Solution of sparse quasi-square rectangular systems by Gaussian elimination

Abstract

We present a general method for the linear least-squares solution of overdetermined and underdetermined systems. The method is particularly efficient when the coefficient matrix is quasi-square, that is when the number of rows and number of columns is almost the same. The numerical methods for linear least-squares problems and minimum-norm solutions do not generally take account of this special characteristic. The proposed method is based on an LU factorization of the original quasi-square matrix A, assuming that A has full rank. In the overdetermined case, the LU factors are used to compute a basis for the null space of A T . The right-hand side vector b is then projected onto this subspace and the least-squares solution is obtained from the solution of this reduced problem. In the case of underdetermined systems, the desired solution is again obtained through the solution of a reduced system. The use of this method may lead to important savings in computational time for both dense and sparse matrices. It is also shown in the paper that, even in cases where the matrices are quite small, sparse solvers perform better than dense solvers. Some practical examples that illustrate the use of the method are included.