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

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Effective Methods in Algebraic Geometry

Part of the book series: Progress in Mathematics ((PM,volume 94))

Abstract

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.

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References

  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.

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  2. Bareiss E. N., Sylvester’s identity and multistep integer-preserving Gaussian elimination, Math. Comput. 22 (1968), 565–578.

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  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.

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  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.

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© 1991 Springer Science+Business Media New York

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Malashonok, G.I. (1991). Algorithms for the Solution of Systems of Linear Equations in Commutative Rings. In: Mora, T., Traverso, C. (eds) Effective Methods in Algebraic Geometry. Progress in Mathematics, vol 94. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0441-1_19

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  • DOI: https://doi.org/10.1007/978-1-4612-0441-1_19

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4612-6761-4

  • Online ISBN: 978-1-4612-0441-1

  • eBook Packages: Springer Book Archive

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