

  8. 8.

    Day, D.: An efficient implementation of the nonsymmetric Lanczos algorithm. SIAM J. Matrix Anal. Appl. 18(3), 566–589 (1997)

  9. 9.

    Hernandez, V., Roman, J.E., Vidal, V.: SLEPc: a scalable and flexible toolkit for the solution of eigenvalue problems. ACM Trans. Math. Softw. 31(3), 351–362 (2005)

  10. 10.

    Hernandez, V., Roman, J.E., Tomas, A.: Parallel Arnoldi eigensolvers with enhanced scalability via global communications rearrangement. Parallel Comput. 33(7–8), 521–540 (2007)

  11. 11.

    Jia, Z., Sun, Y.: A refined variant of SHIRA for the skew-Hamiltonian/Hamiltonian (SHH) pencil eigenvalue problem. Taiwan J. Math. 17(1), 259–274 (2013)

  12. 12.

    Kressner, D., Roman, J.E.: Memory-efficient Arnoldi algorithms for linearizations of matrix polynomials in Chebyshev basis. Numer. Linear Algebra Appl. 21(4), 569–588 (2014)

  13. 13.

    Kressner, D., Pandur, M.M., Shao, M.: An indefinite variant of LOBPCG for definite matrix pencils. Numer. Algorithms 66(4), 681–703 (2014)

  14. 14.

    Lancaster, P.: Linearization of regular matrix polynomials. Electron. J. Linear Algebra 17, 21–27 (2008)

  15. 15.

    Lancaster, P., Ye, Q.: Rayleigh-Ritz and Lanczos methods for symmetric matrix pencils. Linear Algebra Appl. 185, 173–201 (1993)

  16. 16.

    Lu, D., Su, Y.: Two-level orthogonal Arnoldi process for the solution of quadratic eigenvalue problems (2012, manuscript)

  17. 17.

    Meerbergen, K.: The Lanczos method with semi-definite inner product. BIT 41(5), 1069–1078 (2001)

  18. 18.

    Meerbergen, K.: The Quadratic Arnoldi method for the solution of the quadratic eigenvalue problem. SIAM J. Matrix Anal. Appl. 30(4), 1463–1482 (2008)

  19. 19.

    Mehrmann, V., Watkins, D.: Structure-preserving methods for computing eigenpairs of large sparse skew-Hamiltonian/Hamiltonian pencils. SIAM J. Sci. Comput. 22(6), 1905–1925 (2001)

  20. 20.

    Parlett, B.N.: The symmetric Eigenvalue problem. Prentice-Hall, Englewood Cliffs (1980) (reissued with revisions by SIAM, Philadelphia)

  21. 21.

    Parlett, B.N., Chen, H.C.: Use of indefinite pencils for computing damped natural modes. Linear Algebra Appl. 140(1), 53–88 (1990)

  22. 22.

    Parlett, B.N., Taylor, D.R., Liu, Z.A.: A look-ahead Lánczos algorithm for unsymmetric matrices. Math. Comput. 44(169), 105–124 (1985)

  23. 23.

    de Samblanx, G., Bultheel, A.: Nested Lanczos: implicitly restarting an unsymmetric Lanczos algorithm. Numer. Algorithms 18(1), 31–50 (1998)

  24. 24.

    Sleijpen, G.L.G., Booten, A.G.L., Fokkema, D.R., van der Vorst, H.A.: Jacobi-Davidson type methods for generalized eigenproblems and polynomial eigenproblems. BIT 36(3), 595–633 (1996)

  25. 25.

    Stewart, G.W.: A Krylov-Schur algorithm for large eigenproblems. SIAM J. Matrix Anal. Appl. 23(3), 601–614 (2001)

  26. 26.

    Su, Y., Zhang, J., Bai, Z.: A compact Arnoldi algorithm for polynomial eigenvalue problems. In: Presented at RANMEP (2008)

  27. 27.

    Tisseur, F.: Tridiagonal-diagonal reduction of symmetric indefinite pairs. SIAM J. Matrix Anal. Appl. 26(1), 215–232 (2004)

  28. 28.

    Tisseur, F., Meerbergen, K.: The quadratic eigenvalue problem. SIAM Rev. 43(2), 235–286 (2001)

  29. 29.

    Watkins, D.S.: The matrix Eigenvalue problem: GR and Krylov subspace methods. Society for Industrial and Applied Mathematics (2007)

  30. 30.

    Wu, K., Simon, H.: Thick-restart Lanczos method for large symmetric eigenvalue problems. SIAM J. Matrix Anal. Appl. 22(2), 602–616 (2000)

Download references


The authors are grateful to the reviewers for their constructive comments that helped improve the presentation. The computational experiments of Sect. 7 were carried out on the supercomputer Tirant at Universitat de València.

Author information

Correspondence to Jose E. Roman.

Additional information

This work was supported by the Spanish Ministry of Economy and Competitiveness under Grant TIN2013-41049-P. Carmen Campos was supported by the Spanish Ministry of Education, Culture and Sport through an FPU Grant with reference AP2012-0608.

Communicated by Daniel Kressner.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Campos, C., Roman, J.E. Restarted Q-Arnoldi-type methods exploiting symmetry in quadratic eigenvalue problems. Bit Numer Math 56, 1213–1236 (2016).

Download citation


  • Quadratic eigenvalue problem
  • Pseudo-Lanczos
  • Q-Arnoldi
  • TOAR
  • Thick-restart
  • SLEPc

Mathematics Subject Classification

  • 65F15
  • 15A18
  • 65F50