TY - JOUR
T1 - Direct Iterative Methods for Rank Deficient Generalized Least Squares Problems
AU - Yuan , Jin-Yun
AU -  , Xiao-Qing Jin
JO - Journal of Computational Mathematics
VL - 4
SP - 439
EP - 448
PY - 2000
DA - 2000/08
SN - 18
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/jcm/9056.html
KW - Rank deficient generalized LS problem, block SOR method, PCG method, convergence, optimal parameter.
AB - <p style="text-align: justify;">The generalized least squares (LS) problem&nbsp; $$\min_{x\in R^n} (Ax-b)^T W^{-1} (Ax-b)$$ appears in many application areas. Here $W$ is an $m \times m$ symmetric positive definite matrix and $A$ is an $m \times n$ matrix with $m ≥ n$. Since the problem has many solutions in rank deficient case, some special preconditioned techniques are adapted to obtain the minimum 2-norm solution. A block SOR method and the preconditioned conjugate gradient (PCG) method are proposed here. Convergence and optimal relaxation parameter for the block SOR method are studied. An error bound for the PCG method is given. The comparison of these methods is investigated. Some remarks on the implementation of the methods and the operation cost are given as well. &nbsp;</p>