{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Chapter 4 : Truncation Errors and the Taylor Series"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example: 4.1 Page No:79"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The value of f at x=0 : 1.2\n",
      "The value of f at x=1 due to zero order approximation : 1.2\n",
      "Truncation error : -1.0\n",
      "----------------------------------------------\n",
      "The value of first derivative of f at x=0 : -0.4\n",
      "The value of f at x=1 due to first order approximation : 0.8\n",
      "Truncation error : -0.6\n",
      "----------------------------------------------\n",
      "The value of second derivative of f at x=0 : -1.8\n",
      "The value of f at x=1 due to second order approximation : -0.1\n",
      "Truncation error : 0.3\n",
      "----------------------------------------------\n",
      "The value of third derivative of f at x=0 : -0.9\n",
      "The value of f at x=1 due to third order approximation : -0.25\n",
      "Truncation error : 0.45\n",
      "----------------------------------------------\n",
      "The value of fourth derivative of f at x=0 : -2.4\n",
      "The value of f at x=1 due to fourth order approximation : -0.35\n",
      "Truncation error : 0.55\n",
      "----------------------------------------------\n"
     ]
    }
   ],
   "source": [
    "from math import factorial\n",
    "from scipy.misc import derivative\n",
    "def f(x):\n",
    "    y=-0.1*x**4-0.15*x**3-0.5*x**2-0.25*x+1.2#\n",
    "    return y\n",
    "xi=0#\n",
    "xf=1#\n",
    "h=xf-xi#\n",
    "fi=f(xi)##function value at xi\n",
    "ffa=f(xf)##actual function value at xf\n",
    "\n",
    "#for n=0, i.e, zero order approximation\n",
    "ff=fi#\n",
    "Et_1=ffa-ff##truncation error at x=1\n",
    "print \"The value of f at x=0 :\",fi\n",
    "print \"The value of f at x=1 due to zero order approximation :\",ff\n",
    "print \"Truncation error :\",Et_1\n",
    "print \"----------------------------------------------\"\n",
    "\n",
    "#for n=1, i.e, first order approximation\n",
    "def f1(x):\n",
    "    y=derivative(f,x)\n",
    "    return y\n",
    "f1i=f1(xi)##value of first derivative of function at xi\n",
    "f1f=fi+f1i*h##value of first derivative of function at xf\n",
    "Et_2=ffa-f1f##truncation error at x=1\n",
    "print \"The value of first derivative of f at x=0 :\",f1i\n",
    "print \"The value of f at x=1 due to first order approximation :\",f1f\n",
    "print \"Truncation error :\",Et_2\n",
    "print \"----------------------------------------------\"\n",
    "\n",
    "\n",
    "#for n=2, i.e, second order approximation\n",
    "def f2(x):\n",
    "    y=derivative(f1,x)\n",
    "    return y\n",
    "f2i=f2(xi)##value of second derivative of function at xi\n",
    "f2f=f1f+f2i*(h**2)/factorial(2)##value of second derivative of function at xf\n",
    "Et_3=ffa-f2f##truncation error at x=1\n",
    "print \"The value of second derivative of f at x=0 :\",f2i\n",
    "print \"The value of f at x=1 due to second order approximation :\",f2f\n",
    "print \"Truncation error :\",Et_3\n",
    "print \"----------------------------------------------\"\n",
    "\n",
    "#for n=3, i.e, third order approximation\n",
    "def f3(x):\n",
    "    y=derivative(f2,x)\n",
    "    return y\n",
    "f3i=f3(xi)##value of third derivative of function at xi\n",
    "f3f=f2f+f3i*(h**3)/factorial(3)##value of third derivative of function at xf\n",
    "Et_4=ffa-f3f##truncation error at x=1\n",
    "print \"The value of third derivative of f at x=0 :\",f3i\n",
    "print \"The value of f at x=1 due to third order approximation :\",f3f\n",
    "print \"Truncation error :\", Et_4\n",
    "print \"----------------------------------------------\"\n",
    "\n",
    "#for n=4, i.e, fourth order approximation\n",
    "def f4(x):\n",
    "    y=derivative(f3,x)\n",
    "    return y\n",
    "f4i=f4(xi)##value of fourth derivative of function at xi\n",
    "f4f=f3f+f4i*(h**4)/factorial(4)##value of fourth derivative of function at xf\n",
    "Et_5=ffa-f4f##truncation error at x=1\n",
    "print \"The value of fourth derivative of f at x=0 :\",f4i\n",
    "print \"The value of f at x=1 due to fourth order approximation :\",f4f\n",
    "print \"Truncation error :\",Et_5\n",
    "print \"----------------------------------------------\""
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example 4.2:  Page No:82"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The value of f at x=1 due to zero order approximation : 0.707106781187\n",
      "% relative error : -41.4213562373\n",
      "----------------------------------------------\n",
      "The value of f at x=1 due to first order approximation : 0.551333569463\n",
      "% relative error : -10.2667138927\n",
      "----------------------------------------------\n",
      "The value of f at x=1 due to second order approximation : 0.534175415889\n",
      "% relative error : -6.83508317772\n",
      "----------------------------------------------\n",
      "The value of f at x=1 due to third order approximation : 0.535435376789\n",
      "% relative error : -7.08707535775\n",
      "----------------------------------------------\n",
      "The value of f at x=1 due to fourth order approximation : 0.535504768061\n",
      "% relative error : -7.10095361216\n",
      "----------------------------------------------\n",
      "The value of f at x=1 due to fifth order approximation : 0.535501917392\n",
      "% relative error : -7.10038347839\n",
      "----------------------------------------------\n",
      "The value of f at x=1 due to sixth order approximation : 0.535501819651\n",
      "% relative error : -7.10036393016\n",
      "----------------------------------------------\n"
     ]
    }
   ],
   "source": [
    "from math import pi,cos,factorial\n",
    "from scipy.misc import derivative\n",
    "def f(x):\n",
    "    y=cos(x)\n",
    "    return y\n",
    "xi=pi/4#\n",
    "xf=pi/3#\n",
    "h=xf-xi#\n",
    "fi=f(xi)##function value at xi\n",
    "ffa=f(xf)##actual function value at xf\n",
    "\n",
    "#for n=0, i.e, zero order approximation\n",
    "ff=fi#\n",
    "et1=(ffa-ff)*100/ffa##percent relative error at x=1\n",
    "print \"The value of f at x=1 due to zero order approximation :\",ff\n",
    "print \"% relative error :\",et1\n",
    "print \"----------------------------------------------\"\n",
    "\n",
    "#for n=1, i.e, first order approximation\n",
    "def f1(x):\n",
    "    y=derivative(f,x)\n",
    "    return y\n",
    "f1i=f1(xi)##value of first derivative of function at xi\n",
    "f1f=fi+f1i*h##value of first derivative of function at xf\n",
    "et2=(ffa-f1f)*100/ffa##% relative error at x=1\n",
    "print \"The value of f at x=1 due to first order approximation :\",f1f\n",
    "print \"% relative error :\",et2\n",
    "print \"----------------------------------------------\"\n",
    "\n",
    "\n",
    "#for n=2, i.e, second order approximation\n",
    "def f2(x):\n",
    "    y=derivative(f1,x)\n",
    "    return y\n",
    "f2i=f2(xi)##value of second derivative of function at xi\n",
    "f2f=f1f+f2i*(h**2)/factorial(2)##value of second derivative of function at xf\n",
    "et3=(ffa-f2f)*100/ffa##% relative error at x=1\n",
    "print \"The value of f at x=1 due to second order approximation :\",f2f\n",
    "print \"% relative error :\",et3\n",
    "print \"----------------------------------------------\"\n",
    "\n",
    "\n",
    "#for n=3, i.e, third order approximation\n",
    "def f3(x):\n",
    "    y=derivative(f2,x)\n",
    "    return y\n",
    "f3i=f3(xi)##value of third derivative of function at xi\n",
    "f3f=f2f+f3i*(h**3)/factorial(3)##value of third derivative of function at xf\n",
    "et4=(ffa-f3f)*100/ffa##% relative error at x=1\n",
    "print \"The value of f at x=1 due to third order approximation :\",f3f\n",
    "print \"% relative error :\",et4\n",
    "print \"----------------------------------------------\"\n",
    "\n",
    "\n",
    "#for n=4, i.e, fourth order approximation\n",
    "def f4(x):\n",
    "    y=derivative(f3,x)\n",
    "    return y\n",
    "f4i=f4(xi)##value of fourth derivative of function at xi\n",
    "f4f=f3f+f4i*(h**4)/factorial(4)##value of fourth derivative of function at xf\n",
    "et5=(ffa-f4f)*100/ffa##% relative error at x=1\n",
    "print \"The value of f at x=1 due to fourth order approximation :\",f4f\n",
    "print \"% relative error :\",et5\n",
    "print \"----------------------------------------------\"\n",
    "\n",
    "\n",
    "#for n=5, i.e, fifth order approximation\n",
    "f5i=(f4(1.1*xi)-f4(0.9*xi))/(2*0.1)##value of fifth derivative of function at xi (central difference method)\n",
    "f5f=f4f+f5i*(h**5)/factorial(5)##value of fifth derivative of function at xf\n",
    "et6=(ffa-f5f)*100/ffa##% relative error at x=1\n",
    "print \"The value of f at x=1 due to fifth order approximation :\",f5f\n",
    "print \"% relative error :\",et6\n",
    "print \"----------------------------------------------\"\n",
    "\n",
    "\n",
    "#for n=6, i.e, sixth order approximation\n",
    "def f6(x):\n",
    "    y=derivative(f5,x)\n",
    "    return y\n",
    "f6i=(f4(1.1*xi)-2*f4(xi)+f4(0.9*xi))/(0.1**2)##value of sixth derivative of function at xi (central difference method)\n",
    "f6f=f5f+f6i*(h**6)/factorial(6)##value of sixth derivative of function at xf\n",
    "et6=(ffa-f6f)*100/ffa##% relative error at x=1\n",
    "print \"The value of f at x=1 due to sixth order approximation :\",f6f\n",
    "print \"% relative error :\", et6\n",
    "print \"----------------------------------------------\""
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example 4.3 :  Page No:85"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Input value of m:4\n",
      "Input value of h:5\n",
      "\n",
      "Remainder: 21 \n",
      "The value by first order approximation: 1275\n",
      "True Value at x2: 1296\n"
     ]
    }
   ],
   "source": [
    "from math import pi,cos,factorial\n",
    "m=input(\"Input value of m:\")\n",
    "h=input(\"Input value of h:\")\n",
    "def f(x):\n",
    "    y=x**m\n",
    "    return y\n",
    "x1=1#\n",
    "x2=x1+h#\n",
    "fx1=f(x1)#\n",
    "fx2=fx1+m*(fx1**(m-1))*h#\n",
    "if m==1:\n",
    "    R=0#\n",
    "elif m==2 :\n",
    "    R=2*(h**2)/factorial(2)#\n",
    "    \n",
    "elif m==3:\n",
    "    R=(6*(x1)*(h**2)/factorial(2))+(6*(h**3)/factorial(3))#\n",
    "    \n",
    "elif m==4:\n",
    "    R=(12*(x1**2)*(h**2)/factorial(2))+(24*(x1)*(h**3)/factorial(3))+(24*(h**4)/factorial(4))\n",
    "  \n",
    "print \"\\nRemainder:\",fx2,\"\\nThe value by first order approximation:\",R\n",
    "print \"True Value at x2:\",f(x2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example 4.4:  Page No:92"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Input h:1.232323\n",
      "For h= 1.232323\n",
      "and percent error= -2.70944264922 Derivative at x by forward difference method= 114.60931875\n",
      "and percent error= -0.178591334206 Derivative at x by backward difference method= 85.854151746\n",
      "and percent error= -1.44401699172 Derivative at x by central difference method= 14.3775835022\n"
     ]
    }
   ],
   "source": [
    "from scipy.misc import derivative\n",
    "def f(x):\n",
    "    y=-0.1*(x**4)-0.15*(x**3)-0.5*(x**2)-0.25*(x)+1.2\n",
    "    return y\n",
    "x=0.5#\n",
    "h=input(\"Input h:\")\n",
    "x1=x-h#\n",
    "x2=x+h#\n",
    "#forward difference method\n",
    "fdx1=(f(x2)-f(x))/h##derivative at x\n",
    "et1=abs((fdx1-derivative(f,x))/derivative(f,x))*100#\n",
    "#backward difference method\n",
    "fdx2=(f(x)-f(x1))/h##derivative at x\n",
    "et2=abs((fdx2-derivative(f,x))/derivative(f,x))*100#\n",
    "#central difference method\n",
    "fdx3=(f(x2)-f(x1))/(2*h)##derivative at x\n",
    "et3=abs((fdx3-derivative(f,x))/derivative(f,x))*100#\n",
    "print \"For h=\",h\n",
    "print \"and percent error=\",fdx1,\"Derivative at x by forward difference method=\",et1\n",
    "print \"and percent error=\",fdx2,\"Derivative at x by backward difference method=\",et2\n",
    "print \"and percent error=\",fdx3,\"Derivative at x by central difference method=\",et3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example 4.5:  Page No: 95"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "true value is between :  15.4275 and 15.8225\n"
     ]
    }
   ],
   "source": [
    "from scipy.misc import derivative\n",
    "def f(x):\n",
    "    y=x**3\n",
    "    return y\n",
    "x=2.5#\n",
    "delta=0.01#\n",
    "deltafx=abs(derivative(f,x))*delta#\n",
    "fx=f(x)#\n",
    "print \"true value is between : \",fx-deltafx,\"and\",fx+deltafx"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example 4.6:  Page No: 96"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The value of y is between: 0.528721343471 and 0.596278656529\n",
      "ymin is calculated at lower extremes of F, L, E, I values as = 0.524066539965\n",
      "ymax is calculated at higher extremes of F, L, E, I values as = 0.602846335915\n"
     ]
    }
   ],
   "source": [
    "def f(F,L,E,I):\n",
    "    y=(F*(L**4))/(8*E*I)\n",
    "    return y\n",
    "Fbar=50##lb/ft\n",
    "Lbar=30##ft\n",
    "Ebar=1.5*(10**8)##lb/ft**2\n",
    "Ibar=0.06##ft**4\n",
    "deltaF=2##lb/ft\n",
    "deltaL=0.1##ft\n",
    "deltaE=0.01*(10**8)##lb/ft**2\n",
    "deltaI=0.0006##ft**4\n",
    "ybar=(Fbar*(Lbar**4))/(8*Ebar*Ibar)#\n",
    "def f1(F):\n",
    "    y=(F*(Lbar**4))/(8*Ebar*Ibar)\n",
    "    return y\n",
    "def f2(L):\n",
    "    y=(Fbar*(L**4))/(8*Ebar*Ibar)\n",
    "    return y\n",
    "def f3(E):\n",
    "    y=(Fbar*(Lbar**4))/(8*E*Ibar)\n",
    "    return y\n",
    "def f4(I):\n",
    "    y=(Fbar*(Lbar**4))/(8*Ebar*I)\n",
    "    return y\n",
    "\n",
    "deltay=abs(derivative(f1,Fbar))*deltaF+abs(derivative(f2,Lbar))*deltaL+abs(derivative(f3,Ebar))*deltaE+abs(derivative(f4,Ibar))*deltaI#\n",
    "\n",
    "print \"The value of y is between:\",ybar-deltay,\"and\",ybar+deltay\n",
    "ymin=((Fbar-deltaF)*((Lbar-deltaL)**4))/(8*(Ebar+deltaE)*(Ibar+deltaI))#\n",
    "ymax=((Fbar+deltaF)*((Lbar+deltaL)**4))/(8*(Ebar-deltaE)*(Ibar-deltaI))#\n",
    "print \"ymin is calculated at lower extremes of F, L, E, I values as =\",ymin\n",
    "print \"ymax is calculated at higher extremes of F, L, E, I values as =\",ymax"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example 4.7 :  Page No:98"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The condition number of function for x = 0.18201112073 is : 1.72787595947\n",
      "The condition number of function for x = 0.0160083243793 is : 1.58650429006\n"
     ]
    }
   ],
   "source": [
    "from math import pi,tan\n",
    "from scipy.misc import derivative\n",
    "def f(x):\n",
    "    y=tan(x)\n",
    "    return y\n",
    "x1bar=(pi/2)+0.1*(pi/2)#\n",
    "x2bar=(pi/2)+0.01*(pi/2)#\n",
    "#computing condition number for x1bar\n",
    "condnum1=x1bar*derivative(f,x1bar)/f(x1bar)#\n",
    "print \"The condition number of function for x =\",condnum1,\"is :\",x1bar\n",
    "if abs(condnum1)>1:\n",
    "    print \"Function is ill-conditioned for x =\",x1bar\n",
    "\n",
    "#computing condition number for x2bar\n",
    "condnum2=x2bar*derivative(f,x2bar)/f(x2bar)#\n",
    "print \"The condition number of function for x =\",condnum2,\"is :\",x2bar\n",
    "if abs(condnum2)>1:\n",
    "    print \"Function is ill-conditioned for x =\",x2bar"
   ]
  }
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