{"title":"Gauss-Seidel Iterative Methods for Rank Deficient Least Squares Problems","authors":"Davod Khojasteh Salkuyeh, Sayyed Hasan Azizi","volume":55,"journal":"International Journal of Mathematical and Computational Sciences","pagesStart":1010,"pagesEnd":1015,"ISSN":"1307-6892","URL":"https:\/\/publications.waset.org\/pdf\/14436","abstract":"We study the semiconvergence of Gauss-Seidel iterative\r\nmethods for the least squares solution of minimal norm of rank\r\ndeficient linear systems of equations. Necessary and sufficient conditions\r\nfor the semiconvergence of the Gauss-Seidel iterative method\r\nare given. We also show that if the linear system of equations is\r\nconsistent, then the proposed methods with a zero vector as an initial\r\nguess converge in one iteration. Some numerical results are given to\r\nillustrate the theoretical results.","references":"[1] A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical\r\nSciences, SIAM, Philadelphia 1994.\r\n[2] R. E. Cline, Inverses of rank invariant powers of a matrix, SIAM J.\r\nNumer. Anal. 5 (1968) 182-197.\r\n[3] A. Hadjidimos, Accelerated overrelaxation method, Math. Comput. 32\r\n(1978) 149-157.\r\n[4] Y. Huang and Y. Song, AOR iterative methods for rank deficient least\r\nsquares problems, J. Appl. Math. Comput. 26 (2008) 105-124.\r\n[5] V. A. Miller and M. Neumann, Successive overrelaxation methods for\r\nsolving the rank deficient linear squares problem, Linear Algebra Appl\r\n88\/89 (1987) 533-557.\r\n[6] H. Tian, Accelerate overrelaxation methods for rank deficient linear\r\nsystems, Appl. Math. Comput. 140 (2003) 485-499.","publisher":"World Academy of Science, Engineering and Technology","index":"Open Science Index 55, 2011"}