  In this paper, a Gauss-Seidel method with oblique direction (GSO) is proposed
for finding the least-squares solution to a system of linear equations, where
the coefficient matrix may be full rank or rank deficient and the system is
overdetermined or underdetermined. Through this method, the number of iteration
steps and running time can be reduced to a greater extent to find the
least-squares solution, especially when the columns of matrix A are close to
linear correlation. It is theoretically proved that GSO method converges to the
least-squares solution. At the same time, a randomized version--randomized
Gauss-Seidel method with oblique direction (RGSO) is established, and its
convergence is proved. Theoretical proof and numerical results show that the
GSO method and the RGSO method are more efficient than the coordinate descent
(CD) method and the randomized coordinate descent (RCD) method.
