Preconditioned GMRES methods with incomplete Givens orthogonalization method for large sparse least-squares problems

We propose to precondition the GMRES method by using the incomplete Givens orthogonalization (IGO) method for the solution of large sparse linear least-squares problems. Theoretical analysis shows that the preconditioner satisfies the sufficient condition that can guarantee that the preconditioned GMRES method will never break down and always give the least-squares solution of the original problem. Numerical experiments further confirm that the new preconditioner is efficient. We also find that the IGO preconditioned BA-GMRES method is superior to the corresponding CGLS method for ill-conditioned and singular least-squares problems.