Three dimensional x-projection method (with acceleration techniques) for solving systems of linear equations

Abstract

The solution of linear systems of equations using a 3-dimensional x-projection method is presented. At each step of the iterative process the approximate solution vector is projected to a point in the intersection of three of the hyperplanes of the linear system. Nonsingularity of the coefficient matrix is the only requirement for convergence. An algorithm is presented to select triples of hyperplanes to project the approximate solution vector at each step. The algorithm is quasi-optimal since the hyperplanes, which are determined by the row vectors of the coefficient matrix, are selected a priori. This is shown to significantly reduce the number of cycles required for convergence. We observe that in some cases the ratio of the change vectors of the approximate solution vectors after some number of cycles becomes a constant. Thus, when this occurs a simple geometric acceleration can be applied to calculate the solution directly. Geometric acceleration can significantly reduce computation time and improve the accuracy of the solution by orders of magnitude. The 3-dimensional x-projection method was tested against the 2-dimensional x-projection method using random and Hilbert coefficient matrices and proved superior (less C.P.U. time required) in nearly every case.