Algorithms for the solution of systems of linear equations in commutative ring

TL;DR

This paper compares four algorithms for solving linear systems over commutative rings, demonstrating that the one-pass method is most efficient in various computational settings, especially for finite and polynomial rings.

Contribution

It introduces and evaluates a new one-pass solution method, showing its superiority over existing methods in multiple algebraic contexts.

Findings

One-pass method is most efficient for finite rings.

Forward and back-up procedures excel in polynomial rings with classical algorithms.

The paper provides a comparative analysis of four solution methods.

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.