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ME3255S2017/lecture_10/lecture_10.tex
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\section{Gauss Elimination}\label{gauss-elimination} | |
\subsubsection{Solving sets of equations with matrix | |
operations}\label{solving-sets-of-equations-with-matrix-operations} | |
The number of dimensions of a matrix indicate the degrees of freedom of | |
the system you are solving. | |
If you have a set of known output, \(y_{1},~y_{2},~...y_{N}\) and a set | |
of equations that relate unknown inputs, \(x_{1},~x_{2},~...x_{N}\), | |
then these can be written in a vector matrix format as: | |
\(y=Ax\) | |
Consider a problem with 2 DOF: | |
\(x_{1}+3x_{2}=1\) | |
\(2x_{1}+x_{2}=1\) | |
\(\left[ \begin{array}{cc} 1 & 3 \\ 2 & 1 \end{array} \right] \left[\begin{array}{c} x_{1} \\ x_{2} \end{array}\right]= \left[\begin{array}{c} 1 \\ 1\end{array}\right]\) | |
The solution for \(x_{1}\) and \(x_{2}\) is the intersection of two | |
lines: | |
\begin{Verbatim}[commandchars=\\\{\}] | |
{\color{incolor}In [{\color{incolor}3}]:} \PY{n}{x21}\PY{p}{=}\PY{p}{[}\PY{o}{\PYZhy{}}\PY{l+m+mi}{2}\PY{p}{:}\PY{l+m+mi}{2}\PY{p}{]}\PY{p}{;} | |
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\begin{center} | |
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{lecture_10_files/lecture_10_3_0.pdf} | |
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{ \hspace*{\fill} \\} | |
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\PY{n}{A}\PY{o}{\PYZbs{}}\PY{n}{y} \PY{c}{\PYZpc{} matlab\PYZsq{}s Ax=y solution for x} | |
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\begin{Verbatim}[commandchars=\\\{\}] | |
ans = | |
0.40000 | |
0.20000 | |
\end{Verbatim} | |
For a \(3\times3\) matrix, the solution is the intersection of the 3 | |
planes. | |
\(10x_{1}+2x_{2}+x_{3}=1\) | |
\(2x_{1}+x_{2}+x_{3}=1\) | |
\(x_{1}+2x_{2}+10x_{3}=1\) | |
$\left[ \begin{array}{ccc} 10 & 2 & 1\\ 2 & 1 & 1 \\ 1 & 2 & 10\end{array} \right] | |
\left[\begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \end{array}\right]= | |
\left[\begin{array}{c} 1 \\ 1 \\ 1\end{array}\right]$ | |
\begin{Verbatim}[commandchars=\\\{\}] | |
{\color{incolor}In [{\color{incolor}9}]:} \PY{n}{N}\PY{p}{=}\PY{l+m+mi}{25}\PY{p}{;} | |
\PY{n}{x11}\PY{p}{=}\PY{n+nb}{linspace}\PY{p}{(}\PY{o}{\PYZhy{}}\PY{l+m+mi}{2}\PY{p}{,}\PY{l+m+mi}{2}\PY{p}{,}\PY{n}{N}\PY{p}{)}\PY{p}{;} | |
\PY{n}{x12}\PY{p}{=}\PY{n+nb}{linspace}\PY{p}{(}\PY{o}{\PYZhy{}}\PY{l+m+mi}{2}\PY{p}{,}\PY{l+m+mi}{2}\PY{p}{,}\PY{n}{N}\PY{p}{)}\PY{p}{;} | |
\PY{p}{[}\PY{n}{X11}\PY{p}{,}\PY{n}{X12}\PY{p}{]}\PY{p}{=}\PY{n+nb}{meshgrid}\PY{p}{(}\PY{n}{x11}\PY{p}{,}\PY{n}{x12}\PY{p}{)}\PY{p}{;} | |
\PY{n}{X13}\PY{p}{=}\PY{l+m+mi}{1}\PY{o}{\PYZhy{}}\PY{l+m+mi}{10}\PY{o}{*}\PY{n}{X11}\PY{o}{\PYZhy{}}\PY{l+m+mi}{2}\PY{o}{*}\PY{n}{X12}\PY{p}{;} | |
\PY{n}{x21}\PY{p}{=}\PY{n+nb}{linspace}\PY{p}{(}\PY{o}{\PYZhy{}}\PY{l+m+mi}{2}\PY{p}{,}\PY{l+m+mi}{2}\PY{p}{,}\PY{n}{N}\PY{p}{)}\PY{p}{;} | |
\PY{n}{x22}\PY{p}{=}\PY{n+nb}{linspace}\PY{p}{(}\PY{o}{\PYZhy{}}\PY{l+m+mi}{2}\PY{p}{,}\PY{l+m+mi}{2}\PY{p}{,}\PY{n}{N}\PY{p}{)}\PY{p}{;} | |
\PY{p}{[}\PY{n}{X21}\PY{p}{,}\PY{n}{X22}\PY{p}{]}\PY{p}{=}\PY{n+nb}{meshgrid}\PY{p}{(}\PY{n}{x21}\PY{p}{,}\PY{n}{x22}\PY{p}{)}\PY{p}{;} | |
\PY{n}{X23}\PY{p}{=}\PY{l+m+mi}{1}\PY{o}{\PYZhy{}}\PY{l+m+mi}{2}\PY{o}{*}\PY{n}{X11}\PY{o}{\PYZhy{}}\PY{n}{X22}\PY{p}{;} | |
\PY{n}{x31}\PY{p}{=}\PY{n+nb}{linspace}\PY{p}{(}\PY{o}{\PYZhy{}}\PY{l+m+mi}{2}\PY{p}{,}\PY{l+m+mi}{2}\PY{p}{,}\PY{n}{N}\PY{p}{)}\PY{p}{;} | |
\PY{n}{x32}\PY{p}{=}\PY{n+nb}{linspace}\PY{p}{(}\PY{o}{\PYZhy{}}\PY{l+m+mi}{2}\PY{p}{,}\PY{l+m+mi}{2}\PY{p}{,}\PY{n}{N}\PY{p}{)}\PY{p}{;} | |
\PY{p}{[}\PY{n}{X31}\PY{p}{,}\PY{n}{X32}\PY{p}{]}\PY{p}{=}\PY{n+nb}{meshgrid}\PY{p}{(}\PY{n}{x31}\PY{p}{,}\PY{n}{x32}\PY{p}{)}\PY{p}{;} | |
\PY{n}{X33}\PY{p}{=}\PY{l+m+mi}{1}\PY{o}{/}\PY{l+m+mi}{10}\PY{o}{*}\PY{p}{(}\PY{l+m+mi}{1}\PY{o}{\PYZhy{}}\PY{n}{X31}\PY{o}{\PYZhy{}}\PY{l+m+mi}{2}\PY{o}{*}\PY{n}{X32}\PY{p}{)}\PY{p}{;} | |
\PY{n+nb}{mesh}\PY{p}{(}\PY{n}{X11}\PY{p}{,}\PY{n}{X12}\PY{p}{,}\PY{n}{X13}\PY{p}{)}\PY{p}{;} | |
\PY{n+nb}{hold} \PY{n}{on}\PY{p}{;} | |
\PY{n+nb}{mesh}\PY{p}{(}\PY{n}{X21}\PY{p}{,}\PY{n}{X22}\PY{p}{,}\PY{n}{X23}\PY{p}{)} | |
\PY{n+nb}{mesh}\PY{p}{(}\PY{n}{X31}\PY{p}{,}\PY{n}{X32}\PY{p}{,}\PY{n}{X33}\PY{p}{)} | |
\PY{n}{x}\PY{p}{=}\PY{p}{[}\PY{l+m+mi}{10}\PY{p}{,}\PY{l+m+mi}{2}\PY{p}{,} \PY{l+m+mi}{1}\PY{p}{;}\PY{l+m+mi}{2}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{,} \PY{l+m+mi}{1}\PY{p}{;} \PY{l+m+mi}{1}\PY{p}{,} \PY{l+m+mi}{2}\PY{p}{,} \PY{l+m+mi}{10}\PY{p}{]}\PY{o}{\PYZbs{}}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{;}\PY{l+m+mi}{1}\PY{p}{;}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{;} | |
\PY{n}{plot3}\PY{p}{(}\PY{n}{x}\PY{p}{(}\PY{l+m+mi}{1}\PY{p}{)}\PY{p}{,}\PY{n}{x}\PY{p}{(}\PY{l+m+mi}{2}\PY{p}{)}\PY{p}{,}\PY{n}{x}\PY{p}{(}\PY{l+m+mi}{3}\PY{p}{)}\PY{p}{,}\PY{l+s}{\PYZsq{}}\PY{l+s}{o\PYZsq{}}\PY{p}{)} | |
\PY{n+nb}{xlabel}\PY{p}{(}\PY{l+s}{\PYZsq{}}\PY{l+s}{x1\PYZsq{}}\PY{p}{)} | |
\PY{n+nb}{ylabel}\PY{p}{(}\PY{l+s}{\PYZsq{}}\PY{l+s}{x2\PYZsq{}}\PY{p}{)} | |
\PY{n+nb}{zlabel}\PY{p}{(}\PY{l+s}{\PYZsq{}}\PY{l+s}{x3\PYZsq{}}\PY{p}{)} | |
\PY{n+nb}{view}\PY{p}{(}\PY{l+m+mi}{10}\PY{p}{,}\PY{l+m+mi}{45}\PY{p}{)} | |
\end{Verbatim} | |
\begin{center} | |
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{lecture_10_files/lecture_10_6_0.pdf} | |
\end{center} | |
{ \hspace*{\fill} \\} | |
After 3 DOF problems, the solutions are described as \emph{hyperplane} | |
intersections. Which are even harder to visualize | |
\subsection{Gauss elimination}\label{gauss-elimination} | |
\subsubsection{Solving sets of equations | |
systematically}\label{solving-sets-of-equations-systematically} | |
$\left[ | |
\begin{array}{ccc|c} | |
& A & & y \\ | |
10 & 2 & 1 & 1\\ | |
2 & 1 & 1 & 1 \\ | |
1 & 2 & 10 & 1\end{array} | |
\right] $ | |
Ay(2,:)-Ay(1,:)/5 = ({[}2 1 1 1{]}-1/5{[}10 2 1 1{]}) | |
$\left[ | |
\begin{array}{ccc|c} | |
& A & & y \\ | |
10 & 2 & 1 & 1\\ | |
0 & 3/5 & 4/5 & 4/5 \\ | |
1 & 2 & 10 & 1\end{array} | |
\right] $ | |
Ay(3,:)-Ay(1,:)/10 = ({[}1 2 10 1{]}-1/10{[}10 2 1 1{]}) | |
$\left[ | |
\begin{array}{ccc|c} | |
& A & & y \\ | |
10 & 2 & 1 & 1\\ | |
0 & 3/5 & 4/5 & 4/5 \\ | |
0 & 1.8 & 9.9 & 0.9\end{array} | |
\right] $ | |
Ay(3,:)-1.8*5/3*Ay(2,:) = ({[}0 1.8 9.9 0.9{]}-3*{[}0 3/5 4/5 4/5{]}) | |
$\left[ | |
\begin{array}{ccc|c} | |
& A & & y \\ | |
10 & 2 & 1 & 1\\ | |
0 & 3/5 & 4/5 & 4/5 \\ | |
0 & 0 & 7.5 & -1.5\end{array} | |
\right]$ | |
now, \(7.5x_{3}=-1.5\) so \(x_{3}=-\frac{1}{5}\) | |
then, \(3/5x_{2}+4/5(-1/5)=1\) so \(x_{2}=\frac{8}{5}\) | |
finally, \(10x_{1}+2(8/5)+1(-\frac{1}{5})=1\) | |
Consider the problem again from the intro to Linear Algebra, 4 masses | |
are connected in series to 4 springs with K=10 N/m. What are the final | |
positions of the masses? | |
\begin{figure}[htbp] | |
\centering | |
\includegraphics{../lecture_09/mass_springs.svg} | |
\caption{Springs-masses} | |
\end{figure} | |
The masses haves the following amounts, 1, 2, 3, and 4 kg for masses | |
1-4. Using a FBD for each mass: | |
\(m_{1}g+k(x_{2}-x_{1})-kx_{1}=0\) | |
\(m_{2}g+k(x_{3}-x_{2})-k(x_{2}-x_{1})=0\) | |
\(m_{3}g+k(x_{4}-x_{3})-k(x_{3}-x_{2})=0\) | |
\(m_{4}g-k(x_{4}-x_{3})=0\) | |
in matrix form: | |
\(\left[ \begin{array}{cccc} 2k & -k & 0 & 0 \\ -k & 2k & -k & 0 \\ 0 & -k & 2k & -k \\ 0 & 0 & -k & k \end{array} \right] \left[ \begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{array} \right]= \left[ \begin{array}{c} m_{1}g \\ m_{2}g \\ m_{3}g \\ m_{4}g \end{array} \right]\) | |
\begin{Verbatim}[commandchars=\\\{\}] | |
{\color{incolor}In [{\color{incolor}10}]:} \PY{n}{k}\PY{p}{=}\PY{l+m+mi}{10}\PY{p}{;} \PY{c}{\PYZpc{} N/m} | |
\PY{n}{m1}\PY{p}{=}\PY{l+m+mi}{1}\PY{p}{;} \PY{c}{\PYZpc{} kg} | |
\PY{n}{m2}\PY{p}{=}\PY{l+m+mi}{2}\PY{p}{;} | |
\PY{n}{m3}\PY{p}{=}\PY{l+m+mi}{3}\PY{p}{;} | |
\PY{n}{m4}\PY{p}{=}\PY{l+m+mi}{4}\PY{p}{;} | |
\PY{n}{g}\PY{p}{=}\PY{l+m+mf}{9.81}\PY{p}{;} \PY{c}{\PYZpc{} m/s\PYZca{}2} | |
\PY{n}{K}\PY{p}{=}\PY{p}{[}\PY{l+m+mi}{2}\PY{o}{*}\PY{n}{k} \PY{o}{\PYZhy{}}\PY{n}{k} \PY{l+m+mi}{0} \PY{l+m+mi}{0}\PY{p}{;} \PY{o}{\PYZhy{}}\PY{n}{k} \PY{l+m+mi}{2}\PY{o}{*}\PY{n}{k} \PY{o}{\PYZhy{}}\PY{n}{k} \PY{l+m+mi}{0}\PY{p}{;} \PY{l+m+mi}{0} \PY{o}{\PYZhy{}}\PY{n}{k} \PY{l+m+mi}{2}\PY{o}{*}\PY{n}{k} \PY{o}{\PYZhy{}}\PY{n}{k}\PY{p}{;} \PY{l+m+mi}{0} \PY{l+m+mi}{0} \PY{o}{\PYZhy{}}\PY{n}{k} \PY{n}{k}\PY{p}{]} | |
\PY{n}{y}\PY{p}{=}\PY{p}{[}\PY{n}{m1}\PY{o}{*}\PY{n}{g}\PY{p}{;}\PY{n}{m2}\PY{o}{*}\PY{n}{g}\PY{p}{;}\PY{n}{m3}\PY{o}{*}\PY{n}{g}\PY{p}{;}\PY{n}{m4}\PY{o}{*}\PY{n}{g}\PY{p}{]} | |
\end{Verbatim} | |
\begin{Verbatim}[commandchars=\\\{\}] | |
K = | |
20 -10 0 0 | |
-10 20 -10 0 | |
0 -10 20 -10 | |
0 0 -10 10 | |
y = | |
9.8100 | |
19.6200 | |
29.4300 | |
39.2400 | |
\end{Verbatim} | |
\begin{Verbatim}[commandchars=\\\{\}] | |
{\color{incolor}In [{\color{incolor}11}]:} \PY{n}{K1}\PY{p}{=}\PY{p}{[}\PY{n}{K} \PY{n}{y}\PY{p}{]}\PY{p}{;} | |
\PY{n}{K1}\PY{p}{(}\PY{l+m+mi}{2}\PY{p}{,}\PY{p}{:}\PY{p}{)}\PY{p}{=}\PY{n}{K1}\PY{p}{(}\PY{l+m+mi}{1}\PY{p}{,}\PY{p}{:}\PY{p}{)}\PY{o}{/}\PY{l+m+mi}{2}\PY{o}{+}\PY{n}{K1}\PY{p}{(}\PY{l+m+mi}{2}\PY{p}{,}\PY{p}{:}\PY{p}{)} | |
\end{Verbatim} | |
\begin{Verbatim}[commandchars=\\\{\}] | |
K1 = | |
20.00000 -10.00000 0.00000 0.00000 9.81000 | |
0.00000 15.00000 -10.00000 0.00000 24.52500 | |
0.00000 -10.00000 20.00000 -10.00000 29.43000 | |
0.00000 0.00000 -10.00000 10.00000 39.24000 | |
\end{Verbatim} | |
\begin{Verbatim}[commandchars=\\\{\}] | |
{\color{incolor}In [{\color{incolor}12}]:} \PY{n}{K2}\PY{p}{=}\PY{n}{K1}\PY{p}{;} | |
\PY{n}{K2}\PY{p}{(}\PY{l+m+mi}{3}\PY{p}{,}\PY{p}{:}\PY{p}{)}\PY{p}{=}\PY{n}{K1}\PY{p}{(}\PY{l+m+mi}{2}\PY{p}{,}\PY{p}{:}\PY{p}{)}\PY{o}{*}\PY{l+m+mi}{2}\PY{o}{/}\PY{l+m+mi}{3}\PY{o}{+}\PY{n}{K1}\PY{p}{(}\PY{l+m+mi}{3}\PY{p}{,}\PY{p}{:}\PY{p}{)} | |
\end{Verbatim} | |
\begin{Verbatim}[commandchars=\\\{\}] | |
K2 = | |
20.00000 -10.00000 0.00000 0.00000 9.81000 | |
0.00000 15.00000 -10.00000 0.00000 24.52500 | |
0.00000 0.00000 13.33333 -10.00000 45.78000 | |
0.00000 0.00000 -10.00000 10.00000 39.24000 | |
\end{Verbatim} | |
\begin{Verbatim}[commandchars=\\\{\}] | |
{\color{incolor}In [{\color{incolor}13}]:} \PY{n}{K2}\PY{p}{(}\PY{l+m+mi}{4}\PY{p}{,}\PY{p}{:}\PY{p}{)}\PY{p}{=}\PY{o}{\PYZhy{}}\PY{n}{K2}\PY{p}{(}\PY{l+m+mi}{3}\PY{p}{,}\PY{p}{:}\PY{p}{)}\PY{o}{*}\PY{n}{K2}\PY{p}{(}\PY{l+m+mi}{4}\PY{p}{,}\PY{l+m+mi}{3}\PY{p}{)}\PY{o}{/}\PY{n}{K2}\PY{p}{(}\PY{l+m+mi}{3}\PY{p}{,}\PY{l+m+mi}{3}\PY{p}{)}\PY{o}{+}\PY{n}{K2}\PY{p}{(}\PY{l+m+mi}{4}\PY{p}{,}\PY{p}{:}\PY{p}{)} | |
\end{Verbatim} | |
\begin{Verbatim}[commandchars=\\\{\}] | |
K2 = | |
20.00000 -10.00000 0.00000 0.00000 9.81000 | |
0.00000 15.00000 -10.00000 0.00000 24.52500 | |
0.00000 0.00000 13.33333 -10.00000 45.78000 | |
0.00000 0.00000 0.00000 2.50000 73.57500 | |
\end{Verbatim} | |
\begin{Verbatim}[commandchars=\\\{\}] | |
{\color{incolor}In [{\color{incolor}14}]:} \PY{n}{yp}\PY{p}{=}\PY{n}{K2}\PY{p}{(}\PY{p}{:}\PY{p}{,}\PY{l+m+mi}{5}\PY{p}{)}\PY{p}{;} | |
\PY{n}{x4}\PY{p}{=}\PY{n}{yp}\PY{p}{(}\PY{l+m+mi}{4}\PY{p}{)}\PY{o}{/}\PY{n}{K2}\PY{p}{(}\PY{l+m+mi}{4}\PY{p}{,}\PY{l+m+mi}{4}\PY{p}{)} | |
\PY{n}{x3}\PY{p}{=}\PY{p}{(}\PY{n}{yp}\PY{p}{(}\PY{l+m+mi}{3}\PY{p}{)}\PY{o}{+}\PY{l+m+mi}{10}\PY{o}{*}\PY{n}{x4}\PY{p}{)}\PY{o}{/}\PY{n}{K2}\PY{p}{(}\PY{l+m+mi}{3}\PY{p}{,}\PY{l+m+mi}{3}\PY{p}{)} | |
\PY{n}{x2}\PY{p}{=}\PY{p}{(}\PY{n}{yp}\PY{p}{(}\PY{l+m+mi}{2}\PY{p}{)}\PY{o}{+}\PY{l+m+mi}{10}\PY{o}{*}\PY{n}{x3}\PY{p}{)}\PY{o}{/}\PY{n}{K2}\PY{p}{(}\PY{l+m+mi}{2}\PY{p}{,}\PY{l+m+mi}{2}\PY{p}{)} | |
\PY{n}{x1}\PY{p}{=}\PY{p}{(}\PY{n}{yp}\PY{p}{(}\PY{l+m+mi}{1}\PY{p}{)}\PY{o}{+}\PY{l+m+mi}{10}\PY{o}{*}\PY{n}{x2}\PY{p}{)}\PY{o}{/}\PY{n}{K2}\PY{p}{(}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{)} | |
\end{Verbatim} | |
\begin{Verbatim}[commandchars=\\\{\}] | |
x4 = 29.430 | |
x3 = 25.506 | |
x2 = 18.639 | |
x1 = 9.8100 | |
\end{Verbatim} | |
\begin{Verbatim}[commandchars=\\\{\}] | |
{\color{incolor}In [{\color{incolor}15}]:} \PY{n}{K}\PY{o}{\PYZbs{}}\PY{n}{y} | |
\end{Verbatim} | |
\begin{Verbatim}[commandchars=\\\{\}] | |
ans = | |
9.8100 | |
18.6390 | |
25.5060 | |
29.4300 | |
\end{Verbatim} | |
\subsection{Automate Gauss | |
Elimination}\label{automate-gauss-elimination} | |
We can automate Gauss elimination with a function whose input is A and | |
y: | |
\texttt{x=GaussNaive(A,y)} | |
\begin{Verbatim}[commandchars=\\\{\}] | |
{\color{incolor}In [{\color{incolor}16}]:} \PY{n}{x}\PY{p}{=}\PY{n}{GaussNaive}\PY{p}{(}\PY{n}{K}\PY{p}{,}\PY{n}{y}\PY{p}{)} | |
\end{Verbatim} | |
\begin{Verbatim}[commandchars=\\\{\}] | |
x = | |
9.8100 | |
18.6390 | |
25.5060 | |
29.4300 | |
\end{Verbatim} | |
\subsection{Problem (Diagonal element is | |
zero)}\label{problem-diagonal-element-is-zero} | |
If a diagonal element is 0 or very small either: | |
\begin{enumerate} | |
\def\labelenumi{\arabic{enumi}.} | |
\tightlist | |
\item | |
no solution found | |
\item | |
errors are introduced | |
\end{enumerate} | |
Therefore, we would want to pivot before applying Gauss elimination | |
Consider: | |
\begin{enumerate} | |
\def\labelenumi{(\alph{enumi})} | |
\item | |
\(\left[ \begin{array}{cccc} 0 & 2 & 3 \\ 4 & 6 & 7 \\ 2 & -3 & 6 \end{array} \right] \left[ \begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \end{array} \right]= \left[ \begin{array}{c} 8 \\ -3 \\ 5\end{array} \right]\) | |
\item | |
\(\left[ \begin{array}{cccc} 0.0003 & 3.0000 \\ 1.0000 & 1.0000 \end{array} \right] \left[ \begin{array}{c} x_{1} \\ x_{2} \end{array} \right]= \left[ \begin{array}{c} 2.0001 \\ 1.0000 \end{array} \right]\) | |
\end{enumerate} | |
\begin{Verbatim}[commandchars=\\\{\}] | |
{\color{incolor}In [{\color{incolor}17}]:} \PY{n}{format} \PY{n}{short} | |
\PY{n}{Aa}\PY{p}{=}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{2}\PY{p}{,}\PY{l+m+mi}{3}\PY{p}{;}\PY{l+m+mi}{4}\PY{p}{,}\PY{l+m+mi}{6}\PY{p}{,}\PY{l+m+mi}{7}\PY{p}{;}\PY{l+m+mi}{2}\PY{p}{,}\PY{o}{\PYZhy{}}\PY{l+m+mi}{3}\PY{p}{,}\PY{l+m+mi}{6}\PY{p}{]}\PY{p}{;} \PY{n}{ya}\PY{p}{=}\PY{p}{[}\PY{l+m+mi}{8}\PY{p}{;}\PY{o}{\PYZhy{}}\PY{l+m+mi}{3}\PY{p}{;}\PY{l+m+mi}{5}\PY{p}{]}\PY{p}{;} | |
\PY{n}{GaussNaive}\PY{p}{(}\PY{n}{Aa}\PY{p}{,}\PY{n}{ya}\PY{p}{)} | |
\PY{n}{Aa}\PY{o}{\PYZbs{}}\PY{n}{ya} | |
\end{Verbatim} | |
\begin{Verbatim}[commandchars=\\\{\}] | |
warning: division by zero | |
warning: called from | |
GaussNaive at line 16 column 12 | |
warning: division by zero | |
warning: division by zero | |
ans = | |
NaN | |
NaN | |
NaN | |
ans = | |
-5.423913 | |
0.021739 | |
2.652174 | |
\end{Verbatim} | |
\begin{Verbatim}[commandchars=\\\{\}] | |
{\color{incolor}In [{\color{incolor}32}]:} \PY{p}{[}\PY{n}{x}\PY{p}{,}\PY{n}{Aug}\PY{p}{,}\PY{n}{npivots}\PY{p}{]}\PY{p}{=}\PY{n}{GaussPivot}\PY{p}{(}\PY{n}{Aa}\PY{p}{,}\PY{n}{ya}\PY{p}{)} | |
\end{Verbatim} | |
\begin{Verbatim}[commandchars=\\\{\}] | |
x = | |
-5.423913 | |
0.021739 | |
2.652174 | |
Aug = | |
4.00000 6.00000 7.00000 -3.00000 | |
0.00000 -6.00000 2.50000 6.50000 | |
0.00000 0.00000 3.83333 10.16667 | |
npivots = 2 | |
\end{Verbatim} | |
\begin{Verbatim}[commandchars=\\\{\}] | |
{\color{incolor}In [{\color{incolor}33}]:} \PY{n}{format} \PY{n}{long} | |
\PY{n}{Ab}\PY{p}{=}\PY{p}{[}\PY{l+m+mf}{0.3E\PYZhy{}13}\PY{p}{,}\PY{l+m+mf}{3.0000}\PY{p}{;}\PY{l+m+mf}{1.0000}\PY{p}{,}\PY{l+m+mf}{1.0000}\PY{p}{]}\PY{p}{;}\PY{n}{yb}\PY{p}{=}\PY{p}{[}\PY{l+m+mi}{2}\PY{o}{+}\PY{l+m+mf}{0.1e\PYZhy{}13}\PY{p}{;}\PY{l+m+mf}{1.0000}\PY{p}{]}\PY{p}{;} | |
\PY{n}{GaussNaive}\PY{p}{(}\PY{n}{Ab}\PY{p}{,}\PY{n}{yb}\PY{p}{)} | |
\PY{n}{Ab}\PY{o}{\PYZbs{}}\PY{n}{yb} | |
\end{Verbatim} | |
\begin{Verbatim}[commandchars=\\\{\}] | |
ans = | |
0.325665420556713 | |
0.666666666666667 | |
ans = | |
0.333333333333333 | |
0.666666666666667 | |
\end{Verbatim} | |
\begin{Verbatim}[commandchars=\\\{\}] | |
{\color{incolor}In [{\color{incolor}34}]:} \PY{p}{[}\PY{n}{x}\PY{p}{,}\PY{n}{Aug}\PY{p}{,}\PY{n}{npivots}\PY{p}{]}\PY{p}{=}\PY{n}{GaussPivot}\PY{p}{(}\PY{n}{Ab}\PY{p}{,}\PY{n}{yb}\PY{p}{)} | |
\PY{n}{Ab}\PY{o}{\PYZbs{}}\PY{n}{yb} | |
\PY{n}{format} \PY{n}{short} | |
\end{Verbatim} | |
\begin{Verbatim}[commandchars=\\\{\}] | |
x = | |
0.333333333333333 | |
0.666666666666667 | |
Aug = | |
1.000000000000000 1.000000000000000 1.000000000000000 | |
0.000000000000000 2.999999999999970 1.999999999999980 | |
npivots = 1 | |
ans = | |
0.333333333333333 | |
0.666666666666667 | |
\end{Verbatim} | |
\begin{Verbatim}[commandchars=\\\{\}] | |
{\color{incolor}In [{\color{incolor}36}]:} \PY{c}{\PYZpc{} determinant is (\PYZhy{}1)\PYZca{}(number\PYZus{}of\PYZus{}pivots)*diagonal\PYZus{}elements} | |
\PY{n+nb}{det}\PY{p}{(}\PY{n}{Ab}\PY{p}{)} | |
\PY{n}{Aug}\PY{p}{(}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{)}\PY{o}{*}\PY{n}{Aug}\PY{p}{(}\PY{l+m+mi}{2}\PY{p}{,}\PY{l+m+mi}{2}\PY{p}{)} | |
\end{Verbatim} | |
\begin{Verbatim}[commandchars=\\\{\}] | |
ans = -3.0000 | |
ans = 3.0000 | |
\end{Verbatim} | |
\subsubsection{Spring-Mass System again}\label{spring-mass-system-again} | |
Now, 4 masses are connected in series to 4 springs with \(K_{1}\)=10 | |
N/m, \(K_{2}\)=5 N/m, \(K_{3}\)=2 N/m and \(K_{4}\)=1 N/m. What are the | |
final positions of the masses? | |
\begin{figure}[htbp] | |
\centering | |
\includegraphics{../lecture_09/mass_springs.svg} | |
\caption{Springs-masses} | |
\end{figure} | |
The masses have the following amounts, 1, 2, 3, and 4 kg for masses 1-4. | |
Using a FBD for each mass: | |
\(m_{1}g+k_{2}(x_{2}-x_{1})-k_{1}x_{1}=0\) | |
\(m_{2}g+k_{3}(x_{3}-x_{2})-k_{2}(x_{2}-x_{1})=0\) | |
\(m_{3}g+k_{4}(x_{4}-x_{3})-k_{3}(x_{3}-x_{2})=0\) | |
\(m_{4}g-k_{4}(x_{4}-x_{3})=0\) | |
in matrix form: | |
\(\left[ \begin{array}{cccc} k_1+k_2 & -k_2 & 0 & 0 \\ -k_2 & k_2+k_3 & -k_3 & 0 \\ 0 & -k_3 & k_3+k_4 & -k_4 \\ 0 & 0 & -k_4 & k_4 \end{array} \right] \left[ \begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{array} \right]= \left[ \begin{array}{c} m_{1}g \\ m_{2}g \\ m_{3}g \\ m_{4}g \end{array} \right]\) | |
\begin{Verbatim}[commandchars=\\\{\}] | |
{\color{incolor}In [{\color{incolor}24}]:} \PY{n}{k1}\PY{p}{=}\PY{l+m+mi}{10}\PY{p}{;} \PY{n}{k2}\PY{p}{=}\PY{l+m+mi}{5}\PY{p}{;}\PY{n}{k3}\PY{p}{=}\PY{l+m+mi}{2}\PY{p}{;}\PY{n}{k4}\PY{p}{=}\PY{l+m+mi}{1}\PY{p}{;} \PY{c}{\PYZpc{} N/m} | |
\PY{n}{m1}\PY{p}{=}\PY{l+m+mi}{1}\PY{p}{;} \PY{c}{\PYZpc{} kg} | |
\PY{n}{m2}\PY{p}{=}\PY{l+m+mi}{2}\PY{p}{;} | |
\PY{n}{m3}\PY{p}{=}\PY{l+m+mi}{3}\PY{p}{;} | |
\PY{n}{m4}\PY{p}{=}\PY{l+m+mi}{4}\PY{p}{;} | |
\PY{n}{g}\PY{p}{=}\PY{l+m+mf}{9.81}\PY{p}{;} \PY{c}{\PYZpc{} m/s\PYZca{}2} | |
\PY{n}{K}\PY{p}{=}\PY{p}{[}\PY{n}{k1}\PY{o}{+}\PY{n}{k2} \PY{o}{\PYZhy{}}\PY{n}{k2} \PY{l+m+mi}{0} \PY{l+m+mi}{0}\PY{p}{;} \PY{o}{\PYZhy{}}\PY{n}{k2}\PY{p}{,} \PY{n}{k2}\PY{o}{+}\PY{n}{k3}\PY{p}{,} \PY{o}{\PYZhy{}}\PY{n}{k3} \PY{l+m+mi}{0}\PY{p}{;} \PY{l+m+mi}{0} \PY{o}{\PYZhy{}}\PY{n}{k3}\PY{p}{,} \PY{n}{k3}\PY{o}{+}\PY{n}{k4}\PY{p}{,} \PY{o}{\PYZhy{}}\PY{n}{k4}\PY{p}{;} \PY{l+m+mi}{0} \PY{l+m+mi}{0} \PY{o}{\PYZhy{}}\PY{n}{k4} \PY{n}{k4}\PY{p}{]} | |
\PY{n}{y}\PY{p}{=}\PY{p}{[}\PY{n}{m1}\PY{o}{*}\PY{n}{g}\PY{p}{;}\PY{n}{m2}\PY{o}{*}\PY{n}{g}\PY{p}{;}\PY{n}{m3}\PY{o}{*}\PY{n}{g}\PY{p}{;}\PY{n}{m4}\PY{o}{*}\PY{n}{g}\PY{p}{]} | |
\end{Verbatim} | |
\begin{Verbatim}[commandchars=\\\{\}] | |
K = | |
15 -5 0 0 | |
-5 7 -2 0 | |
0 -2 3 -1 | |
0 0 -1 1 | |
y = | |
9.8100 | |
19.6200 | |
29.4300 | |
39.2400 | |
\end{Verbatim} | |
\subsection{Tridiagonal matrix}\label{tridiagonal-matrix} | |
This matrix, K, could be rewritten as 3 vectors e, f and g | |
\(e=\left[ \begin{array}{c} 0 \\ -5 \\ -2 \\ -1 \end{array} \right]\) | |
\(f=\left[ \begin{array}{c} 15 \\ 7 \\ 3 \\ 1 \end{array} \right]\) | |
\(g=\left[ \begin{array}{c} -5 \\ -2 \\ -1 \\ 0 \end{array} \right]\) | |
Where all other components are 0 and the length of the vectors are n and | |
the first component of e and the last component of g are zero | |
\texttt{e(1)=0} | |
\texttt{g(end)=0} | |
No need to pivot and number of calculations reduced enormously. | |
\begin{longtable}[c]{@{}ll@{}} | |
\toprule | |
method & Number of Floating point operations for | |
n\(\times\)n-matrix\tabularnewline | |
\midrule | |
\endhead | |
Gauss & n-cubed\tabularnewline | |
Tridiagonal & n\tabularnewline | |
\bottomrule | |
\end{longtable} | |
\begin{Verbatim}[commandchars=\\\{\}] | |
{\color{incolor}In [{\color{incolor}25}]:} \PY{n+nb}{e}\PY{p}{=}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{;}\PY{o}{\PYZhy{}}\PY{l+m+mi}{5}\PY{p}{;}\PY{o}{\PYZhy{}}\PY{l+m+mi}{2}\PY{p}{;}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{;} | |
\PY{n}{g}\PY{p}{=}\PY{p}{[}\PY{o}{\PYZhy{}}\PY{l+m+mi}{5}\PY{p}{;}\PY{o}{\PYZhy{}}\PY{l+m+mi}{2}\PY{p}{;}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{;}\PY{l+m+mi}{0}\PY{p}{]}\PY{p}{;} | |
\PY{n}{f}\PY{p}{=}\PY{p}{[}\PY{l+m+mi}{15}\PY{p}{;}\PY{l+m+mi}{7}\PY{p}{;}\PY{l+m+mi}{3}\PY{p}{;}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{;} | |
\PY{n}{Tridiag}\PY{p}{(}\PY{n+nb}{e}\PY{p}{,}\PY{n}{f}\PY{p}{,}\PY{n}{g}\PY{p}{,}\PY{n}{y}\PY{p}{)} | |
\PY{n}{K}\PY{o}{\PYZbs{}}\PY{n}{y} | |
\end{Verbatim} | |
\begin{Verbatim}[commandchars=\\\{\}] | |
ans = | |
9.8100 27.4680 61.8030 101.0430 | |
ans = | |
9.8100 | |
27.4680 | |
61.8030 | |
101.0430 | |
\end{Verbatim} | |
\begin{Verbatim}[commandchars=\\\{\}] | |
{\color{incolor}In [{\color{incolor}26}]:} \PY{c}{\PYZpc{} tic ... t=toc } | |
\PY{c}{\PYZpc{} is Matlab timer used for debugging programs} | |
\PY{n}{t\PYZus{}GE} \PY{p}{=} \PY{n+nb}{zeros}\PY{p}{(}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{100}\PY{p}{)}\PY{p}{;} | |
\PY{n}{t\PYZus{}GE\PYZus{}tridiag} \PY{p}{=} \PY{n+nb}{zeros}\PY{p}{(}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{100}\PY{p}{)}\PY{p}{;} | |
\PY{n}{t\PYZus{}TD} \PY{p}{=} \PY{n+nb}{zeros}\PY{p}{(}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{100}\PY{p}{)}\PY{p}{;} | |
\PY{c}{\PYZpc{}for n = 1:200} | |
\PY{k}{for} \PY{n}{n}\PY{p}{=}\PY{l+m+mi}{1}\PY{p}{:}\PY{l+m+mi}{100} | |
\PY{n}{A} \PY{p}{=} \PY{n+nb}{rand}\PY{p}{(}\PY{n}{n}\PY{p}{,}\PY{n}{n}\PY{p}{)}\PY{p}{;} | |
\PY{n+nb}{e} \PY{p}{=} \PY{n+nb}{rand}\PY{p}{(}\PY{n}{n}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{)}\PY{p}{;} \PY{n+nb}{e}\PY{p}{(}\PY{l+m+mi}{1}\PY{p}{)}\PY{p}{=}\PY{l+m+mi}{0}\PY{p}{;} | |
\PY{n}{f} \PY{p}{=} \PY{n+nb}{rand}\PY{p}{(}\PY{n}{n}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{)}\PY{p}{;} | |
\PY{n}{g} \PY{p}{=} \PY{n+nb}{rand}\PY{p}{(}\PY{n}{n}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{)}\PY{p}{;} \PY{n}{g}\PY{p}{(}\PY{k}{end}\PY{p}{)}\PY{p}{=}\PY{l+m+mi}{0}\PY{p}{;} | |
\PY{n}{Atd}\PY{p}{=}\PY{n+nb}{diag}\PY{p}{(}\PY{n}{f}\PY{p}{,} \PY{l+m+mi}{0}\PY{p}{)} \PY{o}{\PYZhy{}} \PY{n+nb}{diag}\PY{p}{(}\PY{n+nb}{e}\PY{p}{(}\PY{l+m+mi}{2}\PY{p}{:}\PY{n}{n}\PY{p}{)}\PY{p}{,} \PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{)} \PY{o}{\PYZhy{}} \PY{n+nb}{diag}\PY{p}{(}\PY{n}{g}\PY{p}{(}\PY{l+m+mi}{1}\PY{p}{:}\PY{n}{n}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{)}\PY{p}{,} \PY{l+m+mi}{1}\PY{p}{)}\PY{p}{;} | |
\PY{n}{b} \PY{p}{=} \PY{n+nb}{rand}\PY{p}{(}\PY{n}{n}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{)}\PY{p}{;} | |
\PY{n+nb}{tic}\PY{p}{;} | |
\PY{n}{x} \PY{p}{=} \PY{n}{GaussPivot}\PY{p}{(}\PY{n}{A}\PY{p}{,}\PY{n}{b}\PY{p}{)}\PY{p}{;} | |
\PY{n}{t\PYZus{}GE}\PY{p}{(}\PY{n}{n}\PY{p}{)} \PY{p}{=} \PY{n+nb}{toc}\PY{p}{;} | |
\PY{n+nb}{tic}\PY{p}{;} | |
\PY{n}{x} \PY{p}{=} \PY{n}{A}\PY{o}{\PYZbs{}}\PY{n}{b}\PY{p}{;} | |
\PY{n}{t\PYZus{}GE\PYZus{}tridiag}\PY{p}{(}\PY{n}{n}\PY{p}{)} \PY{p}{=} \PY{n+nb}{toc}\PY{p}{;} | |
\PY{n+nb}{tic}\PY{p}{;} | |
\PY{n}{x} \PY{p}{=} \PY{n}{Tridiag}\PY{p}{(}\PY{n+nb}{e}\PY{p}{,}\PY{n}{f}\PY{p}{,}\PY{n}{g}\PY{p}{,}\PY{n}{b}\PY{p}{)}\PY{p}{;} | |
\PY{n}{t\PYZus{}TD}\PY{p}{(}\PY{n}{n}\PY{p}{)} \PY{p}{=} \PY{n+nb}{toc}\PY{p}{;} | |
\PY{k}{end} | |
\end{Verbatim} | |
\begin{Verbatim}[commandchars=\\\{\}] | |
{\color{incolor}In [{\color{incolor}28}]:} \PY{n}{n}\PY{p}{=}\PY{l+m+mi}{1}\PY{p}{:}\PY{l+m+mi}{100}\PY{p}{;} | |
\PY{n+nb}{loglog}\PY{p}{(}\PY{n}{n}\PY{p}{,}\PY{n}{t\PYZus{}GE}\PY{p}{,}\PY{n}{n}\PY{p}{,}\PY{n}{t\PYZus{}TD}\PY{p}{,}\PY{n}{n}\PY{p}{,}\PY{n}{t\PYZus{}GE\PYZus{}tridiag}\PY{p}{)} | |
\PY{n+nb}{legend}\PY{p}{(}\PY{l+s}{\PYZsq{}}\PY{l+s}{Gauss elim\PYZsq{}}\PY{p}{,}\PY{l+s}{\PYZsq{}}\PY{l+s}{Matlab \PYZbs{}\PYZsq{}}\PY{p}{,}\PY{l+s}{\PYZsq{}}\PY{l+s}{TriDiag\PYZsq{}}\PY{p}{)} | |
\PY{n+nb}{xlabel}\PY{p}{(}\PY{l+s}{\PYZsq{}}\PY{l+s}{number of elements\PYZsq{}}\PY{p}{)} | |
\PY{n+nb}{ylabel}\PY{p}{(}\PY{l+s}{\PYZsq{}}\PY{l+s}{time (s)\PYZsq{}}\PY{p}{)} | |
\end{Verbatim} | |
\begin{center} | |
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{lecture_10_files/lecture_10_29_0.pdf} | |
\end{center} | |
{ \hspace*{\fill} \\} | |
\begin{Verbatim}[commandchars=\\\{\}] | |
{\color{incolor}In [{\color{incolor}29}]:} \PY{p}{[}\PY{n}{x}\PY{p}{,}\PY{n}{Aug}\PY{p}{,}\PY{n}{npivots}\PY{p}{]}\PY{p}{=}\PY{n}{GaussPivot}\PY{p}{(}\PY{n}{K}\PY{p}{,}\PY{n}{y}\PY{p}{)} | |
\end{Verbatim} | |
\begin{Verbatim}[commandchars=\\\{\}] | |
x = | |
9.8100 | |
27.4680 | |
61.8030 | |
101.0430 | |
Aug = | |
15.00000 -5.00000 0.00000 0.00000 9.81000 | |
0.00000 5.33333 -2.00000 0.00000 22.89000 | |
0.00000 0.00000 2.25000 -1.00000 38.01375 | |
0.00000 0.00000 0.00000 0.55556 56.13500 | |
npivots = 0 | |
\end{Verbatim} | |
\begin{Verbatim}[commandchars=\\\{\}] | |
{\color{incolor}In [{\color{incolor}30}]:} \PY{n}{A}\PY{p}{=}\PY{n}{Aug}\PY{p}{(}\PY{l+m+mi}{1}\PY{p}{:}\PY{l+m+mi}{4}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{:}\PY{l+m+mi}{4}\PY{p}{)} | |
\end{Verbatim} | |
\begin{Verbatim}[commandchars=\\\{\}] | |
A = | |
15.00000 -5.00000 0.00000 0.00000 | |
0.00000 5.33333 -2.00000 0.00000 | |
0.00000 0.00000 2.25000 -1.00000 | |
0.00000 0.00000 0.00000 0.55556 | |
\end{Verbatim} | |
\begin{Verbatim}[commandchars=\\\{\}] | |
{\color{incolor}In [{\color{incolor}31}]:} \PY{n+nb}{det}\PY{p}{(}\PY{n}{A}\PY{p}{)} | |
\PY{n}{detA}\PY{p}{=}\PY{n}{A}\PY{p}{(}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{)}\PY{o}{*}\PY{n}{A}\PY{p}{(}\PY{l+m+mi}{2}\PY{p}{,}\PY{l+m+mi}{2}\PY{p}{)}\PY{o}{*}\PY{n}{A}\PY{p}{(}\PY{l+m+mi}{3}\PY{p}{,}\PY{l+m+mi}{3}\PY{p}{)}\PY{o}{*}\PY{n}{A}\PY{p}{(}\PY{l+m+mi}{4}\PY{p}{,}\PY{l+m+mi}{4}\PY{p}{)} | |
\end{Verbatim} | |
\begin{Verbatim}[commandchars=\\\{\}] | |
ans = 100.00 | |
detA = 100.00 | |
\end{Verbatim} | |
\begin{Verbatim}[commandchars=\\\{\}] | |
{\color{incolor}In [{\color{incolor} }]:} | |
\end{Verbatim} | |
% Add a bibliography block to the postdoc | |
\end{document} |