Numerische Mathematik

, Volume 48, Issue 5, pp 543–560

The rate of convergence of Conjugate Gradients


  • A. van der Sluis
    • Mathematical InstituteUniversity of Utrecht
  • H. A. van der Vorst
    • Department of MathematicsUniversity of Technology

DOI: 10.1007/BF01389450

Cite this article as:
van der Sluis, A. & van der Vorst, H.A. Numer. Math. (1986) 48: 543. doi:10.1007/BF01389450


It has been observed that the rate of convergence of Conjugate Gradients increases when one or more of the extreme Ritz values have sufficiently converged to the corresponding eigenvalues (the “superlinear convergence” of CG). In this paper this will be proved and made quantitative. It will be shown that a very modest degree of convergence of an extreme Ritz value already suffices for an increased rate of convergence to occur.

Subject Classifications

AMS (MOS): 65F10CR: G.1.3
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© Springer-Verlag 1986