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diff --git a/sample_notebooks/Mohdarif/Ch21.ipynb b/sample_notebooks/Mohdarif/Ch21.ipynb new file mode 100644 index 00000000..507735ab --- /dev/null +++ b/sample_notebooks/Mohdarif/Ch21.ipynb @@ -0,0 +1,725 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Chapter 21 Newtin-cotes integration formula" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Ex : 21.1 Pg : 612" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "The Error Et= 1.468\n", + "The percent relative error et= 89.467 %\n", + "The approximate error estimate without using the true value= 2.56\n" + ] + } + ], + "source": [ + "from __future__ import division\n", + "def f(x):\n", + " y=(0.2+25*x-200*x**2+675*x**3-900*x**4+400*x**5)\n", + " return y\n", + "tval=1.640533#\n", + "a=0#\n", + "b=0.8#\n", + "fa=f(a)#\n", + "fb=f(b)#\n", + "l=(b-a)*((fa+fb)/2)#\n", + "Et=tval-l##error\n", + "et=Et*100/tval##percent relative error\n", + "\n", + "#by using approximate error estimate\n", + "\n", + "#the second derivative of f\n", + "def g(x):\n", + " y=-400+4050*x-10800*x**2+8000*x**3\n", + " return y\n", + "\n", + "from sympy.mpmath import quad\n", + "f2x=quad(g,[0,0.8])/(b-a)##average value of second derivative\n", + "Ea=-(1/12)*(f2x)*(b-a)**3#\n", + "print \"The Error Et=\",round(Et,3)\n", + "print \"The percent relative error et=\",round(et,3),\"%\"\n", + "print \"The approximate error estimate without using the true value=\",Ea" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Ex : 21.2 Pg : 613" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "The Error Et= 0.572\n", + "The percent relative error et= 34.85 %\n", + "The approximate error estimate without using the true value= 0.64\n" + ] + } + ], + "source": [ + "def f(x):\n", + " y=(0.2+25*x-200*x**2+675*x**3-900*x**4+400*x**5)\n", + " return y\n", + "a=0#\n", + "b=0.8#\n", + "tval=1.640533#\n", + "n=2#\n", + "h=(b-a)/n#\n", + "fa=f(a)#\n", + "fb=f(b)#\n", + "fh=f(h)#\n", + "l=(b-a)*(fa+2*fh+fb)/(2*n)#\n", + "Et=tval-l##error\n", + "et=Et*100/tval##percent relative error\n", + "\n", + "#by using approximate error estimate\n", + "\n", + "#the second derivative of f\n", + "def g(x):\n", + " y=-400+4050*x-10800*x**2+8000*x**3\n", + " return y\n", + "f2x=quad(g,[0,0.8])/(b-a)##average value of second derivative\n", + "Ea=-(1/12)*(f2x)*(b-a)**3/(n**2)#\n", + "print \"The Error Et=\",round(Et,3)\n", + "print \"The percent relative error et=\",round(et,3),\"%\"\n", + "print \"The approximate error estimate without using the true value=\",Ea" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Ex :21.3 Pg : 614" + ] + }, + { + "cell_type": "code", + "execution_count": 13, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "No. of segments= 10\n", + "Segment size= 1.0\n", + "Estimated d= 288.749146143 m\n", + "0.237014701487 et(%)\n", + "---------------------------------------------------------\n", + "No. of segments= 20\n", + "Segment size= 0.5\n", + "Estimated d= 289.263574224 m\n", + "0.0592795228803 et(%)\n", + "---------------------------------------------------------\n", + "No. of segments= 50\n", + "Segment size= 0.2\n", + "Estimated d= 298.382319223 m\n", + "et(%) -3.09125177877\n", + "---------------------------------------------------------\n", + "No. of segments= 100\n", + "Segment size= 0.1\n", + "Estimated d= 293.915596452 m\n", + "et(%) -1.54799665905\n", + "---------------------------------------------------------\n", + "No. of segments= 100\n", + "Segment size= 0.1\n", + "Estimated d= 293.915596452 m\n", + "et(%) -1.54799665905\n", + "---------------------------------------------------------\n", + "No. of segments= 200\n", + "Segment size= 0.05\n", + "Estimated d= 289.43343055 m\n", + "et(%) 0.000594070904571\n", + "---------------------------------------------------------\n", + "No. of segments= 200\n", + "Segment size= 0.05\n", + "Estimated d= 289.43343055 m\n", + "et(%) 0.000594070904571\n", + "---------------------------------------------------------\n", + "No. of segments= 500\n", + "Segment size= 0.02\n", + "Estimated d= 290.332334709 m\n", + "et(%) -0.309977799375\n", + "---------------------------------------------------------\n", + "No. of segments= 1000\n", + "Segment size= 0.01\n", + "Estimated d= 289.883809248 m\n", + "et(%) -0.155012011658\n", + "---------------------------------------------------------\n", + "No. of segments= 2000\n", + "Segment size= 0.005\n", + "Estimated d= 289.435129352 m\n", + "et(%) 7.13401428866e-06\n", + "---------------------------------------------------------\n", + "No. of segments= 2000\n", + "Segment size= 0.005\n", + "Estimated d= 289.435129352 m\n", + "et(%) 7.13401428866e-06\n", + "---------------------------------------------------------\n", + "No. of segments= 5000\n", + "Segment size= 0.002\n", + "Estimated d= 289.435143766 m\n", + "et(%) 2.15393877364e-06\n", + "---------------------------------------------------------\n", + "No. of segments= 5000\n", + "Segment size= 0.002\n", + "Estimated d= 289.435143766 m\n", + "et(%) 2.15393877364e-06\n", + "---------------------------------------------------------\n", + "No. of segments= 10000\n", + "Segment size= 0.001\n", + "Estimated d= 289.480018962 m\n", + "et(%) -0.0155022506708\n", + "---------------------------------------------------------\n" + ] + } + ], + "source": [ + "g=9.8##m/s**2# acceleration due to gravity\n", + "m=68.1##kg\n", + "c=12.5##kg/sec# drag coefficient\n", + "def f(t):\n", + " from numpy import exp\n", + " v=g*m*(1-exp(-c*t/m))/c\n", + " return v\n", + "tval=289.43515##m\n", + "a=0#\n", + "b=10#\n", + "fa=f(a)#\n", + "fb=f(b)#\n", + "from numpy import arange, exp\n", + "for i in arange(10,21,10):\n", + " n=i#\n", + " h=(b-a)/n#\n", + " print \"No. of segments=\",i\n", + " print \"Segment size=\",h\n", + " j=a+h#\n", + " s=0#\n", + " while j<b:\n", + " s=s+f(j)#\n", + " j=j+h#\n", + " \n", + " l=(b-a)*(fa+2*s+fb)/(2*n)#\n", + " Et=tval-l##error\n", + " et=Et*100/tval##percent relative error\n", + " print \"Estimated d=\",l,\"m\"\n", + " print et,\"et(%)\"\n", + " print \"---------------------------------------------------------\"\n", + "\n", + "for i in arange(50,101,50):\n", + " n=i#\n", + " h=(b-a)/n#\n", + " print \"No. of segments=\",i\n", + " print \"Segment size=\",h\n", + " j=a+h#\n", + " s=0#\n", + " while j<b:\n", + " s=s+f(j)#\n", + " j=j+h#\n", + " \n", + " l=(b-a)*(fa+2*s+fb)/(2*n)#\n", + " Et=tval-l##error\n", + " et=Et*100/tval##percent relative error\n", + " print \"Estimated d=\",l,\"m\"\n", + " print \"et(%)\",et\n", + " print \"---------------------------------------------------------\"\n", + "\n", + "for i in arange(100,201,100):\n", + " n=i#\n", + " h=(b-a)/n#\n", + " print \"No. of segments=\",i\n", + " print \"Segment size=\",h\n", + " j=a+h#\n", + " s=0#\n", + " while j<b:\n", + " s=s+f(j)#\n", + " j=j+h#\n", + " \n", + " l=(b-a)*(fa+2*s+fb)/(2*n)#\n", + " Et=tval-l##error\n", + " et=Et*100/tval##percent relative error\n", + " print \"Estimated d=\",l,\"m\"\n", + " print \"et(%)\",et\n", + " print \"---------------------------------------------------------\"\n", + "\n", + "for i in arange(200,501,300):\n", + " n=i#\n", + " h=(b-a)/n#\n", + " print \"No. of segments=\",i\n", + " print \"Segment size=\",h\n", + " j=a+h#\n", + " s=0#\n", + " while j<b:\n", + " s=s+f(j)#\n", + " j=j+h#\n", + " \n", + " l=(b-a)*(fa+2*s+fb)/(2*n)#\n", + " Et=tval-l##error\n", + " et=Et*100/tval##percent relative error\n", + " print \"Estimated d=\",l,\"m\"\n", + " print \"et(%)\",et\n", + " print \"---------------------------------------------------------\"\n", + "\n", + "for i in arange(1000,2001,1000):\n", + " n=i#\n", + " h=(b-a)/n#\n", + " print \"No. of segments=\",i\n", + " print \"Segment size=\",h\n", + " j=a+h#\n", + " s=0#\n", + " while j<b:\n", + " s=s+f(j)#\n", + " j=j+h#\n", + " \n", + " l=(b-a)*(fa+2*s+fb)/(2*n)#\n", + " Et=tval-l##error\n", + " et=Et*100/tval##percent relative error\n", + " print \"Estimated d=\",l,\"m\"\n", + " print \"et(%)\",et\n", + " print \"---------------------------------------------------------\"\n", + "\n", + "for i in arange(2000,5001,3000):\n", + " n=i#\n", + " h=(b-a)/n#\n", + " print \"No. of segments=\",i\n", + " print \"Segment size=\",h\n", + " j=a+h#\n", + " s=0#\n", + " while j<b:\n", + " s=s+f(j)#\n", + " j=j+h#\n", + " \n", + " l=(b-a)*(fa+2*s+fb)/(2*n)#\n", + " Et=tval-l##error\n", + " et=Et*100/tval##percent relative error\n", + " print \"Estimated d=\",l,\"m\"\n", + " print \"et(%)\",et\n", + " print \"---------------------------------------------------------\"\n", + "\n", + "for i in arange(5000,10001,5000):\n", + " n=i#\n", + " h=(b-a)/n#\n", + " print \"No. of segments=\",i\n", + " print \"Segment size=\",h\n", + " j=a+h#\n", + " s=0#\n", + " while j<b:\n", + " s=s+f(j)#\n", + " j=j+h#\n", + " \n", + " l=(b-a)*(fa+2*s+fb)/(2*n)#\n", + " Et=tval-l##error\n", + " et=Et*100/tval##percent relative error\n", + " print \"Estimated d=\",l,\"m\"\n", + " print \"et(%)\",et\n", + " print \"---------------------------------------------------------\"" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Ex : 21.4 Pg : 618" + ] + }, + { + "cell_type": "code", + "execution_count": 15, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "l= 1.37\n", + "The Error Et= 0.27\n", + "The percent relative error et= 16.645 %\n", + "The approximate error estimate without using the true value= 0.273\n" + ] + } + ], + "source": [ + "def f(x):\n", + " y=(0.2+25*x-200*x**2+675*x**3-900*x**4+400*x**5)\n", + " return y\n", + "a=0#\n", + "b=0.8#\n", + "tval=1.640533#\n", + "n=2#\n", + "h=(b-a)/n#\n", + "fa=f(a)#\n", + "fb=f(b)#\n", + "fh=f(h)#\n", + "l=(b-a)*(fa+4*fh+fb)/(3*n)#\n", + "print\"l=\", round(l,2)\n", + "Et=tval-l##error\n", + "et=Et*100/tval##percent relative error\n", + "\n", + "#by using approximate error estimate\n", + "\n", + "#the fourth derivative of f\n", + "def g(x):\n", + " y=-21600+48000*x\n", + " return y\n", + "\n", + "f4x=quad(g,[0,0.8])/(b-a)##average value of fourth derivative\n", + "Ea=-(1/2880)*(f4x)*(b-a)**5#\n", + "print \"The Error Et=\",round(Et,2)\n", + "print \"The percent relative error et=\",round(et,3),\"%\"\n", + "print \"The approximate error estimate without using the true value=\",round(Ea,3)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Ex : 23.5 Pg : 620" + ] + }, + { + "cell_type": "code", + "execution_count": 16, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "l= 1.62\n", + "The Error Et= 0.02\n", + "The percent relative error et= 1.04 %\n", + "The approximate error estimate without using the true value= 0.017\n" + ] + } + ], + "source": [ + "def f(x):\n", + " y=(0.2+25*x-200*x**2+675*x**3-900*x**4+400*x**5)\n", + " return y\n", + "a=0#\n", + "b=0.8#\n", + "tval=1.640533#\n", + "n=4#\n", + "h=(b-a)/n#\n", + "fa=f(a)#\n", + "fb=f(b)#\n", + "j=a+h#\n", + "s=0#\n", + "count=1#\n", + "while j<b:\n", + " if (-1)**count==-1:\n", + " s=s+4*f(j)#\n", + " else:\n", + " s=s+2*f(j)#\n", + " \n", + " count=count+1#\n", + " j=j+h#\n", + "\n", + "l=(b-a)*(fa+s+fb)/(3*n)#\n", + "print\"l=\", round(l,2)\n", + "Et=tval-l##error\n", + "et=Et*100/tval##percent relative error\n", + "\n", + "#by using approximate error estimate\n", + "\n", + "#the fou:rth derivative of f\n", + "def g(x):\n", + " y=-21600+48000*x\n", + " return y\n", + "f4x=quad(g,[0,0.8])/(b-a)##average value of fourth derivative\n", + "Ea=-(1/(180*4**4))*(f4x)*(b-a)**5#\n", + "print \"The Error Et=\",round(Et,2)\n", + "print \"The percent relative error et=\",round(et,3),\"%\"\n", + "print \"The approximate error estimate without using the true value=\",round(Ea,3)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Ex :23.6 Pg : 625" + ] + }, + { + "cell_type": "code", + "execution_count": 18, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Part A:\n", + "l= 1.519\n", + "The Error Et= 0.12\n", + "The percent relative error et= 7.398 %\n", + "The approximate error estimate without using the true value= 0.121\n", + "---------------------------------------------------\n", + "Part B:\n", + "l= 1.645\n", + "The Error Et= -0.005\n", + "The percent relative error et= -0.277 %\n" + ] + } + ], + "source": [ + "def f(x):\n", + " y=(0.2+25*x-200*x**2+675*x**3-900*x**4+400*x**5)\n", + " return y\n", + "a=0#\n", + "b=0.8#\n", + "tval=1.640533#\n", + "#part a\n", + "n=3#\n", + "h=(b-a)/n#\n", + "fa=f(a)#\n", + "fb=f(b)#\n", + "j=a+h#\n", + "s=0#\n", + "count=1#\n", + "while j<b:\n", + " s=s+3*f(j)#\n", + " count=count+1#\n", + " j=j+h#\n", + "\n", + "l=(b-a)*(fa+s+fb)/(8)#\n", + "print \"Part A:\"\n", + "print \"l=\",round(l,3)\n", + "Et=tval-l##error\n", + "et=Et*100/tval##percent relative error\n", + "\n", + "#by using approximate error estimate\n", + "\n", + "#the fourth derivative of f\n", + "def g(x):\n", + " y=-21600+48000*x\n", + " return y\n", + "f4x=quad(g,[0,0.8])/(b-a)##average value of fourth derivative\n", + "Ea=-(1/6480)*(f4x)*(b-a)**5#\n", + "print \"The Error Et=\",round(Et,2)\n", + "print \"The percent relative error et=\",round(et,3),\"%\"\n", + "print \"The approximate error estimate without using the true value=\",round(Ea,3)\n", + "#part b\n", + "n=5#\n", + "h=(b-a)/n#\n", + "l1=(a+2*h-a)*(fa+4*f(a+h)+f(a+2*h))/6#\n", + "l2=(a+5*h-a-2*h)*(f(a+2*h)+3*(f(a+3*h)+f(a+4*h))+fb)/8#\n", + "l=l1+l2#\n", + "print \"---------------------------------------------------\"\n", + "print \"Part B:\"\n", + "print \"l=\", round(l,3)\n", + "Et=tval-l##error\n", + "et=Et*100/tval##percent relative error\n", + "print \"The Error Et=\", round(Et,3)\n", + "print \"The percent relative error et=\", round(et,3), \"%\"" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Ex : 23.7 Pg : 626" + ] + }, + { + "cell_type": "code", + "execution_count": 22, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "l= 1.59480096\n", + "The Error Et= 0.04573204\n", + "The percent relative error et= 2.78763304365 %\n" + ] + } + ], + "source": [ + "def f(x):\n", + " y=(0.2+25*x-200*x**2+675*x**3-900*x**4+400*x**5)\n", + " return y\n", + "tval=1.640533#\n", + "x=[0, 0.12, 0.22, 0.32, 0.36, 0.4 ,0.44 ,0.54 ,0.64 ,0.7 ,0.8]\n", + "func=[]\n", + "for i in range(0,11):\n", + " func.append(f(x[i]))#\n", + "\n", + "l=0#\n", + "for i in range(0,10):\n", + " l=l+(x[i+1]-x[i])*(func[i]+func[i+1])/2#\n", + "\n", + "print \"l=\",l\n", + "Et=tval-l##error\n", + "et=Et*100/tval##percent relative error\n", + "print \"The Error Et=\",Et\n", + "print \"The percent relative error et=\",et,\"%\"" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Ex : 23.8 Pg : 230" + ] + }, + { + "cell_type": "code", + "execution_count": 23, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "l= 1.60364091733\n", + "The Error Et= 0.0368920826667\n", + "The percent relative error et= 2.2487863802 %\n" + ] + } + ], + "source": [ + "def f(x):\n", + " y=(0.2+25*x-200*x**2+675*x**3-900*x**4+400*x**5)\n", + " return y\n", + "tval=1.640533#\n", + "x=[0, 0.12, 0.22, 0.32, 0.36, 0.4 ,0.44 ,0.54, 0.64, 0.7, 0.8]\n", + "func =[]\n", + "for i in range(0,11):\n", + " func.append(f(x[i]))\n", + "\n", + "l1=(x[1]-x[0])*((f(x[0])+f(x[1]))/2)#\n", + "l2=(x[3]-x[1])*(f(x[3])+4*f(x[2])+f(x[1]))/6#\n", + "l3=(x[6]-x[3])*(f(x[3])+3*(f(x[4])+f(x[5]))+f(x[6]))/8#\n", + "l4=(x[8]-x[6])*(f(x[6])+4*f(x[7])+f(x[8]))/6\n", + "l5=(x[9]-x[8])*((f(x[9])+f(x[8]))/2)#\n", + "l6=(x[10]-x[9])*((f(x[10])+f(x[9]))/2)#\n", + "l=l1+l2+l3+l4+l5+l6#\n", + "print \"l=\",l\n", + "Et=tval-l##error\n", + "et=Et*100/tval##percent relative error\n", + "print \"The Error Et=\",Et\n", + "print \"The percent relative error et=\",et,\"%\"" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Ex : 23.9 Pg : 629" + ] + }, + { + "cell_type": "code", + "execution_count": 25, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "The average termperature is= 53.0\n" + ] + } + ], + "source": [ + "def f(x,y):\n", + " t=2*x*y+2*x-x**2-2*y**2+72\n", + " return t\n", + "Len=8##m,length\n", + "wid=6##m,width\n", + "a=0#\n", + "b=Len#\n", + "n=2#\n", + "h=(b-a)/n#\n", + "a1=0#\n", + "b1=wid#\n", + "h1=(b1-a1)/n#\n", + "\n", + "fa=f(a,0)#\n", + "fb=f(b,0)#\n", + "fh=f(h,0)#\n", + "lx1=(b-a)*(fa+2*fh+fb)/(2*n)#\n", + "\n", + "fa=f(a,h1)#\n", + "fb=f(b,h1)#\n", + "fh=f(h,h1)#\n", + "lx2=(b-a)*(fa+2*fh+fb)/(2*n)#\n", + "\n", + "fa=f(a,b1)#\n", + "fb=f(b,b1)#\n", + "fh=f(h,b1)#\n", + "lx3=(b-a)*(fa+2*fh+fb)/(2*n)#\n", + "\n", + "l=(b1-a1)*(lx1+2*lx2+lx3)/(2*n)#\n", + "\n", + "avg_temp=l/(Len*wid)#\n", + "print\"The average termperature is=\", avg_temp" + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 2", + "language": "python", + "name": "python2" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 2 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython2", + "version": "2.7.9" + } + }, + "nbformat": 4, + "nbformat_minor": 0 +} |