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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