Using Matrix Differential Equations for Solving Systems of Linear Algebraic Equations

Abstract

Various ordinary differential equations of the first order have recently been used by the author for the solution of general, large linear systems of algebraic equations. Exact solutions were derived in terms of a new kind of infinite series of matrices which are truncated and applied repeatedly to approximate the solution. In these matrix series, each new term is obtained from the preceding one by multiplication with a matrix which becomes better and better conditioned tending to the identity matrix. Obviously, this helps the numerical computations. For a more efficient computation of approximate solutions of the algebraic systems, we consider new differential equations which are solved by simple techniques of numerical integration. The solution procedure allows to easily control and monitor the magnitude of the residual vector at each step of integration. A related iterative method is also proposed. The solution methods are flexible, permitting various intervening parameters to be changed whenever necessary in order to increase their efficiency. Efficient computation of a rough approximation of the solution, applicable even to poorly conditioned systems, is also performed based on the alternate application of two different types of minimization of associated functionals. A smaller amount of computation is needed to obtain an approximate solution of large linear systems as compared to existing methods.