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\author{Elizabeth Hunt}
\date{\today}
\title{Homework 5}
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 pdfauthor={Elizabeth Hunt},
 pdftitle={Homework 5},
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\section{Question One}
\label{sec:org88abf18}
See LIZFCM \(\rightarrow\) Matrix Routines \(\rightarrow\) \texttt{lu decomp} \& \texttt{bsubst}.

The test \texttt{UTEST(matrix, lu\_decomp)} is a unit test for the \texttt{lu\_decomp} routine,
and \texttt{UTEST(matrix, bsubst)} verifies back substitution on an upper triangular
3 \texttimes{} 3 matrix with a known solution that can be verified manually.

Both can be found in \texttt{tests/matrix.t.c}.

\section{Question Two}
\label{sec:org098a7f1}
Unless the following are met, the resulting solution will be garbage.

\begin{enumerate}
\item The matrix \(U\) must be not be singular.
\item \(U\) must be square (or it will fail the \texttt{assert}).
\item The system created by \(Ux = b\) must be consistent.
\item \(U\) is in (obviously) upper-triangular form.
\end{enumerate}

Thus, the actual calculation performing the \(LU\) decomposition
(in \texttt{lu\_decomp}) does a sanity
check for 1-3 will fail an assert, should a point along the diagonal (pivot) be
zero, or the matrix be non-factorable.

\section{Question Three}
\label{sec:org40d5983}
See LIZFCM \(\rightarrow\) Matrix Routines \(\rightarrow\) \texttt{fsubst}.

\texttt{UTEST(matrix, fsubst)} verifies forward substitution on a lower triangular 3 \texttimes{} 3
matrix with a known solution that can be verified manually.

\section{Question Four}
\label{sec:orgf7d23bb}

See LIZFCM \(\rightarrow\) Matrix Routines \(\rightarrow\) \texttt{gaussian\_elimination} and \texttt{solve\_gaussian\_elimination}.

\section{Question Five}
\label{sec:org54e966c}
See LIZFCM \(\rightarrow\) Matrix Routines \(\rightarrow\) \texttt{m\_dot\_v}, and the \texttt{UTEST(matrix, m\_dot\_v)} in
\texttt{tests/matrix.t.c}.

\section{Question Six}
\label{sec:org413b527}
See \texttt{UTEST(matrix, solve\_gaussian\_elimination)} in \texttt{tests/matrix.t.c}, which generates a diagonally dominant 10 \texttimes{} 10 matrix
and shows that the solution is consistent with the initial matrix, according to the steps given. Then,
we do a dot product between each row of the diagonally dominant matrix and the solution vector to ensure
it is near equivalent to the input vector.

\section{Question Seven}
\label{sec:orgd3d7443}
See \texttt{UTEST(matrix, solve\_matrix\_lu\_bsubst)} which does the same test in Question Six with the solution according to
\texttt{solve\_matrix\_lu\_bsubst} as shown in the Software Manual.

\section{Question Eight}
\label{sec:orgf8ac9bf}
No, since the time complexity for Gaussian Elimination is always less than that of the LU factorization solution by \(O(n^2)\) operations
(in LU factorization we perform both backwards and forwards substitutions proceeding the LU decomp, in Gaussian Elimination we only need
back substitution).

\section{Question Nine, Ten}
\label{sec:orgb270171}
See LIZFCM Software manual and shared library in \texttt{dist} after compiling.
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