USSOR methods for solving the rank deficient linear least squares problem

In order to find the least squares solution of minimal norm to linear system Ax=b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Ax=b$$\end{document} with A∈Cm×n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A \in \mathcal{C}^{m \times n}$$\end{document} being a matrix of rank r<n≤m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r< n \le m$$\end{document}, b∈Cm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b \in \mathcal{C}^{m}$$\end{document}, Zheng and Wang (Appl Math Comput 169:1305–1323, 2005) proposed a class of symmetric successive overrelaxation (SSOR) methods, which is based on augmenting system to a block 4×4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$4 \times 4$$\end{document} consistent system. In this paper, we construct the unsymmetric successive overrelaxation (USSOR) method. The semiconvergence of the USSOR method is discussed. Numerical experiments illustrate that the number of iterations and CPU time for the USSOR method with the appropriate parameters is respectively less and faster than the SSOR method with optimal parameters.