Lect5.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Physics 404/604\n",
    "\n",
    "## Computational Physics                   |\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# How to solve equations numerically?\n",
    "\n",
    "How to find the solution of 2*sin(x)=x ?\n",
    "\n",
    "or more generally, find the solution of f(x)=0\n",
    "\n",
    "Most important thing: plot f(x) first.\n",
    "\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXIAAAD4CAYAAADxeG0DAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "## Always a good idea to draw the function first\n",
    "# please draw f(x)=2*sin(x) and g(x)=x\n",
    "\n",
    "import numpy as np\n",
    "import matplotlib.pylab as plt\n",
    "\n",
    "x = np.arange(-3., 3.0, 0.01)\n",
    "f=...\n",
    "g=...\n",
    "# please finish these two lines for f and g, the numpy function for sin is np.sin(...)\n",
    "plt.plot(x, f)\n",
    "plt.plot(x,g)\n",
    "plt.show()\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXIAAAD4CAYAAADxeG0DAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "## Always a good idea to draw the function first\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "\n",
    "def f(x):\n",
    "    return 2.*np.sin(x)-x\n",
    "\n",
    "t = np.arange(-3.0, 3.0, 0.01)\n",
    "s = f(t)\n",
    "plt.plot(t, s)\n",
    "s2 = np.zeros(len(t))\n",
    "plt.plot(t,s2)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Always a good idea to draw the function first !!!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "## 1 Trial-and-Error Roots via Bisection\n",
    "At xmin, the function is positive (negative); at xmax, the function is negative(positive); the solution has to be within [xmin,xmax]; search the value at (xmin+xmax)/2; if f((xmin+xmax)/2) has the same sign as f(xmin), then (xmin+xmax)/2 is the new xmin, otherwise it is xmax...."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "def bisection(xmin,xmax, Nmax, eps,f): #xmin is the left bound and xmax is the right bound, Nmax is the iteration limit, eps is the error, and f is the function\n",
    "    for i in range(0,Nmax):\n",
    "        x=(xmin+xmax)/2.\n",
    "        if(f(xmin)*f(x)>0):\n",
    "             # please finish this line (xmin=x or xmax=x???)\n",
    "        else:\n",
    "            # please finish this line\n",
    "        if(abs(f(x))<eps):\n",
    "            print(\"Root is at\",x,\" after \",i,\" iterations, with a precision of \",abs(f(x)))\n",
    "            break\n",
    "    return x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Root is at 1.8954942673444748  after  24  iterations, with a precision of  5.086029375434009e-10\n"
     ]
    }
   ],
   "source": [
    "eps = 1.e-8\n",
    "a=0.5\n",
    "b=3.\n",
    "nmax=100\n",
    "root = bisection(...)# please call bisection function to calculate the root of function f(x) defined before\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2. Newton-Raphson Searching\n",
    "\n",
    "Want to find x so that f(x)=0 :   \n",
    "Starting with a guess at $x_{0}$, then try to correct it with $\\Delta x$, hopeing that $f(x_{0}+\\Delta x)=0$.\n",
    "\\begin{equation}\n",
    "f(x=x_{0}+\\Delta x)\\simeq f(x_{0})+f'(x_{0})\\Delta x=0\n",
    "\\end{equation}\n",
    "So that\n",
    "\\begin{equation}\n",
    "\\Delta x=-\\frac{f(x_{0})}{f'(x_{0})}\n",
    "\\end{equation}\n",
    "Then use $x_{0}=x_{0}+\\Delta x$ to update $x_{0}$.\n",
    "This repeated until it reaches a given precision. \n",
    "\n",
    "Newton-Raphson method requires the first derivative. You can use either analytical solution or numerical derivative we talked about to calculate it (normally forward difference or central difference is sufficient). "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "def newtonraphson(x0,dx, Nmax, eps,f): # update x0 for Nmax times until reaching eps; dx is used in the central difference scheme to calculate the derivative, f is the input function, eps is the  error tolerance \n",
    "    for i in range(0,Nmax):\n",
    "        df =  # using central difference to calculate the derivative\n",
    "        x0 =  # please finish this line\n",
    "        if(abs(f(x0))<eps):\n",
    "            print(\"Root is at\",x0,\" after \",i,\" iterations, with a precision of \",abs(f(x0)))\n",
    "            break\n",
    "    return x0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Root is at 1.8954942670339812  after  4  iterations, with a precision of  4.440892098500626e-16\n"
     ]
    }
   ],
   "source": [
    "eps = 1.e-8\n",
    "dx=1.e-6\n",
    "a=3.\n",
    "nmax=100\n",
    "root = newtonraphson(....)# please call newtonraphson function to find the root and compare the results using bisection method"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Root is at 0.0  after  0  iterations, with a precision of  0.0\n"
     ]
    }
   ],
   "source": [
    "# However, if the initial guess is on the other side of the slope, it will get a different result\n",
    "eps = 1.e-8\n",
    "dx=1.e-6\n",
    "a=0.5\n",
    "nmax=100\n",
    "root = newtonraphson(a,dx,nmax,eps,f)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXMAAAD4CAYAAAAeugY9AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nwDAnInIBDHMiIhfAMCcicgH/B80riJIT7eJKAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# it can even fail, entering infinity loop\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "\n",
    "def f(x):\n",
    "    return x*x*x-2*x+2\n",
    "\n",
    "t = np.arange(-2., 2.0, 0.1)\n",
    "s = f(t)\n",
    "plt.plot(t, s)\n",
    "s2 = np.zeros(len(t))\n",
    "plt.plot(t,s2)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.0 -0.9999999999712443\n",
      "0.9999999999712443 0.9999999998314664\n",
      "1.397779669787269e-10 -0.9999999998314664\n",
      "0.9999999999712443 0.9999999998314664\n",
      "1.397779669787269e-10 -0.9999999998314664\n",
      "0.9999999999712443 0.9999999998314664\n",
      "1.397779669787269e-10 -0.9999999998314664\n",
      "0.9999999999712443 0.9999999998314664\n",
      "1.397779669787269e-10 -0.9999999998314664\n",
      "0.9999999999712443 0.9999999998314664\n",
      "1.397779669787269e-10 -0.9999999998314664\n",
      "0.9999999999712443 0.9999999998314664\n",
      "1.397779669787269e-10 -0.9999999998314664\n",
      "0.9999999999712443 0.9999999998314664\n",
      "1.397779669787269e-10 -0.9999999998314664\n",
      "0.9999999999712443 0.9999999998314664\n",
      "1.397779669787269e-10 -0.9999999998314664\n",
      "0.9999999999712443 0.9999999998314664\n",
      "1.397779669787269e-10 -0.9999999998314664\n",
      "0.9999999999712443 0.9999999998314664\n",
      "1.397779669787269e-10 -0.9999999998314664\n",
      "0.9999999999712443 0.9999999998314664\n",
      "1.397779669787269e-10 -0.9999999998314664\n",
      "0.9999999999712443 0.9999999998314664\n",
      "1.397779669787269e-10 -0.9999999998314664\n",
      "0.9999999999712443 0.9999999998314664\n",
      "1.397779669787269e-10 -0.9999999998314664\n",
      "0.9999999999712443 0.9999999998314664\n",
      "1.397779669787269e-10 -0.9999999998314664\n",
      "0.9999999999712443 0.9999999998314664\n",
      "1.397779669787269e-10 -0.9999999998314664\n",
      "0.9999999999712443 0.9999999998314664\n",
      "1.397779669787269e-10 -0.9999999998314664\n",
      "0.9999999999712443 0.9999999998314664\n",
      "1.397779669787269e-10 -0.9999999998314664\n",
      "0.9999999999712443 0.9999999998314664\n",
      "1.397779669787269e-10 -0.9999999998314664\n",
      "0.9999999999712443 0.9999999998314664\n",
      "1.397779669787269e-10 -0.9999999998314664\n",
      "0.9999999999712443 0.9999999998314664\n",
      "1.397779669787269e-10 -0.9999999998314664\n",
      "0.9999999999712443 0.9999999998314664\n",
      "1.397779669787269e-10 -0.9999999998314664\n",
      "0.9999999999712443 0.9999999998314664\n",
      "1.397779669787269e-10 -0.9999999998314664\n",
      "0.9999999999712443 0.9999999998314664\n",
      "1.397779669787269e-10 -0.9999999998314664\n",
      "0.9999999999712443 0.9999999998314664\n",
      "1.397779669787269e-10 -0.9999999998314664\n",
      "0.9999999999712443 0.9999999998314664\n",
      "1.397779669787269e-10 -0.9999999998314664\n",
      "0.9999999999712443 0.9999999998314664\n",
      "1.397779669787269e-10 -0.9999999998314664\n",
      "0.9999999999712443 0.9999999998314664\n",
      "1.397779669787269e-10 -0.9999999998314664\n",
      "0.9999999999712443 0.9999999998314664\n",
      "1.397779669787269e-10 -0.9999999998314664\n",
      "0.9999999999712443 0.9999999998314664\n",
      "1.397779669787269e-10 -0.9999999998314664\n",
      "0.9999999999712443 0.9999999998314664\n",
      "1.397779669787269e-10 -0.9999999998314664\n",
      "0.9999999999712443 0.9999999998314664\n",
      "1.397779669787269e-10 -0.9999999998314664\n",
      "0.9999999999712443 0.9999999998314664\n",
      "1.397779669787269e-10 -0.9999999998314664\n",
      "0.9999999999712443 0.9999999998314664\n",
      "1.397779669787269e-10 -0.9999999998314664\n",
      "0.9999999999712443 0.9999999998314664\n",
      "1.397779669787269e-10 -0.9999999998314664\n",
      "0.9999999999712443 0.9999999998314664\n",
      "1.397779669787269e-10 -0.9999999998314664\n",
      "0.9999999999712443 0.9999999998314664\n",
      "1.397779669787269e-10 -0.9999999998314664\n",
      "0.9999999999712443 0.9999999998314664\n",
      "1.397779669787269e-10 -0.9999999998314664\n",
      "0.9999999999712443 0.9999999998314664\n",
      "1.397779669787269e-10 -0.9999999998314664\n",
      "0.9999999999712443 0.9999999998314664\n",
      "1.397779669787269e-10 -0.9999999998314664\n",
      "0.9999999999712443 0.9999999998314664\n",
      "1.397779669787269e-10 -0.9999999998314664\n",
      "0.9999999999712443 0.9999999998314664\n",
      "1.397779669787269e-10 -0.9999999998314664\n",
      "0.9999999999712443 0.9999999998314664\n",
      "1.397779669787269e-10 -0.9999999998314664\n",
      "0.9999999999712443 0.9999999998314664\n",
      "1.397779669787269e-10 -0.9999999998314664\n",
      "0.9999999999712443 0.9999999998314664\n",
      "1.397779669787269e-10 -0.9999999998314664\n",
      "0.9999999999712443 0.9999999998314664\n",
      "1.397779669787269e-10 -0.9999999998314664\n",
      "0.9999999999712443 0.9999999998314664\n",
      "1.397779669787269e-10 -0.9999999998314664\n",
      "0.9999999999712443 0.9999999998314664\n",
      "1.397779669787269e-10 -0.9999999998314664\n",
      "0.9999999999712443 0.9999999998314664\n",
      "1.397779669787269e-10 -0.9999999998314664\n",
      "0.9999999999712443 0.9999999998314664\n",
      "1.397779669787269e-10 -0.9999999998314664\n",
      "0.9999999999712443 0.9999999998314664\n"
     ]
    }
   ],
   "source": [
    "def newtonraphson2(x0,dx, Nmax, eps,f):\n",
    "    for i in range(0,Nmax):\n",
    "        df = (f(x0+dx/2)-f(x0-dx/2))/dx\n",
    "        print(x0,f(x0)/df)\n",
    "        x0 = x0-f(x0)/df\n",
    "    return x0\n",
    "\n",
    "eps = 1.e-8\n",
    "dx=1.e-6\n",
    "a=0.\n",
    "nmax=100\n",
    "root = newtonraphson2(a,dx,nmax,eps,f)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3. Newton-Raphson with Backtracking\n",
    "\n",
    "If the code enters such an infinite loop, we need to restart the search with a different guess. \n",
    "\n",
    "If the new guess $x_{0}+\\Delta x$ leads to an increase in the magnitude of the function, $|f(x_{0}+\\Delta x)|>|f(x_{0})|$, we can backtrack somewhat and try a smaller guess, say $x_{0}+1/2 \\Delta x$. If the magnitude still increases, we try $x_{0}+1/4 \\Delta x$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "def newtonraphsonbacktracking(x0,dx,Nmax, eps,f):\n",
    "    for i in range(0,Nmax):        \n",
    "        df = (f(x0+dx/2)-f(x0-dx/2))/dx\n",
    "        dxt = -f(x0)/df\n",
    "        while (abs(f(x0+dxt))>abs(f(x0))):\n",
    "             dxt = ...# please finish this line for adjusting the time interval\n",
    "        print(x0,dxt)\n",
    "        x0 = x0+dxt\n",
    "        if(abs(f(x0))<eps):\n",
    "              print(\"Root is at\",x0,\" after \",i,\" iterations, with a precision of \",abs(f(x0)))\n",
    "              break\n",
    "    return x0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.0 0.9999999999712443\n",
      "0.9999999999712443 -0.2499999999578666\n",
      "0.7500000000133777 0.09218750003582352\n",
      "0.8421875000492013 -0.027896692456292796\n",
      "0.8142908075929085 0.002577244894015586\n",
      "0.8168680524869241 -0.000477474035762443\n",
      "0.8163905784511616 0.0002092166074981222\n",
      "0.8165997950586598 -0.00010742056689300943\n",
      "0.8164923744917667 5.148356490287829e-06\n",
      "0.8164975228482569 -1.4369512100993608e-06\n",
      "0.8164960858970468 6.835451393571088e-07\n",
      "0.8164967694421862 -2.2432932728528058e-07\n",
      "0.8164965451128589 3.6914952591245304e-08\n",
      "0.8164965820278114 -2.5482854441142355\n",
      "-1.7317888620864241 -0.038555343193879904\n",
      "-1.770344205280304 0.0010510573869496104\n",
      "-1.7692931478933545 7.936542706142389e-07\n",
      "Root is at -1.769292354239084  after  16  iterations, with a precision of  3.3448799285906716e-12\n"
     ]
    }
   ],
   "source": [
    "# it can even fail, entering infinity loop\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "\n",
    "def f(x):\n",
    "    return x*x*x-2*x+2\n",
    "\n",
    "eps = 1.e-8\n",
    "dx=1.e-6\n",
    "a=0.\n",
    "nmax=100\n",
    "root = newtonraphsonbacktracking(a,dx,nmax,eps,f)# call newtonraphsonbacktracking and study how x0 changes with iteration"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Newton-Raphson for equations\n",
    "Find the solution for  \n",
    "\\begin{eqnarray}\n",
    "x_{1}^2+x_{1}x_{2}=10\\\\\n",
    "x_{2}+3x_{1}x_{2}^2=57\n",
    "\\end{eqnarray}\n",
    "\n",
    "\n",
    "\n",
    "### Numerical Solutions\n",
    "\\begin{equation}\n",
    "f_{i}(x_{1},x_{2},...,x_{N})=0, \\qquad i=1,2,....,N\n",
    "\\end{equation}\n",
    "\n",
    "\\begin{equation}\n",
    "{\\bf f(x)}=\\left[ \\begin{array}{c} f_{1}(\\bf{x}) \\\\ f_{2}(\\bf{x})  \\end{array}\\right] = \\left[ \\begin{array}{c} x_{1}^2+x_{1}x_{2}-10 \\\\ x_{2}+3x_{1}x_{2}^2-57 \\end{array}\\right] =0 \n",
    "\\end{equation}\n",
    "\n",
    "Make a guess ($x_{1},x_{2}$), and then correct it ($\\Delta x_{1},\\Delta x_{2}$), we have\n",
    "\\begin{equation}\n",
    "f_{i}(x_{1}+\\Delta x_{1}, x_{2}+\\Delta x_{2})=0 \\qquad, i=1,2\n",
    "\\end{equation}\n",
    "We can expand it using Taylor series\n",
    "\\begin{equation}\n",
    "f_{i}(x_{1}+\\Delta x_{1}, x_{2}+\\Delta x_{2})\\simeq f_{i}(x_{1},x_{2})+\\frac{\\partial f_{i}}{\\partial x_{1}}\\Delta x_{1}+\\frac{\\partial f_{i}}{\\partial x_{2}}\\Delta x_{2}=0 \\qquad i=1,2\n",
    "\\end{equation}\n",
    "\n",
    "\\begin{equation}\n",
    "\\left[ \\begin{array}{c} f_{1}\\\\ f_{2} \\end{array}\\right] + \\begin{bmatrix} \\partial f_{1}/\\partial x_{1} & \\partial f_{1}/\\partial x_{2}  \\\\ \\partial f_{2}/\\partial x_{1} & \\partial f_{2}/\\partial x_{2} \\end{bmatrix}\\left[ \\begin{array}{c} \\Delta x_{1} \\\\ \\Delta x_{2}  \\end{array}\\right] =0 \n",
    "\\end{equation}\n",
    "\n",
    "So we want to solve the matrix equation\n",
    "\\begin{equation}\n",
    "F'\\Delta {\\bf{x}}=-\\bf{f}\n",
    "\\end{equation}\n",
    "Here we use bold font for a vector, the captal letter to represent a matrix\n",
    "\n",
    "For 2x2 array,\n",
    "\\begin{equation}\n",
    " \\begin{bmatrix} a & b \\\\c & d \\end{bmatrix}\\left[ \\begin{array}{c} \\Delta x_{1} \\\\ \\Delta x_{2}  \\end{array}\\right] =-\\left[ \\begin{array}{c} f_{1}\\\\ f_{2} \\end{array}\\right] \n",
    "\\end{equation}\n",
    "The solution is\n",
    "\\begin{eqnarray}\n",
    "\\Delta x_1=\\frac{bf_2-df_1}{ad-bc}\\\\\n",
    "\\Delta x_2=\\frac{cf_1-af_2}{ad-bc}\n",
    "\\end{eqnarray}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Newton-Raphson for equations\n",
    "Find the solution for  \n",
    "\\begin{eqnarray}\n",
    "x_{1}^2+x_{1}x_{2}=10\\\\\n",
    "x_{2}+3x_{1}x_{2}^2=57\n",
    "\\end{eqnarray}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "\n",
    "\n",
    "def f1(x1,x2):\n",
    "    return (x1*x1+x1*x2-10.)\n",
    "\n",
    "def f2(x1,x2):\n",
    "    return (x2+3*x1*x2*x2-57)\n",
    "\n",
    "def f1x1(x1,x2):\n",
    "    ....\n",
    "\n",
    "def f1x2(x1,x2):\n",
    "    .....\n",
    "    \n",
    "def f2x1(x1,x2):\n",
    "    ....\n",
    "    \n",
    "def f2x2(x1,x2):\n",
    "    ....\n",
    "    \n",
    "def dx1(f1,f2,f1x1,f1x2,f2x1,f2x2,x1,x2):\n",
    "    ....\n",
    "\n",
    "def dx2(f1,f2,f1x1,f1x2,f2x1,f2x2,x1,x2):\n",
    "    ....\n",
    "    \n",
    "...initial guess x1=...,x2=......\n",
    "for ....\n",
    "...x1=x1+dx1(...)\n",
    "...x2=x2+dx2(...)\n",
    "   until f1(x1,x2)<eps and f2(x1,x2)<eps"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAYIAAAEKCAYAAAAfGVI8AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10f95e2e8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "## Always a good idea to draw the function first\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "\n",
    "def f1(x):\n",
    "    return (10.-x*x)/x\n",
    "\n",
    "def f2(x):\n",
    "    return (-1+np.sqrt(1.+4.*3.*x*57))/6./x\n",
    "\n",
    "## Draw graphics to roughly determine x1 and x2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-7.0, -43.0]\n"
     ]
    }
   ],
   "source": [
    "# alternative way to update an array of values:  return several values using array\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "\n",
    "\n",
    "def F(x,f):\n",
    "    f[0]=x[0]*x[0]+x[0]*x[1]-10.\n",
    "    f[1]=x[1]+3.*x[0]*x[1]*x[1]-57.\n",
    "\n",
    "x=[1.,2.]\n",
    "f=[0.,0.]\n",
    "F(x,f)\n",
    "print(f)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-6.0]\n",
      "0.0\n"
     ]
    }
   ],
   "source": [
    "\n",
    "f=[0.]       #If f is an array or np.array, the main program can notice that it has been changed within a function (it is similar to pass by pointer (C) or reference (C++))\n",
    "def F2(x,f):\n",
    "    f[0]=x-10.\n",
    "\n",
    "F2(4.,f)\n",
    "print(f)\n",
    "\n",
    "f=0.             #If f is a variable, the main program won't notice that it is changed within the program\n",
    "def F3(x,f):\n",
    "    f=x-10.\n",
    "\n",
    "F3(4.,f)\n",
    "print(f)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Homework: using Newton-Raphson method to solve above equation arrays\n",
    "# Hint: first you need to derive the partical derivative of the equation arrays. You can do it either \n",
    "# analytically by hand or numerically using method we learned in the last lecture."
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python [default]",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.8"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}