LQ-Schur projection on large sparse matrix equations

A new paradigm for the solution of non-symmetric large sparse systems of linear equations is proposed. The paradigm is based on an LQ factorization of the matrix of coefficients, i.e. factoring the matrix of coefficients into the product of a lower triangular matrix and an orthogonal matrix. We show how the system of linear equations can be decomposed into a collection of smaller independent problems that can then be used to construct an iterative method for a system of smaller dimensionality. We show that the conditioning of the reduced problem cannot be worse than that of the original, unlike Schur complement methods in the non-symmetric case. The paradigm depends on the existence of an ordering of the rows representing the equations into blocks of rows that are mutually structurally orthogonal, except for a last block row that is coupled to all other rows in a limited way.