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

Chapter
Part of the Progress in Mathematics book series (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.

## Keywords

Commutative Ring Polynomial Ring Linear Equation System Modular Method Forward Procedure
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

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