The Study Of Iterative Methods For Rank Deficient Least Squares Problems

Rank-deficient least squares problems arise from many scientific and engineering computations such as statistics, optimal problem, material and structure mechanics, land measurement, photography measurement problem, signal processing problem and so on. In the practical problems, since the order number of corresponding coefficient matrix of linear equations is larger, matrix rank is a deficit. In other words, A is irreversible. Then solving process is more complex. So it is very important to study of the suitable iterative methods for rank-deficient least squares problems.Recently, many researchers have proposed some effective iterative methods, such as classical iterative methods(Jacobi method, GS method, SOR method [5], AOR method [18]), block iterative methods (BJ method, BGS method, BSOR method [14]), symmetrical iterative methods (SGS method, SSOR method[21], SAOR method), symmetry block iterative methods (SBGS method, SBSOR method[23]), PSD method and various incomplete decomposition methods and so on.Based on the iterative methods (such as2-block SOR method,4-block SOR method and4-block AOR method) solving the least square problems with rank-deficient, at first we give the2-block AOR method by preconditioning technique. Second, we study the convergence analysis of the new AOR method and the choice of optimal relaxation parameters. We get the corresponding theorems. At last, we find the least square solution of minimal norm to the system Ax=b (A∈Crm×n,b∈Cm) by applying the new AOR iterative method. Numerical examples show the effectiveness of our method. It suggests that the new iterative AOR method is simpler, faster in convergence speed, more extensive applicability than the method in [18] by numerical examples and theorems. Meanwhile, we demand that A1is full row rank matrix, it is more convenient than the requirement of A11in [18].