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## 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 is the Rayleigh quotient of , namely, .

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 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 and to an eigenvector which is not closest to the starting vector . 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 at every iteration, and this matrix will be singular when hits an eigenvalue. Hence RQI is practical only if such factorizations can be obtained cheaply at every iteration. See Parlett [353] for more details.

Next: Subspace Iteration Up: Single- and Multiple-Vector Iterations Previous: Inverse Iteration   Contents   Index
Susan Blackford 2000-11-20