= Local cubic convergence of Rayleigh quotient iteration
{c}
Let a real symmetric matrix have simple eigenpairs $(\lambda_i,w_i)$ and let
$$
x=c^{-1}\left(w_1+\epsilon\sum_{i\geq2}a_iw_i\right).
$$
Then its <Rayleigh quotient> is
$$
R_A(x)=\lambda_1+\epsilon^2D+O(\epsilon^4),
\qquad
D=\sum_{i\geq2}a_i^2(\lambda_i-\lambda_1).
$$
After one shifted inverse solve and normalization, the relative coefficients are
$$
-\epsilon^3\frac{Da_i}{\lambda_i-\lambda_1}+O(\epsilon^5).
$$
Thus the eigenvector direction error is cubed at each sufficiently close iterate.
Back to article page
