Numerical Analysis/Inverse iteration 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, ${\textbf {x}}^{(0)}=\left[{\begin{array}{c}1\\1\\1\\\end{array}}\right].$

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

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

Solution, first iteration:

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

 

Solution, second iteration:

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

 

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 ${\textbf {x}}^{(50)}=\left[{\begin{array}{c}0.7750\\-0.9394\\1.0000\end{array}}\right]$.