 Local cubic convergence of Rayleigh quotient iteration

ID: local-cubic-convergence-of-rayleigh-quotient-iteration

Let a real symmetric matrix have simple eigenpairs

(λ_{i}, w_{i})

and let

x = c^{- 1} (w_{1} + ϵ \sum_{i \geq 2} a_{i} w_{i}) .

R_{A} (x) = λ_{1} + ϵ^{2} D + O (ϵ^{4}), D = \sum_{i \geq 2} a_{i}^{2} (λ_{i} - λ_{1}) .

After one shifted inverse solve and normalization, the relative coefficients are

- ϵ^{3} \frac{D a _{i}}{λ _{i} - λ _{1}} + O (ϵ^{5}) .

Thus the eigenvector direction error is cubed at each sufficiently close iterate.

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