Generalized Schultz iterative methods for the computation of outer inverses

We consider a general matrix iterative method of the type Xk+1=Xkp(AXk)Xk+1=Xkp(AXk) for computing an outer inverse AR(G),N(G)(2), for given matrices A∈Cm×nA∈Cm×n and G∈Cn×mG∈Cn×m such that AR(G)⊕N(G)=CmAR(G)⊕N(G)=Cm. Here p(x)p(x) is an arbitrary polynomial of degree dd. The convergence of the method is proven under certain necessary conditions and the characterization of all methods having order rr is given. The obtained results provide a direct generalization of all known iterative methods of the same type. Moreover, we introduce one new method and show a procedure how to improve the convergence order of existing methods. This procedure is demonstrated on one concrete method and the improvement is confirmed by numerical examples.