A New Two-stage Iterative Method for Linear Systems and its Application in Solving the Poisson’s Equation

In the current study, we investigate the two-stage iterative method for solving linear systems. In this regard, a new comparison theorem for the spectral radius of two-stage iterative method with different inner and outer splitting matrices under suitable conditions is presented. Our new theorem shows which splitting generates fast convergence in iterative methods. Finally, based on the Poisson’s equation, we solve the Poisson-Block tridiagonal matrix by two different splittings, which is an issue that is frequently encountered in mechanical engineering and theoretical physics. Based on a particular linear system, numerical computations are presented, which clearly show the reliability and efficiency of the presented algorithm.