= Quadratic Rayleigh-quotient improvement for the power method
For a real symmetric matrix with $|\lambda_1|>|\lambda_2|$ and a starting vector having nonzero component in the leading eigendirection, power-method direction errors are
$$
O\!\left(\left|\frac{\lambda_2}{\lambda_1}\right|^k\right),
$$
whereas the <Rayleigh quotient> error is
$$
O\!\left(\left|\frac{\lambda_2}{\lambda_1}\right|^{2k}\right).
$$
The improvement occurs because orthogonality cancels terms linear in the direction error.
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