From 07507daa4fd622207b94aa55bcdf52370e0824ec Mon Sep 17 00:00:00 2001 From: "Ryan C. Cooper" Date: Tue, 21 Feb 2017 11:47:09 -0500 Subject: [PATCH 1/7] update lecture 10 --- lecture_10/lecture_10.aux | 14 +- lecture_10/lecture_10.ipynb | 497 +++++++++++++----- lecture_10/lecture_10.log | 83 ++- lecture_10/lecture_10.pdf | Bin 162693 -> 190161 bytes lecture_10/lecture_10.tex | 199 ++++--- .../lecture_10_files/lecture_10_29_0.pdf | Bin 0 -> 10972 bytes .../lecture_10_files/lecture_10_29_0.svg | 148 ++++++ .../lecture_10_files/lecture_10_6_0.pdf | Bin 0 -> 30054 bytes .../lecture_10_files/lecture_10_6_0.svg | 397 ++++++++++++++ lecture_10/octave-workspace | Bin 86061 -> 134485 bytes 10 files changed, 1093 insertions(+), 245 deletions(-) create mode 100644 lecture_10/lecture_10_files/lecture_10_29_0.pdf create mode 100644 lecture_10/lecture_10_files/lecture_10_29_0.svg create mode 100644 lecture_10/lecture_10_files/lecture_10_6_0.pdf create mode 100644 lecture_10/lecture_10_files/lecture_10_6_0.svg diff --git a/lecture_10/lecture_10.aux b/lecture_10/lecture_10.aux index 14a6f0d..d8d4c62 100644 --- a/lecture_10/lecture_10.aux +++ b/lecture_10/lecture_10.aux @@ -28,13 +28,13 @@ \@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Springs-masses\relax }}{4}{figure.caption.1}} \@writefile{toc}{\contentsline {subsection}{\numberline {1.2}Automate Gauss Elimination}{6}{subsection.1.2}} \newlabel{automate-gauss-elimination}{{1.2}{6}{Automate Gauss Elimination}{subsection.1.2}{}} -\@writefile{toc}{\contentsline {subsection}{\numberline {1.3}Problem (Diagonal element is zero)}{6}{subsection.1.3}} -\newlabel{problem-diagonal-element-is-zero}{{1.3}{6}{Problem (Diagonal element is zero)}{subsection.1.3}{}} -\@writefile{toc}{\contentsline {subsubsection}{\numberline {1.3.1}Spring-Mass System again}{8}{subsubsection.1.3.1}} -\newlabel{spring-mass-system-again}{{1.3.1}{8}{Spring-Mass System again}{subsubsection.1.3.1}{}} +\@writefile{toc}{\contentsline {subsection}{\numberline {1.3}Problem (Diagonal element is zero)}{7}{subsection.1.3}} +\newlabel{problem-diagonal-element-is-zero}{{1.3}{7}{Problem (Diagonal element is zero)}{subsection.1.3}{}} +\@writefile{toc}{\contentsline {subsubsection}{\numberline {1.3.1}Spring-Mass System again}{9}{subsubsection.1.3.1}} +\newlabel{spring-mass-system-again}{{1.3.1}{9}{Spring-Mass System again}{subsubsection.1.3.1}{}} \@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Springs-masses\relax }}{9}{figure.caption.2}} -\@writefile{toc}{\contentsline {subsection}{\numberline {1.4}Tridiagonal matrix}{9}{subsection.1.4}} -\newlabel{tridiagonal-matrix}{{1.4}{9}{Tridiagonal matrix}{subsection.1.4}{}} \gdef \LT@i {\LT@entry - {2}{67.64778pt}\LT@entry + {2}{61.69104pt}\LT@entry {1}{261.39424pt}} +\@writefile{toc}{\contentsline {subsection}{\numberline {1.4}Tridiagonal matrix}{10}{subsection.1.4}} +\newlabel{tridiagonal-matrix}{{1.4}{10}{Tridiagonal matrix}{subsection.1.4}{}} diff --git a/lecture_10/lecture_10.ipynb b/lecture_10/lecture_10.ipynb index 78e529d..46d4f00 100644 --- a/lecture_10/lecture_10.ipynb +++ b/lecture_10/lecture_10.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "code", - "execution_count": 14, + "execution_count": 1, "metadata": { "collapsed": true }, @@ -13,7 +13,7 @@ }, { "cell_type": "code", - "execution_count": 15, + "execution_count": 2, "metadata": { "collapsed": true }, @@ -58,7 +58,7 @@ }, { "cell_type": "code", - "execution_count": 16, + "execution_count": 3, "metadata": { "collapsed": false }, @@ -223,6 +223,30 @@ "plot(x11,x21,x21,x22)" ] }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "ans =\n", + "\n", + " 0.40000\n", + " 0.20000\n", + "\n" + ] + } + ], + "source": [ + "A=[1,3;2,1]; y=[1;1];\n", + "A\\y % matlab's Ax=y solution for x" + ] + }, { "cell_type": "markdown", "metadata": {}, @@ -251,20 +275,11 @@ }, { "cell_type": "code", - "execution_count": 17, + "execution_count": 9, "metadata": { "collapsed": false }, "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "error: 'X22' undefined near line 1 column 16\n", - "error: 'X13' undefined near line 1 column 14\n", - "error: evaluating argument list element number 3\n" - ] - }, { "data": { "image/svg+xml": [ @@ -312,282 +327,357 @@ "\n", "\n", "\n", - "\t\n", + "\t\n", "\n", - "\t\n", + "\t\n", "\n", - "\t\n", + "\t\n", "\n", - "\t\n", + "\t\n", "\n", - "\t\n", + "\t\n", "\n", - "\t\n", + "\t\n", "\n", - "\t\n", + "\t\n", + "\n", + "\t\n", "\n", - "\t\n", + "\t\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t-2\n", "\t\n", "\n", "\n", "\n", + "\n", + "\t\n", "\n", - "\t\n", + "\t\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t-1.5\n", "\t\n", "\n", "\n", "\n", + "\n", + "\t\n", "\n", - "\t\n", + "\t\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t-1\n", "\t\n", "\n", "\n", "\n", + "\n", + "\t\n", "\n", - "\t\n", + "\t\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t-0.5\n", "\t\n", "\n", "\n", "\n", + "\n", + "\t\n", "\n", - 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"\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n", + "\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n", "\t\n", "\tgnuplot_plot_3a\n", - "\n", + "\n", + "\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n", + "\t\n", + "\tgnuplot_plot_4a\n", + "\n", "\n", "\n", - "\t\n", + "\t\n", "\n", "\t\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\n", - "\t\n", + "\t\n", "\n", - "\t\n", + "\t\t\n", + "\t\tx1\n", + "\t\n", + "\n", "\n", - "\t\n", + "\t\n", + "\t\tx2\n", + "\t\n", + "\n", + "\n", + "\t\n", + "\t\tx3\n", + "\t\n", + "\n", "\n", - "\t\n", + "\n", "\n", "" ], @@ -600,18 +690,19 @@ } ], "source": [ - "x11=linspace(-2,2,5);\n", - "x12=linspace(-2,2,5);\n", + "N=25;\n", + "x11=linspace(-2,2,N);\n", + "x12=linspace(-2,2,N);\n", "[X11,X12]=meshgrid(x11,x12);\n", - "X13=1-10*X11-2*X22;\n", + "X13=1-10*X11-2*X12;\n", "\n", - "x21=linspace(-2,2,5);\n", - "x22=linspace(-2,2,5);\n", + "x21=linspace(-2,2,N);\n", + "x22=linspace(-2,2,N);\n", "[X21,X22]=meshgrid(x21,x22);\n", "X23=1-2*X11-X22;\n", "\n", - "x31=linspace(-2,2,5);\n", - "x32=linspace(-2,2,5);\n", + "x31=linspace(-2,2,N);\n", + "x32=linspace(-2,2,N);\n", "[X31,X32]=meshgrid(x31,x32);\n", "X33=1/10*(1-X31-2*X32);\n", "\n", @@ -621,7 +712,10 @@ "mesh(X31,X32,X33)\n", "x=[10,2, 1;2,1, 1; 1, 2, 10]\\[1;1;1];\n", "plot3(x(1),x(2),x(3),'o')\n", - "view(45,45)" + "xlabel('x1')\n", + "ylabel('x2')\n", + "zlabel('x3')\n", + "view(10,45)" ] }, { @@ -676,7 +770,7 @@ "\n", "then, $3/5x_{2}+4/5(-1/5)=1$ so $x_{2}=\\frac{8}{5}$\n", "\n", - "finally, $10x_{1}+2(8/5)" + "finally, $10x_{1}+2(8/5)+1(-\\frac{1}{5})=1$" ] }, { @@ -720,7 +814,7 @@ }, { "cell_type": "code", - "execution_count": 18, + "execution_count": 10, "metadata": { "collapsed": false }, @@ -759,7 +853,7 @@ }, { "cell_type": "code", - "execution_count": 19, + "execution_count": 11, "metadata": { "collapsed": false }, @@ -785,7 +879,7 @@ }, { "cell_type": "code", - "execution_count": 20, + "execution_count": 12, "metadata": { "collapsed": false }, @@ -811,7 +905,7 @@ }, { "cell_type": "code", - "execution_count": 21, + "execution_count": 13, "metadata": { "collapsed": false }, @@ -836,7 +930,7 @@ }, { "cell_type": "code", - "execution_count": 22, + "execution_count": 14, "metadata": { "collapsed": false }, @@ -862,7 +956,7 @@ }, { "cell_type": "code", - "execution_count": 23, + "execution_count": 15, "metadata": { "collapsed": false }, @@ -898,7 +992,7 @@ }, { "cell_type": "code", - "execution_count": 24, + "execution_count": 16, "metadata": { "collapsed": false }, @@ -962,7 +1056,7 @@ }, { "cell_type": "code", - "execution_count": 25, + "execution_count": 17, "metadata": { "collapsed": false }, @@ -1000,7 +1094,7 @@ }, { "cell_type": "code", - "execution_count": 26, + "execution_count": 32, "metadata": { "collapsed": false }, @@ -1031,7 +1125,7 @@ }, { "cell_type": "code", - "execution_count": 27, + "execution_count": 33, "metadata": { "collapsed": false }, @@ -1062,7 +1156,7 @@ }, { "cell_type": "code", - "execution_count": 28, + "execution_count": 34, "metadata": { "collapsed": false }, @@ -1096,6 +1190,28 @@ "format short" ] }, + { + "cell_type": "code", + "execution_count": 36, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "ans = -3.0000\n", + "ans = 3.0000\n" + ] + } + ], + "source": [ + "% determinant is (-1)^(number_of_pivots)*diagonal_elements\n", + "det(Ab)\n", + "Aug(1,1)*Aug(2,2)" + ] + }, { "cell_type": "markdown", "metadata": {}, @@ -1138,7 +1254,7 @@ }, { "cell_type": "code", - "execution_count": 29, + "execution_count": 24, "metadata": { "collapsed": false }, @@ -1211,13 +1327,13 @@ "\n", "|method |Number of Floating point operations for n$\\times$n-matrix|\n", "|----------------|---------|\n", - "| Naive Gauss | n-cubed |\n", + "| Gauss | n-cubed |\n", "| Tridiagonal | n |" ] }, { "cell_type": "code", - "execution_count": 30, + "execution_count": 25, "metadata": { "collapsed": false }, @@ -1229,6 +1345,13 @@ "ans =\n", "\n", " 9.8100 27.4680 61.8030 101.0430\n", + "\n", + "ans =\n", + "\n", + " 9.8100\n", + " 27.4680\n", + " 61.8030\n", + " 101.0430\n", "\n" ] } @@ -1237,12 +1360,13 @@ "e=[0;-5;-2;-1];\n", "g=[-5;-2;-1;0];\n", "f=[15;7;3;1];\n", - "Tridiag(e,f,g,y)\n" + "Tridiag(e,f,g,y)\n", + "K\\y\n" ] }, { "cell_type": "code", - "execution_count": 12, + "execution_count": 26, "metadata": { "collapsed": true }, @@ -1265,7 +1389,7 @@ " x = GaussPivot(A,b);\n", " t_GE(n) = toc;\n", " tic;\n", - " x = GaussPivot(Atd,b);\n", + " x = A\\b;\n", " t_GE_tridiag(n) = toc;\n", " tic;\n", " x = Tridiag(e,f,g,b);\n", @@ -1275,7 +1399,7 @@ }, { "cell_type": "code", - "execution_count": 13, + "execution_count": 28, "metadata": { "collapsed": false }, @@ -1329,26 +1453,31 @@ "\n", "\n", "\t\t\n", + "\t\t10-5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", "\t\t10-4\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t10-3\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t10-2\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t10-1\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t100\n", "\t\n", "\n", @@ -1358,21 +1487,16 @@ "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t101\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t102\n", "\t\n", "\n", "\n", - "\t\t\n", - "\t\t103\n", - "\t\n", - "\n", - "\n", "\n", "\n", "\t\n", @@ -1388,19 +1512,38 @@ "\n", "\n", "\n", - "\tgnuplot_plot_1a\n", - "\n", + "\n", + "\n", + "\n", + "\t\n", + "\tGauss elim\n", + "\n", "\n", "\n", - "\t\n", + "\t\n", + "\t\tGauss elim\n", "\t\n", - "\tgnuplot_plot_2a\n", + "\n", "\n", - "\t\n", + "\t\n", "\t\n", - "\tgnuplot_plot_3a\n", + "\tMatlab \\\n", + "\n", + "\t\n", + "\t\tMatlab \n", + "\t\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\tTriDiag\n", "\n", - "\t\n", + "\t\n", + "\t\tTriDiag\n", + "\t\n", + "\n", + "\n", + "\t\n", "\t\n", "\n", "\n", @@ -1422,12 +1565,92 @@ } ], "source": [ - "n=1:200;\n", + "n=1:100;\n", "loglog(n,t_GE,n,t_TD,n,t_GE_tridiag)\n", + "legend('Gauss elim','Matlab \\','TriDiag')\n", "xlabel('number of elements')\n", "ylabel('time (s)')" ] }, + { + "cell_type": "code", + "execution_count": 29, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "x =\n", + "\n", + " 9.8100\n", + " 27.4680\n", + " 61.8030\n", + " 101.0430\n", + "\n", + "Aug =\n", + "\n", + " 15.00000 -5.00000 0.00000 0.00000 9.81000\n", + " 0.00000 5.33333 -2.00000 0.00000 22.89000\n", + " 0.00000 0.00000 2.25000 -1.00000 38.01375\n", + " 0.00000 0.00000 0.00000 0.55556 56.13500\n", + "\n", + "npivots = 0\n" + ] + } + ], + "source": [ + "[x,Aug,npivots]=GaussPivot(K,y)" + ] + }, + { + "cell_type": "code", + "execution_count": 30, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "A =\n", + "\n", + " 15.00000 -5.00000 0.00000 0.00000\n", + " 0.00000 5.33333 -2.00000 0.00000\n", + " 0.00000 0.00000 2.25000 -1.00000\n", + " 0.00000 0.00000 0.00000 0.55556\n", + "\n" + ] + } + ], + "source": [ + "A=Aug(1:4,1:4)" + ] + }, + { + "cell_type": "code", + "execution_count": 31, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "ans = 100.00\n", + "detA = 100.00\n" + ] + } + ], + "source": [ + "det(A)\n", + "detA=A(1,1)*A(2,2)*A(3,3)*A(4,4)" + ] + }, { "cell_type": "code", "execution_count": null, diff --git a/lecture_10/lecture_10.log b/lecture_10/lecture_10.log index 088d911..9e3113d 100644 --- a/lecture_10/lecture_10.log +++ b/lecture_10/lecture_10.log @@ -1,4 +1,4 @@ -This is pdfTeX, Version 3.14159265-2.6-1.40.16 (TeX Live 2015/Debian) (preloaded format=pdflatex 2017.1.11) 20 FEB 2017 17:21 +This is pdfTeX, Version 3.14159265-2.6-1.40.16 (TeX Live 2015/Debian) (preloaded format=pdflatex 2017.1.11) 21 FEB 2017 11:46 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -772,28 +772,27 @@ Underfull \hbox (badness 10000) in paragraph at lines 323--324 [] -[2 <./lecture_10_files/lecture_10_3_0.pdf>] -LaTeX Font Info: Try loading font information for TS1+cmtt on input line 359 +LaTeX Font Info: Try loading font information for TS1+cmtt on input line 327 . - 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1131881..633c945 100644 --- a/lecture_10/lecture_10.tex +++ b/lecture_10/lecture_10.tex @@ -277,11 +277,11 @@ \begin{Verbatim}[commandchars=\\\{\}] -{\color{incolor}In [{\color{incolor}14}]:} \PY{c}{\PYZpc{}plot \PYZhy{}\PYZhy{}format svg} +{\color{incolor}In [{\color{incolor}1}]:} \PY{c}{\PYZpc{}plot \PYZhy{}\PYZhy{}format svg} \end{Verbatim} \begin{Verbatim}[commandchars=\\\{\}] -{\color{incolor}In [{\color{incolor}15}]:} \PY{n}{setdefaults} +{\color{incolor}In [{\color{incolor}2}]:} \PY{n}{setdefaults} \end{Verbatim} \section{Gauss Elimination}\label{gauss-elimination} @@ -310,11 +310,11 @@ \subsubsection{Solving sets of equations with matrix lines: \begin{Verbatim}[commandchars=\\\{\}] -{\color{incolor}In [{\color{incolor}16}]:} \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}{;} - \PY{n}{x11}\PY{p}{=}\PY{l+m+mi}{1}\PY{o}{\PYZhy{}}\PY{l+m+mi}{3}\PY{o}{*}\PY{n}{x21}\PY{p}{;} - \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}{;} - \PY{n}{x22}\PY{p}{=}\PY{l+m+mi}{1}\PY{o}{\PYZhy{}}\PY{l+m+mi}{2}\PY{o}{*}\PY{n}{x21}\PY{p}{;} - \PY{n+nb}{plot}\PY{p}{(}\PY{n}{x11}\PY{p}{,}\PY{n}{x21}\PY{p}{,}\PY{n}{x21}\PY{p}{,}\PY{n}{x22}\PY{p}{)} +{\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}{;} + \PY{n}{x11}\PY{p}{=}\PY{l+m+mi}{1}\PY{o}{\PYZhy{}}\PY{l+m+mi}{3}\PY{o}{*}\PY{n}{x21}\PY{p}{;} + \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}{;} + \PY{n}{x22}\PY{p}{=}\PY{l+m+mi}{1}\PY{o}{\PYZhy{}}\PY{l+m+mi}{2}\PY{o}{*}\PY{n}{x21}\PY{p}{;} + \PY{n+nb}{plot}\PY{p}{(}\PY{n}{x11}\PY{p}{,}\PY{n}{x21}\PY{p}{,}\PY{n}{x21}\PY{p}{,}\PY{n}{x22}\PY{p}{)} \end{Verbatim} \begin{center} @@ -322,7 +322,21 @@ \subsubsection{Solving sets of equations with matrix \end{center} { \hspace*{\fill} \\} - For a $3\times3$ matrix, the solution is the intersection of the 3 + \begin{Verbatim}[commandchars=\\\{\}] +{\color{incolor}In [{\color{incolor}4}]:} \PY{n}{A}\PY{p}{=}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{3}\PY{p}{;}\PY{l+m+mi}{2}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{;} \PY{n}{y}\PY{p}{=}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{;}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{;} + \PY{n}{A}\PY{o}{\PYZbs{}}\PY{n}{y} \PY{c}{\PYZpc{} matlab\PYZsq{}s Ax=y solution for x} +\end{Verbatim} + + \begin{Verbatim}[commandchars=\\\{\}] +ans = + + 0.40000 + 0.20000 + + + \end{Verbatim} + + For a \(3\times3\) matrix, the solution is the intersection of the 3 planes. \(10x_{1}+2x_{2}+x_{3}=1\) @@ -331,44 +345,41 @@ \subsubsection{Solving sets of equations with matrix \(x_{1}+2x_{2}+10x_{3}=1\) -$\left[ \begin{array}{ccc} 10 & 2 & 1 \\ 2 & 1 & 1 \\ 1 & 2 & 10\end{array} \right] +$\left[ \begin{array}{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]= 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\begin{Verbatim}[commandchars=\\\{\}] -error: 'X22' undefined near line 1 column 16 -error: 'X13' undefined near line 1 column 14 -error: evaluating argument list element number 3 - - \end{Verbatim} - \begin{center} - \adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{lecture_10_files/lecture_10_5_1.pdf} + \adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{lecture_10_files/lecture_10_6_0.pdf} \end{center} { \hspace*{\fill} \\} @@ -416,13 +427,13 @@ \subsubsection{Solving sets of equations 10 & 2 & 1 & 1\\ 0 & 3/5 & 4/5 & 4/5 \\ 0 & 0 & 7.5 & -1.5\end{array} -\right] $ +\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) +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 @@ -450,7 +461,7 @@ \subsubsection{Solving sets of equations \(\left[ \begin{array}{cccc} 2k & -k & 0 & 0 \\ -k & 2k & -k & 0 \\ 0 & -k & 2k & -k \\ 0 & 0 & -k & k \end{array} \right] \left[ \begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{array} \right]= \left[ \begin{array}{c} m_{1}g \\ m_{2}g \\ m_{3}g \\ m_{4}g \end{array} \right]\) \begin{Verbatim}[commandchars=\\\{\}] -{\color{incolor}In [{\color{incolor}18}]:} \PY{n}{k}\PY{p}{=}\PY{l+m+mi}{10}\PY{p}{;} \PY{c}{\PYZpc{} N/m} +{\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}{;} @@ -479,7 +490,7 @@ \subsubsection{Solving sets of equations \end{Verbatim} \begin{Verbatim}[commandchars=\\\{\}] -{\color{incolor}In [{\color{incolor}19}]:} \PY{n}{K1}\PY{p}{=}\PY{p}{[}\PY{n}{K} \PY{n}{y}\PY{p}{]}\PY{p}{;} +{\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} @@ -495,7 +506,7 @@ \subsubsection{Solving sets of equations \end{Verbatim} \begin{Verbatim}[commandchars=\\\{\}] -{\color{incolor}In [{\color{incolor}20}]:} \PY{n}{K2}\PY{p}{=}\PY{n}{K1}\PY{p}{;} +{\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} @@ -511,7 +522,7 @@ \subsubsection{Solving sets of equations \end{Verbatim} \begin{Verbatim}[commandchars=\\\{\}] 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[{\color{incolor}15}]:} \PY{n}{K}\PY{o}{\PYZbs{}}\PY{n}{y} \end{Verbatim} \begin{Verbatim}[commandchars=\\\{\}] @@ -565,7 +576,7 @@ \subsubsection{Solving sets of equations \texttt{x=GaussNaive(A,y)} \begin{Verbatim}[commandchars=\\\{\}] -{\color{incolor}In [{\color{incolor}24}]:} \PY{n}{x}\PY{p}{=}\PY{n}{GaussNaive}\PY{p}{(}\PY{n}{K}\PY{p}{,}\PY{n}{y}\PY{p}{)} +{\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=\\\{\}] @@ -606,7 +617,7 @@ \subsubsection{Solving sets of equations \end{enumerate} \begin{Verbatim}[commandchars=\\\{\}] -{\color{incolor}In [{\color{incolor}25}]:} \PY{n}{format} \PY{n}{short} +{\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} @@ -634,7 +645,7 @@ \subsubsection{Solving sets of equations \end{Verbatim} \begin{Verbatim}[commandchars=\\\{\}] -{\color{incolor}In [{\color{incolor}26}]:} \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}{)} +{\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=\\\{\}] @@ -655,7 +666,7 @@ \subsubsection{Solving sets of equations \end{Verbatim} \begin{Verbatim}[commandchars=\\\{\}] -{\color{incolor}In [{\color{incolor}27}]:} \PY{n}{format} \PY{n}{long} +{\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} @@ -676,7 +687,7 @@ \subsubsection{Solving sets of equations \end{Verbatim} \begin{Verbatim}[commandchars=\\\{\}] -{\color{incolor}In [{\color{incolor}28}]:} \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}{)} +{\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} @@ -699,6 +710,18 @@ \subsubsection{Solving sets of equations 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} @@ -729,7 +752,7 @@ \subsubsection{Solving sets of equations \(\left[ \begin{array}{cccc} k_1+k_2 & -k_2 & 0 & 0 \\ -k_2 & k_2+k_3 & -k_3 & 0 \\ 0 & -k_3 & k_3+k_4 & -k_4 \\ 0 & 0 & -k_4 & k_4 \end{array} \right] \left[ \begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{array} \right]= \left[ \begin{array}{c} m_{1}g \\ m_{2}g \\ m_{3}g \\ m_{4}g \end{array} \right]\) \begin{Verbatim}[commandchars=\\\{\}] -{\color{incolor}In [{\color{incolor}29}]:} \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} +{\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}{;} @@ -782,16 +805,17 @@ \subsubsection{Solving sets of equations n\(\times\)n-matrix\tabularnewline \midrule \endhead -Naive Gauss & n-cubed\tabularnewline +Gauss & n-cubed\tabularnewline Tridiagonal & n\tabularnewline \bottomrule \end{longtable} \begin{Verbatim}[commandchars=\\\{\}] -{\color{incolor}In [{\color{incolor}30}]:} \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}{;} +{\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=\\\{\}] @@ -799,11 +823,18 @@ \subsubsection{Solving sets of equations 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}12}]:} \PY{c}{\PYZpc{} tic ... t=toc } +{\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}{;} @@ -820,7 +851,7 @@ \subsubsection{Solving sets of equations \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}{GaussPivot}\PY{p}{(}\PY{n}{Atd}\PY{p}{,}\PY{n}{b}\PY{p}{)}\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}{;} @@ -829,17 +860,67 @@ \subsubsection{Solving sets of equations \end{Verbatim} \begin{Verbatim}[commandchars=\\\{\}] -{\color{incolor}In [{\color{incolor}13}]:} \PY{n}{n}\PY{p}{=}\PY{l+m+mi}{1}\PY{p}{:}\PY{l+m+mi}{200}\PY{p}{;} +{\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_27_0.pdf} + \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} diff --git a/lecture_10/lecture_10_files/lecture_10_29_0.pdf b/lecture_10/lecture_10_files/lecture_10_29_0.pdf new file mode 100644 index 0000000000000000000000000000000000000000..20bdd512f2f0927f20e0b03273421012f667618b GIT binary patch literal 10972 zcma)i1z1#F*EZcCC6Y2oH$x2F-QC^I&?(*Bt%P*9($WJeAR*mINQ0z=|Dcc0`@HdA z*LTjGv(LWw+O^g?bFQ`TO|2p!#SCQOM5Aswti46!0I&hTW_D--0svM8ODAi08vrL1 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zhsr@?wtn>ez8+E|cDCN}DJ-^5vgU>P4u8U>P4(PwN4p>}D9Fdu=LJmIFl(Ff%@!Cq zI&k)MOgXrCoMzv)9)Lxjzb(C?_7kT){*J{8|5o@CZMR+aQ455uy5MTo#wFA9H{EWk z>xPAmF9M`hIgq6yudX`0M{4jn!Kd?cH`x*-KDJw;5m-{y{GUzwU{~Q6(+xil?@?Y2 z`^e^1kYZY)Qmce^fcxr*-Fw%LWh*ZW+5h1}9T_kv^Zd`DHsIK?igSF6VIU|{(PcJ^ zEjbvncfrdza{LBw#V4KO?C88LWUQq!2TthDP&8t*)4%kF*=%3T;SW}rerwg_@T@Hx zE-&4}Cc9Hz$4IW_Y`OIN<`*07IHH=TY8I?G$@Z?9tMKg`hjXyj@7&a<&)FRkuji|h zd7RnXF9oC|yk$eO_bpMELQY;N``pn3D>*M$Yzv(6D4Bi8{HGPaH5+qwA8VMYw0IBa zQIh#K(b;R+EUjo(U6d|oT;A!x-`sRL%bxX>Dy~}0Zhth*qsVd+XJcxGzNeTxr)K7@ z7xOce*<*vu(+nT~#xA%l{5-&CE9Z&jf{5QuW^p#hItSnLaAjMLEpv<*TFWVUSY=bw x?#zi)D_prPZ3nwJXTs_|Yff`+ja4Z>{b>ukH0Mfg;J(#t!#Mr0%wiAD{{UU+ll1@q From 3780f0c36732e6e6d24aca6beb2e1b1dbcfe90a1 Mon Sep 17 00:00:00 2001 From: "Ryan C. Cooper" Date: Tue, 21 Feb 2017 12:51:57 -0500 Subject: [PATCH 2/7] added HW4 --- HW4/README.md | 37 ++++++ HW4/collar_mass.svg | 270 ++++++++++++++++++++++++++++++++++++++++++++ 2 files changed, 307 insertions(+) create mode 100644 HW4/README.md create mode 100644 HW4/collar_mass.svg diff --git a/HW4/README.md b/HW4/README.md new file mode 100644 index 0000000..f9c6f19 --- /dev/null +++ b/HW4/README.md @@ -0,0 +1,37 @@ +# Homework #3 +## due 3/1/17 by 11:59pm + + +1. Use your repository 'roots_and_optimization'. Document all the HW4 work under the +heading `# Homework #4` in your `README.md` file + + a. Create a function called 'collar_potential_energy' that computes the total + potential energy of a collar connected to a spring and sliding on a rod. As shown in + the figure given a position, xc, and angle, theta: + + ![Collar-mass on an inclined rod](collar_mass.svg) + + The spring is unstretched when $x_{C}=0.5$. The potential energy due to gravity is: + + $PE_{g}=m x_{C}g\sin\theta$ + + where m=0.5 kg, and g is the acceleration due to gravity, + + and the potential energy due to the spring is: + + $PE_{s}=1/2K (\Delta L)^{2}$ + + where $\Delta L = 0.5 - \sqrt{0.5^{2}+(0.5-x_{C})^{2}}$ + + b. Use the `goldmin.m` function to solve for the minimum potential energy at xc when + theta=0. *create an anonymous function with `@(x) collar_potential_energy(x,theta)` in + the input for goldmin. Be sure to include the script that solves for xc* + + c. Create a for-loop that solves for the minimum potential energy position, xc, at a + given angle, theta, for theta = 0..90 degrees. + + d. Include a plot of xc vs theta. `plot(theta,xc)` with + + `![Steady-state position of collar on rod at angle theta](plot.png)` + +3. Commit your changes to your repository. Sync your local repository with github. diff --git a/HW4/collar_mass.svg b/HW4/collar_mass.svg new file mode 100644 index 0000000..cc18c3d --- /dev/null +++ b/HW4/collar_mass.svg @@ -0,0 +1,270 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + image/svg+xml + + + + + + + + + + xc + + + + + + + + 𝝷 + + + L0=0.5m + + From 4a4ee8b39b30e3407210e9f7cdacdf81591c2d3e Mon Sep 17 00:00:00 2001 From: "Ryan C. Cooper" Date: Tue, 21 Feb 2017 12:54:00 -0500 Subject: [PATCH 3/7] update HW4 --- HW4/README.md | 2 +- HW4/collar_mass.png | Bin 0 -> 6794 bytes 2 files changed, 1 insertion(+), 1 deletion(-) create mode 100644 HW4/collar_mass.png diff --git a/HW4/README.md b/HW4/README.md index f9c6f19..3cacedc 100644 --- a/HW4/README.md +++ b/HW4/README.md @@ -9,7 +9,7 @@ heading `# Homework #4` in your `README.md` file potential energy of a collar connected to a spring and sliding on a rod. As shown in the figure given a position, xc, and angle, theta: - ![Collar-mass on an inclined rod](collar_mass.svg) + ![Collar-mass on an inclined rod](collar_mass.png) The spring is unstretched when $x_{C}=0.5$. 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z&7j>iOt?5M399*a<|o)I^9+4Fb2q8(r`~qzD5oN@uqQ8Xh9ZH^rY=tXX~w@a#`c2D z%62*gJCYK0_o`zhM-552cXb!1duv|u#Z(AaDJZc8QW442Pc<94y|;VlS%t^xXl$(?$~yDx(um2BfJx2*6$Tsf%!~iQ<~I+%T&Is3d#K<3}}|=tUhPNaCUrfGzeL z6ZF1ajD!1zYH4@^wK@nOPW)Mhlj}lQOodeQ9&X^9$7Qt=EJm0jcvk5ox!l>LD^3G9 zen|{~BiyJPOY95*U2Ye%!2+2Ofg>4Jlh331R5B2KilkH<#!##8&ILSmnC5syy-UTj z*UNWbnM5(p_~4{@;Q8;(IQ^$;F0rsrrki~%-LH}@v^=R5duzyIj6~cs zb7;3Ax-`sg+)0~n?NLoEZ2{1&@alP7R4Oh`L&*;&BW>qWKIDxLMrzarwaHeEBNbQ90 zpkIfv=AW>=-UMK^ik8_{v;b;Iud4kIt0E1IsX}23ZtsYTstP94$VnLfch}QMe%7yY zkG#GRm@3__Jzm5qO_sc`#YRJ#A7Jmua^R?SA3n4g9xo2roBn z!eW5hReE1#r2@zolp?gzALO|e=4@J!XNpN Date: Tue, 21 Feb 2017 12:57:11 -0500 Subject: [PATCH 4/7] update HW4 --- HW4/README.md | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/HW4/README.md b/HW4/README.md index 3cacedc..ed5ca61 100644 --- a/HW4/README.md +++ b/HW4/README.md @@ -11,17 +11,17 @@ heading `# Homework #4` in your `README.md` file ![Collar-mass on an inclined rod](collar_mass.png) - The spring is unstretched when $x_{C}=0.5$. The potential energy due to gravity is: + The spring is unstretched when x_C=0.5 m. The potential energy due to gravity is: - $PE_{g}=m x_{C}g\sin\theta$ + PE_g=m x_C*g*sin(theta) where m=0.5 kg, and g is the acceleration due to gravity, and the potential energy due to the spring is: - $PE_{s}=1/2K (\Delta L)^{2}$ + PE_s=1/2*K *(DL)^2$ - where $\Delta L = 0.5 - \sqrt{0.5^{2}+(0.5-x_{C})^{2}}$ + where DL = 0.5 - sqrt(0.5^2+(0.5-x_C)^2) b. Use the `goldmin.m` function to solve for the minimum potential energy at xc when theta=0. *create an anonymous function with `@(x) collar_potential_energy(x,theta)` in From f29d2cd48cf2333e32379a56402ef6b5a19f580e Mon Sep 17 00:00:00 2001 From: "Ryan C. Cooper" Date: Tue, 21 Feb 2017 12:58:13 -0500 Subject: [PATCH 5/7] update HW4 --- HW4/README.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/HW4/README.md b/HW4/README.md index ed5ca61..0b8a5f8 100644 --- a/HW4/README.md +++ b/HW4/README.md @@ -13,13 +13,13 @@ heading `# Homework #4` in your `README.md` file The spring is unstretched when x_C=0.5 m. The potential energy due to gravity is: - PE_g=m x_C*g*sin(theta) + PE_g=m x_C\*g\*sin(theta) where m=0.5 kg, and g is the acceleration due to gravity, and the potential energy due to the spring is: - PE_s=1/2*K *(DL)^2$ + PE_s=1/2\*K \*(DL)^2 where DL = 0.5 - sqrt(0.5^2+(0.5-x_C)^2) From 72ca990cf5c532001f3317577e52f1f54ab1a1b9 Mon Sep 17 00:00:00 2001 From: "Ryan C. Cooper" Date: Tue, 21 Feb 2017 13:34:12 -0500 Subject: [PATCH 6/7] added suggested gauss elimination --- linear_algebra/gauss_suggested.md | 71 +++++++++++++++++++++++++++++++ 1 file changed, 71 insertions(+) create mode 100644 linear_algebra/gauss_suggested.md diff --git a/linear_algebra/gauss_suggested.md b/linear_algebra/gauss_suggested.md new file mode 100644 index 0000000..6b1f6b6 --- /dev/null +++ b/linear_algebra/gauss_suggested.md @@ -0,0 +1,71 @@ +# Linear Algebra Review +## (Gauss Elimination) Suggested problems +### No due date + +1. Solve for x when Ax=b for the following problems: + + a. $A=\left[ \begin{array}{cc} + 1 & 3 \\ + 2 & 1 \end{array} \right] + b= + \left[\begin{array}{c} + 1 \\ + 1\end{array}\right]$ + + a. $A=\left[ \begin{array}{cc} + 1 & 1 \\ + 2 & 3 \end{array} \right] + b= + \left[\begin{array}{c} + 3 \\ + 4\end{array}\right]$ + + a. $A=\left[ \begin{array}{cc} + 1 & 1 \\ + 2 & -2 \end{array} \right] + b= + \left[\begin{array}{c} + 4 \\ + 2\end{array}\right]$ + + b. $A=\left[ \begin{array}{ccc} + 1 & 3 & 1 \\ + -4 & -9 & 2 \\ + 0 & 3 & 6\end{array} \right] + b= + \left[\begin{array}{c} + 0 \\ + 0 \\ + 0\end{array}\right]$ + + c. $A=\left[ \begin{array}{ccc} + 1 & 3 & 1 \\ + -4 & -9 & 2 \\ + 0 & 3 & 6\end{array} \right] + b= + \left[\begin{array}{c} + 1 \\ + -1 \\ + -3\end{array}\right]$ + + d. $A=\left[ \begin{array}{ccc} + 1 & 3 & -5 \\ + 1 & 4 & -8 \\ + -3 & -7 & 9\end{array} \right] + b= + \left[\begin{array}{c} + 1 \\ + -1 \\ + -3\end{array}\right]$ + + d. $A=\left[ \begin{array}{ccc} + 1 & 2 & -1 \\ + 2 & 2 & 2 \\ + 1 & -1 & 2\end{array} \right] + b= + \left[\begin{array}{c} + 2 \\ + 12 \\ + 5\end{array}\right]$ + +2. Calculate the determinant of A from 1a-g. From 543fb4befaf120a39c88cde7fce8a17b07c6798d Mon Sep 17 00:00:00 2001 From: "Ryan C. Cooper" Date: Tue, 21 Feb 2017 13:34:49 -0500 Subject: [PATCH 7/7] added suggested gauss elimination --- linear_algebra/gauss_suggested.pdf | Bin 0 -> 87205 bytes 1 file changed, 0 insertions(+), 0 deletions(-) create mode 100644 linear_algebra/gauss_suggested.pdf diff --git a/linear_algebra/gauss_suggested.pdf b/linear_algebra/gauss_suggested.pdf new file mode 100644 index 0000000000000000000000000000000000000000..a78b2778173adb57f91a5eccafa5b843897da538 GIT binary patch literal 87205 zcmaf(Gn6n&6r{(tZQHhO+qP}n_WWbpwr$(C_w89E8*C+A9dy+93#!f~RS*%QWu#+; zBAs6zS%+d|Ctx73H?o4_;en!;F|{*yu^`~!WFz=rf}$6*v~e+YBA^$uF?2B%F*UX~ zF@@segK~CpGBvb?^4L7ln6leqKS$taLi(`uc{qSBLqrA%Ib85Sv&6F$g9OEmEnBF2|1!!f zsZs>e8dd+`$*xv%e`5C5F_ACcJF=$(Ay%8zg}@@+MzYO}c916h$+BCsX20Xo=b=fv zFjJ&?fOUqqZ+-G?-8E5m|1wvVdAv5fDV`cfs%ozcGH#XX*yo4JF(k;dYkCwKUUEt9 zu#{OTWt7WU^{WjZ9j8c48D$s!u8=2IqBo32r-=?Pm20%p{G}zbj zN*1ug(jUrZcxhk&z89*Wq&^o|QNR!bGD?X-3QPsK4Ftsm8=P@K$tI?>TNPlX3}dHI 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