2023-11-27 17:13:34 -05:00
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#+TITLE: Homework 7
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2023-11-15 16:43:22 -05:00
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#+AUTHOR: Elizabeth Hunt
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#+LATEX_HEADER: \notindent \notag \usepackage{amsmath} \usepackage[a4paper,margin=1in,portrait]{geometry}
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#+LATEX: \setlength\parindent{0pt}
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#+OPTIONS: toc:nil
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* Question One
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See ~UTEST(eigen, dominant_eigenvalue)~ in ~test/eigen.t.c~ and the entry
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~Eigen-Adjacent -> dominant_eigenvalue~ in the LIZFCM API documentation.
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* Question Two
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2023-11-27 12:14:50 -05:00
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See ~UTEST(eigen, leslie_matrix_dominant_eigenvalue)~ in ~test/eigen.t.c~
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and the entry ~Eigen-Adjacent -> leslie_matrix~ in the LIZFCM API
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documentation.
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* Question Three
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2023-11-27 15:32:05 -05:00
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See ~UTEST(eigen, least_dominant_eigenvalue)~ in ~test/eigen.t.c~ which
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finds the least dominant eigenvalue on the matrix:
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2023-11-27 12:14:50 -05:00
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2023-11-27 15:32:05 -05:00
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\begin{bmatrix}
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2 & 2 & 4 \\
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1 & 4 & 7 \\
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0 & 2 & 6
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\end{bmatrix}
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2023-11-27 16:06:16 -05:00
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which has eigenvalues: $5 + \sqrt{17}, 2, 5 - \sqrt{17}$ and should thus produce $5 - \sqrt{17}$.
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2023-11-27 15:32:05 -05:00
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See also the entry ~Eigen-Adjacent -> least_dominant_eigenvalue~ in the LIZFCM API
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documentation.
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2023-11-27 16:06:16 -05:00
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* Question Four
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See ~UTEST(eigen, shifted_eigenvalue)~ in ~test/eigen.t.c~ which
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finds the least dominant eigenvalue on the matrix:
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\begin{bmatrix}
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2 & 2 & 4 \\
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1 & 4 & 7 \\
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0 & 2 & 6
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\end{bmatrix}
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which has eigenvalues: $5 + \sqrt{17}, 2, 5 - \sqrt{17}$ and should thus produce $2.0$.
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With the initial guess: $[0.5, 1.0, 0.75]$.
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See also the entry ~Eigen-Adjacent -> shift_inverse_power_eigenvalue~ in the LIZFCM API
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documentation.
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2023-11-27 16:45:48 -05:00
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* Question Five
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See ~UTEST(eigen, partition_find_eigenvalues)~ in ~test/eigen.t.c~ which
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finds the eigenvalues in a partition of 10 on the matrix:
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\begin{bmatrix}
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2 & 2 & 4 \\
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1 & 4 & 7 \\
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0 & 2 & 6
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\end{bmatrix}
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which has eigenvalues: $5 + \sqrt{17}, 2, 5 - \sqrt{17}$, and should produce all three from
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the partitions when given the guesses $[0.5, 1.0, 0.75]$ from the questions above.
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See also the entry ~Eigen-Adjacent -> partition_find_eigenvalues~ in the LIZFCM API
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documentation.
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* Question Six
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2023-11-27 17:13:34 -05:00
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Consider we have the results of two methods developed in this homework: ~least_dominant_eigenvalue~, and ~dominant_eigenvalue~
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into ~lambda_0~, ~lambda_n~, respectively. Also assume that we have the method implemented as we've introduced,
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~shift_inverse_power_eigenvalue~.
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2023-11-27 17:13:34 -05:00
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Then, we begin at the midpoint of ~lambda_0~ and ~lambda_n~, and compute the
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~new_lambda = shift_inverse_power_eigenvalue~
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with a shift at the midpoint, and some given initial guess.
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1. If the result is equal (or within some tolerance) to ~lambda_n~ then the closest eigenvalue to the midpoint
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is still the dominant eigenvalue, and thus the next most dominant will be on the left. Set ~lambda_n~
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to the midpoint and reiterate.
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2. If the result is greater or equal to ~lambda_0~ we know an eigenvalue of greater or equal magnitude
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exists on the right. So, we set ~lambda_0~ to this eigenvalue associated with the midpoint, and
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re-iterate.
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3. Continue re-iterating until we hit some given maximum number of iterations. Finally we will return
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~new_lambda~.
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