Generalization of the Gauss–Jordan method for solving homogeneous infinite systems of linear algebraic equations

In this paper, we, first, using the reduction method in the narrow sense (the simple reduction method), have generalized the classical Gauss–Jordan method for solving finite systems of linear algebraic equations to inhomogeneous infinite systems. The generalization is based on a new theory of solutions to inhomogeneous infinite systems, proposed by us, which gives an exact analytical solution in the form of a series. Second, we have shown that the application of reduction in the narrow sense in the case of homogeneous systems gives only a trivial solution, therefore, in order to generalize the Gauss–Jordan method for solving infinite homogeneous systems, we used the reduction method in the wide sense. A numerical comparison is given that shows acceptable accuracy.