2023-11-15 16:43:22 -05:00
|
|
|
#+TITLE: Homework 6
|
|
|
|
|
#+AUTHOR: Elizabeth Hunt
|
|
|
|
|
#+LATEX_HEADER: \notindent \notag \usepackage{amsmath} \usepackage[a4paper,margin=1in,portrait]{geometry}
|
|
|
|
|
#+LATEX: \setlength\parindent{0pt}
|
|
|
|
|
#+OPTIONS: toc:nil
|
|
|
|
|
|
|
|
|
|
* Question One
|
|
|
|
|
See ~UTEST(eigen, dominant_eigenvalue)~ in ~test/eigen.t.c~ and the entry
|
|
|
|
|
~Eigen-Adjacent -> dominant_eigenvalue~ in the LIZFCM API documentation.
|
|
|
|
|
* Question Two
|
2023-11-27 12:14:50 -05:00
|
|
|
See ~UTEST(eigen, leslie_matrix_dominant_eigenvalue)~ in ~test/eigen.t.c~
|
|
|
|
|
and the entry ~Eigen-Adjacent -> leslie_matrix~ in the LIZFCM API
|
|
|
|
|
documentation.
|
|
|
|
|
* Question Three
|
2023-11-27 15:32:05 -05:00
|
|
|
See ~UTEST(eigen, least_dominant_eigenvalue)~ in ~test/eigen.t.c~ which
|
|
|
|
|
finds the least dominant eigenvalue on the matrix:
|
2023-11-27 12:14:50 -05:00
|
|
|
|
2023-11-27 15:32:05 -05:00
|
|
|
\begin{bmatrix}
|
|
|
|
|
2 & 2 & 4 \\
|
|
|
|
|
1 & 4 & 7 \\
|
|
|
|
|
0 & 2 & 6
|
|
|
|
|
\end{bmatrix}
|
|
|
|
|
|
2023-11-27 16:06:16 -05:00
|
|
|
which has eigenvalues: $5 + \sqrt{17}, 2, 5 - \sqrt{17}$ and should thus produce $5 - \sqrt{17}$.
|
2023-11-27 15:32:05 -05:00
|
|
|
|
|
|
|
|
See also the entry ~Eigen-Adjacent -> least_dominant_eigenvalue~ in the LIZFCM API
|
|
|
|
|
documentation.
|
2023-11-27 16:06:16 -05:00
|
|
|
* Question Four
|
|
|
|
|
See ~UTEST(eigen, shifted_eigenvalue)~ in ~test/eigen.t.c~ which
|
|
|
|
|
finds the least dominant eigenvalue on the matrix:
|
|
|
|
|
|
|
|
|
|
\begin{bmatrix}
|
|
|
|
|
2 & 2 & 4 \\
|
|
|
|
|
1 & 4 & 7 \\
|
|
|
|
|
0 & 2 & 6
|
|
|
|
|
\end{bmatrix}
|
|
|
|
|
|
|
|
|
|
which has eigenvalues: $5 + \sqrt{17}, 2, 5 - \sqrt{17}$ and should thus produce $2.0$.
|
|
|
|
|
|
|
|
|
|
With the initial guess: $[0.5, 1.0, 0.75]$.
|
|
|
|
|
|
|
|
|
|
See also the entry ~Eigen-Adjacent -> shift_inverse_power_eigenvalue~ in the LIZFCM API
|
|
|
|
|
documentation.
|
