LQ-Schur Projection on Large Sparse Matrix Equations

Abstract

A new paradigm for the solution of nonsymmetric large sparse systems oflinear equations is proposed. The paradigm is based on an LQfactorization of the matrix of coefficients, i.e.~factoring the matrixof coefficients into the product of a lower triangular matrix and anorthogonal matrix. We show how the system of linear equations can bedecomposed into a collection of smaller independent problems which canthen beused to construct an iterative method for a system of smallerdimensionality. We show that the conditioning of the reducedproblem cannot be worse than that of the original, unlike Schurcomplement methods in the nonsymmetric case. The paradigm depends onthe existence of an ordering of the rows representing the equationsinto blocks of rows which are mutually structurely orthogonal, exceptfor a last block row which is coupled to all other rows in a limitedway.