38 lines
1.7 KiB
Org Mode
38 lines
1.7 KiB
Org Mode
#+TITLE: Homework 7
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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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TODO: Update LIZFCM org file with jacobi solve
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* Question One
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See ~UTEST(jacobi, solve_jacobi)~ in ~test/jacobi.t.c~ and the entry
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~Jacobi -> solve_jacobi~ in the LIZFCM API documentation.
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* Question Two
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We cannot just perform the Jacobi algorithm on a Leslie matrix since
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it is obviously not diagonally dominant - which is a requirement. It is
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certainly not always the case, but, if a Leslie matrix $L$ is invertible, we can
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first perform gaussian elimination on $L$ augmented with $n_{k+1}$
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to obtain $n_k$ with the Jacobi method. See ~UTEST(jacobi, leslie_solve)~
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in ~test/jacobi.t.c~ for an example wherein this method is tested on a Leslie
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matrix to recompute a given initial population distribution.
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In terms of accuracy, an LU factorization and back substitution approach will
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always be as correct as possible within the limits of computation; it's a
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direct solution method. It's simply the nature of the Jacobi algorithm being
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a convergent solution that determines its accuracy.
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LU factorization also performs in order $O(n^3)$ runtime for an $n \times n$
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matrix, whereas the Jacobi algorithm runs in order $O(k n^2) = O(n^2)$ but with the
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con that $k$ is given by the convergence criteria, which might end up worse in
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some cases, than LU.
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* Question Three
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See ~UTEST(jacobi, gauss_siedel_solve)~ in ~test/jacobi.t.c~ which runs the same
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unit test as ~UTEST(jacobi, solve_jacobi)~ but using the
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~Jacobi -> gauss_siedel_solve~ method as documented in the LIZFCM API reference.
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* Question Four, Five
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