Numerical Analysis/Power iteration exercises

Exercises

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

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

Solution, first iteration:

$A{\mathbf{\text{x}}}^{(0)}=\left[\begin{array}{ccc}6& 2& -1\\ 2& 5& 1\\ -1& 1& 4\end{array}\right]\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]=\left[\begin{array}{c}7\\ 8\\ 4\end{array}\right],$ so the estimated eigenvalue is ${\mu }_1=8$ and the estimated eigenvector is ${\mathbf{\text{x}}}^{(1)}=\frac{1}{8}A{\mathbf{\text{x}}}^{(0)}=\left[\begin{array}{c}\frac{7}{8}\\ 1\\ \frac{1}{2}\end{array}\right]$.

 

Solution, second iteration:

$A{\mathbf{\text{x}}}^{(1)}=\left[\begin{array}{ccc}6& 2& -1\\ 2& 5& 1\\ -1& 1& 4\end{array}\right]\left[\begin{array}{c}\frac{7}{8}\\ 1\\ \frac{1}{2}\end{array}\right]=\left[\begin{array}{c}\frac{27}{4}\\ \frac{29}{4}\\ \frac{17}{8}\end{array}\right],$ so the estimated eigenvalue is ${\mu }_2=\frac{29}{4}$ and the estimated eigenvector is ${\mathbf{\text{x}}}^{(2)}=\frac{4}{29}A{\mathbf{\text{x}}}^{(1)}=\left[\begin{array}{c}\frac{27}{29}\\ 1\\ \frac{17}{58}\end{array}\right]$.

 

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 ${\mathbf{\text{x}}}^{(50)}=\left[\begin{array}{c}1.0000\\ 0.7507\\ -0.0698\end{array}\right]$