||For least squares problems of minimizing || b - A x ||_2 where A is a large sparse m x n (m >= n) matrix, the common method is to apply the conjugate gradient method to the normal equation A^T A x = A^T b.
However, the condition number of A^T A is square of that of A, and convergence becomes problematic for severely ill-conditioned problems even with preconditioning.
In this paper, we propose two methods for applying the GMRES method to the least squares problem by using a n x m matrix B.
We give the necessary and sufficient condition that B should satisfy in order that the proposed methods give a least squares solution.
Then, for implementations for B, we propose an incomplete QR decomposition IMGS(l).
Numerical experiments show that the simplest case l=0, which is equivalent to B= ( diag (A^T A) )^(-1) A^T, gives best results, and converges faster than previous methods for severely ill-conditioned problems.