(untitled) (Journal of Computational Mathematics, 1997)
RQI is an approach for eigenvectors of matrices. In 1974, B.N Parlett proved that it was a "successful algorithm" with cubic convergent speed for normal matrices. After then, several authors developed relevant theory and put this research into dynamical frame. [3] indicated that RQI failed for non-normal matrices with complex eigenvalues. In this paper, RQI for non-normal matrices with only real spectrum is analyzed. The authers proved that eigenvectors are super-attractive fixed points of RQI. The geometrical and topological behaviours of two periodic orbits are considered in detail. The existence of three or higher periodic orbits and their geometry are considered in detail. The existence of three or higher periodic orbits and their geometry are still open and of interest. It will be reported in our forthcoming paper.