The generalized inverseA T,S (2)of a matrixAis a{2}-inverse ofAwith the prescribed rangeTand null spaceS. A representation for the generalized inverseA T,S (2)has been recently developed with the conditionσ (GA| T)⊂(0,∞), whereGis a matrix withR(G)=TandN(G)=S. In this note, we remove the above condition. Three types of iterative methods forA T,S (2)are presented ifσ(GA|T)is a subset of the open right half-plane and they are extensions of existing computational procedures ofA T,S (2), including

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... pecial cases such as the weighted Moore-Penrose inverseA M,N †and the Drazin inverseAD. Numerical examples are given to illustrate our results.