60 lines
2.5 KiB
Org Mode
60 lines
2.5 KiB
Org Mode
#+TITLE: Homework 5
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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 LIZFCM \rightarrow Matrix Routines \rightarrow ~lu decomp~ & ~bsubst~.
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The test ~UTEST(matrix, lu_decomp)~ is a unit test for the ~lu_decomp~ routine,
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and ~UTEST(matrix, bsubst)~ verifies back substitution on an upper triangular
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3 \times 3 matrix with a known solution that can be verified manually.
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Both can be found in ~tests/matrix.t.c~.
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* Question Two
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Unless the following are met, the resulting solution will be garbage.
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1. The matrix $U$ must be not be singular.
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2. $U$ must be square (or it will fail the ~assert~).
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3. The system created by $Ux = b$ must be consistent.
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4. $U$ is (quite obviously) in upper-triangular form.
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Thus, the actual calculation performing the $LU$ decomposition
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(in ~lu_decomp~) does a sanity
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check for 1-3 will fail an assert, should a point along the diagonal (pivot) be
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zero, or the matrix be non-factorable.
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* Question Three
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See LIZFCM \rightarrow Matrix Routines \rightarrow ~fsubst~.
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~UTEST(matrix, fsubst)~ verifies forward substitution on a lower triangular 3 \times 3
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matrix with a known solution that can be verified manually.
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* Question Four
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See LIZFCM \rightarrow Matrix Routines \rightarrow ~gaussian_elimination~ and ~solve_gaussian_elimination~.
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* Question Five
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See LIZFCM \rightarrow Matrix Routines \rightarrow ~m_dot_v~, and the ~UTEST(matrix, m_dot_v)~ in
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~tests/matrix.t.c~.
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* Question Six
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See ~UTEST(matrix, solve_gaussian_elimination)~ in ~tests/matrix.t.c~, which generates a diagonally dominant 10 \times 10 matrix
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and shows that the solution is consistent with the initial matrix, according to the steps given. Then,
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we do a dot product between each row of the diagonally dominant matrix and the solution vector to ensure
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it is near equivalent to the input vector.
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* Question Seven
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See ~UTEST(matrix, solve_matrix_lu_bsubst)~ which does the same test in Question Six with the solution according to
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~solve_matrix_lu_bsubst~ as shown in the Software Manual.
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* Question Eight
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No, since the time complexity for Gaussian Elimination is always less than that of the LU factorization solution by $O(n^2)$ operations
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(in LU factorization we perform both backwards and forwards substitutions proceeding the LU decomp, in Gaussian Elimination we only need
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back substitution).
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* Question Nine, Ten
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See LIZFCM Software manual and shared library in ~dist~ after compiling.
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