General form of the Gauss-Seidel equation to linearly approximate the Moore-Penrose pseudoinverse in random non-square systems and high order tensors (2503.22273v1)

Abstract: 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.