**1. Introduction**

New matrix series expressions were recently derived by the author [1] for the solution of simple first order differential equations associated with general systems of linear algebraic equations. These differential equations describe the orthogonal trajectories of a family of hypersurfaces that represent a quadratic functional related to the linear algebraic system. The solution of the latter can be obtained by minimizing the functional along an orthogonal trajectory instead of applying various techniques based on minimization along conjugate gradient directions or based on minimized iterations [2]. Since the solutions of the differential equations considered are simply related to the solutions of the corresponding algebraic systems through matrix exponentials, there is the possibility to develop efficient solution methods if only the matrix exponentials could be used in numerical calculations

accurately and with a small computational effort. A survey of various existent algorithms for computing matrix exponentials and a useful discussion of the difficulties involved are presented in [3].

In the present work, we use new formulae for arbitrary matrix exponentials that contain highly convergent infinite series which allow accurate and stable numerical computations. Employing these formulae, two new solution methods are proposed which are particularly efficient for large-scale general linear algebraic systems.
