The Gauss-Seidel method has been used for more than 100 years as the standard method for the solution of linear systems of equations under certain restrictions. This method, as well as Cramer and Jacobi, is widely used in education and engineering, but there is a theoretical gap when we want to solve less restricted systems, or even non-square or non-exact systems of equation. Here, the solution goes through the use of numerical systems, such as the minimization theories or the Moore-Penrose pseudoinverse. In this paper we fill this gap with a global analytical iterative formulation that is capable to reach the solutions obtained with the Moore-Penrose pseudoinverse and the minimization methodologies, but that analytically lies to the solutions of Gauss-Seidel, Jacobi, or Cramer when the system is simplified.

翻译：Gauss-Seidel方法作为特定限制条件下线性方程组求解的标准方法已沿用百余年。该方法与Cramer法、Jacobi法共同广泛应用于教育及工程领域，但在求解限制条件更宽松的方程组乃至非方阵或非精确方程组时存在理论空白。此类问题的解决需借助数值系统方法，如最小化理论或Moore-Penrose伪逆。本文提出一种全局解析迭代公式填补该空白，该公式既能获得与Moore-Penrose伪逆及最小化方法相同的解，又在系统简化时解析收敛至Gauss-Seidel、Jacobi或Cramer法的解。