We present an iterative method for treating extremely large-scale eigenvalue problems. Based on an exact formula and the GMRES method, our approach generates a subspace which has the property that the residual of interior eigenpairs in the subspace is minimized. The result is that the corresponding large matrix is block-diagonalized iteratively. The accuracy of the final eigenpairs of interest is directly controlled by the accuracy of the GMRES procedure. Our method limits the number of Arnoldi iterations involved, and the dimension of the subspace, by including the residual in the subspace and minimizing it at each step of the iteration.