Generalized Block Tuned Preconditioners for SPD Eigensolvers

  • Luca BergamaschiEmail author
  • Ángeles Martínez
Part of the Springer INdAM Series book series (SINDAMS, volume 30)


Given an n × n symmetric positive definite (SPD) matrix A and an SPD preconditioner P, we propose a new class of generalized block tuned (GBT) preconditioners. These are defined as a p-rank correction of P with the property that arbitrary (positive) parameters γ1, …, γp are eigenvalues of the preconditioned matrix. We propose to employ these GBT preconditioners to accelerate the iterative solution of linear systems like (A − θI)s = r in the framework of iterative eigensolvers. We give theoretical evidence that a suitable, and effective, choice of the scalars γj is able to shift p eigenvalues of P(A − θI) very close to one. Numerical experiments on various matrices of very large size show that the proposed preconditioner is able to yield an almost constant number of iterations, for different eigenpairs, irrespective of the relative separation between consecutive eigenvalues. We also give numerical evidence that the GBT preconditioner is always far superior to the spectral preconditioner (Numer. Linear Algebra Appl. 24(3):1–14, 2017), on matrices with highly clustered eigenvalues.


Eigenvalues SPD matrix Newton method Tuned preconditioner Incomplete Cholesky preconditioner 



This work has been supported by the Italian project CPDA155834/15: “Stable and efficient discretizations of the mechanics of faults” and by the Italian INdAM-GNCS Project Metodi numerici per problemi di ottimizzazione vincolata di grandi dimensioni e applicazioni (2017). We wish to thank the anonymous reviewers whose comments and suggestions helped improve the quality of the paper.


  1. 1.
    Bathe, K.J.: Finite Element Procedures in Engineering Analysis. Prentice-Hall, Englewood Cliffs (1982)Google Scholar
  2. 2.
    Saad, Y., Stathopoulos, A., Chelikowsky, J., Wu, K., Öğüt, S.: Solution of large eigenvalue problems in electronic structure calculations. BIT 36(3), 563–578 (1996)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bozzo, E., Franceschet, M.: Approximations of the generalized inverse of the graph Laplacian matrix. Internet Math. 8, 456–481 (2012)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bergamaschi, L., Bozzo, E.: Computing the smallest eigenpairs of the graph Laplacian. SeMA J. 75, 1–16 (2018)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bergamaschi, L., Facca, E., Martínez, A., Putti, M.: Spectral preconditioners for the efficient numerical solution of a continuous branched transport model. J. Comput. Appl. Math. (2018). CrossRefGoogle Scholar
  6. 6.
    Bergamaschi, L., Martínez, A.: Efficiently preconditioned inexact Newton methods for large symmetric eigenvalue problems. Optim. Methods Softw. 30, 301–322 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Sleijpen, G.L.G., van der Vorst, H.A.: A Jacobi-Davidson method for linear eigenvalue problems. SIAM J. Matrix Anal. 17(2), 401–425 (1996)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Martínez, A.: Tuned preconditioners for the eigensolution of large SPD matrices arising in engineering problems. Numer. Linear Algebra Appl. 23(3), 427–443 (2016)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Bergamaschi, L., Gambolati, G., Pini, G.: Asymptotic convergence of conjugate gradient methods for the partial symmetric eigenproblem. Numer. Linear Algebra Appl. 4(2), 69–84 (1997)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Freitag, M.A., Spence, A.: Rayleigh quotient iteration and simplified Jacobi-Davidson method with preconditioned iterative solves. Linear Algebra Appl. 428(8–9), 2049–2060 (2008)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Carpentieri, B., Duff, I.S., Giraud, L.: A class of spectral two-level preconditioners. SIAM J. Sci. Comput. 25(2), 749–765 (2003) (electronic)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Bergamaschi, L., Martínez, A.: Two-stage spectral preconditioners for iterative eigensolvers. Numer. Linear Algebra Appl. 24(3), 1–14 (2017)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Bergamaschi, L., Putti, M.: Numerical comparison of iterative eigensolvers for large sparse symmetric matrices. Comput. Methods App. Mech. Eng. 191(45), 5233–5247 (2002)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Freitag, M.A., Spence, A.: A tuned preconditioner for inexact inverse iteration applied to Hermitian eigenvalue problems. IMA J. Numer. Anal. 28(3), 522–551 (2008)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Notay, Y.: Combination of Jacobi-Davidson and conjugate gradients for the partial symmetric eigenproblem. Numer. Linear Algebra Appl. 9(1), 21–44 (2002)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Civil, Environmental and Architectural EngineeringUniversity of PaduaPadovaItaly
  2. 2.Department of Mathematics “Tullio Levi-Civita”University of PaduaPadovaItaly

Personalised recommendations