An efficient class of iterative methods for computing generalized outer inverse \({M_{T,S}^{(2)}}\)

In this paper, we propose a new matrix iteration scheme for computing the generalized outer inverse for a given complex matrix. The convergence analysis of the proposed scheme is established under certain necessary conditions, which indicates that the methods possess at least fourth-order convergence. The theoretical discussions show that the convergence order improves from 4 to 5 for a particular parameter choice. We prove that the sequence of approximations generated by the family satisfies the commutative property of matrices, provided the initial matrix commutes with the matrix under consideration. Some real-world and academic problems are chosen to validate our methods for solving the linear systems arising from statically determinate truss problems, steady-state analysis of a system of reactors, and elliptic partial differential equations. Moreover, we include a wide variety of large sparse test matrices obtained from the matrix market library. The performance measures used are the number of iterations, computational order of convergence, residual norm, efficiency index, and the computational time. The numerical results obtained are compared with some of the existing robust methods. It is demonstrated that our method gives improved results in terms of computational speed and efficiency.