Regions of convergence of the Rayleigh quotient iteration method

Abstract

AbstractThe Rayleigh quotient iteration method finds an eigenvector and the corresponding eigenvalue of a symmetric matrix. This is a fundamental problem in science and engineering. Parlett and Kahan have shown, in 1968, that for almost any initial vector in the unit sphere, the Rayleigh quotient iteration method converges to some eigenvector. In this paper, the regions of the unit sphere which include all possible initial vectors converging to a specific eigenvector are studied. The generalized eigenvalue problem Ax = λBx is considered. It is shown that the regions do not change when the matrix is shifted or multiplied by a scalar. These regions are completely characterized in the three‐dimensional case. It is shown that, in this case, the area of the region of convergence corresponding to the interior eigenvalue is larger than the area of those corresponding to any extreme one. This can be interpreted, with the appropriate choice of probability distribution, as: the probability of converging to an eigenvector corresponding to the interior eigenvalue is larger than the probability of converging to an eigenvector corresponding to any extreme eigenvalue. It is conjectured that the same is true for matrices of any order. Experiments in higher dimensions are presented which conform with the conjecture.