Advertisement

Properties of Saddle-Point Matrices

  • Miroslav Rozložník
Chapter
Part of the Nečas Center Series book series (NECES)

Abstract

This chapter is devoted to the study of basic algebraic properties of saddle-point matrices such as their invertibility, the existence of their block factorizations, the expressions for their inverses and to the analysis of their spectral properties such as inertia and eigenvalue localization.

References

  1. 12.
    M. Benzi, G.H. Golub, J. Liesen, Numerical solution of saddle point problems. Acta Numerica 14, 1–137 (2005)MathSciNetCrossRefGoogle Scholar
  2. 18.
    Y. Chabrillac, J.P. Crouzeix, Definiteness and semidefiniteness of quadratic forms revisited. Linear Algebra Appl. 63(1), 283–292 (1984)MathSciNetCrossRefGoogle Scholar
  3. 25.
    H. Elman, D.J. Silvester, A.J. Wathen, Finite Elements and Fast Iterative Solvers with Applications in Incompressible Fluid Dynamics (Oxford University Press, Oxford, 2005)zbMATHGoogle Scholar
  4. 26.
    R. Estrin, C. Greif, On nonsingular saddle-point systems with a maximally rank deficient leading block. SIAM J. Matrix Anal. Appl. 36(2), 367–384 (2015)MathSciNetCrossRefGoogle Scholar
  5. 27.
    R. Estrin, C. Greif, Towards an optimal condition number of certain augmented Lagrangian-type saddle-point matrices. Numer. Linear Algebra Appl. 23(4), 693–705 (2016)MathSciNetCrossRefGoogle Scholar
  6. 29.
    B. Fischer, A. Ramage, D.J. Silvester, A.J. Wathen, Minimum residual methods for augmented systems. BIT 38, 527–543 (1998)MathSciNetCrossRefGoogle Scholar
  7. 35.
    G. Golub, C. Greif, On solving block-structured indefinite linear systems. SIAM J. Sci. Comput. 24(6), 2076–2092 (2003)MathSciNetCrossRefGoogle Scholar
  8. 36.
    N.I.M. Gould, V. Simoncini, Spectral analysis of saddle point matrices with indefinite leading blocks. SIAM J. Matrix Anal. Appl. 31(3), 1152–1171 (2009)MathSciNetCrossRefGoogle Scholar
  9. 42.
    R.A. Horn, C.R. Johnson, Matrix Analysis, 2nd edn. (Cambridge University Press, New York, 2012)CrossRefGoogle Scholar
  10. 65.
    I. Perugia, V. Simoncini, Block-diagonal and indefinite symmetric preconditioners for mixed finite element formulations. Numer. Linear Algebra Appl. 7(7–8), 585–616 (2000)MathSciNetCrossRefGoogle Scholar
  11. 73.
    T. Rusten, R. Winther, A preconditioned iterative method for saddlepoint problems. SIAM J. Matrix Anal. Appl. 13(3), 887–904 (1992)MathSciNetCrossRefGoogle Scholar
  12. 76.
    D.J. Silvester, A.J. Wathen, Fast iterative solution of stabilized Stokes systems, part II: using block preconditioners. SIAM J. Numer. Anal. 31, 1352–1367 (1994)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Miroslav Rozložník
    • 1
  1. 1.Institute of MathematicsCzech Academy of SciencesPragueCzech Republic

Personalised recommendations