||The most commonly used iterative method for solving large sparse least squares problems with an m¡ßn coefficient matrix A is the CGLS method, which applies the (preconditioned) conjugate gradient method to the normal equation.
In this paper, we consider alternative methods using an
n¡ßm matrix B to transform the problem to equivalent least squares problems with square coefficient matrices AB or BA, and then applying the Generalized Minimal Residual (GMRES) method, which is a robust Krylov subspace iterative method for solving systems of linear equations with nonsymmetric coefficient matrices.
Next, we give a sufficeint condition concerning B for the proposed methods to give a least squares solution without breakdown for arbitrary right hand side b, for over-determined, under-determined and possibly rank-deficient problems. Then, as an example for B, we propose the IMGS(l) method, which is an incomplete QR decomposition.
Finally, we show by numercial experiments on full-rank over-determined and under-determined problems that, for ill-conditioned problems, the proposed method using the IMGS(0) method, which is equivalent to diagonal scaling, gives a least squares solution faster than previous preconditioned CGLS methods.