Minimal residual-like condition with collinearity for shifted Krylov subspace methods

  • Akira ImakuraEmail author
Special Feature: Original Paper International Workshop on Eigenvalue Problems: Algorithms; Software and Applications, in Petascale Computing (EPASA2018)


In this paper, we consider shifted Krylov subspace methods for solving shifted linear systems. In such methods, the collinearity of the residual vectors plays a very important role. The minimal residual-like condition with collinearity for the shifted Krylov subspace methods was first proposed for the restarted shifted GMRES method by Frommer and Glässner in 1998, and it has been used for several shifted Krylov subspace methods, such as the shifted BiCGSTAB(\(\ell \)) and shifted IDR(s) methods. In this paper, we propose a novel minimal residual-like condition with collinearity for shifted Krylov subspace methods. Numerical experiments indicate that the proposed condition shows a better convergence behavior than the traditional condition.


Shifted linear systems Shifted Krylov subspace methods Minimal residual-like condition Collinearity 

Mathematics Subject Classification

65F10 65F50 



  1. 1.
    Ahmad, I.M., Szyld, B.D., van Gijzen, B.M.: Preconditioned multishift BiCG for \(\cal{H}_2\)-optimal model reduction. SIAM J. Matrix Anal. Appl. 38, 401–424 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Baumann, M., van Gijzen, B.M.: Nested Krylov methods for shifted linear systems. SIAM J. Sci. Comput. 37, 90–112 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Datta, B.N., Saad, Y.: Arnoldi methods for large Sylvester-like observer matrix equations, and an associated algorithm for partial spectrum assignment. Linear Algebra Appl. 154–156, 225–244 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Du, L., Sogabe, T., Zhang, S.-L.: IDR(\(s\)) for solving shifted nonsymmetric linear systems. J. Comput. Appl. Math. 274, 35–43 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Freund, R.W.: On conjugate gradient type methods and polynomial preconditioners for a class of complex non-Hermitian matrices. Numer. Math. 57, 285–312 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Freund, R.W.: Solution of shifted linear systems by quasi-minimal residual iterations. In: Reichel, L., Ruttan, A., Varga, R.S. (eds.) Numerical Linear Algebra, pp. 101–121. W. de Gruyter, Berlin (1993)Google Scholar
  7. 7.
    Frommer, A.: BiCGSTAB(\(l\)) for families of shifted linear systems. Computing 70, 87–109 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Frommer, A., Glässner, U.: Restarted GMRES for shifted linear systems. SIAM J. Sci. Comput. 19, 15–26 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hestenes, M.R., Stiefel, E.: Methods of conjugate gradients for solving linear systems. J. Res. Natl. Bur. Stand. 49, 409–436 (1952)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hoemmen, M.: Communication-avoiding Krylov subspace methods. PhD thesis, University of California, Berkeley (2010)Google Scholar
  11. 11.
    Lagarias, J.C., Reeds, J.A., Wright, M.H., Wright, P.E.: Convergence properties of the Nelder–Mead simplex method in low dimensions. SIAM J. Opt. 9, 112–147 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Matrix Market. Accessed 14 May 2018
  13. 13.
    Saad, Y., Schultz, M.H.: GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7, 856–869 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Simoncini, V.: Restarted full orthogonalization method for shifted linear systems. BIT Numer. Math. 43, 459–466 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Sogabe, T., Hoshi, T., Zhang, S.-L., Fujiwara, T.: A numerical method for calculating the Green’s function arising from electronic structure theory. In: Kaneda, Y., Kawamura, H., Sasai, M. (eds.) Frontiers of Computational Science, pp. 189–195. Springer, Berlin (2007)CrossRefGoogle Scholar
  16. 16.
    van der Vorst, H.A.: Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 13, 631–644 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Sleijpen, G.L.G., Fokkema, D.R.: BiCGStab(\(\ell \)) for linear equations involving unsymmetric matrices with complex spectrum. Electron. Trans. Numer. Anal. 1, 11–32 (1993)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Sonneveld, P., van Gijzen, M.B.: IDR(\(s\)): a family of simple and fast algorithms for solving large nonsymmetric systems of linear equations. SIAM J. Sci. Comput. 31, 1035–1062 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Fletcher, R.: Conjugate gradient methods for indefinite systems. In: Alistair Watros, G. (ed.) Lecture Notes in Mathematics, vol. 506, pp. 73–89. Springer, New York (1976)Google Scholar
  20. 20.
    Saad, Y.: Krylov subspace methods for solving large unsymmetric linear systems. Math. Comput. 37, 105–126 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Paige, C.C., Saunders, M.A.: Solution of sparse indefinite systems of linear equations. SIAM J. Numer. Anal. 12, 617–629 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Freund, R.W., Nachtigal, N.M.: QMR: a quasi-minimal residual method for non-Hermitian linear systems. Numer. Math. 60, 315–339 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Jegerlehner, B.: Krylov space solvers for shifted linear systems (1996). arXiv:hep-lat/9612014

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© The JJIAM Publishing Committee and Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Faculty of Engineering, Information and SystemsUniversity of TsukubaTsukubaJapan

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