Rayleigh Quotient Iteration

A natural extension of inverse iteration is to vary the shift at each step. It turns out that the best shift which can be derived from an eigenvector approximation $x \not = 0$ is the Rayleigh quotient of $x$, namely, $\rho(x) \equiv x^{\ast} \, A \, x / x^{\ast} \, x$.

In general, Rayleigh quotient iteration (RQI) will need fewer iterations to find an eigenvalue than inverse iteration with a constant shift; it ultimately has cubical convergence, while inverse iteration converges linearly. However, it is not obvious how to choose the starting vector $y$ to make RQI converge to any particular eigenvalue/eigenvector pair. For example, the RQI can converge to an eigenvalue which is not the closest to the starting Rayleigh quotient $\rho(y)$ and to an eigenvector which is not closest to the starting vector $y$. Furthermore, there is the tiny but nasty possibility that it may not converge to an eigenvalue/eigenvector pair at all. RQI is more expensive than inverse iteration, requiring a factorization of $A - \rho_k I$ at every iteration, and this matrix will be singular when $\rho_k$ hits an eigenvalue. Hence RQI is practical only if such factorizations can be obtained cheaply at every iteration. See Parlett [353] for more details.