Numerical Integration.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Numerical Integration\n",
    "_By Dhruv Jain_\n",
    "\n",
    "### **Objective: Implementaion of various numerical integration schemes**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Key libraries: Numpy(for mathematical procedures) and matplotlib(to create plots)\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt \n",
    "import copy\n",
    "from scipy.integrate import quad"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Midpoint Rule, Trapezoidal Rule, Simpson's Rule"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "def mid_pt_int(func,a,b):\n",
    "    \"\"\"Dhruv Jain, 18 Oct 2021\n",
    "    Obj: Use Midpoint Rule to compute the integral of a function\n",
    "    Args:\n",
    "        func: function, function that will be integrated from a to b\n",
    "        a: float, lowe limit of the integral\n",
    "        b: float, upper limit of the integral\n",
    "    Output:\n",
    "        I: float, integral of function func from limit a to b\n",
    "    \"\"\"\n",
    "    I = (b-a)*func((b+a)/2)\n",
    "    \n",
    "    return I\n",
    "\n",
    "def trapezoidal_int(func,a,b):\n",
    "    \"\"\"Dhruv Jain, 18 Oct 2021\n",
    "    Obj: Use Trapezoidal Rule to compute the integral of a function\n",
    "    Args:\n",
    "        func: function, function that will be integrated from a to b\n",
    "        a: float, lowe limit of the integral\n",
    "        b: float, upper limit of the integral\n",
    "    Output:\n",
    "        I: float, integral of function func from limit a to b\n",
    "    \"\"\"\n",
    "    I = (b-a)/2*(func(b)+func(a))\n",
    "    \n",
    "    return I\n",
    "\n",
    "def simpson_int(func,a,b):\n",
    "    \"\"\"Dhruv Jain, 18 Oct 2021\n",
    "    Obj: Use Simpson's Rule to compute the integral of a function\n",
    "    Args:\n",
    "        func: function, function that will be integrated from a to b\n",
    "        a: float, lowe limit of the integral\n",
    "        b: float, upper limit of the integral\n",
    "    Output:\n",
    "        I: float, integral of function func from limit a to b\n",
    "    \"\"\"\n",
    "    I = (b-a)/6*(func(b)+4*func((b+a)/2)+func(a))\n",
    "    return I"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# #Example 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Analytical Integral of function: 16.566371\n",
      "\n",
      "Midpoint Rule Integral of function: 17.009254\n",
      "Absol Err of Function: 0.442883\n",
      "\n",
      "Trapezoidal Rule Integral of function: 15.707963\n",
      "Absol Err of Function: 0.858407\n",
      "\n",
      "Simpson's Rule Integral of function: 16.575490\n",
      "Absol Err of Function: 0.009120\n"
     ]
    }
   ],
   "source": [
    "def func2_2(x): # Function 2\n",
    "    return 8+4*np.cos(x)\n",
    "\n",
    "# Using scipy.integrate -> quad to find analytical limit integral\n",
    "a_2_2 = 0 # Lower Limit of Function 1\n",
    "b_2_2 = np.pi/2 # Upper Limit of Function 1\n",
    "I2_a_2 = quad(func2_2, 0, np.pi/2)\n",
    "\n",
    "print('Analytical Integral of function: %0.6f'%I2_a_2[0])\n",
    "\n",
    "# Midpoint Rule\n",
    "I2_b_2 = mid_pt_int(func2_2, a_2_2, b_2_2)\n",
    "print('\\nMidpoint Rule Integral of function: %0.6f'%I2_b_2)\n",
    "err_b_2 = abs(I2_a_2[0]-I2_b_2)\n",
    "print('Absol Err of Function: %0.6f'%err_b_2)\n",
    "\n",
    "# Trapezoidal Rule\n",
    "I2_c_2 = trapezoidal_int(func2_2, a_2_2, b_2_2)\n",
    "print('\\nTrapezoidal Rule Integral of function: %0.6f'%I2_c_2)\n",
    "err_c_2 = abs(I2_a_2[0]-I2_c_2)\n",
    "print('Absol Err of Function: %0.6f'%err_c_2)\n",
    "\n",
    "# Simpson's Rule\n",
    "I2_d_2 = simpson_int(func2_2, a_2_2, b_2_2)\n",
    "print('\\nSimpson\\'s Rule Integral of function: %0.6f'%I2_d_2)\n",
    "err_d_2 = abs(I2_a_2[0]-I2_d_2)\n",
    "print('Absol Err of Function: %0.6f'%err_d_2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Forward Euler Method"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Forward Euler Method (Explicit Method)\n",
    "def forward_euler(t0,tf, h, y0, func):\n",
    "    \"\"\"\"Dhruv Jain, 31 Oct, 2021\n",
    "    Obj: Uses Forward Euler's Method [y(t{k+1}) = y(tk) + h*f(yk,tk)] to solve IVP\n",
    "    Args:\n",
    "        t0: initial time, float\n",
    "        tf: final time, float\n",
    "        h: step size, float\n",
    "        y0: Initial Condition, float\n",
    "        func: evaluation function that describes y' and on which the bisection method is applied \n",
    "    Output:\n",
    "        soln: dict, evaluated y values ['y'] and respective time steps ['t'], ndarray\n",
    "    \"\"\"\n",
    "    # Calculate len(y), time steps, zero array for y\n",
    "    N = int((tf-t0)/h)+1 \n",
    "    tlen = np.linspace(t0,tf,N)\n",
    "    yval = np.zeros((N),dtype = np.float64)\n",
    "    \n",
    "    # Setup count, use IC\n",
    "    count = 1\n",
    "    yval[0] = y0 \n",
    "           \n",
    "    while N > count:       \n",
    "        yval[count] = yval[count-1] + h*func(yval[count-1],tlen[count-1]) # FORWARD EULER's METHOD\n",
    "        count = count + 1\n",
    "    \n",
    "    soln = {}\n",
    "    soln['y'] = yval\n",
    "    soln['t'] = tlen\n",
    "   \n",
    "    return soln"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Backward Euler Method"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Backward Euler Method (Implicit Method)\n",
    "def backward_euler(t0, tf, h, y0, func, dfunc_yk1):\n",
    "    \"\"\"\"Dhruv Jain, 31 Oct, 2021\n",
    "    Obj: Uses Backward Euler's Method [y(t{k+1}) = y(tk) + h*f(yk,tk)] to solve IVP\n",
    "    Args:\n",
    "        t0: initial time, float\n",
    "        tf: final time, float\n",
    "        h: step size, float\n",
    "        y0: Initial Condition, float\n",
    "        func: evaluation function that describes y' and on which the bisection method is applied \n",
    "        dfunc_yk1: \n",
    "    Output:\n",
    "        soln: dict, evaluated y values ['y'] and respective time steps ['t'], ndarray\n",
    "    \"\"\"\n",
    "    \n",
    "    def backward_euler_root_f(y,yprev,t,h): # Use same func as the one passed into backward_euler()\n",
    "        \"\"\"\"Dhruv Jain, 31 Oct, 2021, MSAAE Purdue\n",
    "        Obj: f = yk+1 - (yk + h*f(yk+1,tk+1)) = 0, setup to use nonlinear algebraic solver to get yk+1\n",
    "        Args:\n",
    "            y: yk+1, float  \n",
    "            yprev: yk, float\n",
    "            t: tk+1, float\n",
    "            h: step size, float\n",
    "        Output:\n",
    "           yk+1 - (yk + h*f(yk+1,tk+1))\n",
    "        \"\"\"\n",
    "        return y - (yprev + h*func(y,t))\n",
    "\n",
    "    \n",
    "    # Calculate len(y), time steps, zero array for y\n",
    "    N = int((tf-t0)/h)+1 \n",
    "    tlen = np.linspace(t0,tf,N)\n",
    "    yval = np.zeros((N),dtype = np.float64)\n",
    "    \n",
    "    # Setup count, use IC\n",
    "    count = 1\n",
    "    yval[0] = y0 \n",
    "           \n",
    "    while N > count:       \n",
    "        yval[count] = yval[count-1] # Use yk as the inital guess of yk+1 \n",
    "        \n",
    "        roots,iterat = newton(yval[count], tlen[count], h, 1e-12, 50, backward_euler_root_f, dfunc_yk1) # pass yk+1 = yk (Initial Guess), tk+1 \n",
    "        yval[count] = roots[-1]\n",
    "        \n",
    "        count = count + 1\n",
    "    \n",
    "    soln = {}\n",
    "    soln['y'] = yval\n",
    "    soln['t'] = tlen\n",
    "   \n",
    "    return soln\n",
    "\n",
    "# Newton's Method for Backward Euler's Method\n",
    "def newton(po, ti, h, tol, Nmax, func, dfunc):\n",
    "    \"\"\"Dhruv Jain, 7th Sept 2021\n",
    "    This function uses Newton's method to obtain the zero of the funcion using a set inital guess for a certain tolerance and/or # iterations \n",
    "    Args:\n",
    "        po: inital guess = yk, float\n",
    "        ti: tk+1, float\n",
    "        h: step size, float\n",
    "        tol: absolute tolerance, float\n",
    "        Nmax: Maximum allowable iteration, positive int\n",
    "        func: Function on which the Newton's method will be applied\n",
    "        dfunc: derivative of yk+1 - (yk + h*f(yk+1, tk+1)) wrt yk+1 for Newton's Method \n",
    "    Output: \n",
    "        root: Contains all the intermediate x_k+1 where the last value is the root based on Nmax and tol\n",
    "        count: # iterations for which Newton's method is implemented, positive int\n",
    "    \"\"\"\n",
    "    def g(val):\n",
    "        return val - func(val, po, ti, h)/dfunc(val, ti, h)\n",
    "    pk = np.zeros(Nmax+1)\n",
    "    \n",
    "    count = 0\n",
    "    pk[0] = po \n",
    "    if count < Nmax:\n",
    "        pk[count+1] = g(pk[count])\n",
    "        count = count + 1\n",
    "    else: \n",
    "        return None, None\n",
    "    while count < Nmax and np.abs(pk[count] - pk[count-1]) > tol:\n",
    "        pk[count+1] = g(pk[count])\n",
    "        count = count + 1\n",
    "    roots = pk[:count+1]\n",
    "    return roots, count  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# #Example 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Double Precision \n",
    "h = np.array([0.1, 0.01, 0.001, 0.00001],dtype = np.float64)\n",
    "t0 = 0\n",
    "tf = 1\n",
    "y0 = 0\n",
    "\n",
    "def p1f(y,t):\n",
    "    \"\"\"\"Dhruv Jain, 31 Oct, 2021\n",
    "    Obj: y' expression \n",
    "    Args:\n",
    "        y: yval to evaluate y', float\n",
    "        t: time @ which y' to be evaluated, float\n",
    "    Output:\n",
    "        ydot: y', float\n",
    "    \"\"\"\n",
    "    ydot = (np.sin(y) - np.exp(t))/np.cos(t)\n",
    "    return ydot\n",
    "\n",
    "def d_p1f_yk1(y,t,h):\n",
    "    \"\"\"\"Dhruv Jain, 31 Oct, 2021\n",
    "    Obj: Derivative of yk+1 - (yk + h*f(yk+1, tk+1)) wrt yk+1 for Newton's Method, to find the root of nonlinear algebraic expression calculated in BACKWARD EULER'S METHOD\n",
    "             = 1 - ( 0 +h*cos(yk+1)/cos(tk+1))\n",
    "    Args:\n",
    "        y: yk+1, float\n",
    "        t: tk+1, float\n",
    "        h: step size, float\n",
    "    Output:\n",
    "        yd: derivative of yk+1 - (yk + h*f(yk+1, tk+1)) wrt yk+1, float\n",
    "    \"\"\"\n",
    "    yd = 1-h*np.cos(y)/np.cos(t)\n",
    "    return yd"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Forward Euler Method\n",
    "sol1_a = []\n",
    "\n",
    "for i in range(len(h)):\n",
    "    ya = forward_euler(t0,tf,h[i],y0,p1f) \n",
    "    sol1_a.append(ya)\n",
    "    \n",
    "#Plot\n",
    "for i in range(len(h)-1):\n",
    "    plt.figure(i)\n",
    "    plt.title('IVP-Forward Euler\\'s Method using different h')\n",
    "    plt.plot(sol1_a[i]['t'][:],sol1_a[i]['y'][:],marker='.',label='h='+str(h[i]))\n",
    "    plt.plot(sol1_a[i+1]['t'][:],sol1_a[i+1]['y'][:],marker='.',label='h='+str(h[i+1]))\n",
    "    plt.ylabel('y')\n",
    "    plt.xlabel('t')\n",
    "    plt.grid()\n",
    "    plt.legend()\n",
    "\n",
    "i = i+1\n",
    "plt.figure(i)\n",
    "plt.title('IVP-Forward Euler\\'s Method using different h')\n",
    "plt.plot(sol1_a[0]['t'][:],sol1_a[0]['y'][:],marker='.',label='h='+str(h[0]))\n",
    "plt.plot(sol1_a[i]['t'][:],sol1_a[i]['y'][:],marker='.',label='h='+str(h[i]))\n",
    "plt.ylabel('y')\n",
    "plt.xlabel('t')\n",
    "plt.grid()\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "h,  y(tf){hmin}-y(tf){hi}\n",
      "0.1 0.217156\n",
      "0.01 0.019768\n",
      "0.001 0.001918\n",
      "1e-05 0.0\n"
     ]
    }
   ],
   "source": [
    "print('h,  y(tf){hmin}-y(tf){hi}')\n",
    "for i in range(len(h)):\n",
    "    print(h[i], round(sol1_a[i]['y'][-1]-sol1_a[-1]['y'][-1],6))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "From the above plots and the error in the final term estimated through different steps and the min step decreases. We cannot strictly comment on the accuracy of the solution as no exact solution exists, however, comparing the estimated solution found through various step sizes, we can conclude that as the step stize decreases the values are getting more accurate to a solution close to the one found using the minimum step size(h = 1e-5). The same is true for convergernce and stability as the final estimated value found from different step sizes seem to converge to a value close to the found using h=1e-5 and thus we can even say that the system is stable at the least. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7fd9757aa6a0>"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Backward Euler\n",
    "sol1_b = []\n",
    "\n",
    "for i in range(len(h)):\n",
    "    yb = backward_euler(t0,tf, h[i], y0, p1f, d_p1f_yk1)\n",
    "    sol1_b.append(yb)\n",
    "    \n",
    "#Plot\n",
    "for i in range(len(h)-1):\n",
    "    plt.figure(i)\n",
    "    plt.title('HW 5, P1 (b): IVP-Backward Euler\\'s Method using different h')\n",
    "    plt.scatter(sol1_b[i]['t'][:],sol1_b[i]['y'][:],marker='.',label='h='+str(h[i]))\n",
    "    plt.scatter(sol1_b[i+1]['t'][:],sol1_b[i+1]['y'][:],marker='.',label='h='+str(h[i+1]))\n",
    "    plt.ylabel('y')\n",
    "    plt.xlabel('t')\n",
    "    plt.grid()\n",
    "    plt.legend()\n",
    "\n",
    "i = i+1\n",
    "plt.figure(i)\n",
    "plt.title('HW 5, P1 (b): IVP-Backward Euler\\'s Method using different h')\n",
    "plt.scatter(sol1_b[0]['t'][:],sol1_b[0]['y'][:],marker='.',label='h='+str(h[0]))\n",
    "plt.scatter(sol1_b[i]['t'][:],sol1_b[i]['y'][:],marker='.',label='h='+str(h[i]))\n",
    "plt.ylabel('y')\n",
    "plt.xlabel('t')\n",
    "plt.grid()\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Runge-Kutta $4^{th}$ order Method"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Fourth-Order Runge-Kutta Method for n-order -> n equations\n",
    "def rk4(h, t0, tf, y0,func):\n",
    "    \"\"\"\"Dhruv Jain, 6 Nov, 2021\n",
    "    Obj: Uses Fourth-Order Runge-Kutta Method [y(t{k+1}) = y(tk) + h*(k1 +2k2 + 2k3 + k4)/6] to solve IVP for n equations based on len(y0)\n",
    "    Args:\n",
    "        h: step size, float\n",
    "        t0: startting time, float\n",
    "        tf: final time, float\n",
    "        y0: inital conditions of all teh states, float ndarray\n",
    "        func: state space representation of all the equations(functions)\n",
    "    Output:\n",
    "        soln: dict, evaluated yi values ['y'] and respective time steps ['t'], ndarray\n",
    "    \"\"\"\n",
    "    N = int((tf-t0)/h)+1 \n",
    "    tlen = np.linspace(t0,tf,N)\n",
    "    yval = np.zeros((N,len(y0)),dtype = np.float64)\n",
    "    \n",
    "    # Setup count, use IC\n",
    "    count = 1\n",
    "    yval[0,:] = y0 \n",
    "           \n",
    "    while N > count:\n",
    "        #Calculate the coefficients K1, K2, K3, K4\n",
    "        k1 = func(yval[count-1], tlen[count-1])\n",
    "\n",
    "        k2 = func(yval[count-1] + 0.5*h*k1, tlen[count-1]+0.5*h)\n",
    "\n",
    "        k3 = func(yval[count-1] + 0.5*h*k2, tlen[count-1]+0.5*h)\n",
    "\n",
    "        k4 = func( yval[count-1] + h*k3, tlen[count-1]+h)\n",
    "\n",
    "        yval[count] = yval[count-1] +h*(k1 +2*k2 +2*k3 +k4)/6 # RK 4th Order Method\n",
    "        \n",
    "        count = count +1\n",
    "    soln = {}\n",
    "    soln['y'] = yval\n",
    "    soln['t'] = tlen\n",
    "    return soln"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Example 3\n",
    "EOMs: <br> \n",
    "1) $m\\ddot{x} = -D|v|\\dot{x}$<br>\n",
    "2) $m\\ddot{y} = -D|v|\\dot{y} - mg + F_n$<br>\n",
    "\n",
    "To numerically integrate, we can rewrite the equation as system of first-order DE:<br>\n",
    "$u_1 = x$<br>\n",
    "$u_2 = y$ <br>\n",
    "$u_3 = \\dot{x}$<br>\n",
    "$u_4 = \\dot{y}$ <br>\n",
    "\n",
    "$\\dot{u_1} = \\dot{x} = u_3$<br>\n",
    "$\\dot{u_2} = \\dot{y} = u_4$<br>\n",
    "$\\dot{u_3} = \\frac{-D|v|u_3}{m}$ <br>\n",
    "$\\dot{u_4} = \\frac{-D|v|u_4 - mg + F_n}{m}$ <br>\n",
    "\n",
    "where,<br>\n",
    "$v = \\sqrt{u_3^2+u_4^2}$<br>  \n",
    "$F_n = k(R-u_2(t))$, u2 < R<br>\n",
    "       0, u2 $\\ge$ R"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Given values\n",
    "k = 1000\n",
    "R = 0.2\n",
    "g = 9.81\n",
    "m = 0.3\n",
    "Ddum = [0, 0.01, 0.025]\n",
    "\n",
    "def func1_xy(y,t):\n",
    "    \"\"\"\"Dhruv Jain, 6 Nov, 2021\n",
    "    Obj: State Space Formulation of the problem\n",
    "    Args:\n",
    "        y: yval @ tcurrent, float ndarray\n",
    "        t: time @ which y' to be evaluated, float\n",
    "    Output:\n",
    "        dydt: y' of each state in the setup, float ndarray\n",
    "    \"\"\"\n",
    "    \n",
    "    dydt = np.zeros((len(y)))\n",
    "    dydt[0] = y[2]\n",
    "    dydt[1] = y[3]\n",
    "    \n",
    "    v = np.sqrt(y[2]**2+y[3]**2)\n",
    "    if y[1] < R:\n",
    "        Fn = k*(R-y[1]) \n",
    "    else:\n",
    "        Fn = 0 \n",
    "        \n",
    "    dydt[2] = (-D*v*y[2])/m\n",
    "    dydt[3] = (-D*v*y[3]-m*g + Fn)/m\n",
    "    \n",
    "    return dydt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7fd975721940>"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "t0 = 0\n",
    "tf = 8\n",
    "h = 1e-4\n",
    "ic_xy = np.array([0,2,10,5], dtype=np.float64) # x0 = 0; h = y0 = 2m; vx0 = 10 m/s; vy0 = 5 m/s\n",
    "\n",
    "sol1_xy = []\n",
    "\n",
    "for i in range(len(Ddum)):\n",
    "    D = Ddum[i]\n",
    "    solrk4 = rk4(h, t0, tf, ic_xy,func1_xy)\n",
    "    sol1_xy.append(solrk4)\n",
    "    \n",
    "plt.figure(1)\n",
    "plt.title('Plot of Trajectory for D = [0, 0.01, 0.025]')\n",
    "for i in range(len(Ddum)):\n",
    "    plt.plot(sol1_xy[i]['y'][:,0],sol1_xy[i]['y'][:,1],marker='.',label='D='+str(Ddum[i]))\n",
    "plt.ylabel('y (m)')\n",
    "plt.xlabel('x (m)')\n",
    "plt.grid()\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Composite Trapezoidal Rule"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Newton Interpolating Polynomial Function: \n",
    "def newton_interp(xi,fi,x_interp):\n",
    "    \"\"\"Dhruv Jain, 18 Oct, 2021\n",
    "    Obj: Use Newton Form of the Interpolating Polynomial to find f(x_i) for given x_i using given (x,y) data points  \n",
    "    Args:\n",
    "        xi: (n x 1) vector, x-coordinates of the given data set\n",
    "        fi: (n x 1) vector, y-coordinates of the given data set\n",
    "        x_interp: (n_interp x 1) vector, given x-coordinate points for which we interpolate and finds its y-coordinates\n",
    "    Output:\n",
    "        f_interp: (n_iterp x n) vector, interpolated y-ccordinates of x_interp using Lagrange form of Interpolating Polynomial\n",
    "        coeff: (n-1 x 1) vector, coefficients of Newton form of Interpolating Polynomial: f[x0], f[x0,x1], f[x0,x1,x2].......,f[x0,x1,....,xn]\n",
    "    \"\"\"\n",
    "    n = len(xi)\n",
    "    coeff = np.zeros(n)\n",
    "    coeff[0] = copy.copy(fi[0]) # coeff contains all the f[x0], f[x0,x1]....f[x0,x1,.....,xn] terms \n",
    "    temp = copy.copy(fi)\n",
    "    \n",
    "    # Step 1: Compute f[x0], f[x0,x1] ... f[x0,x1,....xn] coeffecient values; independent of the points we want to interploate for\n",
    "    for i in range(1,n):\n",
    "        for j in range(n-i):\n",
    "            temp[j] = (temp[j+1]-temp[j])/(xi[j+i]-xi[j])\n",
    "        coeff[i] = temp[0]\n",
    "    \n",
    "    # Step 2: Multiply the numerator (x-x0)(x-x1)...(x-xn); dependent on x_interp, thus needs to be called len(x_interp) times\n",
    "    def newton_interp_num_coeff(xval):\n",
    "        \"\"\"Dhruv Jain, 18 Oct, 2021\n",
    "        Obj: Calculate and multiply (x-x0)(x-x1)...(x-xn) based on x_interp[i] value to get f_interp[i]\n",
    "        Args:\n",
    "            xval: float, value for which the interpolated value is calculated\n",
    "        Output:\n",
    "            sum: (n_iterp x n) vector, interpolated y-ccordinates of x_interp using Lagrange form of Interpolating Polynomial\n",
    "        \"\"\"\n",
    "        sum = coeff[0]\n",
    "        for i in range(1,n):\n",
    "            temp = coeff[i]*(xval-xi[i-1])\n",
    "            for j in range(1,i):\n",
    "                temp = temp*(xval-xi[i-j-1])\n",
    "            sum = sum + temp\n",
    "        return sum\n",
    "    \n",
    "    # Initialize and calculate y-coord for each x_interp element\n",
    "    f_interp = np.zeros(len(x_interp))\n",
    "    for i in range(len(x_interp)):\n",
    "        f_interp[i] = newton_interp_num_coeff(x_interp[i])   \n",
    "    \n",
    "    return f_interp, coeff"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "def compo_trap_rule(xdata, fdata):\n",
    "    \"\"\"Dhruv Jain, 18 Oct 2021\n",
    "    Obj: Computes Composite Trapezoidal Rule of a given n (x,y) points\n",
    "    Args:\n",
    "        xdata: nx1 numpy array float, x data\n",
    "        fdata: nx1 numpy array float, y data\n",
    "    Output:\n",
    "        ctr_integral: Composite Trapezoidal Rule (Integral)\n",
    "    \"\"\"\n",
    "    ctr_integral = 0\n",
    "    \n",
    "    n = len(xdata)\n",
    "    for i in range(n-1):\n",
    "        ctr_integral +=  (xdata[i+1]-xdata[i])/2*(fdata[i+1]+fdata[i]) # Compute Trapezoidal Rule Integral for each interval and sum them\n",
    "    \n",
    "    return ctr_integral    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# #Example 4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Composite Trapezoidal Rule with 5 equal-length intervals gives an Integral value:  10.75 Nm\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Sample data\n",
    "xdat = np.array([0,1,2,3,4,5])\n",
    "fdat = np.array([1.5,2.5,3,2.5,1.5,1])\n",
    "x_cont_dat = np.linspace(min(xdat),max(xdat),1000) # To create near-continous data set\n",
    "\n",
    "f_cont_dat, coeff = newton_interp(xdat, fdat, x_cont_dat) # Interpolated result\n",
    "\n",
    "n_trap = 6\n",
    "xdat_b = np.linspace(min(xdat),max(xdat),n_trap) # 6 data points for 5 equal-length intervals\n",
    "fdat_b,_ = newton_interp(xdat, fdat, xdat_b) # 101 f(x) using the Newton interpolating polynomial\n",
    "ctr_integral = compo_trap_rule(xdat_b, fdat_b)\n",
    "\n",
    "# Plot\n",
    "plt.figure(1)\n",
    "plt.title('Newton Interpolation Result')\n",
    "plt.scatter(xdat,fdat,color = 'black',label='Given data')\n",
    "plt.plot(x_cont_dat,f_cont_dat,label='Newton Interpolation Polynomial')\n",
    "plt.ylabel('F (N)')\n",
    "plt.xlabel('x (m)')\n",
    "plt.grid()\n",
    "plt.legend()\n",
    "print('Composite Trapezoidal Rule with',n_trap-1,'equal-length intervals gives an Integral value: ',ctr_integral,'Nm')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2-point Gaussian Quadrature Rule\n",
    "$G$(f) = $w_1f(x_1)$ + $w_2f(x_2)$ <br>\n",
    "We know for [-1,1] -> $w_1$ = $w_2$ = 1, $x_1$ = -$\\sqrt{1/3}$, $x_2$ = $\\sqrt{1/3}$ <br>\n",
    "We first map [a, b] -> [-1, 1] to use the above setup<br>\n",
    "\n",
    "x = $\\frac{(b-a)}{2}t$ + $\\frac{b+a}{2}$\n",
    "\n",
    "Jacobian = $\\frac{dx}{dt}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "def compute_2pt_gauss_quad(func,b,a):\n",
    "    \"\"\"Dhruv Jain, 10 November 2021\n",
    "    Obj: To compute 2 point Gaussian Quadrature Rule\n",
    "    Args: \n",
    "        func: function on which the gaussian quadrature rule will be applied \n",
    "        b: Upper limit of interval\n",
    "        a: Lower limit of interval\n",
    "    \"\"\"\n",
    "    def map_2_sq(b,a,t):\n",
    "        \"\"\"Dhruv Jain, 10 November 2021\n",
    "        Obj: Maps a value in interval [a,b] to [-1,1]\n",
    "        Args: \n",
    "            b: Upper limit of interval\n",
    "            a: Lower limit of interval\n",
    "            t: point to be mapped\n",
    "        \"\"\"\n",
    "        return (b-a)*0.5*t+(b+a)*0.5\n",
    "\n",
    "    def jacobian_2pt_gauss_quad(b,a):\n",
    "        \"\"\" Dhruv Jain, 10 November 2021\n",
    "        Obj: Jacobian to succesfully map to [-1,1] to apply 2-point gaussian quadrature\n",
    "        Args: \n",
    "            b: Upper limit of interval\n",
    "            a: Lower limit of interval\n",
    "        \"\"\"\n",
    "        return (b-a)/2\n",
    "    w1 = 1\n",
    "    w2 = 1\n",
    "    x1 = -np.sqrt(1/3)\n",
    "    x2 = np.sqrt(1/3)\n",
    "    jacobian = jacobian_2pt_gauss_quad(b,a)\n",
    "    return (w1*func(map_2_sq(b,a,x1)) + w2*func(map_2_sq(b,a,x2)))*jacobian"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3-point Gaussian Quadrature Rule\n",
    "For three-point Gauss Quadrature: <br>\n",
    "$G_3$(f) = $w_1f(x_1)$ + $w_2f(x_2)$ + $w_3f(x_3)$ <br>\n",
    "We know for [-1,1] -> $w_1$ = $w_3$ = 5/9, $w_2$ = 8/9, $x_1$ = -$\\sqrt{3/5}$, $x_2$ = 0, $x_3$ = $\\sqrt{3/5}$ <br>\n",
    "We first map [a, b] -> [-1, 1] to use the above setup<br>\n",
    "\n",
    "x = $\\frac{(b-a)}{2}t$ + $\\frac{b+a}{2}$\n",
    "\n",
    "Jacobian = $\\frac{dx}{dt}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "def compute_3pt_gauss_quad(func,b,a):\n",
    "    \"\"\"Dhruv Jain, 10 November 2021\n",
    "    Obj: To compute 3 point Gaussian Quadrature Rule\n",
    "    Args: \n",
    "        func: function on which the gaussian quadrature rule will be applied \n",
    "        b: Upper limit of interval\n",
    "        a: Lower limit of interval\n",
    "    \"\"\"\n",
    "    def map_2_sq(b,a,t):\n",
    "        \"\"\"Dhruv Jain, 10 November 2021\n",
    "        Obj: Maps a value in interval [a,b] to [-1,1]\n",
    "        Args: \n",
    "            b: Upper limit of interval\n",
    "            a: Lower limit of interval\n",
    "            t: point to be mapped\n",
    "        \"\"\"\n",
    "        return (b-a)*0.5*t+(b+a)*0.5\n",
    "\n",
    "    def jacobian_3pt_gauss_quad(b,a):\n",
    "        \"\"\" Dhruv Jain, 10 November 2021\n",
    "        Obj: Jacobian to succesfully map to [-1,1] to apply 2-point gaussian quadrature\n",
    "        Args: \n",
    "            b: Upper limit of interval\n",
    "            a: Lower limit of interval\n",
    "        \"\"\"\n",
    "        return (b-a)/2\n",
    "\n",
    "    w1 = 5/9\n",
    "    w2 = 8/9\n",
    "    w3 = 5/9\n",
    "    x1 = -np.sqrt(3/5)\n",
    "    x2 = 0\n",
    "    x3 = np.sqrt(3/5)\n",
    "    jacobian = jacobian_3pt_gauss_quad(b,a)\n",
    "    \n",
    "    return (w1*func(map_2_sq(b,a,x1)) + w2*func(map_2_sq(b,a,x2)) + w3*func(map_2_sq(b,a,x3)))*jacobian"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# #Example 5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "def func3(x):\n",
    "    \"\"\"\"Dhruv Jain, 6 Nov, 2021\n",
    "    Obj: y(x) in Example 4 using coeffecients found using Newton Interpolation\n",
    "    Args:\n",
    "        y(x): float, function value\n",
    "    \"\"\"\n",
    "    y = coeff[0] + coeff[1]*(x-xdat[0]) + coeff[2]*(x-xdat[0])*(x-xdat[1]) + coeff[3]*(x-xdat[0])*(x-xdat[1])*(x-xdat[2])+ coeff[4]*(x-xdat[0])*(x-xdat[1])*(x-xdat[2])*(x-xdat[3]) + coeff[5]*(x-xdat[0])*(x-xdat[1])*(x-xdat[2])*(x-xdat[3])*(x-xdat[4])\n",
    "    return y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Scipy.quad integral value of the interploated function of data defined in Example 4: 10.807291666666668\n",
      "2 point Gaussian Quadrature Rule of the interploated function of data defined in Example 4: 10.264756944444445\n",
      "3 point Gaussian Quadrature Rule of the interploated function of data defined in Example 4: 10.807291666666664\n"
     ]
    }
   ],
   "source": [
    "# Setup\n",
    "a = 0\n",
    "b = 5\n",
    "\n",
    "# Scipy.quad Integral of function\n",
    "Integral_quad = quad(func3, a, b)\n",
    "\n",
    "Integral_2pt_gq = compute_2pt_gauss_quad(func3,b,a)\n",
    "Integral_3pt_gq = compute_3pt_gauss_quad(func3,b,a)\n",
    "\n",
    "print('Scipy.quad integral value of the interploated function of data defined in Example 4:',Integral_quad[0])\n",
    "print('2 point Gaussian Quadrature Rule of the interploated function of data defined in Example 4:',Integral_2pt_gq)\n",
    "print('3 point Gaussian Quadrature Rule of the interploated function of data defined in Example 4:',Integral_3pt_gq)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There is a significant difference betweeen the integral calculated using 2-point gaussian quarature rule and scipy.qauad integral of the polynomial function. The 2-point gaussian quarature rule does not exactly integrate the polynimal, but provides a good first approximate.<br>\n",
    "\n",
    "The integral calculed using 3-point gaussian quadrature rule and scipy.quad integral of the polynomial function yeild comparable results. Thus 3-point gaussian quadrature rule gives a very good approximate of the definite integral."
   ]
  }
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