%Solving Linear Systems
%If you modify this file, please indicate here and in the footnote
%Modified by Steve Chapin, chapin@math.ohiou.edu
%Math410

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{\Large 
Solving Linear Systems \footnote{Copyright \copyright 2002 Todd Young.
All rights reserved. Please address comments to young@math.ohiou.edu.}}
%Modified for \textsc{MatLab} by Steve Chapin, chapin@math.ohiou.edu.}
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\begin{enumerate}
\item Try the following commands (at the prompt and then press \fbox{Enter}): 
\begin{enumerate}
    \item \verb&A = [1.2969, 0.8648; 0.2161, 0.1441] &
    \item \verb&b1 = [1.2969; 0.2161] &
    \item \verb&x = A\b1 &
    \item Repeat the process but with a vector $b2$ obtained from $b1$
    by rounding off to three decimal places.
    \item Explain exactly what happened.  Why was the first answer so
    simple?  Why do the two answers differ by so much?
\end{enumerate} 
    \item Try the following commands:
\begin{enumerate}
    \item \verb&B = sym(maple(`matrix',`2,2',`(I,J)->sin(I*J)')) &
    \item \verb&c = [1; 2] &
\end{enumerate}
and use \verb& x = B\c & to solve $Bx = c$.  Then change the 2's
to 3's in the first line, change \verb$c$ to \verb$[1; 2; 3]$ and try to solve
again.  Use \verb& x = double(x) & to obtain an approximate numerical value of the
solution.  Try the command \verb& Bn = double(B)&, then
\verb& x = Bn\c&.  When would an exact symbolic solution and when would an
approximate numerical solution be more useful?  For big matrices, which type
of computation would be faster?
\item Input the matrix:
$$ 
\textsf{C = }\left[ 
\begin{array}{rr} 
1 & 2 \\ 
2 & 4
\end{array}
\right] 
$$

and solve $Cx = d$ with \verb$ d1 = [4; 8] $ and \verb$ d2 = [1; 1]$.
Use symbolic and non-symbolic versions of $C$.  Explain the results.  Which
way gives more information?  
    \item Prepare a report as follows:
\begin{enumerate} 
    \item \textbf{Using standard mathematical notation}, write down 
    the results of all the computations, except the symbolic solution 
    to the 3 x 3 system in \#2.  Do {\bf NOT} hand in a printout.
    \item \textbf{Using complete sentences}, briefly answer all of the 
    questions.  This includes giving explanations where requested.
\end{enumerate} 
\end{enumerate} 
\vfill \noindent

\textsf{The matrix in 1.\ is nearly singular, causing the linear system to be
very sensitive to perturbations.  Students are exposed to both
symbolic and numerical solutions.  The ideas of no solutions or
infinitely many solutions are reinforced.} 

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