Gradient methods for computing the Drazin-inverse solution

The problem of computing the vector ADbADb, in the cases when it provides a solution of a singular linear system, is reduced to the problem of finding a least-squares solution of an appropriate linear system. Later the presented ideas are extended to arbitrary linear systems. As a consequence, we present iterative methods for computing the vector ADbADb in general. The functionality of the exposed algorithms is based on a specific representation of the Drazin-inverse solution, as well as the properties that we introduce. Convergence results of the proposed methods are also considered. Illustrative numerical examples are presented.