Numerical Analysis/Inverse iteration exercises

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

The LU decomposition of A is $L=\left[\begin{array}{ccc}1& 0& 0\\ 0.3333& 1& 0\\ -0.167& 0.3077& 1\end{array}\right],U=\left[\begin{array}{ccc}6& 2& -1\\ 0& 4.333& 1.333\\ 0& 0& 3.423\end{array}\right]$.

By hand, use the LU decomposition to do two iterations of the inverse power method (without shift) starting with ${\mathbf{\text{𝐱}}}^{(0)}.$

Solution, first iteration:

$LU{\mathbf{\text{𝐲}}}^{(1)}={\mathbf{\text{𝐱}}}^{(0)}\Rightarrow {\mathbf{\text{𝐲}}}^{(1)}=\left[\begin{array}{c}0.1910\\ 0.0674\\ 0.2809\end{array}\right]$. The estimated eigenvalue is ${\mu }_1=\frac{1}{0.2809}=3.56$ and the estimated eigenvector is ${\mathbf{\text{𝐱}}}^{(1)}=\frac{{\mathbf{\text{𝐲}}}^{(1)}}{0.2809}=\left[\begin{array}{c}0.6800\\ 0.2400\\ 1.0000\end{array}\right]$.

 

Solution, second iteration:

$LU{\mathbf{\text{𝐲}}}^{(2)}={\mathbf{\text{𝐱}}}^{(1)}\Rightarrow {\mathbf{\text{𝐲}}}^{(2)}=\left[\begin{array}{c}0.1996\\ -0.0966\\ 0.3240\end{array}\right]$. The estimated eigenvalue is ${\mu }_2=\frac{1}{0.3240}=3.0864$ and the estimated eigenvector is ${\mathbf{\text{𝐱}}}^{(2)}=\frac{{\mathbf{\text{𝐲}}}^{(2)}}{0.3240}=\left[\begin{array}{c}0.6158\\ -0.2982\\ 1.0000\end{array}\right]$.

 

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

Solution, 50 iterations:

The estimated eigenvalue after 50 iterations is ${\mu }_{50}=\frac{1}{0.4375}=2.2855$ and the estimated eigenvector is ${\mathbf{\text{𝐱}}}^{(50)}=\left[\begin{array}{c}0.7750\\ -0.9394\\ 1.0000\end{array}\right]$.