We develop a general theory for the convergence of the generalized minimal residual method for least squares problems preconditioned with inner iterations. The inner iterations take the form of a stationary iterative method. Also, we present theoretical justifications for using specific inner iterations such as the Jacobi and SOR-type methods. In this paper, the theory is improved particularly in the rank-deficient case. We analyse the spectrum of the preconditioned coefficient matrix, and characterize it with the spectral radius of the iteration matrix for the inner iterations. The analysis is supported by numerical experiments.
Morikuni et al. (Fri,) studied this question.