= Solution
For a nonzero real vector $x$, the <Rayleigh quotient> of the <matrix> $A$ is
$$
\boxed{R_A(x)=\frac{x^TAx}{x^Tx}}.
$$
The <Rayleigh quotient iteration> starts from a unit vector $v_0$. Given $v_k$, compute the shift
$$
\mu_k=R_A(v_k),
$$
solve the <linear system>
$$
(A-\mu_kI)w_k=v_k,
$$
and normalize in the <Euclidean norm>:
$$
v_{k+1}=\frac{w_k}{\|w_k\|_2}.
$$
The next eigenvalue estimate is $\mu_{k+1}=R_A(v_{k+1})$. The process stops when the <eigenpair residual> is sufficiently small.
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