Exercises 5.6 Exercises
1.
Find the dominant eigenvalue of the matrix
\begin{equation*}
A=\begin{pmatrix}\phantom{-}5&-1\cr-1&-1\cr\end{pmatrix}
\end{equation*}
first by hand in exact mathematics and then with a precision of \(10^{-3}\) using the power method.2.
In the problem above, write your own code or modify the code in Section 5.2 to plot the absolute values of the difference between the estimated dominant eigenvalue and its exact value. Assume that this error \(err(k)\) changes with the law \(err(k)=\exp(-\alpha k)\) and estimate roughly \(\alpha\) from the graph.3.
Use the QR method to find all eigenvalues of the matrix \(A\) above. How many iterations are needed to reach a precision of \(10^{-3}\text{?}\)4.
Repeat the analysis of the error of problem (2) for the QR method.5.
Repeat the calculations for the QR decomposition of the matrix
\begin{equation*}
\begin{pmatrix}1&2\cr3&4\cr\end{pmatrix}
\end{equation*}
in Section 5.3 after replacing the Euclidean norm with Taxicab norm.