Numerical Algorithms

, Volume 66, Issue 4, pp 681–703 | Cite as

An indefinite variant of LOBPCG for definite matrix pencils

  • Daniel Kressner
  • Marija Miloloža Pandur
  • Meiyue Shao
Original Paper

Abstract

In this paper, we propose a novel preconditioned solver for generalized Hermitian eigenvalue problems. More specifically, we address the case of a definite matrix pencil \(A-\lambda B\), that is, A, B are Hermitian and there is a shift \(\lambda _{0}\) such that \(A-\lambda _{0} B\) is definite. Our new method can be seen as a variant of the popular LOBPCG method operating in an indefinite inner product. It also turns out to be a generalization of the recently proposed LOBP4DCG method by Bai and Li for solving product eigenvalue problems. Several numerical experiments demonstrate the effectiveness of our method for addressing certain product and quadratic eigenvalue problems.

Keywords

Eigenvalue Definite matrix pencil Minimization principle LOBPCG 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    HSL.: A collection of Fortran codes for large scale scientific computation. Available from http://www.hsl.rl.ac.uk/catalogue/ (2011)
  2. 2.
    Arbenz, P., Drmač, Z.: On positive semidefinite matrices with known null space. SIAM J. Matrix Anal. Appl. 24(1), 132–149 (2002)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Bai, Z., Demmel, J.W., Dongarra, J.J., Ruhe, A., van der Vorst, H. (eds.): Templates for the solution of algebraic eigenvalue problems. Software, Environments, and Tools. SIAM, Philadelphia (2000)Google Scholar
  4. 4.
    Bai, Z., Li, R.-C.: Minimization principles for the linear response eigenvalue problem I: theory. SIAM J. Matrix Anal. Appl. 33(4), 1075–1100 (2012)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Bai, Z., Li, R.-C.: Minimization principles for the linear response eigenvalue problem II: computation. SIAM J. Matrix Anal. Appl. 34(2), 392–416 (2013)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Bai, Z., Li, R.-C.: Minimization principles for the linear response eigenvalue problem III: general case. Mathematics preprint series. The University of Texas, Arlington (2013)Google Scholar
  7. 7.
    Benner, P., Kressner, D., Mehrmann, V.: Skew-Hamiltonian and Hamiltonian eigenvalue problems: theory, algorithms and applications. In: Drmač, Z., Marušić, M., Tutek, Z. (eds.) Proceedings of the Conference on Applied Mathematics and Scientific Computing, Brijuni (Croatia), June 23-27, 2003, pp. 3–39. Springer-Verlag (2005)Google Scholar
  8. 8.
    Betcke, T., Higham, N.J., Mehrmann, V., Schröder, C., Tisseur, F.: NLEVP: a collection of nonlinear eigenvalue problems. ACM Trans. Math. Software 39(2), 7:1–7:28 (2013). Also available from http://www.mims.manchester.ac.uk/research/numerical-analysis/nlevp.html. CrossRefGoogle Scholar
  9. 9.
    D′yakonov, E.G.: Optimization in Solving Elliptic Problems. CRC Press, Boca Raton (1996)MATHGoogle Scholar
  10. 10.
    Fan, H.-Y., Lin, W.-W., Van Dooren, P.: Normwise scaling of second order polynomial matrices. SIAM J. Matrix Anal. Appl. 26(1), 252–256 (2004)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Gohberg, I., Lancaster, P., Rodman, L.: Matrices and indefinite scalar products. Operator Theory: Advances and Applications, vol. 8. Birkhäuser Verlag, Basel (1983)Google Scholar
  12. 12.
    Hansen, P.C., Yalamov, P.Y.: Symmetric rank revealing factorizations. In: Recent Advances in Numerical Methods and Applications, II (Sofia, 1998), pp. 687–695. World Sci. Publ., River Edge (1999)Google Scholar
  13. 13.
    Hari, V., Singer, S., Singer, S.: Block-oriented J-Jacobi methods for Hermitian matrices. Linear Algebra Appl. 433(8–10), 1491–1512 (2010)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Hetmaniuk, U., Lehoucq, R.: Basis selection in LOBPCG. J. Comput. Phys. 218(1), 324–332 (2006)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Higham, N.J., Tisseur, F., Van Dooren, P.: Detecting a definite Hermitian pair and a hyperbolic or elliptic quadratic eigenvalue problem, and associated nearness problems. Linear Algebra Appl. 351/352, 455–474 (2002)CrossRefGoogle Scholar
  16. 16.
    Knyazev, A.V.: Toward the optimal preconditioned eigensolver: Locally optimal block preconditioned conjugate gradient method. SIAM J. Sci. Comput. 23(2), 517–541 (2001)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Knyazev, A.V., Neymeyr, K.: Efficient solution of symmetric eigenvalue problems using multigrid preconditioners in the locally optimal block conjugate gradient method. Electron. Trans. Numer. Anal. 15, 38–55 (2003)MATHMathSciNetGoogle Scholar
  18. 18.
    Knyazev, A.V., Neymeyr, K.: A geometric theory for preconditioned inverse iteration. III: a short and sharp convergence estimate for generalized eigenvalue problems. Linear Algebra Appl. 358, 95–114 (2003)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Knyazev, A.V., Neymeyr, K.: Gradient flow approach to geometric convergence analysis of preconditioned eigensolvers. SIAM J. Matrix Anal. Appl. 31(2), 621–628 (2009)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Kovač-Striko, J., Veselić, K.: Trace minimization and definiteness of symmetric pencils. Linear Algebra Appl. 216, 139–158 (1995)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Lancaster, P., Rodman, L.: Canonical forms for Hermitian matrix pairs under strict equivalence and congurence. SIAM Rev. 47(3), 407–443 (2005)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Lancaster, P., Ye, Q.: Variational and numerical methods for symmetric matrix pencils. Bull. Austral. Math. Soc. 43(1), 1–17 (1991)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Lashuk, I., Argentati, M., Ovtchinnikov, E., Knyazev, A.: Preconditioned eigensolver LOBPCG in hypre and PETSc. In: Widlund, O., Keyes, D. (eds.) Domain Decomposition Methods in Science and Engineering XVI, vol. 55 , pp. 635–642. Lecture Notes Computation Science Engineering, (2007)Google Scholar
  24. 24.
    Liang, X., Li, R.-C., Bai, Z.: Trace minimization principles for positive semi-definite pencils. Linear Algebra Appl. 438(7), 3085–3106 (2013)CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Mathias, R.: Quadratic residual bounds for the Hermitian eigenvalue problem. SIAM J. Matrix Anal. Appl. 19(2), 541–550 (1998)CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Miloloža Pandur, M.: Some iterative methods for solving the symmetric generalized eigenvelue problem. PhD thesis, Department of Mathematics, University of Zagreb, in preparationGoogle Scholar
  27. 27.
    Neymeyr, K.: A geometric theory for preconditioned inverse iteration. I: extrema of the Rayleigh quotient. Linear Algebra Appl. 322(1–3), 61–85 (2001)CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    Neymeyr, K.: A geometric theory for preconditioned inverse iteration. II: convergence estimates. Linear Algebra Appl. 322(1–3), 87–104 (2001)CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Neymeyr, K.: A geometric theory for preconditioned inverse iteration applied to a subspace. Math. Comp. 71(237), 197–216 (2002)CrossRefMATHMathSciNetGoogle Scholar
  30. 30.
    Neymeyr, K.: A geometric convergence theory for the preconditioned steepest descent iteration. SIAM Numer. Anal. 50(6), 3188–3207 (2012)CrossRefMATHMathSciNetGoogle Scholar
  31. 31.
    Neymeyr, K., Ovtchinnikov, E., Zhou, M.: Convergence analysis of gradient iterations for the symmetric eigenvalue problem. SIAM J. Matrix Anal. Appl. 32(2), 443–456 (2011)CrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    Parlett, B.N.: The Symmetric Eigenvalue Problem, Classics in Applied Mathematics, vol. 20. Corrected reprint of the 1980 original. SIAM, Philadelphia (1998)Google Scholar
  33. 33.
    Stewart, G.W.: Basic decompositions. Matrix Algorithms, vol. I. SIAM, Philadelphia (1998)CrossRefGoogle Scholar
  34. 34.
    Stewart, G.W., Sun, J.-G.: Matrix Perturbation Theory. Academic, New York (1990)Google Scholar
  35. 35.
    Truhar, N.: Relative Perturbation Theory for Matrix Spectral Decompositions. PhD thesis, Department of Mathematics, University of Zagreb (2000)Google Scholar
  36. 36.
    Veselić, K.: A Jacobi eigenreduction algorithm for definite matrix pairs. Numer. Math. 64(2), 241–269 (1993)CrossRefMATHMathSciNetGoogle Scholar
  37. 37.
    Veselić, K.: A mathematical introduction. Damped Oscillations of Linear Systems, vol. 2023. Lecture Notes in Mathematics.Springer, Heidelberg (2011)Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Daniel Kressner
    • 1
  • Marija Miloloža Pandur
    • 1
    • 2
  • Meiyue Shao
    • 1
  1. 1.ANCHP, MATHICSE, EPF LausanneLausanneSwitzerland
  2. 2.Department of MathematicsJ. J. Strossmayer University of OsijekOsijekCroatia

Personalised recommendations