Algorithms for the Solution of Systems of Linear Equations in Commutative Rings

  • Gennadi I. Malashonok
Part of the Progress in Mathematics book series (PM, volume 94)


Solution methods for linear equation systems in a commutative ring are discussed. Four methods are compared, in the setting of several different rings: Dodgson’s method [1], Bareiss’s method [2] and two methods of the author — method by forward and back-up procedures [3] and a one-pass method [4].

We show that for the number of coefficient operations, or for the number of operations in the finite rings, or for modular computation in the polynomial rings the one-pass method [4] is the best. The method of forward and back-up procedures [3] is the best for the polynomial rings when we make use of classical algorithms for polynomial operations.


Commutative Ring Polynomial Ring Linear Equation System Modular Method Forward Procedure 
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  1. [1]
    Dodgson C. L., Condensation of determinants, being a new and brief method for computing their arithmetic values, Proc. Royal Soc. Lond. A. 15 (1866), 150–155.CrossRefGoogle Scholar
  2. [2]
    Bareiss E. N., Sylvester’s identity and multistep integer-preserving Gaussian elimination, Math. Comput. 22 (1968), 565–578.MathSciNetMATHGoogle Scholar
  3. [3]
    Malashonok G. I., On the solution of a linear equation system over commutative ring, Math. Notes of the Acad. Sci. USSR 42, N4 (1987), 543–548.MathSciNetGoogle Scholar
  4. [4]
    Malashonok G. I., A new solution method for linear equation systems over the commutative ring, in “Int. Algebraic Conf., Novosibirsk,” Aug. 21-26, 1989, Theses on the ring theory, algebras and modules, 1989, p. 82.Google Scholar

Copyright information

© Springer Science+Business Media New York 1991

Authors and Affiliations

  • Gennadi I. Malashonok
    • 1
  1. 1.Institute for Applied Problems of Mechanics and MathematicsAcademy of SciencesLvovUSSR

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