95 lines
3.5 KiB
TeX
95 lines
3.5 KiB
TeX
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% Created 2023-10-30 Mon 19:05
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% Intended LaTeX compiler: pdflatex
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\documentclass[11pt]{article}
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\usepackage[utf8]{inputenc}
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\usepackage[T1]{fontenc}
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\usepackage{graphicx}
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\usepackage{longtable}
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\usepackage{wrapfig}
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\usepackage{rotating}
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\usepackage[normalem]{ulem}
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\usepackage{amsmath}
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\usepackage{amssymb}
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\usepackage{capt-of}
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\usepackage{hyperref}
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\notindent \notag \usepackage{amsmath} \usepackage[a4paper,margin=1in,portrait]{geometry}
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\author{Elizabeth Hunt}
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\date{\today}
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\title{Homework 5}
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\hypersetup{
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pdfauthor={Elizabeth Hunt},
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pdftitle={Homework 5},
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pdfkeywords={},
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pdfsubject={},
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pdfcreator={Emacs 28.2 (Org mode 9.7-pre)},
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pdflang={English}}
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\begin{document}
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\maketitle
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\setlength\parindent{0pt}
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\section{Question One}
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\label{sec:org88abf18}
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See LIZFCM \(\rightarrow\) Matrix Routines \(\rightarrow\) \texttt{lu decomp} \& \texttt{bsubst}.
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The test \texttt{UTEST(matrix, lu\_decomp)} is a unit test for the \texttt{lu\_decomp} routine,
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and \texttt{UTEST(matrix, bsubst)} verifies back substitution on an upper triangular
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3 \texttimes{} 3 matrix with a known solution that can be verified manually.
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Both can be found in \texttt{tests/matrix.t.c}.
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\section{Question Two}
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\label{sec:org098a7f1}
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Unless the following are met, the resulting solution will be garbage.
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\begin{enumerate}
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\item The matrix \(U\) must be not be singular.
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\item \(U\) must be square (or it will fail the \texttt{assert}).
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\item The system created by \(Ux = b\) must be consistent.
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\item \(U\) is in (obviously) upper-triangular form.
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\end{enumerate}
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Thus, the actual calculation performing the \(LU\) decomposition
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(in \texttt{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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\section{Question Three}
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\label{sec:org40d5983}
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See LIZFCM \(\rightarrow\) Matrix Routines \(\rightarrow\) \texttt{fsubst}.
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\texttt{UTEST(matrix, fsubst)} verifies forward substitution on a lower triangular 3 \texttimes{} 3
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matrix with a known solution that can be verified manually.
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\section{Question Four}
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\label{sec:orgf7d23bb}
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See LIZFCM \(\rightarrow\) Matrix Routines \(\rightarrow\) \texttt{gaussian\_elimination} and \texttt{solve\_gaussian\_elimination}.
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\section{Question Five}
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\label{sec:org54e966c}
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See LIZFCM \(\rightarrow\) Matrix Routines \(\rightarrow\) \texttt{m\_dot\_v}, and the \texttt{UTEST(matrix, m\_dot\_v)} in
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\texttt{tests/matrix.t.c}.
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\section{Question Six}
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\label{sec:org413b527}
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See \texttt{UTEST(matrix, solve\_gaussian\_elimination)} in \texttt{tests/matrix.t.c}, which generates a diagonally dominant 10 \texttimes{} 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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\section{Question Seven}
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\label{sec:orgd3d7443}
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See \texttt{UTEST(matrix, solve\_matrix\_lu\_bsubst)} which does the same test in Question Six with the solution according to
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\texttt{solve\_matrix\_lu\_bsubst} as shown in the Software Manual.
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\section{Question Eight}
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\label{sec:orgf8ac9bf}
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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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\section{Question Nine, Ten}
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\label{sec:orgb270171}
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See LIZFCM Software manual and shared library in \texttt{dist} after compiling.
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\end{document}
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