Numerical Analysis/Power iteration exercises

Exercises

Consider the matrix, $A=\left[{\begin{array}{c c c}6&2&-1\\2&5&1\\-1&1&4\end{array}}\right]$, and the vector, ${\textbf {x}}^{(0)}=\left[{\begin{array}{c}1\\1\\1\\\end{array}}\right].$

By hand, do two iterations of the power method starting with ${\textbf {x}}^{(0)}.$

Solution, first iteration:

$A{\textbf {x}}^{(0)}=\left[{\begin{array}{c c c}6&2&-1\\2&5&1\\-1&1&4\end{array}}\right]\left[{\begin{array}{c}1\\1\\1\end{array}}\right]=\left[{\begin{array}{c}7\\8\\4\end{array}}\right],$ so the estimated eigenvalue is $\mu _{1}=8$ and the estimated eigenvector is ${\textbf {x}}^{(1)}={\frac {1}{8}}A{\textbf {x}}^{(0)}=\left[{\begin{array}{c}{\frac {7}{8}}\\1\\{\frac {1}{2}}\end{array}}\right]$.

 

Solution, second iteration:

$A{\textbf {x}}^{(1)}=\left[{\begin{array}{c c c}6&2&-1\\2&5&1\\-1&1&4\end{array}}\right]\left[{\begin{array}{c}{\frac {7}{8}}\\1\\{\frac {1}{2}}\end{array}}\right]=\left[{\begin{array}{c}{\frac {27}{4}}\\{\frac {29}{4}}\\{\frac {17}{8}}\end{array}}\right],$ so the estimated eigenvalue is $\mu _{2}={\frac {29}{4}}$ and the estimated eigenvector is ${\textbf {x}}^{(2)}={\frac {4}{29}}A{\textbf {x}}^{(1)}=\left[{\begin{array}{c}{\frac {27}{29}}\\1\\{\frac {17}{58}}\end{array}}\right]$.

 

Use a computational software package to do 50 iterations. What was the result?

 

Solution, 50 iterations:

The estimated eigenvalue after 50 iterations is $\mu =7.5712$ with estimated eigenvector ${\textbf {x}}^{(50)}=\left[{\begin{array}{c}1.0000\\0.7507\\-0.0698\end{array}}\right]$