On an iterative method for solving the least squares problem of rank-deficient systems

摘要

In this paper, we first show an iterative method for finding the least squares (LS) solution to the inconsistent system Ax = b, where A is an m × n matrix of rank r. The method is an iteration scheme for consistent system of linear equations Mx = b which is an augmented system associated with Ax = b. It denotes that under some conditions, the sequence, x_0,x_1, x_2,..., converges to the LS solution of the system Ax = b for every initial vector x_0, where (M + γE)x_i = γEx_(i-1) + b, for i = 1,2,... In our numerical test, we propose to find E without using decomposition methods. The improved timings are shown with matrices of substantial size.