The Conjugate Gradient Method

  • T. Ginsburg
Part of the Die Grundlehren der mathematischen Wissenschaften book series (GL, volume 186)


The CG-algorithm is an iterative method to solve linear systems
$$Ax + b = 0$$
where A is a symmetric and positive definite coefficient matrix of order n. The method has been described first by Stiefel and Hesteness [1, 2] and additional information is contained in [3] and [4]. The notations used here coincide partially with those used in [5] where also the connexions with other methods and generalisations of the CG-algorithm are presented.




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  1. 1.
    Hestenes, M. R., Stiefel, E.: Methods of Conjugate Gradients for Solving Linear Systems. Nat. Bur. Standards J. of Res. 49, 409–436 (1952).MathSciNetMATHGoogle Scholar
  2. 2.
    Stiefel, E.: Einige Methoden der Relaxationsrechnung. Z. Angew. Math. Phys. 3, 1–33 (1952).MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Stiefel, E. Relaxationsmethoden bester Strategie zur Lösung linearer Gleichungssysteme. Comment. Math. Helv. 29, 157–179 (1955).MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Stiefel, E. Kernel Polynomials in Linear Algebra and their Numerical Applications, Nat. Bur. Standards. Appl. Math. Ser. 49, 1–22 (1958).MathSciNetGoogle Scholar
  5. 5.
    Engeli, M., Ginsburg, Th., Rutishauser, H., Stiefel, E.: Refined Iterative Methods for Computation of the Solution and the Eigenvalues of Self-adjoint Boundary Value Problems. Mitt. Inst, angew. Math. ETH Zürich, No. 8 (Basel: Birkhäuser 1959).Google Scholar

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© Springer-Verlag Berlin · Heidelberg 1971

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  • T. Ginsburg

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