On Randomized Block Gauss-Seidel Algorithms for Solving Inner Inverses of a Matrix

In the field of operator equations, solving the inner inverses is essential. In this paper, based on the Lagrangian function a class of block Gauss-Seidel methods is proposed, which contains three pseudoinverse-free algorithms. Among them, there is one randomized Gauss-Seidel method with one column and one with multi-columns. Moreover, one variant of randomized Gauss-Seidel methods is obtained by employing random Gaussian vectors instead of coordinate vectors. With the use of some inequalities and the properties of norms, the convergence analysis of new algorithms is investigated. Finally, the effectiveness of the new numerical methods is verified through some numerical experiments including real-world data and inconsistent problems.