The inverse, shifted inverse, and Rayleigh quotient iterations are well-known
algorithms for computing an eigenvector of a symmetric matrix. In this talk
we demonstrate that each one of these three algorithms can be viewed as a
standard form of Newton's method from the nonlinear programming literature,
involving an norm projection. This provides an explanation for their good
behavior despite the need to solve systems with nearly singular coefficient
matrices. Our equivalence result also leads us naturally to a new proof that
the convergence of the Rayleigh quotient iteration is q-cubic
with rate constant at worst 1.
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