The Conjugate Gradient Method

  • Tom Lyche
Part of the Texts in Computational Science and Engineering book series (TCSE, volume 22)


The conjugate gradient method was published by Hestenes and Stiefel in 1952, as a direct method for solving linear systems. Today its main use is as an iterative method for solving large sparse linear systems. On a test problem we show that it performs as well as the SOR method with optimal acceleration parameter, and we do not have to estimate any such parameter. However the conjugate gradient method is restricted to positive definite systems. We also consider a mathematical formulation of the preconditioned conjugate gradient method. It is used to speed up convergence of the conjugate gradient method. We only give one example of a possible preconditioner.


  1. 1.
    O. Axelsson, Iterative Solution Methods (Cambridge University Press, 1994)Google Scholar
  2. 6.
    M. Hestenes, E. Stiefel, Methods of conjugate gradients for solving linear systems. J. Res. Natl. Bur. Stand. 29, 409–439 (1952)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Tom Lyche
    • 1
  1. 1.Blindern, University of OsloOsloNorway

Personalised recommendations