course | course_year | question_number | tags | title | year | |||
---|---|---|---|---|---|---|---|---|
Numerical Analysis |
II |
102 |
|
Paper 4, Section II, B |
2016 |
(a) Describe an implementation of the power method for determining the eigenvalue of largest modulus and its associated eigenvector for a matrix that has a unique eigenvalue of largest modulus.
Now let
(b) Consider the following variant of the power method, called the two-stage power method, applied to the matrix
-
Pick
$\mathbf{x}^{(0)} \in \mathbb{R}^{n}$ satisfying$\left|\mathbf{x}^{(0)}\right|=1$ . Let$0<\varepsilon \ll 1$ . Set$k=0$ and$\mathbf{x}^{(1)}=A \mathbf{x}^{(0)}$ . -
Calculate
$\mathbf{x}^{(k+2)}=A \mathbf{x}^{(k+1)}$ and calculate$\alpha, \beta$ that minimise
-
If
$f(\alpha, \beta) \leqslant \varepsilon$ , solve$\lambda^{2}+\alpha \lambda+\beta=0$ and let the roots be$\lambda_{1}$ and$\lambda_{2}$ . They are accepted as eigenvalues of$A$ , and the corresponding eigenvectors are estimated as$\mathbf{x}^{(k+1)}-\lambda_{2} \mathbf{x}^{(k)}$ and$\mathbf{x}^{(k+1)}-\lambda_{1} \mathbf{x}^{(k)}$ . -
Otherwise, divide
$\mathbf{x}^{(k+2)}$ and$\mathbf{x}^{(k+1)}$ by the current value of$\left|\mathbf{x}^{(k+1)}\right|$ , increase$k$ by 1 and return to Step$\mathbf{1}$ .
Explain the justification behind Step 2 of the algorithm.
(c) Let
$$\mathbf{y}{k}=\mathbf{x}^{(k)}=\left(\begin{array}{l} 1 \ 1 \ 1 \end{array}\right), \quad \mathbf{y}{k+1}=A \mathbf{x}^{(k)}=\left(\begin{array}{c} 2 \ 3 \ 4 \end{array}\right), \quad \mathbf{y}_{k+2}=A^{2} \mathbf{x}^{(k)}=\left(\begin{array}{c} 2 \ 4 \ 6 \end{array}\right)$$
Find two eigenvalues of