A note on the Rayleigh quotient iteration for symmetric eigenvalue problems

The Rayleigh quotient iteration is a famous algorithm for solving symmetric eigenvalue problems but suffers a serious limitation: it does not converge in a few peculiar cases. In the present study we show that the Rayleigh quotient iteration always converges when the iterative vector is replaced. The main benefit of our proposed algorithm is that, unlike the existing modification that also guarantees convergence, it admits a direct convergence proof.