Let $A$ be a real symmetric $n \times n$ matrix with real and distinct eigenvalues $0=\lambda_{1}<\cdots<\lambda_{n-1}=1<\lambda_{n}$ and a corresponding orthogonal basis of normalized real eigenvectors $\left(\mathbf{w}_{i}\right)_{i=1}^{n}$.

To estimate the eigenvector $\mathbf{w}_{n}$ of $A$ whose eigenvalue is $\lambda_{n}$, the power method with shifts is employed which has the following form:

$$\mathbf{y}=\left(A-s_{k} I\right) \mathbf{x}^{(k)}, \quad \mathbf{x}^{(k+1)}=\mathbf{y} /\|\mathbf{y}\|, \quad s_{k} \in \mathbb{R}, \quad k=0,1,2, \ldots$$

Three versions of this method are considered:

(i) no shift: $s_{k} \equiv 0$;

(ii) single shift: $s_{k} \equiv \frac{1}{2}$;

(iii) double shift: $s_{2 \ell} \equiv s_{0}=\frac{1}{4}(2+\sqrt{2}), s_{2 \ell+1} \equiv s_{1}=\frac{1}{4}(2-\sqrt{2})$.

Assume that $\lambda_{n}=1+\epsilon$, where $\epsilon>0$ is very small, so that the terms $\mathcal{O}\left(\epsilon^{2}\right)$ are negligible, and that $\mathbf{x}^{(0)}$ contains substantial components of all the eigenvectors.

By considering the approximation after $2 m$ iterations in the form

$$\mathbf{x}^{(2 m)}=\pm \mathbf{w}_{n}+\mathcal{O}\left(\rho^{2 m}\right) \quad(m \rightarrow \infty)$$

find $\rho$ as a function of $\epsilon$ for each of the three versions of the method.

Compare the convergence rates of the three versions of the method, with reference to the number of iterations needed to achieve a prescribed accuracy.