Lecture 19

  • Eugene E. Tyrtyshnikov


If \( f(x) = \tfrac{1} {2}(Ax,x) - \operatorname{Re} (b,x), A = A^* \in \mathbb{C}^{n \times n} \), then the boundedness of f from below is equivalent to the nonnegative definiteness of A (prove this). Let us assume that A > 0. In this case, a linear system Ax = b has a unique solution z, and, for any x,
$$ f(x) - f(z) = \frac{1} {2}(A(x - z),x - z) \equiv E(x). \Rightarrow $$
z is the single minimum point for f (x). ⇒ A minimization method for f can equally serve as a method of solving a linear system with the Hermitian positively definite coefficient matrix.


Conjugate Gradient Method Krylov Subspace Arnoldi Method Minimal Residual Hessenberg Matrix 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A.W. Chou. On the optimality of Krylov information. J. of Complexity 3: 26–40 (1987).MATHCrossRefGoogle Scholar
  2. 2.
    G. I. Marchuk and Yu. A. Kuznetsov. Iterative methods and quadratic functional. Methods of Numerical Mathematics. Novosibirsk, 1975, pp. 4–143.Google Scholar
  3. 3.
    Y. Saad and M.H. Schultz. GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Scientific and Stat. Comp. 7: 856–869 (1986).MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    I first heard about this striking relation from S. A. Goreinov.Google Scholar
  5. 5.
    L. Zhou and H. F. Walker. Residual smoothing techniques for iterative methods. SIAM 3. on Sci. Comput. 15(2): 297–312 (1994).MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    R. W. Freund and N. M. Nachtigal. QMR: a quasi-minimal residual method for non-Hermitian linear systems. Numer. Math. 60: 315–339 (1991).MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • Eugene E. Tyrtyshnikov
    • 1
  1. 1.Institute of Numerical MathematicsRussian Academy of SciencesMoscowRussia

Personalised recommendations