On partial solution of systems of linear algebraic equations

Abstract—The problem of partial solution of systems of linear algebraic equations is formulated and consists in finding groups of components of the vector of the solution in the case of the right-hand side of the system belonging to a subspace. Such a problem arises, for example, in realization of block relaxation methods in subspaces. The problem is investigated in detail in the case of Jacobi matrices. The algorithms proposed are applied to constructing efficient direct methods for solving systems of difference equations approximating boundary value problems for Poisson's equation in the rectangle.