||In this paper, we construct a preconditioner for least squares
problems min||b-Ax||, where A can be matrices with any shape
and any rank. The preconditioner itself is a sparse approximation to the Moore-Penrose inverse of the coefficient matrix A. For this preconditioner, we give theoretical analysis to show that under certain assumption, the problem preconditioned by this preconditioner is equivalent to the original problem, and the GMRES method can determine a solution to the preconditioned problem before breakdown happens.