The power method generates iterates $v^{(k+1)} = Av^{(k)} / \|Av^{(k)}\|$ starting from $v^{(0)}$. If $A$ has a dominant eigenvalue $\lambda_1$ with $|\lambda_1| > |\lambda_2| \geq \cdots \geq |\lambda_n|$, and $v^{(0)}$ has a nonzero component in the $\lambda_1$-eigenspace, then $v^{(k)} \to \pm v_1$ (the dominant eigenvector) and the Rayleigh quotient $\rho^{(k)} = (v^{(k)})^T A v^{(k)} / (v^{(k)})^T v^{(k)} \to \lambda_1$. The convergence rate is $|\lambda_2/\lambda_1|^k$.

Inverse iteration applies the power method to $(A - \sigma I)^{-1}$: solve $(A - \sigma I)w^{(k+1)} = v^{(k)}$, then normalize $v^{(k+1)} = w^{(k+1)}/\|w^{(k+1)}\|$. This converges to the eigenvector of $A$ whose eigenvalue $\lambda_j$ is closest to the shift $\sigma$, with rate $|\lambda_j - \sigma|/\min_{i \neq j}|\lambda_i - \sigma|$. When $\sigma$ is a good eigenvalue approximation, convergence is very rapid.

Rayleigh quotient iteration (RQI) updates the shift at each step: $\sigma_k = \rho(v^{(k)})$ (the Rayleigh quotient), then performs inverse iteration with this shift. For symmetric $A$, RQI converges cubically: $|\lambda - \sigma_{k+1}| = O(|\lambda - \sigma_k|^3)$. This means roughly tripling the number of correct digits per iteration -- just 2-3 iterations typically suffice.