From 545ef3262fa6e61236b8aaa3d5394f9cebd3985e Mon Sep 17 00:00:00 2001 From: "Ryan C. Cooper" Date: Tue, 4 Apr 2017 09:17:26 -0400 Subject: [PATCH 01/10] added lecture 19 --- lecture_17/octave-workspace | Bin 23760 -> 5146 bytes lecture_18/Newtint.m | 67 +- lecture_18/lecture_18.ipynb | 631 +++++-- lecture_18/octave-workspace | Bin 0 -> 7470 bytes .../lecture 19-checkpoint.ipynb | 497 ++++++ lecture_19/Newtint.m | 34 + lecture_19/lecture 19.ipynb | 1488 +++++++++++++++++ lecture_19/newton_cotes.png | Bin 0 -> 78112 bytes lecture_19/q1.png | Bin 0 -> 30855 bytes lecture_19/q2.png | Bin 0 -> 22150 bytes lecture_19/q3.png | Bin 0 -> 30938 bytes lecture_19/q4.png | Bin 0 -> 34995 bytes lecture_19/simpson3.m | 23 + lecture_19/trap.m | 22 + 14 files changed, 2569 insertions(+), 193 deletions(-) create mode 100644 lecture_18/octave-workspace create mode 100644 lecture_19/.ipynb_checkpoints/lecture 19-checkpoint.ipynb create mode 100644 lecture_19/Newtint.m create mode 100644 lecture_19/lecture 19.ipynb create mode 100644 lecture_19/newton_cotes.png create mode 100644 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"\t\tresiduals of function\n", + "\t\n", + "\n", + "\n", + "\n", + "\tgnuplot_plot_1a\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "[sse,yhat]=sse_nonlin_exp(a,data(:,1),data(:,2));\n", + "plot(data(:,1),data(:,2)-yhat)\n", + "title('residuals of function')" + ] + }, { "cell_type": "markdown", "metadata": {}, @@ -506,12 +701,12 @@ "\n", "$\\sigma(t)=\\frac{1}{1+e^{-t}}$\n", "\n", - "We can use this function to describe the likelihood of failure (1) or success (0). When z=0, the probability of failure is 50%. " + "We can use this function to describe the likelihood of failure (1) or success (0). When t=0, the probability of failure is 50%. " ] }, { "cell_type": "code", - "execution_count": 30, + "execution_count": 131, "metadata": { "collapsed": false }, @@ -691,7 +886,7 @@ }, { "cell_type": "code", - "execution_count": 25, + "execution_count": 132, "metadata": { "collapsed": false }, @@ -739,7 +934,7 @@ }, { "cell_type": "code", - "execution_count": 27, + "execution_count": 134, "metadata": { "collapsed": false }, @@ -786,115 +981,125 @@ "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", "\n", "\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t0\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t0.2\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t0.4\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t0.6\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t0.8\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t1\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t50\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t55\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t60\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t65\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t70\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t75\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t80\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t85\n", "\t\n", "\n", "\n", "\n", "\n", - "\t\n", + "\t\n", + "\n", + "\t\n", + "\t\tfailure (1)/ pass (0)\n", + "\t\n", + "\n", + "\n", + "\t\n", + "\t\tTemp (F)\n", + "\t\n", + "\n", "\n", "\n", "\tgnuplot_plot_1a\n", - "\n", + "\n", "\n", "\n", "\t \n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", "\n", "\t\n", "\n", @@ -917,19 +1122,21 @@ } ], "source": [ - "plot(oring(:,2),oring(:,3),'o')" + "plot(oring(:,2),oring(:,3),'o')\n", + "xlabel('Temp (F)')\n", + "ylabel('failure (1)/ pass (0)')" ] }, { "cell_type": "code", - "execution_count": 54, + "execution_count": 135, "metadata": { "collapsed": true }, "outputs": [], "source": [ "function J=sse_logistic(a,x,y)\n", - " % Create function to calculate SSE of logistic function\n", + " % Create function to calculate cost of logistic function\n", " % t = a0+a1*x\n", " % sigma(t) = 1./(1+e^(-t))\n", " sigma=@(t) 1./(1+exp(-t));\n", @@ -942,7 +1149,7 @@ }, { "cell_type": "code", - "execution_count": 61, + "execution_count": 142, "metadata": { "collapsed": false }, @@ -1006,36 +1213,41 @@ "\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t0\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t0.2\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t0.4\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t0.6\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t0.8\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t1\n", "\t\n", "\n", "\n", + "\t\t\n", + "\t\t1.2\n", + "\t\n", + "\n", + "\n", "\t\t\n", "\t\t50\n", "\t\n", @@ -1086,36 +1298,40 @@ "\n", "\n", "\t \n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", "\n", "\t\n", "\tgnuplot_plot_2a\n", "\n", - "\t\n", + "\t\n", "\t\n", - "\n", + "\tgnuplot_plot_3a\n", + "\n", + "\t\n", + "\t\n", + "\n", "\n", "\n", "\n", @@ -1139,12 +1355,13 @@ "a=fminsearch(@(a) sse_logistic(a,oring(:,2),oring(:,3)),[0,-3])\n", "\n", "T=linspace(50,85);\n", - "plot(oring(:,2),oring(:,3),'o',T,sigma(a(1)+a(2)*T))" + "plot(oring(:,2),oring(:,3),'o',T,sigma(a(1)+a(2)*T),T,a(1)+a(2)*T)\n", + "axis([50,85,-0.1,1.2])" ] }, { "cell_type": "code", - "execution_count": 75, + "execution_count": 139, "metadata": { "collapsed": false }, @@ -1154,13 +1371,15 @@ "output_type": "stream", "text": [ "probability of failure when 70 degrees is 23.00% \n", - "probability of failure when 60 degrees is 75.25%\n" + "probability of failure when 60 degrees is 75.25%\n", + "probability of failure when 36 degrees is 99.87%\n" ] } ], "source": [ "fprintf('probability of failure when 70 degrees is %1.2f%% ',100*sigma(a(1)+a(2)*70))\n", - "fprintf('probability of failure when 60 degrees is %1.2f%%',100*sigma(a(1)+a(2)*60))\n" + "fprintf('probability of failure when 60 degrees is %1.2f%%',100*sigma(a(1)+a(2)*60))\n", + "fprintf('probability of failure when 36 degrees is %1.2f%%',100*sigma(a(1)+a(2)*36))\n" ] }, { @@ -1196,7 +1415,7 @@ }, { "cell_type": "code", - "execution_count": 84, + "execution_count": 149, "metadata": { "collapsed": false }, @@ -1207,6 +1426,7 @@ "text": [ "ln(2)~0.358352\n", "ln(2)~0.462098\n", + "ln(2)~0.549306\n", "ln(2)=0.693147\n" ] } @@ -1215,14 +1435,16 @@ "ln2_16=log(1)+(log(6)-log(1))/(6-1)*(2-1);\n", "fprintf('ln(2)~%f\\n',ln2_16)\n", "ln2_14=log(1)+(log(4)-log(1))/(4-1)*(2-1);\n", + "ln2_13=log(1)+(log(3)-log(1))/(3-1)*(2-1);\n", "fprintf('ln(2)~%f\\n',ln2_14)\n", + "fprintf('ln(2)~%f\\n',ln2_13)\n", "ln2=log(2);\n", "fprintf('ln(2)=%f\\n',ln2)" ] }, { "cell_type": "code", - "execution_count": 87, + "execution_count": 147, "metadata": { "collapsed": false }, @@ -1269,98 +1491,108 @@ "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", "\n", "\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t0\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t0.5\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t1\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t1.5\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t2\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t1\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t2\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t3\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t4\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t5\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t6\n", "\t\n", "\n", "\n", "\n", "\n", - "\t\n", + "\t\n", + "\n", + "\t\n", + "\t\tln(x)\n", + "\t\n", + "\n", + "\n", + "\t\n", + "\t\tx\n", + "\t\n", + "\n", "\n", "\n", "\tgnuplot_plot_1a\n", - "\n", + "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_2a\n", "\n", "\t \n", - "\t\n", + "\t\n", "\n", "\t\n", "\tgnuplot_plot_3a\n", "\n", - "\t\t \n", - "\t\n", - "\t\n", - "\t\n", + "\t\t \n", + "\t\n", + "\t\n", + "\t\n", "\n", "\t\n", "\tgnuplot_plot_4a\n", "\n", - "\t\t \n", - "\t\n", - "\t\n", - "\t\n", + "\t\t \n", + "\t\n", + "\t\n", + "\t\n", "\n", "\t\n", "\n", @@ -1386,7 +1618,9 @@ "x=linspace(1,6);\n", "plot(x,log(x),2,log(2),'*',...\n", "[1,2,6],[log(1),ln2_16,log(6)],'o-',...\n", - "[1,2,4],[log(1),ln2_14,log(4)],'s-')" + "[1,2,4],[log(1),ln2_14,log(4)],'s-')\n", + "ylabel('ln(x)')\n", + "xlabel('x')" ] }, { @@ -1417,7 +1651,41 @@ }, { "cell_type": "code", - "execution_count": 89, + "execution_count": 154, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Z =\n", + "\n", + " 1 1 1\n", + " 1 4 16\n", + " 1 600 360000\n", + "\n", + "ans = 5.1766e+05\n", + "ans =\n", + "\n", + " -4.6513e-01\n", + " 4.6589e-01\n", + " -7.5741e-04\n", + "\n" + ] + } + ], + "source": [ + "x=[1,4,600]';\n", + "Z=[x.^0,x.^1,x.^2]\n", + "cond(Z)\n", + "Z\\log(x)" + ] + }, + { + "cell_type": "code", + "execution_count": 155, "metadata": { "collapsed": false }, @@ -1441,18 +1709,25 @@ "f3=log(x3);\n", "\n", "b1=f1\n", - "b2=(f2-f1)/(x2-x1)\n", + "b2=(f2-b1)/(x2-x1)\n", "b3=(f3-f2)/(x3-x2)-b2;\n", "b3=b3/(x3-x1)" ] }, { "cell_type": "code", - "execution_count": 91, + "execution_count": 157, "metadata": { "collapsed": false }, "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "ans = 0.56584\r\n" + ] + }, { "data": { "image/svg+xml": [ @@ -1615,7 +1890,8 @@ "f=@(x) b1+b2*(x-x1)+b3*(x-x1).*(x-x2);\n", "plot(x,log(x),2,log(2),'*',...\n", "[1,4,6],[log(1),log(4),log(6)],'ro',...\n", - "x,f(x),'r-',2,f(2),'s')" + "x,f(x),'r-',2,f(2),'s')\n", + "f(2)" ] }, { @@ -1649,7 +1925,41 @@ }, { "cell_type": "code", - "execution_count": 105, + "execution_count": 160, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "b =\n", + "\n", + " 0.00000 0.54931 -0.08721 0.01178\n", + " 1.09861 0.28768 -0.02832 0.00000\n", + " 1.38629 0.20273 0.00000 0.00000\n", + " 1.79176 0.00000 0.00000 0.00000\n", + "\n", + "ans = 0.66007\n", + "ans =\n", + "\n", + " 0.00000\n", + " 1.09861\n", + " 1.38629\n", + " 1.79176\n", + "\n" + ] + } + ], + "source": [ + "Newtint([1,3,4,6],log([1,3,4,6]),2)\n", + "log([1,3,4,6]')" + ] + }, + { + "cell_type": "code", + "execution_count": 162, "metadata": { "collapsed": false }, @@ -1659,7 +1969,7 @@ "output_type": "stream", "text": [ "ln(2)=0.693147\n", - "ln(2)~0.722462\n" + "ln(2)~0.366349\n" ] }, { @@ -1704,113 +2014,113 @@ "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", "\n", "\n", "\n", "\n", - "\t\t\n", - "\t\t-1\n", + "\t\t\n", + "\t\t-2\n", "\t\n", "\n", "\n", - "\t\t\n", - "\t\t-0.5\n", + "\t\t\n", + "\t\t0\n", "\t\n", "\n", "\n", - "\t\t\n", - "\t\t0\n", + "\t\t\n", + "\t\t2\n", "\t\n", "\n", "\n", - "\t\t\n", - "\t\t0.5\n", + "\t\t\n", + "\t\t4\n", "\t\n", "\n", "\n", - "\t\t\n", - "\t\t1\n", + "\t\t\n", + "\t\t6\n", "\t\n", "\n", "\n", - "\t\t\n", - "\t\t1.5\n", + "\t\t\n", + "\t\t8\n", "\t\n", "\n", "\n", - "\t\t\n", - "\t\t2\n", + "\t\t\n", + "\t\t10\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t0\n", "\t\n", "\n", "\n", - "\t\t\n", - "\t\t1\n", + "\t\t\n", + "\t\t10\n", "\t\n", "\n", "\n", - "\t\t\n", - "\t\t2\n", + "\t\t\n", + "\t\t20\n", "\t\n", "\n", "\n", - "\t\t\n", - "\t\t3\n", + "\t\t\n", + "\t\t30\n", "\t\n", "\n", "\n", - "\t\t\n", - "\t\t4\n", + "\t\t\n", + "\t\t40\n", "\t\n", "\n", "\n", - "\t\t\n", - "\t\t5\n", + "\t\t\n", + "\t\t50\n", "\t\n", "\n", "\n", "\t\t\n", - "\t\t6\n", + "\t\t60\n", "\t\n", "\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\n", "\n", "\tgnuplot_plot_1a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_2a\n", "\n", "\t \n", - "\t\n", + "\t\n", "\n", "\t\n", "\tgnuplot_plot_3a\n", "\n", "\t \n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", "\n", "\t\n", "\tgnuplot_plot_4a\n", "\n", - "\t\n", + "\t\n", "\t\n", "\n", "\n", @@ -1833,10 +2143,11 @@ ], "source": [ "\n", - "yy=zeros(size(xx));\n", - "x=[0.5,1,3,4,5,6]; % define independent var's\n", + "\n", + "x=[0.2,3,10,20,50,60]; % define independent var's\n", "y=log(x); % define dependent var's\n", "xx=linspace(min(x),max(x));\n", + "yy=zeros(size(xx));\n", "for i=1:length(xx)\n", " yy(i)=Newtint(x,y,xx(i));\n", "end\n", diff --git a/lecture_18/octave-workspace b/lecture_18/octave-workspace new file mode 100644 index 0000000000000000000000000000000000000000..c65169697346d5d351f3f9767bf0ba8da01d8669 GIT binary patch literal 7470 zcmeHLdst0b|3Bv>={72*QaC4-3Q@TYu|LVJ`?(Z`&|Q&|l8U!U?q-nUCz^RD#x<7^ zxpy`%5;ZffgJH%sxBLc;OS!#!t-Vg>70>T7&+m`l^St|c_W7Q**7v@?Ypu^-A<=1( z>9O5Bx&=yFN+c4W@lmmnlZ8JL$ve`X9GNyXA>-}4ckhgZ7I_VXmpq`qlE+$+{?*?m zJ^$7#rt;r;_$f*YF2SnEdCf0FeG`QI)?T|P^PxzzJ3~4rg z^!m8Naeur3kq&)7`hIniff(UstJ4p9d$QM~n;-NizVV+m+7GQd&U}1iz8{RvH?ENF z@q_8Y%>KSN{a`lm<@MRE{b5m`KJS>fKdd%A>!Lr=AJ${j&J0=Uk9KsxANG|g_LWck zLDU#@41j}2@`T}m0dTB+WU??b0G*2xcU5f&fUEN1NTc%s=t>Mr1L1z)-rBF+13}oU z#{{A$6$nJ{%P|E(<$?Ho$*sOiZUw^IH|p0P%z{AO85jhgZR1^)lY&6H^9qA7lsI%e z2*ay?v9+iVLcpxiTD3eF!BkN&sM)MwXo@eET5bwP7>VOft2 zK?HGhQ3$B|y&;G#2en9>wz!j8ZlpeG`KCyTHj!e zn70@WXc^4VAf8N6z6SS&SwFBz16qDs4M%B)*=jK8ppeafi4<^(G@hBHazb4A+3*F$VK zSox4887aKYPEs-;6J7&z>|9@G?vB6N#8+39)o6+N$fQ)^x0y@}p^M1rPG5@xNG1K1UA=T297ad+`1+kyyPh!d1HjLUcDreMzQAd=VubVX@ z;tCgQdvla^AALS(O?|tAmhSj&Nu^2X2}^Xao0TzETdUP@$!_XMR~R@|?|$Po01xg4 z4RIS4gctcI;@jUH0jpcDgIxB-!ePsp0ULWvp%oD;Ap;$V9ogxi-O` z#f@&hBM$H8tOq3CtWllk>VwUt|7cswZA`Z>c)0s1chdc}v&+<1T+zK*bq|Niu;o}v z??vldV(`|3Tu7D?!ZUXHo|tBeuA6FutVXw{X5-$QjhmJfotBWCB#)0wib;(9Oc;T8 zWKwcc=H%pQsRQM`$gIF6!&M&TA@3rO@|3&DT{7g|{${}X3774t%>r^OpgnrZj5$i)Sb~DjCwHY$*32jJsIuA zXm3XQh&0Ea(LmN8#M&W@D&a+*GNst&m8hvE4{J+QVn0+OpKy9oet}ZHuisM%ebkJf zQT{!pa>A)c~{qORLM@T9ar+eK8 zn~sSv^5;#*hO}MMT;(2e_wI5RvhT!ZOp&{EOLKK4b(xwlVKO@l$jnIuNG^X zz|-tpBxhob@Um0Kkr(k|C!W9rg-0iz;;1SIp5omo2cF`@Mg>pcUQB_4r?{w#f){bF zoEPz-oF}M3`LH8TahZKbUc{Llc#5mC+w&ApmfG_a7hSjGDekeb<0-E3ZO4na#g?bI zWW5bf;A-(jYo6k818bhb-o9;l5pP@Z1U@?#S@INDSGVCQ41Z(6cc(zpf~R*tIeVK>bf?9CiI+E zLIK@=9pzX*$M`rV$C-(GIHs3lb~t92lZ$?E%rB1l3FbGh5`5~_0zVPldq!Wx)5!)R zPB$^cRxuCwM$8Kcw~!~WyfNHvhTRk(nzQc|3$VP4-#685uv(lK2CVo#D zSR+TASG*?pbG;42tG1{k_|~_bh=(ogK;Y`j>vjmI9|ra!KF@BCK>GdC0VfImx9{R&VsQ4ZqwLX`tp-D5EQu;~P0Z3vGeJ?_%!{||6) z6Tm5|+BW9_U%{2u%zQ4*t>Gqb^*LSj-FdEHyQ=QO*-PB);&m!hy#4=8twMlbg=^ z%Nbt=lZ#{a7c;(iW_K0yC!6uDXL5&`{V$nc*-URL)3fYfc^k>%n8ft273H+^m|e!lvG}x=OwR_Ue+ILw%j165U(EF9F}*p=&Uq&9 z&8V97k7s<@Ol~F96T|#i!rD52(^(!TGW*OAZY`5vz~X!;%4vUP_Q+9_@v#*|1_+Wf zSiOooRqV@AhrS#^g(Nx@>309FU^HB@fa#w~n95qDYiJSzJ!p~r`vCWC4<>sh<5I53RVTrHl zB~M!fPBqQy`rHO48nU2o;>c5=|8LaD&qnB8BW4mjjoHc|9MfgzRin`k;y=;#(hf z=j4O^R*S~0={pD>*}E4%ojVAdYk$@Ho*M*(5Y=+IuQDdAmN@06zkb02*m8z*dQ!ed7)RQU0fjp zUt;mw@vAa)x##uj@p&2gl@~?U?~q|-z<}B_nKG<1RctubM+OtQq4SuhQY`qxDrM*n zDJEQ**r#Z^6sjd>>-Gmpaic2w?&v;J3_s_Ty}Yj!dZv3i-w2f=WA5ETr=?O9%}cpe zeqV|`PK%aD#K~~#?9y`+Tk2uiy{$RLr}S{X|4dC-Sqo^kY-^ar>%*tCq*wTpmUwor z-VGQG9#vOYOueCL52vn@39XeKaJ9w!gjlX4R*r9f z>|JX)>_-G$)XR|L!S>X!s~6?yH~qk-9Rf<+yzcG6vS|uz?VCB@X}1C`V&hka)+*pv zy073nV+WX6ZLdu4loS92`~W;(!&WEW^%NIpEIe{u|HME1)`ERQcmk z1!N0ipC8Rt;I7w^>Z4u?950-HOi!@)vR{$*a)KN~Uq{JPf9r^W%l5uY`>6veLZqeL zH?>FI-TO5$@%9M3>u2lO#tzw%8CgG+*uqg$($?0~21m-=#+hwxi!1df?Ck8V5bk0* z=gr(UF#qEpgRed^M^g3k0&bKUS}w5vvg>kF3@Cn^k+RGP!A^dOm&y!q4~qV$4fJs@ zpHEq}T86}8n9&8EAIM|Y@?BqcU+I!sJrty8Hyi{GoIF' to search the manual index.\n", + "\n", + "Help and information about Octave is also available on the WWW\n", + "at http://www.octave.org and via the help@octave.org\n", + "mailing list.\n" + ] + } + ], + "source": [ + "help interp1" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "'interp2' is a function from the file /usr/share/octave/4.0.0/m/general/interp2.m\n", + "\n", + " -- Function File: ZI = interp2 (X, Y, Z, XI, YI)\n", + " -- Function File: ZI = interp2 (Z, XI, YI)\n", + " -- Function File: ZI = interp2 (Z, N)\n", + " -- Function File: ZI = interp2 (Z)\n", + " -- Function File: ZI = interp2 (..., METHOD)\n", + " -- Function File: ZI = interp2 (..., METHOD, EXTRAP)\n", + "\n", + " Two-dimensional interpolation.\n", + "\n", + " Interpolate reference data X, Y, Z to determine ZI at the\n", + " coordinates XI, YI. The reference data X, Y can be matrices, as\n", + " returned by 'meshgrid', in which case the sizes of X, Y, and Z must\n", + " be equal. If X, Y are vectors describing a grid then 'length (X)\n", + " == columns (Z)' and 'length (Y) == rows (Z)'. In either case the\n", + " input data must be strictly monotonic.\n", + "\n", + " If called without X, Y, and just a single reference data matrix Z,\n", + " the 2-D region 'X = 1:columns (Z), Y = 1:rows (Z)' is assumed.\n", + " This saves memory if the grid is regular and the distance between\n", + " points is not important.\n", + "\n", + " If called with a single reference data matrix Z and a refinement\n", + " value N, then perform interpolation over a grid where each original\n", + " interval has been recursively subdivided N times. This results in\n", + " '2^N-1' additional points for every interval in the original grid.\n", + " If N is omitted a value of 1 is used. As an example, the interval\n", + " [0,1] with 'N==2' results in a refined interval with points at [0,\n", + " 1/4, 1/2, 3/4, 1].\n", + "\n", + " The interpolation METHOD is one of:\n", + "\n", + " \"nearest\"\n", + " Return the nearest neighbor.\n", + "\n", + " \"linear\" (default)\n", + " Linear interpolation from nearest neighbors.\n", + "\n", + " \"pchip\"\n", + " Piecewise cubic Hermite interpolating\n", + " polynomial--shape-preserving interpolation with smooth first\n", + " derivative.\n", + "\n", + " \"cubic\"\n", + " Cubic interpolation (same as \"pchip\").\n", + "\n", + " \"spline\"\n", + " Cubic spline interpolation--smooth first and second\n", + " derivatives throughout the curve.\n", + "\n", + " EXTRAP is a scalar number. It replaces values beyond the endpoints\n", + " with EXTRAP. Note that if EXTRAPVAL is used, METHOD must be\n", + " specified as well. If EXTRAP is omitted and the METHOD is\n", + " \"spline\", then the extrapolated values of the \"spline\" are used.\n", + " Otherwise the default EXTRAP value for any other METHOD is \"NA\".\n", + "\n", + " See also: interp1, interp3, interpn, meshgrid.\n", + "\n", + "Additional help for built-in functions and operators is\n", + "available in the online version of the manual. Use the command\n", + "'doc ' to search the manual index.\n", + "\n", + "Help and information about Octave is also available on the WWW\n", + "at http://www.octave.org and via the help@octave.org\n", + "mailing list.\n" + ] + } + ], + "source": [ + "help interp2" + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "Gnuplot\n", + "Produced by GNUPLOT 5.0 patchlevel 3 \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\t\n", + "\t\t-1.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-1\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-0.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t1\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t1.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-3\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-2\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-1\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t1\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t2\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t3\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\tdata\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\tdata\n", + "\t\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\n", + "\t\n", + "\tlinear\n", + "\n", + "\t\n", + "\t\tlinear\n", + "\t\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\tcubic spline\n", + "\n", + "\t\n", + "\t\tcubic spline\n", + "\t\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\tpiecewise cubic\n", + "\n", + "\t\n", + "\t\tpiecewise cubic\n", + "\t\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "x=linspace(-pi,pi,9);\n", + "xi=linspace(-pi,pi,100);\n", + "y=sin(x);\n", + "yi_lin=interp1(x,y,xi,'linear');\n", + "yi_spline=interp1(x,y,xi,'spline'); \n", + "yi_cubic=interp1(x,y,xi,'cubic');\n", + "plot(x,y,'o',xi,yi_lin,xi,yi_spline,xi,yi_cubic)\n", + "axis([-pi,pi,-1.5,1.5])\n", + "legend('data','linear','cubic spline','piecewise cubic','Location','NorthWest')\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Example: Accelerate then hold velocity\n", + "\n", + "Here the time is given as vector t in seconds and the " + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "t=[0 2 40 56 68 80 84 96 104 110]';\n", + "v=[0 20 20 38 80 80 100 100 125 125]';\n" + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Octave", + "language": "octave", + "name": "octave" + }, + "language_info": { + "file_extension": ".m", + "help_links": [ + { + "text": "MetaKernel Magics", + "url": "https://github.com/calysto/metakernel/blob/master/metakernel/magics/README.md" + } + ], + "mimetype": "text/x-octave", + "name": "octave", + "version": "0.19.14" + } + }, + "nbformat": 4, + "nbformat_minor": 2 +} diff --git a/lecture_19/Newtint.m b/lecture_19/Newtint.m new file mode 100644 index 0000000..e4c6c83 --- /dev/null +++ b/lecture_19/Newtint.m @@ -0,0 +1,34 @@ +function yint = Newtint(x,y,xx) +% Newtint: Newton interpolating polynomial +% yint = Newtint(x,y,xx): Uses an (n - 1)-order Newton +% interpolating polynomial based on n data points (x, y) +% to determine a value of the dependent variable (yint) +% at a given value of the independent variable, xx. +% input: +% x = independent variable +% y = dependent variable +% xx = value of independent variable at which +% interpolation is calculated +% output: +% yint = interpolated value of dependent variable + +% compute the finite divided differences in the form of a +% difference table +n = length(x); +if length(y)~=n, error('x and y must be same length'); end +b = zeros(n,n); +% assign dependent variables to the first column of b. +b(:,1) = y(:); % the (:) ensures that y is a column vector. +for j = 2:n + for i = 1:n-j+1 + b(i,j) = (b(i+1,j-1)-b(i,j-1))/(x(i+j-1)-x(i)); + end +end +%b +% use the finite divided differences to interpolate +xt = 1; +yint = b(1,1); +for j = 1:n-1 + xt = xt*(xx-x(j)); + yint = yint+b(1,j+1)*xt; +end diff --git a/lecture_19/lecture 19.ipynb b/lecture_19/lecture 19.ipynb new file mode 100644 index 0000000..e6f9f76 --- /dev/null +++ b/lecture_19/lecture 19.ipynb @@ -0,0 +1,1488 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 1, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "setdefaults" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "%plot --format svg" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Questions from last class\n", + "\n", + "![q1](q1.png)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "![q2](q2.png)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "![q3](q3.png)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "![q4](q4.png)\n", + "\n", + "#### Other:\n", + "\n", + "Twice the amount of points needed\n", + "\n", + "depends on what order polynomial it is and how far the data needs to be extrapolated \n", + "\n", + "As man you as possible \n", + "\n", + "Never extrapolate unless linear interpolation.\n", + "\n", + "You shouldn't. 2 if the linear is a good fit for the region, and you absolutely have to.\n", + "\n", + "Wait can you do that?\n", + "\n", + "Don't use extrapolation\n", + "\n", + "do not extrapolate\n", + "\n", + "As many data points as you have\n", + "\n", + "the more the better so that the best polynomial can be made through the data points\n", + "\n", + "Twice the amount of points needed" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Questions from you\n", + "\n", + "- when will the project assignment be finalized? Also do you pronounce it \"jiff\" or \"gif\"?\n", + "\n", + "- If blue is red and red is blue, then what is purple? \n", + "\n", + "- How do we open the .ipynb lecture files? Or will the lectures continue to be also saved in pdf (last few have not).\n", + "\n", + "- When will we be put on teams for the final project?\n", + "\n", + "- What is the grading rubric for the project?\n", + "\n", + "- How to sync repository with files from laptop like hw without using Github desktop \n", + "\n", + "- Are there any upcoming deadlines for the project?\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Splines (Brief introduction before next section)\n", + "\n", + "Following interpolation discussion, instead of estimating 9 data points with an eighth-order polynomial, it makes more sense to fit sections of the curve to lower-order polynomials:\n", + "\n", + "0. zeroth-order (nearest neighbor)\n", + "\n", + "1. first-order (linear interpolation)\n", + "\n", + "2. third-order (cubic interpolation)\n", + "\n", + "Matlab and Octave have built-in functions for 1D and 2D interpolation:\n", + "\n", + "`interp1`\n", + "\n", + "`interp2`" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "'interp1' is a function from the file /usr/share/octave/4.0.0/m/general/interp1.m\n", + "\n", + " -- Function File: YI = interp1 (X, Y, XI)\n", + " -- Function File: YI = interp1 (Y, XI)\n", + " -- Function File: YI = interp1 (..., METHOD)\n", + " -- Function File: YI = interp1 (..., EXTRAP)\n", + " -- Function File: YI = interp1 (..., \"left\")\n", + " -- Function File: YI = interp1 (..., \"right\")\n", + " -- Function File: PP = interp1 (..., \"pp\")\n", + "\n", + " One-dimensional interpolation.\n", + "\n", + " Interpolate input data to determine the value of YI at the points\n", + " XI. If not specified, X is taken to be the indices of Y ('1:length\n", + " (Y)'). If Y is a matrix or an N-dimensional array, the\n", + " interpolation is performed on each column of Y.\n", + "\n", + " The interpolation METHOD is one of:\n", + "\n", + " \"nearest\"\n", + " Return the nearest neighbor.\n", + "\n", + " \"previous\"\n", + " Return the previous neighbor.\n", + "\n", + " \"next\"\n", + " Return the next neighbor.\n", + "\n", + " \"linear\" (default)\n", + " Linear interpolation from nearest neighbors.\n", + "\n", + " \"pchip\"\n", + " Piecewise cubic Hermite interpolating\n", + " polynomial--shape-preserving interpolation with smooth first\n", + " derivative.\n", + "\n", + " \"cubic\"\n", + " Cubic interpolation (same as \"pchip\").\n", + "\n", + " \"spline\"\n", + " Cubic spline interpolation--smooth first and second\n", + " derivatives throughout the curve.\n", + "\n", + " Adding '*' to the start of any method above forces 'interp1' to\n", + " assume that X is uniformly spaced, and only 'X(1)' and 'X(2)' are\n", + " referenced. This is usually faster, and is never slower. The\n", + " default method is \"linear\".\n", + "\n", + " If EXTRAP is the string \"extrap\", then extrapolate values beyond\n", + " the endpoints using the current METHOD. If EXTRAP is a number,\n", + " then replace values beyond the endpoints with that number. When\n", + " unspecified, EXTRAP defaults to 'NA'.\n", + "\n", + " If the string argument \"pp\" is specified, then XI should not be\n", + " supplied and 'interp1' returns a piecewise polynomial object. This\n", + " object can later be used with 'ppval' to evaluate the\n", + " interpolation. There is an equivalence, such that 'ppval (interp1\n", + " (X, Y, METHOD, \"pp\"), XI) == interp1 (X, Y, XI, METHOD, \"extrap\")'.\n", + "\n", + " Duplicate points in X specify a discontinuous interpolant. There\n", + " may be at most 2 consecutive points with the same value. If X is\n", + " increasing, the default discontinuous interpolant is\n", + " right-continuous. If X is decreasing, the default discontinuous\n", + " interpolant is left-continuous. The continuity condition of the\n", + " interpolant may be specified by using the options \"left\" or \"right\"\n", + " to select a left-continuous or right-continuous interpolant,\n", + " respectively. Discontinuous interpolation is only allowed for\n", + " \"nearest\" and \"linear\" methods; in all other cases, the X-values\n", + " must be unique.\n", + "\n", + " An example of the use of 'interp1' is\n", + "\n", + " xf = [0:0.05:10];\n", + " yf = sin (2*pi*xf/5);\n", + " xp = [0:10];\n", + " yp = sin (2*pi*xp/5);\n", + " lin = interp1 (xp, yp, xf);\n", + " near = interp1 (xp, yp, xf, \"nearest\");\n", + " pch = interp1 (xp, yp, xf, \"pchip\");\n", + " spl = interp1 (xp, yp, xf, \"spline\");\n", + " plot (xf,yf,\"r\", xf,near,\"g\", xf,lin,\"b\", xf,pch,\"c\", xf,spl,\"m\",\n", + " xp,yp,\"r*\");\n", + " legend (\"original\", \"nearest\", \"linear\", \"pchip\", \"spline\");\n", + "\n", + " See also: pchip, spline, interpft, interp2, interp3, interpn.\n", + "\n", + "Additional help for built-in functions and operators is\n", + "available in the online version of the manual. Use the command\n", + "'doc ' to search the manual index.\n", + "\n", + "Help and information about Octave is also available on the WWW\n", + "at http://www.octave.org and via the help@octave.org\n", + "mailing list.\n" + ] + } + ], + "source": [ + "help interp1" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "'interp2' is a function from the file /usr/share/octave/4.0.0/m/general/interp2.m\n", + "\n", + " -- Function File: ZI = interp2 (X, Y, Z, XI, YI)\n", + " -- Function File: ZI = interp2 (Z, XI, YI)\n", + " -- Function File: ZI = interp2 (Z, N)\n", + " -- Function File: ZI = interp2 (Z)\n", + " -- Function File: ZI = interp2 (..., METHOD)\n", + " -- Function File: ZI = interp2 (..., METHOD, EXTRAP)\n", + "\n", + " Two-dimensional interpolation.\n", + "\n", + " Interpolate reference data X, Y, Z to determine ZI at the\n", + " coordinates XI, YI. The reference data X, Y can be matrices, as\n", + " returned by 'meshgrid', in which case the sizes of X, Y, and Z must\n", + " be equal. If X, Y are vectors describing a grid then 'length (X)\n", + " == columns (Z)' and 'length (Y) == rows (Z)'. In either case the\n", + " input data must be strictly monotonic.\n", + "\n", + " If called without X, Y, and just a single reference data matrix Z,\n", + " the 2-D region 'X = 1:columns (Z), Y = 1:rows (Z)' is assumed.\n", + " This saves memory if the grid is regular and the distance between\n", + " points is not important.\n", + "\n", + " If called with a single reference data matrix Z and a refinement\n", + " value N, then perform interpolation over a grid where each original\n", + " interval has been recursively subdivided N times. This results in\n", + " '2^N-1' additional points for every interval in the original grid.\n", + " If N is omitted a value of 1 is used. As an example, the interval\n", + " [0,1] with 'N==2' results in a refined interval with points at [0,\n", + " 1/4, 1/2, 3/4, 1].\n", + "\n", + " The interpolation METHOD is one of:\n", + "\n", + " \"nearest\"\n", + " Return the nearest neighbor.\n", + "\n", + " \"linear\" (default)\n", + " Linear interpolation from nearest neighbors.\n", + "\n", + " \"pchip\"\n", + " Piecewise cubic Hermite interpolating\n", + " polynomial--shape-preserving interpolation with smooth first\n", + " derivative.\n", + "\n", + " \"cubic\"\n", + " Cubic interpolation (same as \"pchip\").\n", + "\n", + " \"spline\"\n", + " Cubic spline interpolation--smooth first and second\n", + " derivatives throughout the curve.\n", + "\n", + " EXTRAP is a scalar number. It replaces values beyond the endpoints\n", + " with EXTRAP. Note that if EXTRAPVAL is used, METHOD must be\n", + " specified as well. If EXTRAP is omitted and the METHOD is\n", + " \"spline\", then the extrapolated values of the \"spline\" are used.\n", + " Otherwise the default EXTRAP value for any other METHOD is \"NA\".\n", + "\n", + " See also: interp1, interp3, interpn, meshgrid.\n", + "\n", + "Additional help for built-in functions and operators is\n", + "available in the online version of the manual. Use the command\n", + "'doc ' to search the manual index.\n", + "\n", + "Help and information about Octave is also available on the WWW\n", + "at http://www.octave.org and via the help@octave.org\n", + "mailing list.\n" + ] + } + ], + "source": [ + "help interp2" + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "Gnuplot\n", + "Produced by GNUPLOT 5.0 patchlevel 3 \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\t\n", + "\t\t-1.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-1\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-0.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t1\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t1.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-3\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-2\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-1\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t1\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t2\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t3\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\tdata\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\tdata\n", + "\t\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\n", + "\t\n", + "\tlinear\n", + "\n", + "\t\n", + "\t\tlinear\n", + "\t\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\tcubic spline\n", + "\n", + "\t\n", + "\t\tcubic spline\n", + "\t\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\tpiecewise cubic\n", + "\n", + "\t\n", + "\t\tpiecewise cubic\n", + "\t\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "x=linspace(-pi,pi,9);\n", + "xi=linspace(-pi,pi,100);\n", + "y=sin(x);\n", + "yi_lin=interp1(x,y,xi,'linear');\n", + "yi_spline=interp1(x,y,xi,'spline'); \n", + "yi_cubic=interp1(x,y,xi,'cubic');\n", + "plot(x,y,'o',xi,yi_lin,xi,yi_spline,xi,yi_cubic)\n", + "axis([-pi,pi,-1.5,1.5])\n", + "legend('data','linear','cubic spline','piecewise cubic','Location','NorthWest')\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Example: Accelerate then hold velocity\n", + "\n", + "Here the time is given as vector t in seconds and the velocity is in mph. " + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "Gnuplot\n", + "Produced by GNUPLOT 5.0 patchlevel 3 \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t20\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t40\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t60\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t80\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t100\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t120\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t140\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t20\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t40\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t60\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t80\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t100\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t120\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\t\n", + "\t\tv (mph)\n", + "\t\n", + "\n", + "\n", + "\t\n", + "\t\tt (s)\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\tdata\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\tdata\n", + "\t\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\n", + "\t\n", + "\tlinear\n", + "\n", + "\t\n", + "\t\tlinear\n", + "\t\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\tcubic spline\n", + "\n", + "\t\n", + "\t\tcubic spline\n", + "\t\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\tpiecewise cubic\n", + "\n", + "\t\n", + "\t\tpiecewise cubic\n", + "\t\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "t=[0 20 40 56 68 80 84 96 104 110]';\n", + "v=[0 20 20 38 80 80 100 100 125 125]';\n", + "tt=linspace(0,110)';\n", + "v_lin=interp1(t,v,tt);\n", + "v_spl=interp1(t,v,tt,'spline');\n", + "v_cub=interp1(t,v,tt,'cubic');\n", + "\n", + "plot(t,v,'o',tt,v_lin,tt,v_spl,tt,v_cub)\n", + "xlabel('t (s)')\n", + "ylabel('v (mph)')\n", + "legend('data','linear','cubic spline','piecewise cubic','Location','NorthWest')" + ] + }, + { + "cell_type": "code", + "execution_count": 14, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "Gnuplot\n", + "Produced by GNUPLOT 5.0 patchlevel 3 \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\t\n", + "\t\t-4\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-2\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t2\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t4\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t6\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t8\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t20\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t40\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t60\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t80\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t100\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t120\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\t\n", + "\t\tdv/dt (mph/s)\n", + "\t\n", + "\n", + "\n", + "\t\n", + "\t\tt (s)\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\tlinear\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\tlinear\n", + "\t\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\tcubic spline\n", + "\n", + "\t\n", + "\t\tcubic spline\n", + "\t\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\tpiecewise cubic\n", + "\n", + "\t\n", + "\t\tpiecewise cubic\n", + "\t\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "t=[0 20 40 56 68 80 84 96 104 110]';\n", + "v=[0 20 20 38 80 80 100 100 125 125]';\n", + "tt=linspace(0,110)';\n", + "v_lin=interp1(t,v,tt);\n", + "v_spl=interp1(t,v,tt,'spline');\n", + "v_cub=interp1(t,v,tt,'cubic');\n", + "\n", + "\n", + "plot(tt(2:end),diff(v_lin)./diff(tt),tt(2:end),diff(v_spl)./diff(tt),tt(2:end),diff(v_cub)./diff(tt))\n", + "xlabel('t (s)')\n", + "ylabel('dv/dt (mph/s)')\n", + "legend('linear','cubic spline','piecewise cubic','Location','NorthWest')" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Choose spline wisely\n", + "\n", + "For example of sin(x), not very important\n", + "\n", + "For stop-and-hold examples, the $C^{2}$-continuity should not be preserved. You don't need smooth curves.\n", + "\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Numerical Integration\n", + "\n", + "A definite integral is defined by \n", + "\n", + "$I=\\int_{a}^{b}f(x)dx$\n", + "\n", + "To determine the mass of an object with varying density, you can perform a summation\n", + "\n", + "mass=$\\sum_{i=1}^{n}\\rho_{i}\\Delta V_{i}$\n", + "\n", + "or taking the limit as $\\Delta V \\rightarrow dV=dxdydz$\n", + "\n", + "mass=$\\int_{0}^{h}\\int_{0}^{w}\\int_{0}^{l}\\rho(x,y,z)dxdydz$\n", + "\n", + "## Newton-Cotes Formulas\n", + "\n", + "$I=\\int_{a}^{b}f(x)dx=\\int_{a}^{b}f_{n}(x)dx$\n", + "\n", + "where $f_{n}$ is an n$^{th}$-order polynomial approximation of f(x)\n", + "\n", + "## First-Order: Trapezoidal Rule\n", + "\n", + "$I=\\int_{a}^{b}f(x)dx\\approx \\int_{a}^{b}\\left(f(a)+\\frac{f(b)-f(a)}{b-a}(x-a)\\right)dx$\n", + "\n", + "$I\\approx(b-a)\\frac{f(a)+f(b)}{2}$" + ] + }, + { + "cell_type": "code", + "execution_count": 20, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "I_trap = 0.78540\n", + "I_act = 1.00000\n" + ] + }, + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "Gnuplot\n", + "Produced by GNUPLOT 5.0 patchlevel 3 \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.2\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.4\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.6\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.8\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t1\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t1\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t1.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t2\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t2.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t3\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t3.5\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\n", + "\tgnuplot_plot_1a\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\tgnuplot_plot_2a\n", + "\n", + "\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "x=linspace(0,pi)';\n", + "plot(x,sin(x),[0,pi/2],sin([0,pi/2]))\n", + "I_trap=mean(sin(([0,pi/2]))*(diff([0,pi/2])))\n", + "I_act = -(cos(pi/2)-cos(0))\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Improve estimate with more points\n", + "\n", + "$I=\\int_{a}^{b}f(x)dx=\\int_{a}^{a+\\Delta x}f(x)dx+\\int_{a+\\Delta x}^{a+2\\Delta x}f(x)dx+ \\cdots \\int_{b-\\Delta x}^{b}f(x)dx$\n", + "\n", + "$I\\approx\\Delta x\\frac{f(a)+f(a+\\Delta x)}{2}+\\Delta x\\frac{f(a+\\Delta x)+f(a+2\\Delta x)}{2}\n", + "+\\cdots \\Delta x\\frac{f(b-\\Delta x)+f(b)}{2}$\n", + "\n", + "$I\\approx \\frac{\\Delta x}{2}\\left(f(a)+2\\sum_{i=1}^{n-1}f(a+i\\Delta x) +f(b)\\right)$" + ] + }, + { + "cell_type": "code", + "execution_count": 48, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "For 5 steps\n", + "trapezoid approximation of integral is 0.79 \n", + " actual integral is 1.00\n" + ] + } + ], + "source": [ + "N=5;\n", + "I_trap=trap(@(x) sin(x),0,pi/2,N);\n", + "fprintf('For %i steps\\ntrapezoid approximation of integral is %1.2f \\n actual integral is %1.2f',N,I_trap,I_act)" + ] + }, + { + "cell_type": "code", + "execution_count": 43, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "Gnuplot\n", + "Produced by GNUPLOT 5.0 patchlevel 3 \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.2\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.4\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.6\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.8\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t1\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t1\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t1.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t2\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t2.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t3\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t3.5\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\n", + "\tgnuplot_plot_1a\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\tgnuplot_plot_2a\n", + "\n", + "\t \n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "\n", + "plot(x,sin(x),linspace(0,pi/2,N),sin(linspace(0,pi/2,N)),'o')" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Increase accuracy = Increase polynomial order\n", + "\n", + "### Simpson's Rules\n", + "\n", + "When integrating f(x) and using a second order polynomial, this is known as **Simpson's 1/3 Rule**\n", + "\n", + "$I=\\frac{h}{3}(f(x_{0})+4f(x_{1})+f(x_{2}))$\n", + "\n", + "where a=$x_{0}$, b=$x_{2}$, and $x_{1}=\\frac{a+b}{2}$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "This can be used with n=3 or multiples of 2 intervals\n", + "\n", + "$I=\\int_{x_{0}}^{x_{2}}f(x)dx+\\int_{x_{2}}^{x_{4}}f(x)dx+\\cdots +\\int_{x_{n-2}}^{x_{n}}f(x)dx$\n", + "\n", + "$I=(b-a)\\frac{f(x_{0})+4\\sum_{i=1,3,5}^{n-1}f(x_{i})+2\\sum_{i=2,4,6}^{n-2}f(x_{i})+f(x_{n})}{3n}$" + ] + }, + { + "cell_type": "code", + "execution_count": 68, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "ans = 1.6235\n", + "Is_1_3 = 1.0023\n" + ] + } + ], + "source": [ + "f=@(x) 0.2+25*x-200*x.^2+675*x.^3-900*x.^4+400*x.^5;\n", + "simpson3(f,0,0.8,4)\n", + "Is_1_3=simpson3(@(x) sin(x),0,pi/2,2)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## General Newton-Cotes formulae\n", + "\n", + "![Newton-Cotes Table](newton_cotes.png)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Octave", + "language": "octave", + "name": "octave" + }, + "language_info": { + "file_extension": ".m", + "help_links": [ + { + "text": "MetaKernel Magics", + "url": "https://github.com/calysto/metakernel/blob/master/metakernel/magics/README.md" + } + ], + "mimetype": "text/x-octave", + "name": "octave", + "version": "0.19.14" + } + }, + "nbformat": 4, + "nbformat_minor": 2 +} diff --git a/lecture_19/newton_cotes.png b/lecture_19/newton_cotes.png new file mode 100644 index 0000000000000000000000000000000000000000..b734d114512b221d5e1cc484da08e59afce27fb1 GIT binary patch literal 78112 zcma&Oby!tjyEVK~q)Q~EC6w-NC8a@9Is~Lcx*KU}De3O+E&=Hd>F(}$=l(tCJm;ME 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zX`jq^H?-VDZVn?tnffW9&mA_{e_Nj=HEEj+z4#0|BOUKil%Q929TF$sET^RyIu`?F zYjmf+OxB}@rLh7gFP;n949Ekl6nU%^6;@wQ`zNwx$9O)H;ebvNx2qlj4ABRlj#5;- zVDr|mg>XO1P@6DK)1JX_70|uAQ9~*I3bdW22kM_7F>pUD6;RK3Oj`Q=Q}r5G%(d_#P+LN>7&Y;;U~7vQ!`B8z)Za{1K-SS6=>0u5v-AKq?|K$<%LltJ zDSg`@$)P)9fr>As@;PuYW)|V?+H0as&!oV!sGcP)|6UJ>RpeR3l*RFlGa6Z&kOjLXvh-GO@>$a),error('upper bound must be greater than lower'),end +if nargin<4|isempty(n),n=100;end +x = a; h = (b - a)/n; + +xvals=linspace(a,b,n+1); +fvals=func(xvals,varargin{:}); +s=fvals(1); +s = s + 4*sum(fvals(2:2:end-1)); +s = s + 2*sum(fvals(3:2:end-2)); +s = s + fvals(end); +I = (b - a) * s/(3*n); diff --git a/lecture_19/trap.m b/lecture_19/trap.m new file mode 100644 index 0000000..85b8685 --- /dev/null +++ b/lecture_19/trap.m @@ -0,0 +1,22 @@ +function I = trap(func,a,b,n,varargin) +% trap: composite trapezoidal rule quadrature +% I = trap(func,a,b,n,pl,p2,...): +% composite trapezoidal rule +% input: +% func = name of function to be integrated +% a, b = integration limits +% n = number of segments (default = 100) +% pl,p2,... = additional parameters used by func +% output: +% I = integral estimate +if nargin<3,error('at least 3 input arguments required'),end +if ~(b>a),error('upper bound must be greater than lower'),end +if nargin<4|isempty(n),n=100;end + +x = a; h = (b - a)/n; +xvals=linspace(a,b,n); +fvals=func(xvals,varargin{:}); +s=func(a,varargin{:}); +s = s + 2*sum(fvals(2:n-1)); +s = s + func(b,varargin{:}); +I = (b - a) * s/(2*n); From 598519d94dca50d2e7d576009741145143fed347 Mon Sep 17 00:00:00 2001 From: "Ryan C. Cooper" Date: Tue, 4 Apr 2017 12:56:30 -0400 Subject: [PATCH 02/10] added hw6 --- HW6/.~lock.Primary_Energy_monthly.csv# | 1 + HW6/README.md | 86 +++++ HW6/README.pdf | Bin 0 -> 168643 bytes HW6/boussinesq_lookup.m | 22 ++ HW6/cost_logistic.m | 27 ++ HW6/problem_1_data.m | 15 + lecture_18/lecture_18.ipynb | 90 +++--- lecture_18/octave-workspace | Bin 7470 -> 2001 bytes lecture_19/lecture 19.ipynb | 426 ++++++++++++++++++++++--- lecture_19/octave-workspace | Bin 0 -> 10815 bytes 10 files changed, 588 insertions(+), 79 deletions(-) create mode 100644 HW6/.~lock.Primary_Energy_monthly.csv# create mode 100644 HW6/README.md create mode 100644 HW6/README.pdf create mode 100644 HW6/boussinesq_lookup.m create mode 100644 HW6/cost_logistic.m create mode 100644 HW6/problem_1_data.m create mode 100644 lecture_19/octave-workspace diff --git a/HW6/.~lock.Primary_Energy_monthly.csv# b/HW6/.~lock.Primary_Energy_monthly.csv# new file mode 100644 index 0000000..eef216f --- /dev/null +++ b/HW6/.~lock.Primary_Energy_monthly.csv# @@ -0,0 +1 @@ +,ryan,fermi,31.03.2017 16:47,file:///home/ryan/.config/libreoffice/4; \ No newline at end of file diff --git a/HW6/README.md b/HW6/README.md new file mode 100644 index 0000000..1f3c86c --- /dev/null +++ b/HW6/README.md @@ -0,0 +1,86 @@ +# Homework #6 +## due 4/14 by 11:59pm + + +0. Create a new github repository called 'curve_fitting'. + + a. Add rcc02007 and pez16103 as collaborators. + + b. Clone the repository to your computer. + + +1. Create a least-squares function called `least_squares.m` that accepts a Z-matrix and +dependent variable y as input and returns the vector of best-fit constants, a, the +best-fit function evaluated at each point $f(x_{i})$, and the coefficient of +determination, r2. + +```matlab +[a,fx,r2]=least_squares(Z,y); +``` + + Test your function on the sets of data in script `problem_1_data.m` and show that the + following functions are the best fit lines: + + a. y=0.3745+0.98644x+0.84564/x + + b. y=-11.4887+7.143817x-1.04121 x^2+0.046676 x^3 + + c. y=4.0046e^(-1.5x)+2.9213e^(-0.3x)+1.5647e^(-0.05x) + + +2. Use the Temperature and failure data from the Challenger O-rings in lecture_18 +(challenger_oring.csv). Your independent variable is temerature and your dependent +variable is failure (1=fail, 0=pass). Create a function called `cost_logistic.m` that +takes the vector `a`, and independent variable `x` and dependent variable `y`. Use the +function, $\sigma(t)=\frac{1}{1+e^{-t}}$ where $t=a_{0}+a_{1}x$. Use the cost function, + + $J(a_{0},a_{1})=\sum_{i=1}^{n}\left[-y_{i}\log(\sigma(t_{i}))-(1-y_{i})\log((1-\sigma(t_{i})))\right]$ + + and gradient + + $\frac{\partial J}{\partial a_{i}}= + 1/m\sum_{k=1}^{N}\left(\sigma(t_{k})-y_{k}\right)t_{k}$ + + a. edit `cost_logistic.m` so that the output is `[J,grad]` or [cost, gradient] + + b. use the following code to solve for a0 and a1 + +```matlab +% Set options for fminunc +options = optimset('GradObj', 'on', 'MaxIter', 400); +% Run fminunc to obtain the optimal theta +% This function will return theta and the cost +[theta, cost] = ... +fminunc(@(a)(costFunction(a, x, y)), initial_a, options); +``` + + c. plot the data and the best-fit logistic regression model + +```matlab +plot(x,y, x, sigma(a(1)+a(2)*x)) +``` + +3. The vertical stress under a corner of a rectangular area subjected to a uniform load of +intensity $q$ is given by the solution of the Boussinesq's equation: + + $\sigma_{z} = + \frac{q}{4\pi}\left(\frac{2mn\sqrt{m^{2}+n^{2}+1}}{m^{2}+n^{2}+1+m^{2}n^{2}}\frac{m^{2}+n^{2}+2}{m^{2}+n^{2}+1}+sin^{-1}\left(\frac{2mn\sqrt{m^{2}+n^{2}+1}}{m^{2}+n^{2}+1+m^{2}n^{2}}\right)\right)$ + + Typically, this equation is solved as a table of values where: + + $\sigma_{z}=q f(m,n)$ + + where $f(m,n)$ is the influence value, q is the uniform load, m=a/z, n=b/z, a and b are + width and length of the rectangular area and z is the depth below the area. + + a. Finish the function `boussinesq_lookup.m` so that when you enter a force, q, + dimensions of rectangular area a, b, and depth, z, it uses a third-order polynomial + interpolation of the four closest values of m to determine the stress in the vertical + direction, sigma_z=$\sigma_{z}$. Use a $0^{th}$-order, polynomial interpolation for + the value of n (i.e. round to the closest value of n). + + b. Copy the `boussinesq_lookup.m` to a file called `boussinesq_spline.m` and use a + cubic spline to interpolate in two dimensions, both m and n, that returns sigma_z. + + + diff --git a/HW6/README.pdf b/HW6/README.pdf new file mode 100644 index 0000000000000000000000000000000000000000..3cc89911b274f1e5fcca0d59039ce6695c387b84 GIT binary patch literal 168643 zcma&NL$EMh(4={7+qP}nwr$(CZQI6sZQHhOYret1>WSz+tE|;oRaHEhQKSkYVzi8O ztWc!$%OmShjLZZK1olQ&P&_Vgh{SNIo>L(cp`fDake746G!!PS+wu(3v8@%MB1JyKoIk;^C< zqT=J`$GYTGhDoY&zVTt@#`ssZdHhBmG*kU~Ga+65c#6D@Gdml=R~CslxR5S-@kJe^?fL%VB*|I2nV6pHx1pHY zXC9?^5WnbgS@puz<)X5Nsjwr{Ktmykgs9|%dA8h1#aq0Y?E^2qIl1r_QKG)pS#09m zW&7k6g|9)F0t0D~ESr*nQYT<;CWa{dBBLm00rlAwc#{j`nNT@cG5&!wYcNi$Z0AwT6Ki@Ad(ysiI5EXH5yEc{pMBRNvsqq5h+L3(#EcNMX5{o zwAJSLY-V>eHbQ27bvQs@6yWH2vXPTs-7Qk*g>WZ_w62Q2al$38<#iY+EmhV$tT32B zH*8Zn?kYC&6ZbX;O<}AWi zj`w>ZVq0Wib8*n5)N| zv-DFrb=aK-YUx z@-XHD^+2_DsQA?7A1c@E_z8khvVQn5Qp$k`A(rOHclqE)gWQON!OE{uZs&-xtxbMM 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The calculated solutions are in the fmn data + % m=fmn(:,1) + % in column 2, fmn(:,2), n=1.2 + % in column 3, fmn(:,2), n=1.4 + % in column 4, fmn(:,2), n=1.6 + + fmn= [0.1,0.02926,0.03007,0.03058 + 0.2,0.05733,0.05894,0.05994 + 0.3,0.08323,0.08561,0.08709 + 0.4,0.10631,0.10941,0.11135 + 0.5,0.12626,0.13003,0.13241 + 0.6,0.14309,0.14749,0.15027 + 0.7,0.15703,0.16199,0.16515 + 0.8,0.16843,0.17389,0.17739]; + + m=a/z; + n=b/z; + + %... +end diff --git a/HW6/cost_logistic.m b/HW6/cost_logistic.m new file mode 100644 index 0000000..169718f --- /dev/null +++ b/HW6/cost_logistic.m @@ -0,0 +1,27 @@ +function [J, grad] = cost_logistic(a, x, y) +% cost_logistic Compute cost and gradient for logistic regression +% J = cost_logistic(theta, X, y) computes the cost of using theta as the +% parameter for logistic regression and the gradient of the cost +% w.r.t. to the parameters. + +% Initialize some useful values +N = length(y); % number of training examples + +% You need to return the following variables correctly +J = 0; +grad = zeros(size(theta)); + +% ====================== YOUR CODE HERE ====================== +% Instructions: Compute the cost of a particular choice of a. +% Compute the partial derivatives and set grad to the partial +% derivatives of the cost w.r.t. each parameter in theta +% +% Note: grad should have the same dimensions as theta +% + + + +% ============================================================= + +end + diff --git a/HW6/problem_1_data.m b/HW6/problem_1_data.m new file mode 100644 index 0000000..6af2956 --- /dev/null +++ b/HW6/problem_1_data.m @@ -0,0 +1,15 @@ + +% part a +xa=[1 2 3 4 5]'; +yb=[2.2 2.8 3.6 4.5 5.5]'; + +% part b + +xb=[3 4 5 7 8 9 11 12]'; +yb=[1.6 3.6 4.4 3.4 2.2 2.8 3.8 4.6]'; + +% part c + +xc=[0.5 1 2 3 4 5 6 7 9]; +yc=[6 4.4 3.2 2.7 2.2 1.9 1.7 1.4 1.1]; + diff --git a/lecture_18/lecture_18.ipynb b/lecture_18/lecture_18.ipynb index bf7731c..f1fd1b3 100644 --- a/lecture_18/lecture_18.ipynb +++ b/lecture_18/lecture_18.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "code", - "execution_count": 146, + "execution_count": 2, "metadata": { "collapsed": true }, @@ -13,7 +13,7 @@ }, { "cell_type": "code", - "execution_count": 107, + "execution_count": 3, "metadata": { "collapsed": true }, @@ -1959,7 +1959,7 @@ }, { "cell_type": "code", - "execution_count": 162, + "execution_count": 8, "metadata": { "collapsed": false }, @@ -1969,7 +1969,7 @@ "output_type": "stream", "text": [ "ln(2)=0.693147\n", - "ln(2)~0.366349\n" + "ln(2)~0.693147\n" ] }, { @@ -2014,113 +2014,117 @@ "\n", "\n", "\t\n", - "\t\t\n", + "\t\t\n", "\t\n", "\n", "\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t-2\n", "\t\n", "\n", "\n", - "\t\t\n", - "\t\t0\n", + "\t\t\n", + "\t\t-1.5\n", "\t\n", "\n", "\n", - "\t\t\n", - "\t\t2\n", + "\t\t\n", + "\t\t-1\n", "\t\n", "\n", "\n", - "\t\t\n", - "\t\t4\n", + "\t\t\n", + "\t\t-0.5\n", "\t\n", "\n", "\n", - "\t\t\n", - "\t\t6\n", + "\t\t\n", + "\t\t0\n", "\t\n", "\n", "\n", - "\t\t\n", - "\t\t8\n", + "\t\t\n", + "\t\t0.5\n", "\t\n", "\n", "\n", - "\t\t\n", - "\t\t10\n", + "\t\t\n", + "\t\t1\n", "\t\n", "\n", "\n", - "\t\t\n", - "\t\t0\n", + "\t\t\n", + "\t\t1.5\n", "\t\n", "\n", "\n", - "\t\t\n", - "\t\t10\n", + "\t\t\n", + "\t\t2\n", "\t\n", "\n", "\n", - "\t\t\n", - "\t\t20\n", + "\t\t\n", + "\t\t0\n", "\t\n", "\n", "\n", - "\t\t\n", - "\t\t30\n", + "\t\t\n", + "\t\t1\n", "\t\n", "\n", "\n", - "\t\t\n", - "\t\t40\n", + "\t\t\n", + "\t\t2\n", "\t\n", "\n", "\n", - "\t\t\n", - "\t\t50\n", + "\t\t\n", + "\t\t3\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t4\n", "\t\n", "\n", "\n", "\t\t\n", - "\t\t60\n", + "\t\t5\n", "\t\n", "\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\n", "\n", "\tgnuplot_plot_1a\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_2a\n", "\n", "\t \n", - "\t\n", + "\t\n", "\n", "\t\n", "\tgnuplot_plot_3a\n", "\n", "\t \n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", "\n", "\t\n", "\tgnuplot_plot_4a\n", "\n", - "\t\n", + "\t\n", "\t\n", "\n", "\n", @@ -2144,9 +2148,9 @@ "source": [ "\n", "\n", - "x=[0.2,3,10,20,50,60]; % define independent var's\n", + "x=[0.2,1,2,3,4]; % define independent var's\n", "y=log(x); % define dependent var's\n", - "xx=linspace(min(x),max(x));\n", + "xx=linspace(min(x),max(x)+1);\n", "yy=zeros(size(xx));\n", "for i=1:length(xx)\n", " yy(i)=Newtint(x,y,xx(i));\n", diff --git a/lecture_18/octave-workspace b/lecture_18/octave-workspace index c65169697346d5d351f3f9767bf0ba8da01d8669..d40f9acca1ca8de96bef3f1163fd0113738bc80b 100644 GIT binary patch literal 2001 zcmb7DdpMM76dz`Cn=r^DMcHjtDpGp1UC+yQu~AYzr8X6ktT7peR6OajC0#bDNTTh& zl?UmzwS2k=u|+Bok%rM2ncFumVfKB$sr_sBk3G-xzTf4%@A;i`e!t@zEDDSbah~Gr z!(y{otUy6D-dQXy2XDb4eEy(TtJTBr`gob)HJki&4&jGHga|})@eh2~7;n+S_0giJ zFhS_o3naiMz9{^ET{6U%LIp9A{4F8^8ip5+Z-BSx;6Q$06!BwQjfgjnd1t&4ffzgP zEA$ZTFw8hbtQ?p_Ecurw3+cX%zC3w|6c+cNp*CXgKcSvyxAw1o_8jzUhPxits{y;$ 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"\t\t3000\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t4000\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t5000\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t6000\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t10\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t15\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t20\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\n", + "\tgnuplot_plot_1a\n", + "\n", + "\n", + "\n", + "\t \n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\n", + "\t\n", + "\tgnuplot_plot_2a\n", + "\n", + "\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "% for linear regression\n", + "x=linspace(0,20)';\n", + "y=x.^2 +10*exp(-2*x)+0.5*sinh(x/2)+rand(size(x))*200-100;\n", + "Z=[x.^2 exp(-2*x) sinh(x/2)];\n", + "a=Z\\y\n", + "\n", + "plot(x,y,'o',x,a(1)*x.^2+a(2)*exp(-2*x)+a(3)*sinh(x/2))" + ] + }, { "cell_type": "markdown", "metadata": {}, @@ -51,6 +328,8 @@ "source": [ "![q4](q4.png)\n", "\n", + "# Answer: It depends\n", + "\n", "#### Other:\n", "\n", "Twice the amount of points needed\n", @@ -101,7 +380,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "# Splines (Brief introduction before next section)\n", + "# Splines (Brief introduction before integrals)\n", "\n", "Following interpolation discussion, instead of estimating 9 data points with an eighth-order polynomial, it makes more sense to fit sections of the curve to lower-order polynomials:\n", "\n", @@ -315,7 +594,7 @@ }, { "cell_type": "code", - "execution_count": 9, + "execution_count": 75, "metadata": { "collapsed": false }, @@ -532,12 +811,12 @@ "source": [ "### Example: Accelerate then hold velocity\n", "\n", - "Here the time is given as vector t in seconds and the velocity is in mph. " + "Test driving a car, the accelerator is pressed, then released, then pressed again for 20-second intervals, until speed is 120 mph. Here the time is given as vector t in seconds and the velocity is in mph. " ] }, { "cell_type": "code", - "execution_count": 10, + "execution_count": 82, "metadata": { "collapsed": false }, @@ -683,12 +962,12 @@ "\n", "\n", "\n", - "\t\n", + "\t\n", "\tdata\n", "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\tdata\n", "\t\n", "\n", @@ -696,7 +975,6 @@ "\t\n", "\t\n", "\t\n", - "\t\n", "\t\n", "\t\n", "\t\n", @@ -706,34 +984,45 @@ "\t\n", "\n", "\t\n", - "\tlinear\n", + "\tremoved data point\n", "\n", - "\t\n", + "\t\n", + "\t\tremoved data point\n", + "\t\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\n", + "\t\n", + "\tlinear\n", + "\n", + "\t\n", "\t\tlinear\n", "\t\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", - "\tcubic spline\n", + "\tcubic spline\n", "\n", - "\t\n", + "\t\n", "\t\tcubic spline\n", "\t\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", - "\tpiecewise cubic\n", + "\tpiecewise cubic\n", "\n", - "\t\n", + "\t\n", "\t\tpiecewise cubic\n", "\t\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", - "\n", + "\n", "\n", "\n", "\n", @@ -753,22 +1042,22 @@ } ], "source": [ - "t=[0 20 40 56 68 80 84 96 104 110]';\n", - "v=[0 20 20 38 80 80 100 100 125 125]';\n", + "t=[0 20 40 68 80 84 96 104 110]';\n", + "v=[0 20 20 80 80 100 100 125 125]';\n", "tt=linspace(0,110)';\n", "v_lin=interp1(t,v,tt);\n", "v_spl=interp1(t,v,tt,'spline');\n", "v_cub=interp1(t,v,tt,'cubic');\n", "\n", - "plot(t,v,'o',tt,v_lin,tt,v_spl,tt,v_cub)\n", + "plot(t,v,'o',56,38,'s',tt,v_lin,tt,v_spl,tt,v_cub)\n", "xlabel('t (s)')\n", "ylabel('v (mph)')\n", - "legend('data','linear','cubic spline','piecewise cubic','Location','NorthWest')" + "legend('data','removed data point','linear','cubic spline','piecewise cubic','Location','NorthWest')" ] }, { "cell_type": "code", - "execution_count": 14, + "execution_count": 83, "metadata": { "collapsed": false }, @@ -1206,7 +1495,7 @@ }, { "cell_type": "code", - "execution_count": 48, + "execution_count": 86, "metadata": { "collapsed": false }, @@ -1215,21 +1504,21 @@ "name": "stdout", "output_type": "stream", "text": [ - "For 5 steps\n", - "trapezoid approximation of integral is 0.79 \n", + "For 70 steps\n", + "trapezoid approximation of integral is 0.99 \n", " actual integral is 1.00\n" ] } ], "source": [ - "N=5;\n", + "N=70;\n", "I_trap=trap(@(x) sin(x),0,pi/2,N);\n", "fprintf('For %i steps\\ntrapezoid approximation of integral is %1.2f \\n actual integral is %1.2f',N,I_trap,I_act)" ] }, { "cell_type": "code", - "execution_count": 43, + "execution_count": 92, "metadata": { "collapsed": false }, @@ -1365,11 +1654,76 @@ "\t\n", "\tgnuplot_plot_2a\n", "\n", - "\t \n", + "\t\t \n", "\t\n", - "\t\n", - "\t\n", - "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", "\t\n", "\n", "\t\n", @@ -1393,8 +1747,8 @@ } ], "source": [ - "\n", - "plot(x,sin(x),linspace(0,pi/2,N),sin(linspace(0,pi/2,N)),'o')" + "x=linspace(0,pi);\n", + "plot(x,sin(x),linspace(0,pi/2,N),sin(linspace(0,pi/2,N)),'-o')" ] }, { @@ -1425,7 +1779,7 @@ }, { "cell_type": "code", - "execution_count": 68, + "execution_count": 93, "metadata": { "collapsed": false }, diff --git a/lecture_19/octave-workspace b/lecture_19/octave-workspace new file mode 100644 index 0000000000000000000000000000000000000000..4bd89dd249beee3bc2d071dccd02246ca43bf138 GIT binary patch literal 10815 zcmd5?2{_f;+TT=)CWS;qDn*gl=DqiN31ukrJY_5m=CL}_s8dpDq>0i&8ih2j-5jMs zsfdouGpW$TxA$Jl>D+VfeeQj}@A;nR?x(E&Yj}s>`}@8B6|DUteWC-@=BrsTMlcu* 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lecture_20/gauss_weights.png create mode 100644 lecture_20/lecture_20.ipynb create mode 100644 lecture_20/q1.png create mode 100644 lecture_20/q2.png create mode 100644 lecture_20/stainless_steel_psi.jpg.dat create mode 100644 lecture_20/steel_psi.jpg create mode 100644 lecture_20/structural_steel_psi.jpg.dat create mode 100644 lecture_20/trap_example.png diff --git a/lecture_20/.ipynb_checkpoints/lecture_20-checkpoint.ipynb b/lecture_20/.ipynb_checkpoints/lecture_20-checkpoint.ipynb new file mode 100644 index 0000000..2fd6442 --- /dev/null +++ b/lecture_20/.ipynb_checkpoints/lecture_20-checkpoint.ipynb @@ -0,0 +1,6 @@ +{ + "cells": [], + "metadata": {}, + "nbformat": 4, + "nbformat_minor": 2 +} diff --git a/lecture_20/gauss_weights.png b/lecture_20/gauss_weights.png new file mode 100644 index 0000000000000000000000000000000000000000..9e3f29d918f9b8bf9eb795796fe53c539855b96c GIT binary patch literal 83452 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{ + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Questions from last class\n", + "\n", + "The interp2 function uses splines to interpolate between data points. What are three options for interpolation:\n", + "\n", + "- cubic spline\n", + "\n", + "- piecewise cubic spline\n", + "\n", + "- linear spline\n", + "\n", + "- quadratic spline\n", + "\n", + "- fourth-order spline\n", + "\n", + "![q1](q1.png)\n", + "\n", + "Numerical integration is a general application of the Newton-Cotes formulas. What is the first order approximation of the Newton-Cotes formula? *\n", + "\n", + "- trapezoidal rule\n", + "\n", + "- Simpson's 1/3 rule\n", + "\n", + "- Simpson's 3/8 rule\n", + "\n", + "- linear approximation of integral\n", + "\n", + "- constant approximation of integral (sum(f(x)*dx))\n", + "\n", + "![q2](q2.png)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Questions from you\n", + "\n", + "- Is spring here to stay?\n", + " \n", + " - [Punsxatawney Phil](http://www.groundhog.org/)\n", + "\n", + "- The time is now.\n", + "\n", + "- Final Project Sheet?\n", + " \n", + " - coming this evening/tomorrow\n", + "\n", + "- can you provide some sort of hw answer key?\n", + " \n", + " - we can go through some of the HW \n", + "\n", + "- What's the most 1337 thing you've ever done?\n", + "\n", + " - sorry, I'm n00b to this\n", + "\n", + "- Can we do out more examples by hand (doc cam or drawing on computer notepad) instead of with pre-written code?\n", + "\n", + " - forthcoming\n", + "\n", + "- Favorite movie?\n", + " \n", + " - Big Lebowski\n", + " " + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Integrals in practice\n", + "\n", + "### Example: Compare toughness of two steels\n", + "\n", + "![Stress-strain plot of steel](steel_psi.jpg)\n", + "\n", + "Use the plot shown to determine the toughness of stainless steel and the toughness of structural steel.\n", + "\n" + ] + }, + { + "cell_type": "code", + "execution_count": 49, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "toughness of structural steel is 10.2 psi\n", + "toughness of stainless steel is 18.6 psi\n" + ] + } + ], + "source": [ + "fe_c=load('structural_steel_psi.jpg.dat');\n", + "fe_cr =load('stainless_steel_psi.jpg.dat');\n", + "\n", + "fe_c_toughness=trapz(fe_c(:,1),fe_c(:,2));\n", + "fe_cr_toughness=trapz(fe_cr(:,1),fe_cr(:,2));\n", + "\n", + "fprintf('toughness of structural steel is %1.1f psi\\n',fe_c_toughness)\n", + "fprintf('toughness of stainless steel is %1.1f psi',fe_cr_toughness)\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Gauss Quadrature (for functions)\n", + "\n", + "Evaluating an integral, we assumed a polynomial form for each Newton-Cotes approximation.\n", + "\n", + "If we can evaluate the function at any point, it makes more sense to choose points more wisely rather than just using endpoints\n", + "\n", + "![trapezoidal example](trap_example.png)\n", + "\n", + "Let us set up two unknown constants, $c_{0}$ and $x_{0}$ and determine a *wise* place to evaluate f(x) such that \n", + "\n", + "$I=c_{0}f(x_{0})$\n", + "\n", + "and I is exact for polynomial of n=0, 1\n", + "\n", + "$\\int_{a}^{b}1dx=b-a=c_{0}$\n", + "\n", + "$\\int_{a}^{b}xdx=\\frac{b^2-a^2}{2}=c_{0}x_{0}$\n", + "\n", + "so $c_{0}=b-a$ and $x_{0}=\\frac{b+a}{2}$\n", + "\n", + "$I=\\int_{a}^{b}f(x)dx \\approx (b-a)f\\left(\\frac{b+a}{2}\\right)$\n", + "\n" + ] + }, + { + "cell_type": "code", + "execution_count": 147, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "f1 =\n", + "\n", + "@(x) x + 1\n", + "\n", + "f2 =\n", + "\n", + "@(x) 1 / 2 * x .^ 2 + x + 1\n", + "\n", + "f3 =\n", + "\n", + "@(x) 1 / 6 * x .^ 3 + 1 / 2 * x .^ 2 + x\n", + "\n" + ] + }, + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "Gnuplot\n", + "Produced by GNUPLOT 5.0 patchlevel 3 \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\t\n", + "\t\t-1000\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-500\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t500\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t1000\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t1500\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-20\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-15\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-10\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t10\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t15\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t20\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\n", + "\tgnuplot_plot_1a\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "f1=@(x) x+1\n", + "f2=@(x) 1/2*x.^2+x+1\n", + "f3=@(x) 1/6*x.^3+1/2*x.^2+x\n", + "plot(linspace(-18,18),f3(linspace(-18,18)))" + ] + }, + { + "cell_type": "code", + "execution_count": 148, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "integral of f1 from 2 to 3 = 3.500000\n", + "integral of f1 from 2 to 3 ~ 3.500000\n", + "integral of f2 from 2 to 3 = 6.666667\n", + "integral of f2 from 2 to 3 ~ 6.625000\n" + ] + } + ], + "source": [ + "fprintf('integral of f1 from 2 to 3 = %f',f2(3)-f2(2))\n", + "fprintf('integral of f1 from 2 to 3 ~ %f',(3-2)*f1(3/2+2/2))\n", + "\n", + "fprintf('integral of f2 from 2 to 3 = %f',f3(3)-f3(2))\n", + "fprintf('integral of f2 from 2 to 3 ~ %f',(3-2)*f2(3/2+2/2))\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "This process is called **Gauss Quadrature**. Usually, the bounds are fixed at -1 and 1 instead of a and b\n", + "\n", + "$I=c_{0}f(x_{0})$\n", + "\n", + "and I is exact for polynomial of n=0, 1\n", + "\n", + "$\\int_{-1}^{1}1dx=b-a=c_{0}$\n", + "\n", + "$\\int_{-1}^{1}xdx=\\frac{1^2-(-1)^2}{2}=c_{0}x_{0}$\n", + "\n", + "so $c_{0}=2$ and $x_{0}=0$\n", + "\n", + "$I=\\int_{-1}^{1}f(x)dx \\approx 2f\\left(0\\right)$\n", + "\n", + "Now, integrals can be performed with a change of variable\n", + "\n", + "a=2\n", + "\n", + "b=3\n", + "\n", + "x= 2 to 3\n", + "\n", + "or $x_{d}=$ -1 to 1\n", + "\n", + "$x=a_{1}+a_{2}x_{d}$\n", + "\n", + "at $x_{d}=-1$, x=a\n", + "\n", + "at $x_{d}=1$, x=b\n", + "\n", + "so \n", + "\n", + "$x=\\frac{(b+a) +(b-a)x_{d}}{2}$\n", + "\n", + "$dx=\\frac{b-a}{2}dx_{d}$\n", + "\n", + "$\\int_{2}^{3}x+1dx=\\int_{-1}^{1}\\left(\\frac{(2+3) +(3-2)x_{d}}{2}\n", + "+1\\right)\n", + "\\frac{3-2}{2}dx_{d}$\n", + "\n", + "$\\int_{2}^{3}x+1dx=\\int_{-1}^{1}\\left(\\frac{5 +x_{d}}{2}\n", + "+1\\right)\n", + "\\frac{3-2}{2}dx_{d}$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$\\int_{2}^{3}x+1dx=\\int_{-1}^{1}\\left(\\frac{7}{4}+\\frac{1}{4}x_{d}\\right)dx_{d}=2\\frac{7}{4}=3.5$" + ] + }, + { + "cell_type": "code", + "execution_count": 15, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "function I=gauss_1pt(func,a,b)\n", + " % Gauss quadrature using single point\n", + " % exact for n<1 polynomials\n", + " c0=2;\n", + " xd=0;\n", + " dx=(b-a)/2;\n", + " x=(b+a)/2+(b-a)/2*xd;\n", + " I=func(x).*dx*c0;\n", + "end" + ] + }, + { + "cell_type": "code", + "execution_count": 13, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "ans = 3.5000\r\n" + ] + } + ], + "source": [ + "gauss_1pt(f1,2,3)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## General Gauss weights and points\n", + "\n", + "![Gauss quadrature table](gauss_weights.png)\n", + "\n", + "### If you need to evaluate an integral, to increase accuracy, increase number of Gauss points\n", + "\n", + "### Adaptive Quadrature\n", + "\n", + "Matlab/Octave built-in functions use two types of adaptive quadrature to increase accuracy of integrals of functions. \n", + "\n", + "1. `quad`: Simpson quadrature good for nonsmooth functions\n", + "\n", + "2. `quadl`: Lobatto quadrature good for smooth functions\n", + "\n", + "```matlab\n", + "q = quad(fun, a, b, tol, trace, p1, p2, …)\n", + "fun : function to be integrates\n", + "a, b: integration bounds\n", + "tol: desired absolute tolerance (default: 10-6)\n", + "trace: flag to display details or not\n", + "p1, p2, …: extra parameters for fun\n", + "quadl has the same arguments\n", + "```\n" + ] + }, + { + "cell_type": "code", + "execution_count": 149, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "ans = 6.6667\n", + "ans = 6.6667\n", + "ans = 6.6667\n" + ] + } + ], + "source": [ + "% integral of quadratic\n", + "quad(f2,2,3)\n", + "quadl(f2,2,3)\n", + "f3(3)-f3(2)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Numerical Differentiation\n", + "\n", + "Expanding the Taylor Series:\n", + "\n", + "$f(x_{i+1})=f(x_{i})+f'(x_{i})h+\\frac{f''(x_{i})}{2!}h^2+\\cdots$" + ] + }, + { + "cell_type": "code", + "execution_count": 82, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "Gnuplot\n", + "Produced by GNUPLOT 5.0 patchlevel 3 \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\t\n", + "\t\t-1.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-1\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-0.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t1\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t1.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-4\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-3\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-2\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-1\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t1\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t2\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t3\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t4\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\t\n", + "\t\tLow noise in sin wave\n", + "\t\n", + "\n", + "\n", + "\n", + "\tgnuplot_plot_1a\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\tgnuplot_plot_2a\n", + "\n", + "\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "x=linspace(-pi,pi);\n", + "y_smooth=sin(x);\n", + "y_noise =y_smooth+rand(size(x))*0.1-0.05;\n", + "plot(x,y_smooth,x,y_noise)\n", + "title('Low noise in sin wave')" + ] + }, + { + "cell_type": "code", + "execution_count": 98, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "Gnuplot\n", + "Produced by GNUPLOT 5.0 patchlevel 3 \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\t\n", + "\t\t-1.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-1\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-0.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t1\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t1.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-4\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-3\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-2\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-1\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t1\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t2\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t3\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t4\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\n", + "\t\n", + "\t\tNoise Amplified with derivative\n", + "\t\n", + "\n", + "\n", + "\n", + "\tgnuplot_plot_1a\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\tgnuplot_plot_2a\n", + "\n", + "\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "% Central Difference derivative\n", + "\n", + "dy_smooth=zeros(size(x));\n", + "dy_smooth([1,end])=NaN;\n", + "dy_smooth(2:end-1)=(y_smooth(3:end)-y_smooth(1:end-2))/2/(x(2)-x(1));\n", + "\n", + "dy_noise=zeros(size(x));\n", + "dy_noise([1,end])=NaN;\n", + "dy_noise(2:end-1)=(y_noise(3:end)-y_noise(1:end-2))/2/(x(2)-x(1));\n", + "\n", + "plot(x,dy_smooth,x,dy_noise)\n", + "title('Noise Amplified with derivative')" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Reduce noise\n", + "\n", + "Options:\n", + "\n", + "1. Fit a function and take derivative\n", + " \n", + " a. splines won't help much\n", + " \n", + " b. best fit curve (better)\n", + " \n", + "2. Smooth data (does not matter if you smooth before/after derivative)" + ] + }, + { + "cell_type": "code", + "execution_count": 99, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "Gnuplot\n", + "Produced by GNUPLOT 5.0 patchlevel 3 \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\t\n", + "\t\t-1.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-1\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t-0.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t0.5\n", + "\t\n", + "\n", + "\n", + "\t\t\n", + "\t\t1\n", + "\t\n", + "\n", + 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} + ], + "source": [ + "y_spline_i1=interp1(x,y_noise,x+0.1);\n", + "y_spline_in1=interp1(x,y_noise,x-0.1);\n", + "dy_spline=(y_spline_i1-y_spline_in1)/0.2;\n", + "plot(x,dy_spline,x,dy_noise)\n", + "legend('deriv of spline','no change')" + ] + }, + { + "cell_type": "code", + "execution_count": 100, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "a = 0.99838\r\n" + ] + }, + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "Gnuplot\n", + "Produced by GNUPLOT 5.0 patchlevel 3 \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\t\n", + "\t \n", + "\t \n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\n", + "\t\t\n", + "\t\n", + "\n", + "\n", + "\n", + "\n", + "\t\t\n", + "\t\t-1.5\n", + "\t\n", + "\n", + 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Cooper" Date: Fri, 7 Apr 2017 17:19:32 -0400 Subject: [PATCH 04/10] added final project --- final_project/README.md | 103 ++++++ final_project/beam.png | Bin 0 -> 11561 bytes final_project/beam.svg | 615 ++++++++++++++++++++++++++++++++++++ final_project/rubric.md | 27 ++ lecture_20/lecture_20.ipynb | 543 ++++++++++++++++--------------- 5 files changed, 1014 insertions(+), 274 deletions(-) create mode 100644 final_project/README.md create mode 100644 final_project/beam.png create mode 100644 final_project/beam.svg create mode 100644 final_project/rubric.md diff --git a/final_project/README.md b/final_project/README.md new file mode 100644 index 0000000..cee3ee3 --- /dev/null +++ b/final_project/README.md @@ -0,0 +1,103 @@ +# ME 3255 - Final Project +## Due May 1 by 11:59pm + +In this project you are going to solve for the shape of a beam under different loading +conditions. The shape of the beam varies along the x-axis and as a function of time. + +Notes: Label the plots with legends, x- and y-axis labels and make sure the plots are easy +to read (you can use the `setdefaults.m` script we have used in class). All functions +should have a help file and your README.md should describe each file in your repository +and provide a description of each problem and each solution (use `#`-headings in your file +to show the start of new problems) + +You will be graded both on documentation and implementation of the solutions. + +![Diagram of beam and loading conditions](beam.png) + +We will use the Euler-Bernoulli beam equation to describe the shape of the beam, the +differential equation that governs the solution is: + +$\frac{\partial^4 w}{\partial x^4}-\frac{P}{EI}\frac{\partial^2 w}{\partial +x^2}+\frac{\rho A}{EI}\frac{\partial^2 w}{\partial t^2}=q(x)$ (1) + +Where w(x,t) is the displacement of the beam away from the neutral axis as a function of +position along the beam, x, and time, t, P is the transverse loading of the beam, E is the +Young's modulus, I is the second moment of Inertia of the beam, $\rho$ is the density, A +is the cross-sectional area, and q(x) is the transverse distributed load (for a uniform +pressure, it is the applied pressure times the width of the beam, in units of +force/length). + +We can separate the function $w(x,t)=w(x)e^{i\omega t}$, now equation (1) becomes + +$\left(\frac{\partial^4 w}{\partial x^4}-\frac{P}{EI}\frac{\partial^2 w}{\partial +x^2}-\frac{\rho A \omega^{2}}{EI}w\right)e^{i\omega t}=\frac{q(x)}{EI}$ (2) + +For the simply-supported beam shown in Figure 1, the boundary conditions are: + +$w(0)=w(L)=0$ + +$w''(0)=w''(L)=0$ + +The material is aluminum, E=70 GPa, $\rho$=2700 kg/m$^3$. The bar is 1-m-long with a base +width, b=0.1 m, and height, h=0.01 m, and the second moment of inertia, +I=$\frac{bh^3}{12}$. + +1. Analytically solve for the shape of the beam if q(x)=cst, P=0, and $\omega$=0 and +create a function called `shape_simple_support.m` that returns the displacement w(x) given +q and x + +``` +w=shape_simple_support(x,q); +``` + +a. Plot q vs the maximum deflection, $\delta x$, of the beam + +b. Use a Monte Carlo model to determine the mean and standard deviation for the +maximum deflection $\delta x$ if b and h are normally distributed random variables +with 0.1 % standard deviations at q=50 N/m. + +3. Now use the central difference approximation to set up a system of equations for the +beam for q(x)=cst, P=0, and $\omega=0$. Use the boundary conditions with a numerical +differentiation to determine the valuea of the end points + + a. set up the system of equations for 6 segments as a function of q + + b. set up the system of equations for 10 segments as a function of q + + c. set up the system of equations for 20 segments as a function of q + + d. solve a-c for q=1,10,20,30,50 and plot the numerical results of q vs $\delta x$ + + e. Comment on the results from the analytical and numerical approaches (if you used + functions then provide help files, if you used scripts, then describe the steps used) + +4. Now set up the system of equations using a central difference method if P>0 and +$\omega=0$ + + a. set up the system of equations for 6 segments as a function of q and P + + b. set up the system of equations for 10 segments as a function of q and P + + c. set up the system of equations for 20 segments as a function of q and P + + d. solve a-c for q=1,10,20,30,50 and plot the numerical results of q vs $\delta x$ for + P=0, 100, 200, 300 (4 lines, labeled as P=0,P=100,...) + +5. Now set up an eigenvalue problem to solve for the natural frequencies of the simply +supported beam if P=0 and q=0. + + a. set up the system of equations for 6 segments + + b. set up the system of equations for 10 segments + + c. set up the system of equations for 20 segments + + d. solve for the natural frequencies ($\omega_{1}$, $\omega_{2}$,...) + + e. Plot the shape of the beam for the first 3 natural frequencies + +6. (Bonus 5pt) Create a function to return the system of equations for the eigenvalue +problem as a function of P, if P>0. 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b/final_project/beam.svg new file mode 100644 index 0000000..b02b62d --- /dev/null +++ b/final_project/beam.svg @@ -0,0 +1,615 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + image/svg+xml + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + q(x) + P + P + + + w(x) + + + x + y + + diff --git a/final_project/rubric.md b/final_project/rubric.md new file mode 100644 index 0000000..08248bd --- /dev/null +++ b/final_project/rubric.md @@ -0,0 +1,27 @@ +# Grading Rubric + +| Component | Grade | Description | +| --------- |------ |--------------------------------- | +|Solutions | 50 % | Problems are solved correctly and solution makes sense | +|Documentation| 30 % | All files have comments and all functions have help files | +|Problem Statements | 10% | Descriptions of each problem and the approach | +|Team Work | 10% | Each group member made commits to repository| +|Github Bonus | 5% | If a group member opens an [issue](https://guides.github.com/features/issues/) and you commit code to close it| + +The Documentation and Problem Statements will be assessed based upon the included m-files +and the main README.md documentation. This is a final report, so make sure the README.md +is easy to understand with headers (`#`, `##`, ...) to start each problem and sub-problem. + +The Solutions will be graded based upon correctness and the approach. The plots will be +graded based upon the axis labels, titles, legend, correctness, and design. +If you are asked to solve for a value, you can choose to put it in a table or in a +paragraph, but it must be easy to read/understand. + +The Team Work section is meant to demonstrate working knowledge of Github (now as a +collaborative project). If everyone makes commits to the project, then you will receive +credit. + +The Github Bonus requires extra reading. Create an issue and tag your group member. That +group member can then respond and *fix* the issue with a commit. This is a bonus 5pts (on +an individual basis, for everyone to get credit, everyone in the group would need to close +an issue) diff --git a/lecture_20/lecture_20.ipynb b/lecture_20/lecture_20.ipynb index 50f858a..da8c4ac 100644 --- a/lecture_20/lecture_20.ipynb +++ b/lecture_20/lecture_20.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "code", - "execution_count": 1, + "execution_count": 152, "metadata": { "collapsed": true }, @@ -13,7 +13,7 @@ }, { "cell_type": "code", - "execution_count": 2, + "execution_count": 153, "metadata": { "collapsed": true }, @@ -97,17 +97,19 @@ "source": [ "# Integrals in practice\n", "\n", - "### Example: Compare toughness of two steels\n", + "### Example: Compare toughness of Stainless steel to Structural steel\n", "\n", "![Stress-strain plot of steel](steel_psi.jpg)\n", "\n", + "### Step 1 - G3Data to get points \n", + "\n", "Use the plot shown to determine the toughness of stainless steel and the toughness of structural steel.\n", "\n" ] }, { "cell_type": "code", - "execution_count": 49, + "execution_count": 154, "metadata": { "collapsed": false }, @@ -162,7 +164,7 @@ }, { "cell_type": "code", - "execution_count": 147, + "execution_count": 155, "metadata": { "collapsed": false }, @@ -347,7 +349,7 @@ }, { "cell_type": "code", - "execution_count": 148, + "execution_count": 156, "metadata": { "collapsed": false }, @@ -429,7 +431,7 @@ }, { "cell_type": "code", - "execution_count": 15, + "execution_count": 157, "metadata": { "collapsed": true }, @@ -448,7 +450,7 @@ }, { "cell_type": "code", - "execution_count": 13, + "execution_count": 159, "metadata": { "collapsed": false }, @@ -457,12 +459,12 @@ "name": "stdout", "output_type": "stream", "text": [ - "ans = 3.5000\r\n" + "ans = 6.6250\r\n" ] } ], "source": [ - "gauss_1pt(f1,2,3)" + "gauss_1pt(f2,2,3)" ] }, { @@ -496,7 +498,7 @@ }, { "cell_type": "code", - "execution_count": 149, + "execution_count": 164, "metadata": { "collapsed": false }, @@ -507,7 +509,13 @@ "text": [ "ans = 6.6667\n", "ans = 6.6667\n", - "ans = 6.6667\n" + "ans = 6.6667\n", + "f_c =\n", + "\n", + "@(x) cosh (x / 30) + exp (-10 * x) .* erf (x)\n", + "\n", + "ans = 2.0048\n", + "ans = 2.0048\n" ] } ], @@ -515,7 +523,11 @@ "% integral of quadratic\n", "quad(f2,2,3)\n", "quadl(f2,2,3)\n", - "f3(3)-f3(2)" + "f3(3)-f3(2)\n", + "f_c=@(x) cosh(x/30)+exp(-10*x).*erf(x)\n", + "\n", + "quad(f_c,1,3)\n", + "quadl(f_c,1,3)" ] }, { @@ -531,7 +543,7 @@ }, { "cell_type": "code", - "execution_count": 82, + "execution_count": 165, "metadata": { "collapsed": false }, @@ -682,7 +694,7 @@ "\t\n", "\tgnuplot_plot_2a\n", "\n", - "\t\n", + "\t\n", "\t\n", "\n", "\n", @@ -713,7 +725,7 @@ }, { "cell_type": "code", - "execution_count": 98, + "execution_count": 166, "metadata": { "collapsed": false }, @@ -767,31 +779,36 @@ "\n", "\n", "\t\t\n", + "\t\t-2\n", + "\t\n", + "\n", + "\n", + "\t\t\n", "\t\t-1.5\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t-1\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t-0.5\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t0\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t0.5\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t1\n", "\t\n", "\n", @@ -860,11 +877,11 @@ "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_2a\n", "\n", - "\t\n", + "\t\n", "\t\n", "\n", "\n", @@ -919,7 +936,7 @@ }, { "cell_type": "code", - "execution_count": 99, + "execution_count": 168, "metadata": { "collapsed": false }, @@ -973,31 +990,36 @@ "\n", "\n", "\t\t\n", + "\t\t-2\n", + "\t\n", + "\n", + "\n", + "\t\t\n", "\t\t-1.5\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t-1\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t-0.5\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t0\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t0.5\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t1\n", "\t\n", "\n", @@ -1070,7 +1092,7 @@ "\t\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tno change\n", "\n", @@ -1079,7 +1101,7 @@ "\t\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\n", "\n", @@ -1110,7 +1132,7 @@ }, { "cell_type": "code", - "execution_count": 100, + "execution_count": 169, "metadata": { "collapsed": false }, @@ -1119,7 +1141,7 @@ "name": "stdout", "output_type": "stream", "text": [ - "a = 0.99838\r\n" + "a = 1.0007\r\n" ] }, { @@ -1259,11 +1281,11 @@ "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_2a\n", "\n", - "\t\n", + "\t\n", "\t\n", "\n", "\n", @@ -1554,39 +1576,7 @@ }, { "cell_type": "code", - "execution_count": 120, - "metadata": { - "collapsed": false - }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "a =\n", - "\n", - " 1 2 3 4 5\n", - "\n", - "pa =\n", - "\n", - " 2 1 1 2 3 4 5 5 4\n", - "\n", - "ans =\n", - "\n", - " 1 9\n", - "\n" - ] - } - ], - "source": [ - "a=[1,2,3,4,5]\n", - "pa=[a(4/2:-1:1) a a(end:-1:end-4/2+1)]\n", - "size(pa)" - ] - }, - { - "cell_type": "code", - "execution_count": 137, + "execution_count": 170, "metadata": { "collapsed": false }, @@ -1738,111 +1728,111 @@ "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_2a\n", "\n", "\t \n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", "\n", "\t\n", "\n", @@ -1879,7 +1869,7 @@ }, { "cell_type": "code", - "execution_count": 138, + "execution_count": 171, "metadata": { "collapsed": false }, @@ -1933,31 +1923,36 @@ "\n", "\n", "\t\t\n", + "\t\t-2\n", + "\t\n", + "\n", + "\n", + "\t\t\n", "\t\t-1.5\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t-1\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t-0.5\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t0\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t0.5\n", "\t\n", "\n", "\n", - "\t\t\n", + "\t\t\n", "\t\t1\n", "\t\n", "\n", @@ -2026,113 +2021,113 @@ "\n", "\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_2a\n", "\n", - "\t\n", + "\t\n", "\t\n", "\tgnuplot_plot_3a\n", "\n", "\t \n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - "\t\n", - 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a/HW6/problem_1_data.m b/HW6/problem_1_data.m index 6af2956..3606608 100644 --- a/HW6/problem_1_data.m +++ b/HW6/problem_1_data.m @@ -1,7 +1,7 @@ % part a xa=[1 2 3 4 5]'; -yb=[2.2 2.8 3.6 4.5 5.5]'; +ya=[2.2 2.8 3.6 4.5 5.5]'; % part b @@ -10,6 +10,6 @@ % part c -xc=[0.5 1 2 3 4 5 6 7 9]; -yc=[6 4.4 3.2 2.7 2.2 1.9 1.7 1.4 1.1]; +xc=[0.5 1 2 3 4 5 6 7 9]'; +yc=[6 4.4 3.2 2.7 2.2 1.9 1.7 1.4 1.1]'; From ee2c3e2428aee8be6685299da362311d8a2a4d43 Mon Sep 17 00:00:00 2001 From: "Ryan C. Cooper" Date: Mon, 10 Apr 2017 16:34:08 -0400 Subject: [PATCH 07/10] fix HW6 gradient of cost equation --- HW6/README.md | 6 +- lecture_17/octave-workspace | Bin 5146 -> 153 bytes lecture_18/octave-workspace | Bin 2001 -> 153 bytes lecture_19/octave-workspace | Bin 10815 -> 153 bytes lecture_20/octave-workspace | Bin 0 -> 11 bytes .../lecture_21-checkpoint.ipynb | 6 + lecture_21/lecture_21.ipynb | 430 ++++++++++++++++++ lecture_21/q1.png | Bin 0 -> 28691 bytes lecture_21/q2.png | Bin 0 -> 36399 bytes lecture_21/reduce_noise.png | Bin 0 -> 7390788 bytes 10 files changed, 440 insertions(+), 2 deletions(-) create mode 100644 lecture_20/octave-workspace create mode 100644 lecture_21/.ipynb_checkpoints/lecture_21-checkpoint.ipynb create mode 100644 lecture_21/lecture_21.ipynb create mode 100644 lecture_21/q1.png create mode 100644 lecture_21/q2.png create mode 100644 lecture_21/reduce_noise.png diff --git a/HW6/README.md b/HW6/README.md index 1f3c86c..c72ba05 100644 --- a/HW6/README.md +++ b/HW6/README.md @@ -34,12 +34,14 @@ variable is failure (1=fail, 0=pass). Create a function called `cost_logistic.m` takes the vector `a`, and independent variable `x` and dependent variable `y`. Use the function, $\sigma(t)=\frac{1}{1+e^{-t}}$ where $t=a_{0}+a_{1}x$. Use the cost function, - $J(a_{0},a_{1})=\sum_{i=1}^{n}\left[-y_{i}\log(\sigma(t_{i}))-(1-y_{i})\log((1-\sigma(t_{i})))\right]$ + $J(a_{0},a_{1})=1/m\sum_{i=1}^{n}\left[-y_{i}\log(\sigma(t_{i}))-(1-y_{i})\log((1-\sigma(t_{i})))\right]$ and gradient $\frac{\partial J}{\partial a_{i}}= - 1/m\sum_{k=1}^{N}\left(\sigma(t_{k})-y_{k}\right)t_{k}$ + 1/m\sum_{k=1}^{N}\left(\sigma(t_{k})-y_{k}\right)x_{k}^{i}$ + + where $x_{k}^{i} is the k-th value of temperature raised to the i-th power (0, and 1) a. edit `cost_logistic.m` so that the output is `[J,grad]` or [cost, gradient] diff --git a/lecture_17/octave-workspace b/lecture_17/octave-workspace index c9ec2aa3f1753ebfc595762a3c1c99b8573c1bf4..8c437bb6e55a5d1b6115661b3a20e86870909d32 100644 GIT binary patch delta 20 bcmbQGF_V$oKe;5aELGP~*N0*9e6~ISLiq+~ literal 5146 zcmb7Hd05O_A8$lOO1l=)q6H0-5OEIOibyI%MMAPWZCYohJzBiyG=2SXpZERa?RkDP&n&+)=X}1OExQfA?tUIzW9||$j+mI3Grh#b zeoD}X?e4zbTm1Wf{`|R-z8XR=NqP;-oxil1W@)YvXq~PO@_SqlMzE0DI 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+ ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "```matlab\n", + "% Define function ...\n", + "function collar_potential_energy(x,theta)\n", + " % function to evaluate E_total\n", + " Etotal = m*x*g.*sin(theta)+K/2*(0.5-sqrt(0.5^2+(0.5-x).^2);\n", + "end\n", + "\n", + "% solve for x\n", + "\n", + "x=goldmin(@(x) collar_potential_energy(x,theta),xl,xu)\n", + "```" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## HW #5\n", + "\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Initial Value Problem" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Octave", + "language": "octave", + "name": "octave" + }, + "language_info": { + "file_extension": ".m", + "help_links": [ + { + "text": "MetaKernel Magics", + "url": "https://github.com/calysto/metakernel/blob/master/metakernel/magics/README.md" + } + ], + "mimetype": "text/x-octave", + "name": "octave", + "version": "0.19.14" + } + }, + "nbformat": 4, + "nbformat_minor": 2 +} diff --git a/lecture_21/q1.png b/lecture_21/q1.png new file mode 100644 index 0000000000000000000000000000000000000000..3145b8d50745f1e76b825a3abe1d4de86684ecd5 GIT binary patch literal 28691 zcmdpdcTkgG)FxO!L_m=uU8G9~=|u(Uy(1+ENDIA&j;KiQy-Ozq2rYyb0@9>QFQG{9 zAe}&fjlbXbeLFj|yED79e{E)%T#`5Uz4yN7o_o&oJSSXJU4iH!^+OyS93myfciK2O zcaS(ZxbhG1u%B2uoabZT?zzh-={~@|{2zP_!~Rd@A!q2J<818#GIz7Wv2k*CwBmNR zbhEN@a<_H%IJk?H#=&`xqx9~ru6Ne%e1JdG#{13d+?p+5LY!g|ueP%G)~h1z#VT!4 zMKO-#?_`jY`vgXD8KbjHWAvO^clEt%N8Rc9{*c)(th*O9D`j?oPXQ96iR_1kFo8}V8&kP-^N!_K4vueKi}eiV@CedSZ<43 zHTd_p&k`^X|8^2~S2n@yZ{vaM2gM`KzYRv>NB`B?h#0z}&?Qqf&7Z1_j;%D|Fr6=b zj=3+>^kcHgbWATbv(C&e2(D3)#!BI0DTMt})mEKhG#@>lAMmJQtUPx~GZ-0phjl}KQJP_(A#=dxi?Vw4?%emNnvJyzBT zn1j!?&$D?*#!Fvb`982^iJ3nINRI5jB)H`nf#_1^8C5vC_UQj)eabv~U)bxxBs=R} 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