1_Solving_Equations.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# ★ Solving Equations ★"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 1.1 The Bisection Method"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## DEFINITION 1.1\n",
    "\n",
    "The function $f(x)$ has a root at $x = r$ if $f(r) = 0$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## THEOREM 1.2\n",
    "\n",
    "Let $f$ be a continuous function on $[a, b]$, satisfying $f(a)f(b) < 0$. Then $f$ has a root between $a$ and $b$, that is, that exists a number $r$ satisfying $a < r < b$ and $f(r) = 0$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Import modules\n",
    "import traceback\n",
    "import numpy as np\n",
    "import scipy\n",
    "import scipy.linalg as LA\n",
    "import matplotlib.pyplot as plt\n",
    "import sympy\n",
    "import sympy.abc\n",
    "\n",
    "from scipy import optimize\n",
    "from IPython.display import display\n",
    "\n",
    "# Initialize sympy\n",
    "sympy.init_printing(use_unicode=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "def bisect(f, a, b, tol):\n",
    "    \"\"\"\n",
    "    Computes approximate solution of f(x)=0\n",
    "    \n",
    "    Arguments:\n",
    "        f (function prototype) : function handle \n",
    "        a (real number) : left bound of the interval\n",
    "        b (real number) : right bound of the interval\n",
    "        tol (real number) : tolerance\n",
    "        \n",
    "    Returns:\n",
    "        Approximate solution x\n",
    "        \n",
    "    Raises:\n",
    "        ValueError:\n",
    "            - a * b must be smaller than zero\n",
    "            - a > b will be considered to be wrong\n",
    "    \"\"\"\n",
    "    try:\n",
    "        if a > b :\n",
    "            raise ValueError('a must be <= b')\n",
    "        \n",
    "        fa = f(a)\n",
    "        fb = f(b)\n",
    "        \n",
    "        if np.sign(fa) * np.sign(fb) >= 0 :\n",
    "            raise ValueError('It must be verified that f(a) * f(b) < 0')\n",
    "            \n",
    "        while (b - a) / 2 > tol :\n",
    "            # Find the intermediate point  \n",
    "            c = (a + b) / 2\n",
    "            fc = f(c)\n",
    "            if fc == 0 :\n",
    "                return c\n",
    "            elif fa * fc < 0 :\n",
    "                b = c\n",
    "                fb = fc\n",
    "            else :\n",
    "                a = c\n",
    "                fa = fc\n",
    "                \n",
    "        return (a + b) / 2\n",
    "            \n",
    "    except ValueError as e :\n",
    "        print('ValueError Exception : ', e)\n",
    "        traceback.print_exception()\n",
    "    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example \n",
    "\n",
    "Find a root of $f(x) = x^3 + x - 1$ by using the Bisection Method on the interval $[0, 1]$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.68232780\n"
     ]
    }
   ],
   "source": [
    "root = scipy.optimize.bisect(lambda x : np.power(x, 3) + x - 1, 0, 1)\n",
    "print('{:>.8f}'.format(root))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## DEFINITION 1.3\n",
    "A solution is correct within  $p$ decimal places if the error is less than $0.5 \\times 10^{-p}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example\n",
    "\n",
    "Use the Bisection Method to find a root of $f(x) = cosx - x$ in the interval $[0, 1]$ to within six correct places."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.739085\n"
     ]
    }
   ],
   "source": [
    "root = scipy.optimize.bisect(lambda x : np.cos(x) - x, 0, 1)\n",
    "print('{:>.6f}'.format(root))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1. Apply two steps of the Bisection Method on the interval $[1, 2]$ to find the approximate root of $f(x) = 2x^3 - x - 7$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.627889\n"
     ]
    }
   ],
   "source": [
    "root = scipy.optimize.bisect(lambda x : 2 * np.power(x, 3) - x - 7, 1, 2)\n",
    "print('{:>.6f}'.format(root))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2. Use the ~bisect.m code~ (Bisection Method) to find the solution of $e^x = 3$ correct to six decimal places."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.098612\n"
     ]
    }
   ],
   "source": [
    "root = scipy.optimize.bisect(lambda x : np.exp(x) - 3, 1, 2)\n",
    "print('{:>.6f}'.format(root))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1.1 Computer Problems"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1. Use the Bisection Method to ﬁnd the root to six correct decimal places."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(a) $x^3 = 9$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2.080084\n"
     ]
    }
   ],
   "source": [
    "root = scipy.optimize.bisect(lambda x : np.power(x, 3) - 9, 2, 3)\n",
    "print('{:>.6f}'.format(root))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(b) $3x^3 + x^2 = x + 5$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.169726\n"
     ]
    }
   ],
   "source": [
    "root = scipy.optimize.bisect(lambda x : 3 * np.power(x, 3) + np.power(x, 2) - x - 5, 1, 2)\n",
    "print('{:>.6f}'.format(root))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(c) $\\cos^2{x} + 6 = x$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "6.776092\n"
     ]
    }
   ],
   "source": [
    "root = scipy.optimize.bisect(lambda x : np.power(np.cos(x), 2) - x + 6, 0, 10)\n",
    "print('{:>.6f}'.format(root))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "2. Use the Bisection Method to ﬁnd the root to eight correct decimal places."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(a) $x^5 + x = 1$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.75487767\n"
     ]
    }
   ],
   "source": [
    "root = scipy.optimize.bisect(lambda x : np.power(x, 5) + x - 1, 0, 1)\n",
    "print('{:>.8f}'.format(root))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(b) $sin{x} = 6x + 5$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-0.97089892\n"
     ]
    }
   ],
   "source": [
    "root = scipy.optimize.bisect(lambda x : np.sin(x) - 6 * x - 5, -5, 5)\n",
    "print('{:>.8f}'.format(root))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(c) $ln{x} + x^2 = 3$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.59214294\n"
     ]
    }
   ],
   "source": [
    "root = scipy.optimize.bisect(lambda x : np.log(x) + np.power(x, 2) - 3, 1, 2)\n",
    "print('{:>.8f}'.format(root))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "3. Use the Bisection Method to locate all solutions of the following equations. Sketch the function by using MATLAB’s plot command and identify three intervals of length one that contain a root. Then find the roots to six correct decimal places."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "def myplot(xpts, ypts, title = '', xlabel = 'x', ylabel = 'y'):\n",
    "    plt.plot(xpts, ypts, color='black')\n",
    "    plt.xlabel(xlabel)\n",
    "    plt.ylabel(ylabel)\n",
    "    plt.title(title)\n",
    "    plt.axhline(0, color='black', linestyle=':', linewidth=1)\n",
    "    plt.show()\n",
    "    plt.close()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(a) $2x^3 - 6x - 1 = 0$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "root = -1.641784\n",
      "root = -0.168254\n",
      "root = +1.810038\n"
     ]
    }
   ],
   "source": [
    "f = lambda x : 2 * np.power(x, 3) - 6 * x - 1\n",
    "xpts = np.linspace(-2, 2, 41)\n",
    "ypts = f(xpts)\n",
    "\n",
    "myplot(xpts, ypts, '$f(x) = 2x^3 - 6x - 1$')\n",
    "print('root = {:>+.6f}'.format(scipy.optimize.bisect(f, -2, -1)))\n",
    "print('root = {:>+.6f}'.format(scipy.optimize.bisect(f, -0.5, 0)))\n",
    "print('root = {:>+.6f}'.format(scipy.optimize.bisect(f, 1.5, 2)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(b) $e^{x-2} + x^3 - x = 0$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "root = -1.023482\n",
      "root = +0.163822\n",
      "root = +0.788941\n"
     ]
    }
   ],
   "source": [
    "f = lambda x : np.exp(x - 2) + np.power(x, 3) - x\n",
    "xpts = np.linspace(-1.2, 1.2, 41)\n",
    "ypts = f(xpts)\n",
    "\n",
    "myplot(xpts, ypts, '$f(x) = e^{x - 2} + x^3 - x$')\n",
    "print('root = {:>+.6f}'.format(scipy.optimize.bisect(f, -1.5, -0.5)))\n",
    "print('root = {:>+.6f}'.format(scipy.optimize.bisect(f, -0.5, 0.5)))\n",
    "print('root = {:>+.6f}'.format(scipy.optimize.bisect(f, 0.5, 1.0)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(c) $1 + 5x - 6x^3 - e^{2x} = 0$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "root = -0.818094\n",
      "root = -0.000000\n",
      "root = +0.506308\n"
     ]
    }
   ],
   "source": [
    "f = lambda x : 1 + 5 * x - 6 * np.power(x, 3) - np.exp(2 * x)\n",
    "xpts = np.linspace(-1, 0.75, 41)\n",
    "ypts = f(xpts)\n",
    "\n",
    "myplot(xpts, ypts, '$f(x) = 1 + 5x - 6x^3 - e^{2x}$')\n",
    "print('root = {:>+.6f}'.format(scipy.optimize.bisect(f, -1.0, -0.5)))\n",
    "print('root = {:>+.6f}'.format(scipy.optimize.bisect(f, -0.5, 0.25)))\n",
    "print('root = {:>+.6f}'.format(scipy.optimize.bisect(f, 0.25, 0.75)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "4. Calculate the square roots of the following numbers to eight correct decimal places by using the Bisection Method to solve $x^2 - A = 0$, where A is (a) 2 (b) 3 (c) 5. State your starting interval and the number of steps needed."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(a) $f(x) = x^2 - 2$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x is 1.41421356 in [1, 2]\n"
     ]
    }
   ],
   "source": [
    "ranges = [1, 2]\n",
    "f = lambda x : np.power(x, 2) - 2\n",
    "root = scipy.optimize.bisect(f, *ranges)\n",
    "print('x is {:>.8f} in {:}'.format(root, ranges))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(b) $f(x) = x^2 - 3$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x is 1.73205081 in [1, 2]\n"
     ]
    }
   ],
   "source": [
    "ranges = [1, 2]\n",
    "f = lambda x : np.power(x, 2) - 3\n",
    "root = scipy.optimize.bisect(f, *ranges)\n",
    "print('x is {:>.8f} in {:}'.format(root, ranges))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(c) $f(x) = x^2 - 5$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x is 2.23606798 in [2, 3]\n"
     ]
    }
   ],
   "source": [
    "ranges = [2, 3]\n",
    "f = lambda x : np.power(x, 2) - 5\n",
    "root = scipy.optimize.bisect(f, *ranges)\n",
    "print('x is {:>.8f} in {:}'.format(root, ranges))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "5. Calculate the cube roots of the following numbers to eight correct decimal places by using the Bisection Method to solve $x^3 - A = 0$, where A is (a) 2 (b) 3 (c) 5. State your starting interval and the number of steps needed."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(a) $f(x) = x^3 - 2$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x is 1.25992105 in [1, 2]\n"
     ]
    }
   ],
   "source": [
    "ranges = [1, 2]\n",
    "f = lambda x : np.power(x, 3) - 2\n",
    "root = scipy.optimize.bisect(f, *ranges)\n",
    "print('x is {:>.8f} in {:}'.format(root, ranges))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(b) $f(x) = x^3 - 3$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x is 1.44224957 in [1, 2]\n"
     ]
    }
   ],
   "source": [
    "ranges = [1, 2]\n",
    "f = lambda x : np.power(x, 3) - 3\n",
    "root = scipy.optimize.bisect(f, *ranges)\n",
    "print('x is {:>.8f} in {:}'.format(root, ranges))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(c) $f(x) = x^3 - 5$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x is 1.70997595 in [1, 2]\n"
     ]
    }
   ],
   "source": [
    "ranges = [1, 2]\n",
    "f = lambda x : np.power(x, 3) - 5\n",
    "root = scipy.optimize.bisect(f, *ranges)\n",
    "print('x is {:>.8f} in {:}'.format(root, ranges))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "6. Use the Bisection Method to calculate the solution of $cos{x} = sin{x}$ in the interval $[0, 1]$ within six correct decimal places."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x is 0.785398\n"
     ]
    }
   ],
   "source": [
    "ranges = [0, 1]\n",
    "f = lambda x : np.cos(x) - np.sin(x)\n",
    "root = scipy.optimize.bisect(f, *ranges)\n",
    "print('x is {:>.6f}'.format(root))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "7. Use the Bisection Method to find the two real numbers x, within six correct decimal places, that make the determinant of the matrix \n",
    "\n",
    "$A = \n",
    "\\begin{bmatrix}\n",
    "1 & 2 & 3 & x \\\\ \n",
    "4 & 5 & x & 6 \\\\ \n",
    "7 & x & 8 & 9 \\\\ \n",
    "x & 10 & 11 & 12 \n",
    "\\end{bmatrix}\n",
    "$\n",
    "\n",
    "equal to 1000. For each solution you find, test it by computing the corresponding determinant and reporting how many correct decimal places (after the decimal point) the determinant has when your solution x is used. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x is -17.188498 and 9.708299\n"
     ]
    }
   ],
   "source": [
    "def A(x):\n",
    "    template_A = '1 2 3 {0} ; 4 5 {0} 6 ; 7 {0} 8 9 ; {0} 10 11 12'.format(x)\n",
    "    _A = np.array(np.mat(template_A))\n",
    "    return _A\n",
    "\n",
    "def As(x):\n",
    "    _A = None\n",
    "    for _x in x:\n",
    "        template_A = '1 2 3 {0} ; 4 5 {0} 6 ; 7 {0} 8 9 ; {0} 10 11 12'.format(_x)\n",
    "        if _A is None:\n",
    "            _A = np.array(np.mat(template_A))\n",
    "        else:\n",
    "            newA = np.array(np.mat(template_A))\n",
    "            _A = np.concatenate((_A, newA))\n",
    "    _A = np.reshape(_A, (int(_A.size / 16), 4, 4))\n",
    "    return _A\n",
    "\n",
    "# define f(x) = det(A(x)) - 1000\n",
    "f = lambda x : np.linalg.det(A(x)) - 1000\n",
    "fs = lambda x : np.linalg.det(As(x)) - 1000\n",
    "\n",
    "# plot f(x)\n",
    "xpts = np.linspace(-17.9, 13, 41)\n",
    "ypts = fs(xpts)\n",
    "myplot(xpts, ypts, '$f(x) = det(A(x)) - 1000$', ylabel = 'f(x)')\n",
    "\n",
    "# Use bisection method to find root\n",
    "root1 = scipy.optimize.bisect(f, -20, -15)\n",
    "root2 = scipy.optimize.bisect(f, 5, 15)\n",
    "print('x is {:>.6f} and {:>.6f}'.format(root1, root2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "8. The **Hilbert Matrix**  is the n × n matrix whose ijth entry is $\\frac{1}{i + j - 1}$. Let $A$ denote the\n",
    "5 × 5 Hilbert matrix. Its largest eigenvalue is about 1.567. Use the Bisection Method to decide\n",
    "how to change the upper left entry $A_{11}$ to make the largest eigenvalue of $A$ equal to $\\pi$. Determine $A_\n",
    "{11}$ within six correct decimal places."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Make A[1,1] to be 2.948011 to make the largest eigenvalue equal to pi\n"
     ]
    }
   ],
   "source": [
    "def hilbert_matrix_eigvals(upper_left_element):\n",
    "    hilbert_matrix = LA.hilbert(5)\n",
    "    hilbert_matrix[0][0] = upper_left_element\n",
    "    return LA.eigvals(hilbert_matrix)\n",
    "\n",
    "f = lambda x : np.max(hilbert_matrix_eigvals(x)).real - scipy.pi\n",
    "\n",
    "def fs(f, xpts):\n",
    "    ypts = np.zeros(xpts.size)\n",
    "    for i in range(len(xpts)):\n",
    "        ypts[i] = f(xpts[i])\n",
    "    return ypts\n",
    "\n",
    "# Generate samples\n",
    "xpts = np.linspace(-2, 4, 21)\n",
    "ypts = fs(f, xpts)\n",
    "\n",
    "# Plot f(x)\n",
    "plt.plot(xpts,ypts)\n",
    "myplot(xpts, ypts, 'Eigenvalue of Hilbert Matrix $f(x) = eigval(H(x)) - \\pi$', ylabel = 'f(x)')\n",
    "\n",
    "root = scipy.optimize.bisect(f, 2, 4)\n",
    "print('Make A[1,1] to be {:>.6f} to make the largest eigenvalue equal to pi'.format(root))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "9. Find the height reached by 1 cubic meter of water stored in a spherical tank of radius 1 meter. Give your answer ±1 mm. (Hint: First note that the sphere will be less than half full. The volume of the bottom $H$ meters of a hemisphere of radius $R$ is $\\pi H^2(R - 1/3H)$.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.635501 m\n"
     ]
    }
   ],
   "source": [
    "# First note implies that the range will be [0, 1]\n",
    "ranges = [0, 1]\n",
    "\n",
    "# Define f(x) = V(x, 1) - 1 where V(H, R) = pi * H^2 * (R - H / 3)\n",
    "volume = lambda H, R : scipy.pi * np.power(H, 2) * (R - H / 3) \n",
    "f = lambda x : volume(x, 1) - 1\n",
    "\n",
    "root = scipy.optimize.bisect(f, *ranges)\n",
    "print('{:>.6f} m'.format(root))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "10. A planet orbiting the sun traverses an ellipse. The eccentricity $e$ of the ellipse is the distance between the center of the ellipse and either of its foci divided by the length of the semimajor axis. The perihelion is the nearest point of the orbit to the sun. *Kepler’s equation* $M = E - e\\sin{E}$ relates the *eccentric anomaly* $E$, the true angular distance (in radians) from perihelion, to the *mean anomaly* $M$, the fictitious angular distance from perihelion if it were on a circular orbit with the same period as the ellipse."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(a) Assume $e = 0.1$. Use the Bisection Method to find the eccentric anomalies $E$ when $M = \\pi/6$ and $M = \\pi/2$. Begin by finding a starting interval and explain why it works. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "eccentric anomalies is 0.578255 while M = π/6\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"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "eccentric anomalies is 1.670302 while M = π/2\n"
     ]
    }
   ],
   "source": [
    "e = 0.1\n",
    "\n",
    "# define f(x) = M - E + e * sin(E)\n",
    "f = lambda E, e, M : M - E + e * np.sin(E)\n",
    "\n",
    "# when M = π/6\n",
    "M = scipy.pi / 6\n",
    "xpts = np.linspace(-1, 1, 21)\n",
    "ypts = f(xpts, e, M)\n",
    "myplot(xpts, ypts, '$M = \\pi/6, e = 0.1$', xlabel = 'E', ylabel = 'f(x)')\n",
    "root = scipy.optimize.bisect(lambda x : f(x, e, M), 0.25, 0.75)\n",
    "print('eccentric anomalies is {:>.6f} while M = π/6'.format(root))\n",
    "\n",
    "# when M = π/2\n",
    "M = scipy.pi / 2\n",
    "xpts = np.linspace(1, 2, 21)\n",
    "ypts = f(xpts, e, M)\n",
    "myplot(xpts, ypts, '$M = \\pi/2, e = 0.1$', xlabel = 'E', ylabel = 'f(x)')\n",
    "root = scipy.optimize.bisect(lambda x : f(x, e, M), 1, 2)\n",
    "print('eccentric anomalies is {:>.6f} while M = π/2'.format(root))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(b) How do the answers to (a) change if the eccentricity is changed to $e = 0.2$?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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kli1bxhNPPLH/Cohjxoxh9+7dYUcTkVxUQCTuVK5cmb/85S/k5OTQoUMHbr31Vpo3b46u5SISX1RAJG7VrVuXWbNmMXPmTD799FNatmzJzTffrFN+ReKECojENTOjW7durFu3jr59+/LII4+QlJTE/Pnzw44mUuGpgEiZsO+U32XLlnHYYYfRqVMnLr/8cj799NOwo4lUWCogUqacddZZvPfee9x999288sorJCQkMGnSJJ3yKxICFRApc6pVq8add95JRkYGSUlJ9O7dm3bt2rFhw4awo4lUKKEWEDPraGbvm9lGMxuSx/xHzGxNcNtgZl9FzdsTNW9OqQaXuNCwYUOWLl3KhAkTeO+992jSpAn33nsvu3btCjuaSIVgYXX9zawysAE4D9gMrAIud/ecfJb/K3C6u18TTO909xqF2WZqaqrrVNDyadu2bQwcOJAZM2bQqFEjJkyYwB/+8IewY4mUC2aW7u6pudvD7IG0ADa6+yZ33wVMA7ocYPnLgamlkkzKnDp16jB9+nTmzp3LN998w1lnncX111/PV199FXY0kXIrzALyW+C/UdObg7ZfMbN6QH1gcVRzdTNLM7MVZtY1v42YWZ9gubQdO3YUQ2yJZxdeeCE5OTncdNNNTJgwgYSEBP7xj39okF2kBJSVQfTLgJfdfU9UW72gS9UdeNTMfpfXiu4+wd1T3T21du3apZFVQlajRg0efvhhVq5cyQknnMCll17KRRddxMcffxx2NJFyJcwCsgU4MWq6btCWl8vIdfjK3bcEPzcBS4HTiz+ilGUpKSm8++67PPzwwyxZsoRGjRrx8MMP63u1RIpJmAVkFXCqmdU3s6pEisSvzqYys4ZALWB5VFstM6sW3D8GOAvIc/BdKrZDDjmEm266iZycHNq2bcstt9xCy5YtSU9PDzuaSJkXWgFx991Af2ABsA6Y4e7ZZjbSzDpHLXoZMM1/eRA7AUgzswxgCXB/fmdviQDUq1ePuXPnMmPGDLZu3UqLFi30vVoiRRTaabxh0Gm8AvDVV18xdOhQxo8fz0knncS4ceO48MILw44lErfi8TRekVAceeSRPPXUUyxbtowaNWpw0UUXcemll7Jt27awo4mUKSogUmHt+16te+65hzlz5tCwYUOeeuop9u7dG3Y0kTJBBUQqtKpVqzJs2DCysrJITU3lhhtuoFWrVmRlZYUdTSTuqYCIAKeeeioLFy7kueee44MPPqBZs2YMHTqU77//PuxoInFLBUQkYGb06NGD9evX06NHD+6//34aN27MggULwo4mEpdUQERyOfroo5k0aRJLliyhSpUqdOzYke7du+viVSK5qICI5KNt27ZkZGQwYsQIZs6cScOGDXn66ac1yC4SUAEROYBq1apx1113kZmZSXJyMn369KFNmzbk5OhzqyIqICIF0KBBAxYvXsyzzz5LTk4OycnJ3HHHHfzwww9hRxMJjQqISAGZGb169WL9+vVcfvnl3HvvvTRp0oSFCxeGHU0kFCogIoVUu3ZtpkyZwqJFizAzzjvvPHr06MH27dvDjiZSqlRARGLUrl07MjMzGT58ONOnTychIYFJkybp4lVSYaiAiBRB9erVGTlyJBkZGTRq1IjevXvTtm1b1q1bF3Y0kRKnAiJSDBISEli6dCkTJ04kKyuLpk2bcuedd/Ljjz+GHU2kxKiAiBSTSpUqcc0117B+/Xouu+wyRo0aRZMmTVi0aFHY0URKRIEKiJkda2Z/NLO/mNk1ZtbCzFR8RPJw7LHH8txzz/H666/j7rRv356ePXuyY8eOsKOJFKsDFgEzO8fMFgDzgAuAOkAicAeQZWZ3m9kRJR9TpOxp3749WVlZDB8+nGnTptGwYUMNsku5csArEprZGOBv7v6fPOYdAlwIVHb3mSUXsfjoioQSlnXr1tG3b1/eeustWrduzfjx40lISAg7lkiBxHRFQncfnFfxCObtdvfZZaV4iIRJg+xSHhV0DOR5M6sZNX2ymWlkUKQQNMgu5U1BB8KXAe+aWSczuw74f8CjRd24mXU0s/fNbKOZDcljfi8z22Fma4LbtVHzrjKzD4LbVUXNIlJaNMgu5cUBx0B+saBZK2AJ8Blwurt/UqQNm1UGNgDnAZuBVcDl7p4TtUwvINXd++da9yggDUgFHEgHUtz9ywNtU2MgEm9+/PFH7r33Xh544AEOP/xwxowZw9VXX42ZhR1NZL+YxkCiVu4BTAJ6ApOBf5lZ0yJmagFsdPdN7r4LmAZ0KeC65wOvu/sXQdF4HehYxDwipa569eqMGjVKn2SXMqmgh7AuBlq5+1R3Hwr0I1JIiuK3wH+jpjcHbb/atpllmtnLZnZiIdcVKRM0yC5lUYEKiLt3dfftUdMrgZYllup/5gInu3sTIr2MKYV9ADPrY2ZpZpamY8wSzzTILmXNwT5IeEcw3vAr7r7LzNqZ2YUxbnsLcGLUdN2gLXobn7v7T8HkM0BKQdeNeowJ7p7q7qm1a9eOMapI6dEgu5QVB+uBZAFzzWyRmY0xs1vN7M7gtN4s4CLg3Ri3vQo41czqm1lV4DJgTvQCZlYnarIzsO/A8AKgg5nVMrNaQIegTaTc0CfZJd4drIBc4u5nEfnnnA1UBr4BXgBauPtN7h7T2yJ33w30Dx57HTDD3bPNbKSZdQ4WG2Bm2WaWAQwAegXrfgGMIlKEVgEjgzaRckVfFy/x7GBfZZIDtAfmA+fknl/W/mnrNF4py/bu3cvkyZMZNGgQO3fuZMiQIdx+++1Ur1497GhSzsV6Gu94YBHQkMjnLvbd0oOfIlJK8hpkb9y4sa7JLqE52HdhPe7uCcAkdz8l6lbf3U8ppYwiEiV6kB3Yf012DbJLaSvoabzXl3QQESmcfYPsd9xxB9OnT9cgu5Q6XRRKpAzb90n2NWvWkJiYuH+Qff369WFHkwpABUSkHEhMTOSNN97g6aefJjMzk6ZNmzJixAh9kl1KlAqISDlRqVIlrr32WtavX88ll1zC3XffTdOmTVmyZEnY0aScUgERKWeOO+44XnzxRRYsWMDu3btp164dvXr14rPPPgs7mpQzKiAi5VSHDh3Iyspi6NChvPjiizRs2JApU6ZokF2KjQqISDl22GGHMXr0aFavXs1pp51Gr169OPfcc9mwYUPY0aQcUAERqQAaN27MsmXLeOqpp1i9ejVNmjRh1KhR/PTTTwdfWSQfKiAiFUSlSpXo168f69ato0uXLtx5550kJyfz5ptvhh1NyigVEJEKpk6dOkyfPp158+bxww8/0KZNG6677jq+/PKAV4QW+RUVEJEKqlOnTmRnZzNo0CCeffZZGjZsyNSpUzXILgWmAiJSgf3mN79hzJgxpKWlcdJJJ9G9e3cuuOACPvzww7CjSRmgAiIiJCcns2LFCh577DHefvttGjVqxJgxY/j555/DjiZxTAVERACoXLkyAwYMICcnh/POO49bb72V5s2bs2rVqrCjSZxSARGRXzjxxBOZPXs2M2fOZMeOHbRs2ZIBAwbw7bffhh1N4owKiIj8ipnRrVs3cnJyuOGGG3jiiSdITEzk1VdfDTuaxBEVEBHJV82aNXniiSd45513qFWrFl27dqVbt25s2bIl7GgSB1RAROSgzjjjDNLT07n//vt57bXXSEhIYNy4cezZsyfsaBIiFRARKZAqVapw2223sXbtWs4880z69+/PWWedRVZWVtjRJCShFhAz62hm75vZRjMbksf8m80sx8wyzWyRmdWLmrfHzNYEtzmlm1yk4jrllFN47bXXeOGFF9i0aRPNmjVj2LBh/PDDD2FHk1IWWgExs8rAOOACIBG43MwScy32HpDq7k2Al4EHo+b94O7Jwa1zqYQWESAyyH7FFVewbt06rrzySkaPHk2TJk108aoKJsweSAtgo7tvcvddwDSgS/QC7r7E3b8PJlcAdUs5o4gcwNFHH82zzz7LwoULcXfatWvHNddcw+effx52NCkFYRaQ3wL/jZreHLTlpzcwP2q6upmlmdkKM+taAvlEpIDOPffc/Revev7550lISND3alUAZWIQ3cyuBFKBMVHN9dw9FegOPGpmv8tn3T5BoUnbsWNHKaQVqZgOPfRQRo8eTXp6OvXr16d79+506tSJjz76KOxoUkLCLCBbgBOjpusGbb9gZu2BYUBnd99/9Rt33xL83AQsBU7PayPuPsHdU909tXbt2sWXXkTy1KRJE9555x0ef/xxli1bRqNGjXj44YfZvXt32NGkmIVZQFYBp5pZfTOrClwG/OJsKjM7Hfg7keKxPaq9lplVC+4fA5wF5JRachE5oMqVK/PXv/6VnJwc2rVrxy233ELLli1ZvXp12NGkGIVWQNx9N9AfWACsA2a4e7aZjTSzfWdVjQFqAP/IdbpuApBmZhnAEuB+d1cBEYkzJ554InPmzGHGjBls2bKFFi1aMHjwYL777ruwo0kxsIo0yJWamuppaWlhxxCpkL788ktuu+02nn76aerXr89TTz3F+eefH3YsKQAzSw/GnH+hTAyii0jZV6tWLSZMmMAbb7xB1apV6dixIz179uSzzz4LO5rESAVEREpV69atWbNmDcOHD2fq1KkkJCTw4osv6pTfMkgFRERKXfXq1Rk5ciSrV6/md7/7HVdeeSWdOnXi448/DjuaFIIKiIiEpnHjxrz99ts89thjvPXWWzRq1IjHHntM3/JbRqiAiEio9l1KNzs7m9atWzNw4ED+8Ic/6Ft+ywAVEBGJC/Xq1WPevHm89NJL+7/ld/jw4fz4449hR5N8qICISNwwMy6//HLWrVtH9+7dueeee0hOTuatt94KO5rkQQVEROLOMcccw5QpU1iwYAE//fQTrVu3pl+/fnz99ddhR5MoKiAiErc6dOjA2rVrufnmm3n66adJTExk9uzZYceSgAqIiMS13/zmN4wdO5YVK1ZwzDHH8Mc//pFLLrmETz75JOxoFZ4KiIiUCc2bNyctLY3Ro0fzz3/+k4SEBCZPnqwPIIZIBUREyowqVaowdOhQMjIySEpK4uqrr+b888/nww8/DDtahaQCIiJlToMGDXjjjTd48sknWb58OUlJSfoAYghUQESkTKpUqRLXX3892dnZtGnThoEDB9KqVStycnRlh9KiAiIiZdpJJ53EvHnzeOGFF/jggw9ITk5m5MiR7Nq1K+xo5Z4KiIiUeWbGFVdcQU5ODhdffDF33XUXKSkprFq1Kuxo5ZoKiIiUG8ceeyxTp05lzpw5fPnll5xxxhkMGjSI77//Puxo5ZIKiIiUOxdddBHZ2dlcd911jB07lsaNG7N48eKwY5U7KiAiUi7VrFmT8ePHs2TJEipVqsS5557Lddddx1dffRV2tHJDBUREyrW2bduSkZHB4MGDmTRpEomJicyZMyfsWOWCCoiIlHuHHXYYDz74IO+++y61a9emS5cudO/eXddjL6JQC4iZdTSz981so5kNyWN+NTObHsx/18xOjpo3NGh/38zOL9XgIlImpaamsmrVKu6++25efvllEhMTmTFjhr4OJUahFRAzqwyMAy4AEoHLzSwx12K9gS/d/ffAI8ADwbqJwGVAI6Aj8GTweCIiB1S1alXuvPNO0tPTqVevHn/+85+5+OKL9eWMMQizB9IC2Ojum9x9FzAN6JJrmS7AlOD+y8C5ZmZB+zR3/8ndPwQ2Bo8nIlIgjRs3Zvny5TzwwAP861//IjExkeeff169kUIIs4D8Fvhv1PTmoC3PZdx9N/A1cHQB1wXAzPqYWZqZpW3cuJERI0YAcNppp7FhwwbS09NJSUkB4JZbbmHs2LEAnHDCCWzdupWlS5fStm1bAPr06cOECRMAOPzww/n222+ZO3cuF110EQDdu3fnpZde2rddAF566SW6d+8ORE4tnDt3Lt9++y2HH344ABMmTKBPnz5AZLBv6dKlbN26lRNOOAGAsWPHcssttwCQkpJCeno6GzZs4LTTTgNgxIgRek56TnpOMT6njIwMunbtyvHHH09CQgI9e/bktNNOY/PmzWX2OZXE7yk/Fla1NbNLgI7ufm0w3QNo6e79o5ZZGyyzOZj+N9ASGAGscPcXgvaJwHx3f/lA20xNTfW0tLSSeDoiUsbt2bOHcePGMXToUA455BAeeughrr322v3/6CsyM0t399Tc7WH2QLYAJ0ZN1w3a8lzGzA4BagKfF3BdEZECq1y5MgMGDCArK4uUlBT69OnDeeedp6+KP4AwC8gq4FQzq29mVYkMiuc+OXsOcFVw/xJgsUe6THOAy4KztOoDpwIrSym3iJRjp5xyCgsXLmT8+PGsXLmSpKQk/va3v7F3796wo8Wd0ApIMKbRH1gArANmuHu2mY00s87BYhOBo81sI3AzMCRYNxuYAeQArwF/cXddCEBEikWlSpXo27cva9eupXXr1gwYMIA2bdqwYcOGsKPFldDGQMKgMRARKSx357nnnmPgwIH8+OOPjBo1iptuuonKlSvOJwficQxERCTumRlXXXUVOTk5nH/++QwePJhWrVqxfv36sKOFTgVERKQA6tSpw6xZs3jhhRd4//33SU5O5qGHHqrQl9FVARERKaDoC1d17NixwvdGVEBERArp+OOPV28EFRARkZioN6ICIiJSJBW5N6ICIiJSRBW1N6ICIiJSTCpab0QFRESkGO3rjWRnZ5f7z42ogIiIlIA6deowe/bsX/RGxo4dW656IyogIiIlJHdvZNCgQbRp04aNGzeGHa1YqICIiJSwfb2RKVOmsHbtWpo2bcq4cePK/Df8qoCIiJQCM6Nnz56sXbuWVq1a0b9/fzp06MB//vOfsKPFTAVERKQU1a1bl9dee42///3vrFixgqSkJCZNmlQmr8WuAiIiUsrMjD59+pCZmUmzZs3o3bs3F154IVu3bg07WqGogIiIhOSUU05h8eLFPProoyxevJikpCReeumlMtMbUQEREQlRpUqVuPHGG1mzZg0NGjTgiiuu4E9/+hM7duwIO9pBqYCIiMSBBg0a8NZbb3Hfffcxd+5cGjVqxKxZs8KOdUAqICIiceKQQw5hyJAhpKenU7duXbp160aPHj348ssvw46WJxUQEZE4k5SUxLvvvstdd93FtGnTSEpKYv78+WHH+pVQCoiZHWVmr5vZB8HPWnksk2xmy80s28wyzezPUfMmm9mHZrYmuCWX6hMQESlhVapUYcSIEaxYsYJatWrRqVMn+vXrx86dO8OOtl9YPZAhwCJ3PxVYFEzn9j3Q090bAR2BR83syKj5g909ObitKenAIiJhSElJIS0tjUGDBjFhwgSSk5N55513wo4FhFdAugBTgvtTgK65F3D3De7+QXB/K7AdqF1aAUVE4kX16tUZM2YMS5YsYffu3Zx99tncfvvt7Nq1K9RcYRWQ49x9W3D/E+C4Ay1sZi2AqsC/o5rvDQ5tPWJm1Uoop4hI3GjTpg2ZmZn06tWL++67jxYtWrB27drQ8pRYATGzhWa2No9bl+jlPPKJmXw/NWNmdYDngavdfd83jw0FGgLNgaOA2w6wfh8zSzOztLJwXrWIyIEcccQRTJw4kdmzZ7N161ZSUlJCu2iVhfGJRzN7H2jr7tuCArHU3RvksdwRwFJgtLu/nM9jtQUGufuFB9tuamqqp6WlFSW6iEjc2L59O3379mX27Nm0bt2aKVOmcPLJJxf7dsws3d1Tc7eHdQhrDnBVcP8q4NXcC5hZVWAW8Fzu4hEUHczMiIyfhNeHExEJybHHHssrr7zCs88+y3vvvUeTJk149tlnS+2rUMIqIPcD55nZB0D7YBozSzWzZ4JlLgVaA73yOF33RTPLArKAY4B7SjW9iEicMDN69epFVlYWKSkpXHPNNfzxj39k+/btJb/tsvKlXcVBh7BEpDzbu3cvjz76KLfffjtHHHEETz/9NF26dDn4igcRb4ewRESkmFWqVImbb755/1ehdO3alauvvppvvvmmZLZXIo8qIiKhadSoEStWrGDYsGE899xzNGnShOzs7GLfjgqIiEg5VLVqVe655x7efvttGjRowEknnVTs2zik2B9RRETixhlnnMGCBQtK5LHVAxERkZiogIiISExUQEREJCYqICIiEhMVEBERiYkKiIiIxEQFREREYqICIiIiMalQX6ZoZjuAj2Nc/Rjgs2KMU1yUq3CUq3CUq3DKa6567v6rS4pXqAJSFGaWlte3UYZNuQpHuQpHuQqnouXSISwREYmJCoiIiMREBaTgJoQdIB/KVTjKVTjKVTgVKpfGQEREJCbqgYiISExUQKKY2Z/MLNvM9ppZvmcsmFlHM3vfzDaa2ZCo9vpm9m7QPt3MqhZTrqPM7HUz+yD4WSuPZc4xszVRtx/NrGswb7KZfRg1L7m0cgXL7Yna9pyo9jD3V7KZLQ9+35lm9ueoecW6v/J7vUTNrxY8/43B/jg5at7QoP19Mzu/KDliyHWzmeUE+2eRmdWLmpfn77SUcvUysx1R2782at5Vwe/9AzO7qpRzPRKVaYOZfRU1r0T2l5lNMrPtZrY2n/lmZo8HmTPNrFnUvKLvK3fXLbgBCUADYCmQms8ylYF/A6cAVYEMIDGYNwO4LLg/Hri+mHI9CAwJ7g8BHjjI8kcBXwCHBdOTgUtKYH8VKBewM5/20PYXcBpwanD/BGAbcGRx768DvV6ilrkBGB/cvwyYHtxPDJavBtQPHqdyKeY6J+o1dP2+XAf6nZZSrl7AE3msexSwKfhZK7hfq7Ry5Vr+r8CkUthfrYFmwNp85ncC5gMGnAG8W5z7Sj2QKO6+zt3fP8hiLYCN7r7J3XcB04AuZmZAO+DlYLkpQNdiitYleLyCPu4lwHx3/76Ytp+fwubaL+z95e4b3P2D4P5WYDvwqw9KFYM8Xy8HyPsycG6wf7oA09z9J3f/ENgYPF6p5HL3JVGvoRVA3WLadpFyHcD5wOvu/oW7fwm8DnQMKdflwNRi2na+3P1NIm8W89MFeM4jVgBHmlkdimlfqYAU3m+B/0ZNbw7ajga+cvfdudqLw3Huvi24/wlw3EGWv4xfv3jvDbqwj5hZtVLOVd3M0sxsxb7DasTR/jKzFkTeVf47qrm49ld+r5c8lwn2x9dE9k9B1i3JXNF6E3knu09ev9PSzHVx8Pt52cxOLOS6JZmL4FBffWBxVHNJ7a+DyS93seyrCndNdDNbCByfx6xh7v5qaefZ50C5oifc3c0s31PngncXjYHoiyAPJfKPtCqR0/luA0aWYq567r7FzE4BFptZFpF/kjEr5v31PHCVu+8NmmPeX+WRmV0JpAJtopp/9Tt193/n/QjFbi4w1d1/MrO+RHpv7Upp2wVxGfCyu++Jagtzf5WYCldA3L19ER9iC3Bi1HTdoO1zIt3DQ4J3kfvai5zLzD41szruvi34h7f9AA91KTDL3X+Oeux978Z/MrNngUGlmcvdtwQ/N5nZUuB0YCYh7y8zOwKYR+TNw4qox455f+Uhv9dLXstsNrNDgJpEXk8FWbckc2Fm7YkU5Tbu/tO+9nx+p8XxD/Ggudz986jJZ4iMee1bt22udZcWQ6YC5YpyGfCX6IYS3F8Hk1/uYtlXOoRVeKuAUy1yBlFVIi+WOR4ZmVpCZPwB4CqguHo0c4LHK8jj/urYa/BPdN+4Q1cgzzM2SiKXmdXadwjIzI4BzgJywt5fwe9uFpHjwy/nmlec+yvP18sB8l4CLA72zxzgMoucpVUfOBVYWYQshcplZqcDfwc6u/v2qPY8f6elmKtO1GRnYF1wfwHQIchXC+jAL3viJZoryNaQyKD08qi2ktxfBzMH6BmcjXUG8HXwBql49lVJnBlQVm/AH4kcC/wJ+BRYELSfAPwrarlOwAYi7yCGRbWfQuQPfCPwD6BaMeU6GlgEfAAsBI4K2lOBZ6KWO5nIO4tKudZfDGQR+Uf4AlCjtHIBfwi2nRH87B0P+wu4EvgZWBN1Sy6J/ZXX64XIIbHOwf3qwfPfGOyPU6LWHRas9z5wQTG/3g+Wa2Hwd7Bv/8w52O+0lHLdB2QH218CNIxa95pgP24Eri7NXMH0COD+XOuV2P4i8mZxW/Ba3kxkrKof0C+Yb8C4IHMWUWeXFse+0ifRRUQkJjqEJSIiMVEBERGRmKiAiIhITFRAREQkJiogIiISkwr3QUKReGJme4icXrnPNHe/P6w8IoWh03hFQmRmO929Rtg5RGKhQ1giIhITFRCRcB1qv7wQ2J8PvopIfNAhLJEQ6RCWlGXqgYiISExUQEREJCY6hCUSojxO433N3YeElUekMFRAREQkJjqEJSIiMVEBERGRmKiAiIhITFRAREQkJiogIiISExUQERGJiQqIiIjERAVERERi8v8BQ4lHP7nYiv0AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "eccentric anomalies is 0.643617 while M = π/6\n"
     ]
    },
    {
     "data": {
      "image/png": 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lISGBLVu2MGHCBBITE4mOjtZlzEWCwLOiMLMSwBSgHRAF3GxmUXmMKwsMBpKDm1AKo/DwcIYOHUpmZiZNmzZlyJAhNGrUiNTUVK+jiRRbXm5R1Af2Ouc+ds79BrwGdM5j3OPAOOBIMMNJ4VatWjVWrFjBggUL+PTTT6lXr57u2S0SIF4WxaXAZzmmP/fNO8nM6gKXOef83oTZzPqZWYqZpRw4cKDgk0qhZGbcdNNN7Ny5k969ezNu3DhiY2N59913vY4mUqwU2oPZZhYGPAMMPd1Y59w051y8cy6+cuXKgQ8nhUrFihV5+eWXef/99wkPD6d169b06tULfWkQKRheFsUXwGU5piN8804oC8QAa83sE6AhsEwHtOVUmjdvTmpqKiNHjuS1114jMjKSOXPm6FRakXPkZVFsAa42syvMrCRwE7DsxELn3A/OuUrOuWrOuWpAEtDJOaebL8splSpVitGjR7N9+3Zq1KjB7bffTqtWrdi7d6/X0USKLM+Kwjl3DBgIvAPsBN5wzmWa2Wgz6+RVLikeoqOjWb9+Pf/85z/ZsmULsbGxjB07VteNEjkLVtw2y+Pj411KijY65P/bv38/gwYN4s033yQ2NpZp06bRsGFDr2OJFCpmttU5l+eu/UJ7MFukoFStWpVFixaxZMkSvv32W/7yl78waNAgfvrpJ6+jiRQJKgoJGSduktS/f39eeOEFoqOjWbHC75nXIoKKQkJMuXLleOGFF/jwww8pW7YsHTp04Oabb+brr7/2OppIoaWikJDUqFEjtm/fzqhRo1i8eDGRkZHMnj1bp9KK5EFFISGrZMmSPProo+zYsYPIyEh69+5N69at+eijj7yOJlKoqCgk5EVGRrJu3TpefPFFkpOTiY2NZcKECRw7dszraCKFgopCBAgLC+Puu+8mKyuLVq1aMWzYMBo0aMC2bdu8jibiORWFSA4REREsWbKEhQsXsn//furXr88DDzygq9JKSFNRiORiZnTt2pWsrCz69OnD+PHjiY2NJTEx0etoIp5QUYicwoUXXsj06dN5//33KVGiBC1btqRPnz58++23XkcTCSoVhchpnLgq7YgRI5g7dy6RkZEsXLhQp9JKyFBRiORD6dKlGTNmDFu2bCEiIoLu3bvz17/+lf3793sdTSTgVBQiZ6B27dokJyfz9NNPs3r1aqKiopg+fbq2LqRYU1GInKHw8HCGDRtGeno6derUoV+/flx33XW654UUWyoKkbP05z//mcTERKZNm8bWrVv1Qz0ptlQUIucgLCyMvn37kpWVRZs2bRg2bBgNGzYkNTXV62giBUZFIVIALr30Ut566y3eeOMNPvvsM+Lj43n44Yc5cuSI19FEzpmKQqSAmBndunUjKyuLnj17MmbMGGrXrs2GDRu8jiZyTlQUIgXsj3/8I6+88gqrV6/myJEjNG3alIEDB+qOelJk5asozKyKmd1oZgPM7A4zq29mKhkRP9q0aUNGRgaDBw/mxRdfJDo6mlWrVnkdS+SM+f2wN7MWZvYOsAJoB1wCRAGPAOlmNsrMygU+pkjRVKZMGSZNmnTyjnrXX389vXr14uDBg15HE8m3020VXA/0dc4lOOf6Oececc7d75zrBNQCtgOtAp5SpIhr1KgR27ZtY+TIkSxYsICoqCgWLVrkdSyRfPFbFM65Yc65fadYdsw5t8Q592ZgookUL+effz6jR48mJSWFiIgIunXrRpcuXfjqq6+8jibiV36PUbxqZuVzTFczM11zWeQs1KpVi+TkZJ566ilWrFhBVFSU7tcthVp+D0hvAJLN7Hoz6wv8C5gUsFQixVx4eDgPPvggqampREVF0bt3b9q1a8e+fXluwIt4Kl9F4ZybCtwFLAVGA9c4594OZDCRUFCjRg3WrVvH888/z4YNG4iOjubFF1/k+PHjXkcTOSm/u55uA2YCvYBXgJVmViuAuURCRlhYGAMHDiQjI4NGjRoxYMAAWrRowZ49e7yOJgLkf9dTF6CJc26Bc24E8HeyC0NECki1atV45513mDlzJmlpacTFxTF+/HhdZFA8l99dTzc4577OMb0ZaBCwVCIhyszo06cPmZmZtG3blgceeIBGjRqRnp7udTQJYaf7wd0jZlYxr2XOud/M7Foz6xCYaCKhq2rVqixevJjXX3+dTz/9lLp16/LYY4/x22+/eR1NQtDptijSgbfNLNHMxpvZA2b2qO902XSgI5Ac+JgiocfM6N69O1lZWfTo0YNRo0YRHx/P1q1bvY4mIeZ0RdHVOdcYeAfIBEoAPwJzgfrOuXudcwfO9s3NrK2Z/dvM9prZ8DyW32dmWWaW5iurP53te4kUVZUqVWLu3LksW7aMgwcP0qBBAx566CFdwlyC5nRFUc/MqgI9gWXAVGAOsAUofS5vbGYlgClkX0MqCrjZzKJyDdsOxDvn4oBFwNPn8p4iRVnHjh3JzMykV69ejB07lrp165KUlOR1LAkBpyuKl4BEoCaQkuOx1ffPc1Ef2Ouc+9g59xvwGtA55wDn3PvOuZ99k0lAxDm+p0iRVqFCBWbOnMmqVas4dOgQjRs35v777+eXX37xOpoUY6e71tNzzrlIYKZz7socjyucc1ee43tfCnyWY/pz37xTuRPI8xrNZtbPzFLMLOXAgbPeEyZSZLRt25aMjAz69u3LxIkTqVWrFuvXr/c6lhRT+T099u5AB/HHzG4F4oHxeS13zk1zzsU75+IrV64c3HAiHilXrhwvvfQSiYmJHDt2jGbNmjFo0CAOHTrkdTQpZry8+dAXwGU5piN88/6LmbUEHgY6Oed+DVI2kSLj2muvJS0tjYEDB/L8888TFxfHe++953UsKUa8LIotwNVmdoWZlQRuIvuA+UlmVofsA+idcv7gT0T+W5kyZXjuuedYt24d4eHhXHfddfz973/nxx9/9DqaFAOeFYVz7hgwkOxTb3cCbzjnMs1stJl18g0bD5QBFprZDjNbdoqXExGgadOm7Nixg6FDhzJ9+nRiYmJ45513vI4lRZwVt2vgx8fHu5SUcz0hS6ToS0pK4o477mDnzp306dOHZ555hgoVKngdSwopM9vqnIvPa5mXu55EJIAaNmzItm3bGDFiBHPmzCEmJoaVK1d6HUuKIBWFSDFWqlQpxowZQ1JSEhUqVKB9+/b06dOH77//3utoUoSoKERCwIlrRD388MO8+uqrREdHs2LFCq9jSRGhohAJEeeffz5PPPEESUlJVKxYkQ4dOtC7d2++++47r6NJIaeiEAkx8fHxpKSk8MgjjzB37lxiYmK0dSF+qShEQtD555/P448/TnJy8smti9tvv11bF5InFYVICKtXr97JrYt58+YRHR3N8uXLvY4lhYyKQiTEndi62Lx5M5UqVaJjx4706tVLWxdykopCRACoW7cuKSkpjBw5kvnz5xMdHc3bb7/tdSwpBFQUInJSyZIlGT16NJs3b6Zy5cp06tRJWxeiohCR/1W3bl22bNnCo48+yoIFC/S7ixCnohCRPJUsWZJRo0aRnJxMpUqV6NChA3fccYd+1R2CVBQi4teJrYuHH36YOXPmEBsbqyvShhgVhYic1olfdW/atIly5crRtm1b+vXrp/tdhAgVhYjkW0JCAlu3buXBBx9kxowZxMbGsmbNGq9jSYCpKETkjJQqVYqnnnqKDz/8kNKlS9OqVSv69++ve3UXYyoKETkrDRs2ZPv27dx333289NJLxMbGsnbtWq9jSQCoKETkrJUuXZqJEyeevFd3ixYtGDRoEIcPH/Y6mhQgFYWInLMmTZqwY8cOBg0axPPPP0+tWrVYv36917GkgKgoRKRA/OEPf2Dy5MmsXbuW48eP06xZM+69915+/vlnr6PJOVJRiEiBatasGWlpadx9991MmjSJOnXqkJSU5HUsOQcqChEpcGXKlGHKlCmsWbOGI0eO0LhxYx566CF+/fVXr6PJWVBRiEjAXHfddaSnp9OnTx/Gjh1LQkIC27dv9zqWnCEVhYgEVLly5Xj55ZdZvnw5Bw4coH79+jz++OMcPXrU62iSTyoKEQmK9u3bk5mZSffu3Xn00Uf5y1/+QlZWltexJB9UFCISNBUrVmTevHksXLiQTz75hLp16zJhwgR+//13r6OJHyoKEQm6rl27kpGRQbt27Rg2bBjNmjVj7969XseSU1BRiIgnLrroIhYvXsycOXPIyMigVq1aTJkyhePHj3sdTXJRUYiIZ8yM2267jYyMDJo2bcrAgQNp3bo1+/bt8zqa5KCiEBHPRUREsGrVKqZNm0ZycjKxsbHMmjUL55zX0QQVhYgUEmZG3759SUtLo06dOtxxxx106tSJr776yutoIc/TojCztmb2bzPba2bD81h+vpm97luebGbVPIgpIkF0xRVX8N577zFp0iTWrFlDTEwMCxcu9DpWSPOsKMysBDAFaAdEATebWVSuYXcC3znn/gw8C4wLbkoR8UJYWBiDBw9m+/btXHnllXTv3p2ePXvy3XffeR0tJHm5RVEf2Ouc+9g59xvwGtA515jOwGzf80XAdWZmQcwoIh6qWbMmGzduZPTo0bzxxhvExMTwzjvveB0r5HhZFJcCn+WY/tw3L88xzrljwA/AH3O/kJn1M7MUM0vZu3cvjz32GADVq1dn9+7dbN26lXr16gEwdOhQJk6cCEDVqlXZv38/a9eupXnz5gD069ePadOmAVC2bFl++ukn3n77bTp27AjALbfcwvz580+8LwDz58/nlltuAaBjx468/fbb/PTTT5QtWxaAadOm0a9fPwCaN2/O2rVr2b9/P1WrVgVg4sSJDB06FIB69eqxdetWdu/eTfXq1QF47LHHtE5ap5Bdp1WrVjFkyBBKlixJhQoVaNu2LVFRURw6dKjIrlNh/Dv5Y16dVWBmXYG2zrm7fNO3AQ2ccwNzjMnwjfncN/2Rb8w3p3rd+Ph4l5KSEtjwIuKJI0eOMHLkSCZOnMgVV1zB7NmzadKkidexigUz2+qci89rmZdbFF8Al+WYjvDNy3OMmYUD5YGDQUknIoVOqVKlGD9+PB988AHOOa655hoeeOABjhw54nW0Ys3LotgCXG1mV5hZSeAmYFmuMcuA233PuwLvOZ1YLRLymjZtSmpqKn379mX8+PEkJCSwY8cOr2MVW54Vhe+Yw0DgHWAn8IZzLtPMRptZJ9+wGcAfzWwvcB/wP6fQikhoKlu2LFOnTmXFihUcPHiQhIQEnnzySY4dO+Z1tGLHs2MUgaJjFCKh59tvv2XAgAG89tprNGjQgNmzZ1OjRg2vYxUphfUYhYhIgahYsSILFizgtddeY8+ePdSpU4fnnntOFxgsICoKESk2evToQUZGBi1atGDw4MG0adOGzz//3OtYRZ6KQkSKlUsuuYTly5czdepUNm3aRExMDPPmzdMFBs+BikJEih0zo1+/fqSmphIdHc2tt95Kjx49OHhQZ9efDRWFiBRbV111FevWrWPMmDEsWbKE2NhYVq9e7XWsIkdFISLFWokSJRgxYgSbN2+mYsWKtGvXjv79+3P48GGvoxUZKgoRCQm1a9cmJSWFoUOH8tJLL1GnTh2SkpK8jlUkqChEJGSUKlWKCRMm8N577/Hrr7/SuHFjRo4cydGjR72OVqipKEQk5DRv3py0tDR69erFE088QcOGDcnKyvI6VqGlohCRkFS+fHlmzZrF4sWL2bdvH3Xr1mXSpEn6kV4eVBQiEtJuvPFGMjIyaN26Nffeey+tWrVi3759XscqVFQUIhLyLrroIpYuXcr06dPZvHkzcXFxzJ07Vz/S81FRiIiQ/SO9u+66i9TUVGJiYrjtttu46aab+Pbbb72O5jkVhYhIDldeeSUffPABY8aMYfHixcTGxrJmzRqvY3lKRSEiksuJH+klJydTvnx5WrVqxZAhQ/jll1+8juYJFYWIyCnUrVuXrVu3MmjQICZPnkx8fDzbt2/3OlbQqShERPwoXbo0kydP5l//+hfff/89DRo0YOzYsfz+++9eRwsaFYWISD60atWK9PR0brjhBh566CGaNWvGf/7zH69jBYWKQkQknypWrMjrr7/Oq6++Snp6OnFxccyaNavYn0arohAROQNmxq233kp6ejrx8fHccccddOnShQMHDngdLWBUFCIiZ+Hyyy8nMTGR8ePHs2LFCmJjY1m5cqXXsQJCRSEicpbCwsK4//772bJlC1WqVKF9+/bcfffdxe5eFyoKEZFzFBcXx+bNm7n//vuZOnUqdevWJSUlxetYBUZFISJSAEqVKsX48eNJTEzk559/plGjRjz55JPF4jRaFYWISAFq0aIFaWlpdOnShUceeaRYnEarohARKWAXXnghCxYsYO7cuaSnp1OrVi3mzJlTZE+jVVGIiASAmdGzZ09SU1OpXbs2t99+e5G9Gq2KQkQkgKpVq8b777/P2LFjWbx4MXFxcSQmJnod64yoKEREAqxEiRIMHz6cpKQkypQpQ8uWLRk6dChHjhzxOlq+qChERIKkXr16bNu2jf79+/PMM89Qv3590tPTvY51WioKEZEguuCCC5gyZQrLly/n//7v/0hISGDSpEkcP37c62in5ElRmFlFM3vXzPb4/nlhHmNqm9kmM8s0szQz6+FFVhGRQGjfvj3p6em0bt2ae++9lzZt2vDFF194HStPXm1RDAcSnXNXA4m+6dx+Bno556KBtsAkM6sQvIgiIoFVpUoVli5dytSpU9m4cSOxsbG8+eabXsf6H14VRWdgtu/5bOCG3AOcc7udc3t8z/cDXwOVgxVQRCQYzIx+/fqxfft2rrrqKrp27cqdd97JoUOHvI52kldFcZFz7kvf86+Ai/wNNrP6QEngo1Ms72dmKWaWUpwv9SsixVf16tXZuHEjDz30ELNmzaJ27dokJyd7HQsIYFGY2Rozy8jj0TnnOJf9U8VT/lzRzC4BXgX6OOfyPNrjnJvmnIt3zsVXrqyNDhEpms477zyefPJJ1q5dy9GjR2ncuDFPPPGE59eLClhROOdaOudi8ngsBf7PVwAniuDrvF7DzMoBK4CHnXNJgcoqIlKYXHPNNaSmptKtWzdGjhxJ8+bN+eSTTzzL49Wup2XA7b7ntwNLcw8ws5LAW8Ac59yiIGYTEfFchQoVmD9/Pq+++iqpqanUqlWLefPmeZLFq6J4CmhlZnuAlr5pzCzezF72jekOXAP0NrMdvkdtT9KKiHjgxG1XU1NTiY2N5dZbb6Vnz5788MMPwc1RVK9meCrx8fGuON0wREQE4NixY4wdO5ZRo0YRERHBq6++StOmTQvs9c1sq3MuPq9l+mW2iEgREB4ezsiRI9mwYQPh4eE0b96ckSNHcvTo0YC/t4pCRKQIadiwIdu3b+f222/niSeeoEmTJuzZsyeg76miEBEpYsqWLcvMmTNZuHAhe/bsoU6dOsycOTNgN0ZSUYiIFFFdu3YlLS2N+vXrc+edd9KjR4+AXFwwvMBfUUREgiYiIoI1a9YwceJEfvjhB8LCCv77v4pCRKSICwsLY9iwYYF7/YC9soiIFAsqChER8UtFISIifqkoRETELxWFiIj4paIQERG/VBQiIuKXikJERPwqdpcZN7MDwKfn8BKVgG8KKE5REWrrHGrrC1rnUHEu6/wn51ye95IudkVxrsws5VTXZC+uQm2dQ219QescKgK1ztr1JCIifqkoRETELxXF/5rmdQAPhNo6h9r6gtY5VARknXWMQkRE/NIWhYiI+BWSRWFmM83sazPLOMVyM7PnzGyvmaWZWd1gZyxo+Vjnnr51TTezjWZWK9gZC9rp1jnHuAQzO2ZmXYOVLRDys75m1tzMdphZppl9EMx8gZCP/6/Lm9nbZpbqW+c+wc5Y0MzsMjN738yyfOs0OI8xBfoZFpJFAbwCtPWzvB1wte/RD/hnEDIF2iv4X+f/AM2cc7HA4xSP/buv4H+dMbMSwDjgX8EIFGCv4Gd9zawC8CLQyTkXDXQLTqyAegX/f+MBQJZzrhbQHJhoZiWDkCuQjgFDnXNRQENggJlF5RpToJ9hIVkUzrl1wLd+hnQG5rhsSUAFM7skOOkC43Tr7Jzb6Jz7zjeZBEQEJVgA5ePvDHAP8CbwdeATBVY+1vcWYLFzbp9vfCisswPKmpkBZXxjjwUjW6A45750zm3zPf8J2AlcmmtYgX6GhWRR5MOlwGc5pj/nf/8QxdmdwCqvQwSamV0K3Ejx2GLMj+rAhWa21sy2mlkvrwMFwQtAJLAfSAcGO+eOexup4JhZNaAOkJxrUYF+hume2fJfzKwF2UXRxOssQTAJeNA5dzz7C2exFw7UA64DSgObzCzJObfb21gB1QbYAVwLXAW8a2brnXM/epqqAJhZGbK3hocEen1UFHn7Argsx3SEb16xZmZxwMtAO+fcQa/zBEE88JqvJCoB15vZMefcEk9TBc7nwEHn3GHgsJmtA2oBxbko+gBPuezfAew1s/8ANYHN3sY6N2Z2HtklMc85tziPIQX6GaZdT3lbBvTynTnQEPjBOfel16ECycwuBxYDtxXzb5gnOeeucM5Vc85VAxYB/YtxSQAsBZqYWbiZXQA0IHv/dnG2j+wtKMzsIqAG8LGnic6R73jLDGCnc+6ZUwwr0M+wkNyiMLMFZJ8BUcnMPgf+AZwH4Jx7CVgJXA/sBX4m+1tJkZaPdX4U+CPwou8b9rGifkG1fKxzsXK69XXO7TSz1UAacBx42Tnn99Thwi4ff+PHgVfMLB0wsnc1FvUryjYGbgPSzWyHb95DwOUQmM8w/TJbRET80q4nERHxS0UhIiJ+qShERMQvFYWIiPilohAREb9C8vRYkWAzs9/JvoTECa85557yKo/ImdDpsSJBYGaHnHNlvM4hcja060lERPxSUYgER2nfDYNOPHp4HUgkv7TrSSQItOtJijJtUYiIiF8qChER8Uu7nkSCII/TY1c754Z7lUfkTKgoRETEL+16EhERv1QUIiLil4pCRET8UlGIiIhfKgoREfFLRSEiIn6pKERExC8VhYiI+PX/AFPc65Rp3YA2AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "eccentric anomalies is 1.766961 while M = π/2\n"
     ]
    }
   ],
   "source": [
    "e = 0.2\n",
    "\n",
    "# define f(x) = M - E + e * sin(E)\n",
    "f = lambda E, e, M : M - E + e * np.sin(E)\n",
    "\n",
    "# when M = π/6\n",
    "M = scipy.pi / 6\n",
    "xpts = np.linspace(-1, 1, 21)\n",
    "ypts = f(xpts, e, M)\n",
    "myplot(xpts, ypts, '$M = \\pi/6, e = 0.2$', xlabel = 'E', ylabel = 'f(x)')\n",
    "root = scipy.optimize.bisect(lambda x : f(x, e, M), 0.25, 0.75)\n",
    "print('eccentric anomalies is {:>.6f} while M = π/6'.format(root))\n",
    "\n",
    "# when M = π/2\n",
    "M = scipy.pi / 2\n",
    "xpts = np.linspace(1, 2, 21)\n",
    "ypts = f(xpts, e, M)\n",
    "myplot(xpts, ypts, '$M = \\pi/2, e = 0.2$', xlabel = 'E', ylabel = 'f(x)')\n",
    "root = scipy.optimize.bisect(lambda x : f(x, e, M), 1, 2)\n",
    "print('eccentric anomalies is {:>.6f} while M = π/2'.format(root))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "# 1.2 Fixed-Point Iteration"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## DEFINITION 1.4\n",
    "\n",
    "The real number r is a **fixed point** of the function g if $g(r) = r$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [],
   "source": [
    "def fpi(g, x0, k, tol = 0):\n",
    "    \"\"\"\n",
    "    Computes approximate solution of g(x)=x by Fixed-Point Iteration\n",
    "    \n",
    "    Args:\n",
    "        g (function prototype) : function handle\n",
    "        x0 (real number) : starting guess \n",
    "        k (integer) : number of iteration steps\n",
    "        tol (real number) : tolerance (default : 0)\n",
    "        \n",
    "    Returns:\n",
    "        Approximate solution xc\n",
    "    \n",
    "    Raises:\n",
    "        ValueError :\n",
    "            - g is None\n",
    "            - k is not the type of positive integer \n",
    "            - tol is negative\n",
    "    \"\"\"\n",
    "    try:\n",
    "        if g is None :\n",
    "            raise ValueError('f is null')\n",
    "        if k != int(k) :\n",
    "            raise ValueError('k must be type of integer')\n",
    "        if k < 0 :\n",
    "            raise ValueError('k must be positive integer')\n",
    "        if tol < 0 :\n",
    "            raise ValueError('tol must be positive')\n",
    "            \n",
    "        x = x0\n",
    "        for _ in range(k) :\n",
    "            tx = g(x)\n",
    "            if tol > 0 and abs(tx - x) / abs(tx) < tol :\n",
    "                return tx\n",
    "            else :\n",
    "                x = tx\n",
    "        \n",
    "        return x\n",
    "    except ValueError as e:\n",
    "        print('ValueError Exception : ', e)\n",
    "        traceback.print_exception()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## DEFINITION 1.5 \n",
    "\n",
    "Let $e_i$ denote the error at step $i$ of an iterative method. If \n",
    "\n",
    "$$\\lim_{i \\rightarrow \\infty} \\frac{e_{i+1}}{e_i} = S < 1$$\n",
    "\n",
    "the method is said to obey linear convergence with rate $S$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## THEOREM 1.6\n",
    "\n",
    "Assume that $g$ is continuously differentiable, that $g(r) = r$, and that $S = \\left | {g}'(r) \\right | < 1$. Then Fixed-Point Iteration converges linearly with rate $S$ to the fixed point $r$ for initial guesses sufficiently close to $r$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## DEFINITION 1.7\n",
    "\n",
    "An iterative method is called **locally convergent** to r if the method converges to r for initial guesses sufficiently close to r."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example\n",
    "Use Fixed-Point Iteration to find a root of $cosx = sinx$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$x + \\cos{x} - \\sin{x} = x $\n",
    "\n",
    "$\\Rightarrow g(x) = x + \\cos{x} - \\sin{x}$ "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.78539816\n"
     ]
    }
   ],
   "source": [
    "g = lambda x : x + np.cos(x) - np.sin(x)\n",
    "root = scipy.optimize.fixed_point(g, 0, method='iteration')\n",
    "print('{:>.8f}'.format(root))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example\n",
    "Find the fixed points of $g(x)=2.8x - x^2$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.80000001\n"
     ]
    }
   ],
   "source": [
    "g = lambda x : 2.8 * x - np.power(x, 2)\n",
    "root = scipy.optimize.fixed_point(g, 0.1, method='iteration')\n",
    "print('{:>.8f}'.format(root))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example\n",
    "Calculate $\\sqrt{2}$ by using **FPI**."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$x = \\sqrt{2} $\n",
    "\n",
    "$\\Rightarrow x - \\sqrt{2} = 0 $\n",
    "\n",
    "$\\Rightarrow x^2 - 2\\sqrt{2}x + 2 = 0 $\n",
    "\n",
    "$\\Rightarrow x^2 + 2 = 2\\sqrt{2}x $\n",
    "\n",
    "$\\Rightarrow g(x) = x = \\sqrt{2} = \\frac{x^2 + 2}{2x} $"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.41421356\n"
     ]
    }
   ],
   "source": [
    "g = lambda x : (np.power(x, 2) + 2) / (2 * x)\n",
    "root = scipy.optimize.fixed_point(g, 2.0, method='iteration')\n",
    "print('{:>.8f}'.format(root))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "## 1.2 Computer Problems"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1. Apply Fixed-Point Iteration to find the solution of each equation to eight correct decimal places."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(a) $ x^3 = 2x + 2 $"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$ x^3 = 2x + 2 $\n",
    "\n",
    "$\\Rightarrow x = \\sqrt[3]{2x + 2}$\n",
    "\n",
    "$\\Rightarrow g(x) = \\sqrt[3]{2x + 2}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.76929236\n"
     ]
    }
   ],
   "source": [
    "root = scipy.optimize.fixed_point(lambda x : np.power(2 * x + 2, 1 / 3), 2.0, method='iteration')\n",
    "print('{:>.8f}'.format(root))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(b) $ e^x + x = 7 $"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$ e^x + x = 7 $\n",
    "\n",
    "$ \\Rightarrow e^x = 7 - x $\n",
    "\n",
    "$ \\Rightarrow x = \\ln{7 - x} $\n",
    "\n",
    "$ \\Rightarrow g(x) = \\ln{7 - x} $"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.67282170\n"
     ]
    }
   ],
   "source": [
    "root = scipy.optimize.fixed_point(lambda x : np.log(7 - x), 0, method='iteration')\n",
    "print('{:>.8f}'.format(root))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(c) $ e^x + \\sin{x} = 4 $"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$ e^x + \\sin{x} = 4 $\n",
    "\n",
    "$ \\Rightarrow e^x = 4 - \\sin{x} $\n",
    "\n",
    "$ \\Rightarrow x = \\ln{(4 - \\sin{x})} $\n",
    "\n",
    "$ \\Rightarrow g(x) = \\ln{(4 - \\sin{x})} $"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.12998050\n"
     ]
    }
   ],
   "source": [
    "root = scipy.optimize.fixed_point(lambda x : np.log(4 - np.sin(x)), 0, method='iteration')\n",
    "print('{:>.8f}'.format(root))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "2. Apply Fixed-Point Iteration to find the solution of each equation to eight correct decimal places."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(a) $x^5 + x = 1$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$ x^5 + x = 1 $\n",
    "\n",
    "$ \\Rightarrow x^5 = 1 - x $\n",
    "\n",
    "$ \\Rightarrow x = \\sqrt[5]{1 - x} $\n",
    "\n",
    "$ \\Rightarrow g(x) = \\sqrt[5]{1 - x} $"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.75487766\n"
     ]
    }
   ],
   "source": [
    "root = scipy.optimize.fixed_point(lambda x : np.power(1 - x, 1 / 5), 0.1, method='iteration')\n",
    "print('{:>.8f}'.format(root))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(b) $\\sin{x} = 6x + 5$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$ \\sin{x} = 6x + 5 $\n",
    "\n",
    "$ \\Rightarrow x = \\frac{\\sin{x} - 5}{6} $\n",
    "\n",
    "$ \\Rightarrow g(x) = \\frac{\\sin{x} - 5}{6} $"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-0.97089892\n"
     ]
    }
   ],
   "source": [
    "root = scipy.optimize.fixed_point(lambda x : (np.sin(x) - 5) / 6, 0.1, method='iteration')\n",
    "print('{:>.8f}'.format(root))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "3. Calculate the square roots of the following numbers to eight correct decimal places by using Fixed-Point Iteration as in Example 1.6: (a) 3 (b) 5. State your initial guess and the number of steps needed."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can conclude that $g(x, c) = x = \\sqrt{c} = \\frac{x^2 + c}{2x}$ from Example 1.6. Then"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [],
   "source": [
    "g = lambda x, c : (np.power(x, 2) + c) / (2 * x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(a) $g(x, 3) = \\frac{x^2 + 3}{2x}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.73205081\n"
     ]
    }
   ],
   "source": [
    "root = scipy.optimize.fixed_point(lambda x : g(x, 3), 0.1, method='iteration')\n",
    "print('{:>.8f}'.format(root))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(b) $g(x, 5) = \\frac{x^2 + 5}{2x}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2.23606798\n"
     ]
    }
   ],
   "source": [
    "root = scipy.optimize.fixed_point(lambda x : g(x, 5), 0.1, method='iteration')\n",
    "print('{:>.8f}'.format(root))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "4. Calculate the cube roots of the following numbers to eight correct decimal places, by using Fixed-Point Iteration with $g(x) = (2x + A/x^2) / 3$, where A is (a) 2 (b) 3 (c) 5. State your initial guess and the number of steps needed."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let $y^3 = x$ and $g(x, A) = (2y^3 + A/y^6)/3$\n",
    "\n",
    "$\\Rightarrow y^3 = (2y^3 + A/y^6)/3$\n",
    "\n",
    "$\\Rightarrow y = \\sqrt[3]{(2y^3 + A/y^6)/3}$\n",
    "\n",
    "$\\Rightarrow h(y, A) = \\sqrt[3]{(2y^3 + A/y^6)/3}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [],
   "source": [
    "h = lambda y, A : np.power((2 * np.power(y, 3) + A / np.power(y, 6)) / 3, 1 / 3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(a) $h(y, 2) = \\sqrt[3]{(2y^3 + 2/y^6)/3}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.08005974\n"
     ]
    }
   ],
   "source": [
    "root = scipy.optimize.fixed_point(lambda y : h(y, 2), 1.0, method='iteration')\n",
    "print('{:>.8f}'.format(root))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(b) $h(y, 3) = \\sqrt[3]{(2y^3 + 3/y^6)/3}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.12983096\n"
     ]
    }
   ],
   "source": [
    "root = scipy.optimize.fixed_point(lambda y : h(y, 3), 1.0, method='iteration')\n",
    "print('{:>.8f}'.format(root))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(c) $h(y, 5) = \\sqrt[3]{(2y^3 + 5/y^6)/3}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.19581317\n"
     ]
    }
   ],
   "source": [
    "root = scipy.optimize.fixed_point(lambda y : h(y, 5), 1.0, method='iteration')\n",
    "print('{:>.8f}'.format(root))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "5. Example 1.3 showed that $g(x) = \\cos{x}$ is a convergent FPI. Is the same true for $g(x) = \\cos^2{x}$? Find the fixed point to six correct decimal places, and report the number of FPI steps needed. Discuss local convergence, using Theorem 1.6."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.641714\n"
     ]
    }
   ],
   "source": [
    "root = scipy.optimize.fixed_point(lambda x : np.power(np.cos(x), 2), 1.0, method='iteration')\n",
    "print('{:>.6f}'.format(root))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - 2 \\sin{\\left(x \\right)} \\cos{\\left(x \\right)}$"
      ],
      "text/plain": [
       "-2⋅sin(x)⋅cos(x)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "S = 0.958993\n"
     ]
    }
   ],
   "source": [
    "dg = sympy.diff(sympy.cos(sympy.abc.x) ** 2)\n",
    "display(dg)\n",
    "S = np.abs(sympy.lambdify(sympy.abc.x, dg)(root))\n",
    "print('S = {:>.6f}'.format(S))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Because $g'(x) = -2\\sin{x}\\cos{x}$ and $S = \\left | g'(x = 0.641714) \\right | = \\left | -0.958993 \\right | = 0.958993 < 1$. It is locally convergent."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "6. Derive three different $g(x)$ for finding roots to six correct decimal places of the following $f(x) = 0$ by Fixed-Point Iteration. Run FPI for each $g(x)$ and report results, convergence or divergence. Each equation $f(x) = 0$ has three roots. Derive more $g(x)$ if necessary until all roots are found by FPI. For each convergent run, determine the value of S from the errors $e_{i+1}/e_{i}$ , and compare with $S$ determined from calculus as in (1.11)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(a) $f(x) = 2x^3 - 6x - 1$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(1) $6x = 2x^3 - 1$\n",
    "\n",
    "$ \\Rightarrow g(x) = x = \\frac{2x^3 - 1}{6} $"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1st root is -0.168254\n",
      "S = 0.028310\n"
     ]
    }
   ],
   "source": [
    "f = lambda x : (2 * np.power(x, 3) - 1) / 6\n",
    "root = scipy.optimize.fixed_point(f, 1.0, method='iteration')\n",
    "print('1st root is {:>.6f}'.format(root))\n",
    "\n",
    "dg = sympy.diff((2 * sympy.abc.x ** 3 - 1) / 6)\n",
    "S = np.abs(sympy.lambdify(sympy.abc.x, dg)(root))\n",
    "print('S = {:>.6f}'.format(S))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It is locally convergent due to S = 0.028310 < 1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(2) $ 2x^3 = 6x + 1 $\n",
    "\n",
    "$ \\Rightarrow g(x) = x = \\sqrt[3]{\\frac{6x + 1}{2}} $"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2nd root is 1.810038\n",
      "S = 0.305228\n"
     ]
    }
   ],
   "source": [
    "f = lambda x : np.power((6 * x + 1) / 2, 1 / 3)\n",
    "root = scipy.optimize.fixed_point(f, 1.0, method='iteration')\n",
    "print('2nd root is {:>.6f}'.format(root))\n",
    "\n",
    "dg = sympy.diff( ((6 * sympy.abc.x + 1) / 2) ** (1 / 3) )\n",
    "S = np.abs(sympy.lambdify(sympy.abc.x, dg)(root))\n",
    "print('S = {:>.6f}'.format(S))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It is locally convergent due to S = 0.305228 < 1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(3) $ 2x \\cdot x^2 = 6x + 1 $\n",
    "\n",
    "$ \\Rightarrow g(x) = x = \\frac{\\sqrt{6/x + 1/x^2}}{2} $\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3rd root is 1.195609\n",
      "S = 0.561172\n"
     ]
    }
   ],
   "source": [
    "f = lambda x : np.sqrt(6 / x + 1 / np.power(x, 2)) / 2\n",
    "root = scipy.optimize.fixed_point(f, 2.0, method='iteration')\n",
    "print('3rd root is {:>.6f}'.format(root))\n",
    "\n",
    "dg = sympy.diff( ((6 / sympy.abc.x + 1 / sympy.abc.x ** 2) ** 0.5) / 2 )\n",
    "S = np.abs(sympy.lambdify(sympy.abc.x, dg)(root))\n",
    "print('S = {:>.6f}'.format(S))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It is locally convergent due to S = 0.561172 < 1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(b) $f(x) = e^{x-2} + x^3 - x$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(1) $x = e^{x-2} + x^3$\n",
    "\n",
    "$ \\Rightarrow g(x) = e^{x-2} + x^3 $"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1st root is 0.163822\n",
      "S = 0.464096\n"
     ]
    }
   ],
   "source": [
    "g = lambda x : np.exp(x - 2) + np.power(x, 3)\n",
    "root = scipy.optimize.fixed_point(g, 0, method='iteration')\n",
    "print('1st root is {:>.6f}'.format(root))\n",
    "\n",
    "dg = sympy.diff(sympy.log(sympy.abc.x - 2) + sympy.abc.x ** 3)\n",
    "S = np.abs(sympy.lambdify(sympy.abc.x, dg)(root))\n",
    "print('S = {:>.6f}'.format(S))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It is locally convergent due to S = 0.464096 < 1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(2) $x^3 = x - e^{x-2} $\n",
    "\n",
    "$ \\Rightarrow x = \\sqrt[3]{x - e^{x-2}} $\n",
    "\n",
    "$ \\Rightarrow g(x) = \\sqrt[3]{x - e^{x-2}} $"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2nd root is 0.788941\n",
      "S = 0.376010\n"
     ]
    }
   ],
   "source": [
    "g = lambda x : np.power(x - np.exp(x - 2), 1 / 3)\n",
    "root = scipy.optimize.fixed_point(g, 1, method='iteration')\n",
    "print('2nd root is {:>.6f}'.format(root))\n",
    "\n",
    "dg = sympy.diff((sympy.abc.x - sympy.exp(sympy.abc.x - 2)) ** (1 / 3))\n",
    "S = np.abs(sympy.lambdify(sympy.abc.x, dg)(root))\n",
    "print('S = {:>.6f}'.format(S))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It is locally convergent due to S = 0.376010 < 1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(3) $e^{x-2} = x - x^3$\n",
    "\n",
    "$ \\Rightarrow x - 2 = \\ln{(x - x^3)} $\n",
    "\n",
    "$ \\Rightarrow g(x) = 2 + \\ln{(x - x^3)} $"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Diverge, cannot find 3rd root by FPI\n",
      "3rd root is -1.023482\n",
      "S = 44.056711\n"
     ]
    }
   ],
   "source": [
    "g = lambda x : np.exp(x - 2) + x ** 3 - x\n",
    "print('Diverge, cannot find 3rd root by FPI')\n",
    "\n",
    "# We use other method to find 3rd root\n",
    "root = scipy.optimize.fsolve(g, -1)[0]\n",
    "print('3rd root is {:>.6f}'.format(root))\n",
    "\n",
    "dg = sympy.diff(2 + sympy.log(sympy.abc.x - sympy.abc.x ** 3))\n",
    "S = np.abs(sympy.lambdify(sympy.abc.x, dg)(root))\n",
    "print('S = {:>.6f}'.format(S))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(c) $f(x) = 1 + 5x - 6x^3 - e^{2x}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(1) $f(x) = 1 + 5x - 6x^3 - e^{2x}$\n",
    "\n",
    "$ \\Rightarrow 5x = 6x^3 + e^{2x} - 1$\n",
    "\n",
    "$ \\Rightarrow g(x) = x = \\frac{6x^3 + e^{2x} - 1}{5}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1st root is 0.000000\n",
      "S = 0.400000\n"
     ]
    }
   ],
   "source": [
    "g = lambda x : (6 * np.power(x, 3) + np.exp(2 * x) - 1) / 5\n",
    "root = scipy.optimize.fixed_point(g, 0.5, method='iteration')\n",
    "print('1st root is {:>.6f}'.format(root))\n",
    "\n",
    "dg = sympy.diff((6 * sympy.abc.x ** 3 + sympy.exp(2 * sympy.abc.x) - 1) / 5)\n",
    "S = np.abs(sympy.lambdify(sympy.abc.x, dg)(root))\n",
    "print('S = {:>.6f}'.format(S))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It is locally convergent due to S = 0.400000 < 1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(2) $f(x) = 1 + 5x - 6x^3 - e^{2x}$\n",
    "\n",
    "$ \\Rightarrow 6x^3 = 1 + 5x - e^{2x}$\n",
    "\n",
    "$ \\Rightarrow x^3 = \\frac{1 + 5x - e^{2x}}{6}$\n",
    "\n",
    "$ \\Rightarrow g(x) = x =  \\sqrt[3]{\\frac{1 + 5x - e^{2x}}{6}}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2nd root is 0.506308\n",
      "S = 0.199103\n"
     ]
    }
   ],
   "source": [
    "g = lambda x : np.power((1 + 5 * x - np.exp(2 * x)) / 6, 1 / 3)\n",
    "root = scipy.optimize.fixed_point(g, 0.5, method='iteration')\n",
    "print('2nd root is {:>.6f}'.format(root))\n",
    "\n",
    "dg = sympy.diff((1 + 5 * sympy.abc.x - sympy.exp(2 * sympy.abc.x)) ** (1 / 3))\n",
    "S = np.abs(sympy.lambdify(sympy.abc.x, dg)(root))\n",
    "print('S = {:>.6f}'.format(S))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It is locally convergent due to S = 0.506308 < 1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(3) $f(x) = 1 + 5x - 6x^3 - e^{2x}$\n",
    "\n",
    "$ \\Rightarrow e^{2x} = 1 + 5x - 6x^3$\n",
    "\n",
    "$ \\Rightarrow 2x = \\ln{(1 + 5x - 6x^3)}$\n",
    "\n",
    "$ \\Rightarrow g(x) = x = \\frac{\\ln{(1 + 5x - 6x^3)}}{2} $"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Diverge, cannot find 3rd root by FPI\n",
      "3rd root is -0.912871\n",
      "S = 5.000000\n"
     ]
    }
   ],
   "source": [
    "g = lambda x : np.log(1 + 5 * x - 6 * np.power(x, 3)) / 2\n",
    "print('Diverge, cannot find 3rd root by FPI')\n",
    "\n",
    "# We use other method to find 3rd root\n",
    "root = scipy.optimize.fsolve(g, -1)[0]\n",
    "print('3rd root is {:>.6f}'.format(root))\n",
    "\n",
    "dg = sympy.diff(sympy.log(1 + 5 * sympy.abc.x - 6 * sympy.abc.x ** 3) / 2)\n",
    "S = np.abs(sympy.lambdify(sympy.abc.x, dg)(root))\n",
    "print('S = {:>.6f}'.format(S))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "# 1.3 Limits of Accuracy"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example\n",
    "\n",
    "Use the Bisection Method to find the root of $f(x) = x^3 - 2x^2 + \\frac{4}{3}x - \\frac{8}{27}$ to within six correct significant digits"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.666664123535\n"
     ]
    }
   ],
   "source": [
    "root = scipy.optimize.bisect(lambda x : np.power(x, 3) - 2 * np.power(x, 2) + 4 / 3 * x - 8 / 27, 0, 2)\n",
    "print('{:>.12f}'.format(root))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## DEFINITION 1.8\n",
    "\n",
    "Assume that $f$ is a function and that $r$ is a root, meaning that it satisfies $f(r) = 0$. Assume that $x_a$ is an approximation to $r$. For the root-finding problem, the **backward error** of the approximation $x_a$ is $\\left | f(x_a) \\right |$ and  the **forward error**  is $\\left | r - x_a \\right |$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## DEFINITION 1.9\n",
    "\n",
    "Assume that $r$ is a root of the differentiable function $f$ ; that is, assume that $f(r) = 0$. Then if $0 = f(r) = f'(r) = f''(r) = \\dots = f^{(m-1)}(r)$, but $f^{(m)}(r) \\neq 0$, we say that $f$ has a **root** of **multiplicity** $m$ at $r$. We say that $f$ has a **multiple root** at $r$ if the multiplicity is greater than one. The root is called **simple** if the multiplicity is one."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example\n",
    "\n",
    "The function $f(x) = sinx - x$ has  a triple root at $r = 0$. Find the forward and backward error of the approximate root $x_a = 0.001$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$f(0) = \\sin0 - 0 = 0$<br/>\n",
    "$f'(0) = \\cos0 - 1 = 0$<br/>\n",
    "$f''(0) = -\\sin0 - 0 = 0$<br/>\n",
    "$f'''(0) = -\\cos0 - 0 = -1$<br/>\n",
    "\n",
    "The forward error is $\\left | r - x_a \\right | = 10^{-3}$<br/>\n",
    "The backward error is $\\left | f(x_a) \\right | = \\left | sin(0.001) - 0.001 \\right | \\approx 1.6667 \\times 10^{-10}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Sensitivity Formula for Roots\n",
    "   \n",
    "Assume that $r$ is a root of $f(x)$ and $r + \\Delta r$ is a root of $f(x) + \\epsilon g(x)$. Then \n",
    "$$ \\Delta r \\approx -\\frac{\\epsilon g(r)}{f'(r)}$$\n",
    "if $\\epsilon \\ll  f'(r)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Error Magnification Factor\n",
    "\n",
    "$$\\textbf{error magnification factor} = \\frac{\\text{relative forward error}}{\\text{relative backward error}}$$\n",
    "\n",
    "$$ \n",
    "\\text{error magnification factor} \n",
    "= \\left | \\frac{\\Delta{r}/r}{\\epsilon{g(r)/g(r)}} \\right |\n",
    "= \\left | \\frac{-\\epsilon{g(r)/(rf'(r))}}{\\epsilon} \\right |\n",
    "= \\frac{\\left | g(r) \\right |}{\\left | rf'(r) \\right |}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Additional Examples\n",
    "\n",
    "1. Find the multiplicity of the root $r = 0$ of $f(x) = 6x - 6\\sin{x} - x^3$.\n",
    "\n",
    "$f(x) = 6x - 6\\sin{(x)} - x^3, f(x = 0) = 0$\n",
    "\n",
    "$f'(x) = -3x^2 - 6\\cos{(x)} + 6, f'(x = 0) = 0$\n",
    "\n",
    "$f''(x) = -6x + 6\\sin{(x)}, f''(x = 0) = 0$\n",
    "\n",
    "$f^{(3)}(x) = -6 + 6\\cos{(x)}, f^{(3)}(x = 0) = 0$\n",
    "\n",
    "$f^{(4)}(x) = - 6\\sin{(x)}, f^{(4)}(x = 0) = 0$\n",
    "\n",
    "$f^{(5)}(x) = - 6\\cos{(x)}, f^{(5)}(x = 0) = -6$\n",
    "\n",
    "The root at 0 has multiplicity **five**."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2. Use the MATLAB command fzero with initial guess 0.001 to approximate the root of $f(x) = 6x - 6\\sin{x} - x^3$. Compute the forward and backward errors of the approximate root."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.000000\n"
     ]
    }
   ],
   "source": [
    "f = lambda x : 6 * x - 6 * np.sin(x) - np.power(x, 3)\n",
    "root = scipy.optimize.fsolve(g, 0.001)[0]\n",
    "print('{:>.6f}'.format(root))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "## 1.3 Computer Problems"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1. Let $f(x) = sinx - x$.\n",
    "\n",
    "(a) Find the multiplicity of the root r = 0.\n",
    "\n",
    "$f(0) = \\sin{0} - 0 = 0$\n",
    "\n",
    "$f'(0) = \\cos0 - 1 = 0$\n",
    "\n",
    "$f''(0) = -\\sin0 = 0$\n",
    "\n",
    "$f'''(0) = -\\cos0 = -1$\n",
    "\n",
    "The multiplicity is 3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(b) Use initial guess x = 0.1 to locate a root. What are the forward and backward errors."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "forward error  = 0.0000000213751396\n",
      "backward error = 0.0000000000000000\n"
     ]
    }
   ],
   "source": [
    "f = lambda x : np.sin(x) - x\n",
    "root = scipy.optimize.fsolve(f, 0.1)[0]\n",
    "print('forward error  = %.16f' %(np.abs(root)))\n",
    "print('backward error = %.16f' %(np.abs(f(root))))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "2. Carry out Computer Problem 1 for $f(x) = \\sin{x^3} - x^3$.\n",
    "\n",
    "(a) Find the multiplicity of the root r = 0."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - x^{3} + \\sin{\\left(x^{3} \\right)}$"
      ],
      "text/plain": [
       "   3      ⎛ 3⎞\n",
       "- x  + sin⎝x ⎠"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 0.0$"
      ],
      "text/plain": [
       "0.0"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 3 x^{2} \\cos{\\left(x^{3} \\right)} - 3 x^{2}$"
      ],
      "text/plain": [
       "   2    ⎛ 3⎞      2\n",
       "3⋅x ⋅cos⎝x ⎠ - 3⋅x "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 0.0$"
      ],
      "text/plain": [
       "0.0"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - 9 x^{4} \\sin{\\left(x^{3} \\right)} + 6 x \\cos{\\left(x^{3} \\right)} - 6 x$"
      ],
      "text/plain": [
       "     4    ⎛ 3⎞          ⎛ 3⎞      \n",
       "- 9⋅x ⋅sin⎝x ⎠ + 6⋅x⋅cos⎝x ⎠ - 6⋅x"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 0.0$"
      ],
      "text/plain": [
       "0.0"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - 27 x^{6} \\cos{\\left(x^{3} \\right)} - 54 x^{3} \\sin{\\left(x^{3} \\right)} + 6 \\cos{\\left(x^{3} \\right)} - 6$"
      ],
      "text/plain": [
       "      6    ⎛ 3⎞       3    ⎛ 3⎞        ⎛ 3⎞    \n",
       "- 27⋅x ⋅cos⎝x ⎠ - 54⋅x ⋅sin⎝x ⎠ + 6⋅cos⎝x ⎠ - 6"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 0.0$"
      ],
      "text/plain": [
       "0.0"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 81 x^{8} \\sin{\\left(x^{3} \\right)} - 324 x^{5} \\cos{\\left(x^{3} \\right)} - 180 x^{2} \\sin{\\left(x^{3} \\right)}$"
      ],
      "text/plain": [
       "    8    ⎛ 3⎞        5    ⎛ 3⎞        2    ⎛ 3⎞\n",
       "81⋅x ⋅sin⎝x ⎠ - 324⋅x ⋅cos⎝x ⎠ - 180⋅x ⋅sin⎝x ⎠"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 0.0$"
      ],
      "text/plain": [
       "0.0"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 243 x^{10} \\cos{\\left(x^{3} \\right)} + 1620 x^{7} \\sin{\\left(x^{3} \\right)} - 2160 x^{4} \\cos{\\left(x^{3} \\right)} - 360 x \\sin{\\left(x^{3} \\right)}$"
      ],
      "text/plain": [
       "     10    ⎛ 3⎞         7    ⎛ 3⎞         4    ⎛ 3⎞            ⎛ 3⎞\n",
       "243⋅x  ⋅cos⎝x ⎠ + 1620⋅x ⋅sin⎝x ⎠ - 2160⋅x ⋅cos⎝x ⎠ - 360⋅x⋅sin⎝x ⎠"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACEAAAASCAYAAADVCrdsAAABXUlEQVR4nM3Vv0pcQRTH8c9KbJRltUklRFn88wCiolVSiGVeIbWghY3lprBWG0sRfAJbG6tI3sDEKAhCUgmKKNiIxZ2FcbLLzliop/ld5s753h9n7pxTa7Va3jr6OqyNYBd/8YALbGH4Bfws1ockqYljfMQBfmEGq1jCAq4yDWSz0krshKQVfMU6vmATk9jINFDEqkX/RBNnqpI18RgB6/iHWgDf9TBQxIor8TnoYZIEt/iBAcz1MFDMik1MBj3tAv4TdCLDRBErNtEIetMlsb0+lGGiiNXpir56xCba7hqdNkbr1xncIlZs4nfQbmc+HrTbOcdRxIpNHAVd9P8x1VXN5R4/M0wUseIN56orNYrlJPE7BrHveY9oYgr9yf4iVi0ZYGmrPcGs6t6fYt7ztn2BTxgLzy9ipaU6xzT2QsJagG2rGkvu3ChipQMMLvEt80OjPd5nsd5dn3izeAKTe2S+Jy2elwAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$\\displaystyle 0.0$"
      ],
      "text/plain": [
       "0.0"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - 729 x^{12} \\sin{\\left(x^{3} \\right)} + 7290 x^{9} \\cos{\\left(x^{3} \\right)} + 17820 x^{6} \\sin{\\left(x^{3} \\right)} - 9720 x^{3} \\cos{\\left(x^{3} \\right)} - 360 \\sin{\\left(x^{3} \\right)}$"
      ],
      "text/plain": [
       "       12    ⎛ 3⎞         9    ⎛ 3⎞          6    ⎛ 3⎞         3    ⎛ 3⎞      \n",
       "- 729⋅x  ⋅sin⎝x ⎠ + 7290⋅x ⋅cos⎝x ⎠ + 17820⋅x ⋅sin⎝x ⎠ - 9720⋅x ⋅cos⎝x ⎠ - 360\n",
       "\n",
       "    ⎛ 3⎞\n",
       "⋅sin⎝x ⎠"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 0.0$"
      ],
      "text/plain": [
       "0.0"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - 2187 x^{14} \\cos{\\left(x^{3} \\right)} - 30618 x^{11} \\sin{\\left(x^{3} \\right)} + 119070 x^{8} \\cos{\\left(x^{3} \\right)} + 136080 x^{5} \\sin{\\left(x^{3} \\right)} - 30240 x^{2} \\cos{\\left(x^{3} \\right)}$"
      ],
      "text/plain": [
       "        14    ⎛ 3⎞          11    ⎛ 3⎞           8    ⎛ 3⎞           5    ⎛ 3⎞\n",
       "- 2187⋅x  ⋅cos⎝x ⎠ - 30618⋅x  ⋅sin⎝x ⎠ + 119070⋅x ⋅cos⎝x ⎠ + 136080⋅x ⋅sin⎝x ⎠\n",
       "\n",
       "          2    ⎛ 3⎞\n",
       " - 30240⋅x ⋅cos⎝x ⎠"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 0.0$"
      ],
      "text/plain": [
       "0.0"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAABF8AAAAYCAYAAAAsyFKjAAAVb0lEQVR4nO2de7gdVXmH3xMDJIBNFAnxAhxAg6FgjtRYUAKJEpRLNVFBHirlFAhF7qWgRUVOilwtKaBy0VQSQCAVCdCCJCgJCCkgkBRSAuGSQBIDEuIJ14Ch6R+/Nc+ePXtmn7XWnpm9N3u9z7OfOWf2mpk1M9/v+769Zq01XX19fQQCgUAgEAgEAoFAIBAIBIphULMr0CLsDdwKrAI2Ar0Z5T4IzAReAtYDjwP7lFC/sjkeeBR4xXz+GziwqTUKNIuBtNHpttLp5x+oYBtHjgOWoRjyMDCujMo1gaCNgC/LkYaSn9uaWKcyCdoJJDkDaeDHza5IQQSbD7QLDdtqaHwRWwKLgZOBNzPKDAfuA7rQRR4NnAj8sYT6lc1K4NvA7sCngLuAm4FPNLFOgeYwkDY63VY6/fwDFWziyNeBS4BzgU8CC4BfA9uVUcGSCdoI+DIWPeyKPrujH57/0cxKlUjQTiDOHsAx6Affu5Vg84F2oWFbDY0v4nbgO8CNwP9llPkWsBr4O+BB9OTyt8ASz2N2o2Rihuf2acxEjUFbNLifW9APgqeBpcB3gVeBPWNl/grV/+gGjxVobQbSho2ttCp56MX2/INe3v3YxJFTkc//GYodJ6K48k3PY3YT4kjg3cdLwAuxzwHoCWOrN74E7QTyZhjwC+BI4E8N7qubEC8CnU1L5P1ZjS+fB2ajoPcW8AdgDgqAcZaT3jV0o9m2HrbHAPga8CPgdygAbwSuHWD/eTMJeACYhW7cIuAE1BOmFRgLHA6cD7ye437fAxyKnuouiK1/GLX0nW2+awaudrEVEsFsJJo3gXXAvcBRpOvBZ5s0vkFFG0kh9pKto+jzTkH1ypMsW2lFitBLvfNvBb1EuPjeLmAK8n2voWv1EHAsdjZWz+7jfAT4uanLWyi2XAy8L+dtDgTmoicXbwLPAr+knMbCTVEwnptYPxf4TAnHtyHEEbv84gL08GUFsqO1wELgLOSb6+GiP3C3cx8bL0t/PnShmHYt2T3KWoFO1E4vbrmLa/kIV42WkRtGFK2Dn6IG/Xk57S9POtHm4xTty+PY5lJl+f92pGXy/sEpO7oQOB3duFuBNcDWKGkcj57uxVmHblKS1+pU1vUY3wPGmH2uBD5eZ99FsSMaq/9v6Mb1IOcOfmMwV6GhS+vyqBxwDgoyl+e0v93QOLYh6LpPBh5LlDkP/Sg7CXWjLxtXuzgYXZ/VKJA9D2wDfAWYDuxvymxscJsk2yIbeY30gLEImJqx7Tjgc6iVtdFzKQobW2k18tSL7fk3Wy/g7nuvBQ5DDc7XA28AE9F1+wzqCZjFQHYfsRMKWiPQE4UngE+j4TtfBD4LvJzDNhegHowvo4C4Bvgo8GXgq+ZcimzU/wAK1C8m1r8I7Ou5zxBHGscnv/hH4BHgTipP0fYA+tDwgD1Qw0wSV/252rmPjZelP18mAjug3mKtTCdqZxFuuYtr+QhXjZaRG0LxOpiC9PuNBvYRJ8SL/Cjal8exzaXK8v/tSsvk/V2Jtx1NQa2sM1EC8XZiJ5sAf479v9wsux0q7HoMgAnIwJ9GE9zOQ93w8nJIcV5DPVpmJNa/jZ74xp9Qnosu+OgC6uHCKCSY6eia5sGmaB6CYegpwhTkUBYnyi0BNkfJUVZX+6JwtYvPoST5NqrrOhINJdsWneuvGtwmThdK0HcAbgJOQ9dyusX5gcS9B3Ket+ZYLx+ytGFrK61C3npxOf9m6sXV905GNrsMBeM1Zv2myK4OQgH9ppRjudj9HGA/FJx+FFs/Df3IvRL1tGlkm5Eo8XwJjcuNz9U1AY3ZXYYa2fMgTSsfMnXYB7gntv77wN8CO+d0bF/aOY70AleheznfcVuf/GIImjA5yTlo6Nnl6GFNHJ/cx8XOfW28DP01wi+B7ZEPalU6VTv1yMpdfMq7arSM3BCK1cHOqOfNXsCTZt18dP9P8NxnnrSzzTdK0b48jm0uVab/b0daKu+Pd6XbDCUOz5NuTFBrTK74HmMe8BT2T/HnmrJfTazvQsnwRtR7xYXV6O1GcZZQO1Hil1B35NVUuqHdTW0i1k3t2Mv4um7gBvSDZz1q+Dkoo25HonOblfG9z/V4GwWuh9Es64uQEJPcgK7BxIxjF4mrXdwF/Ce1zvoF4Arz9/gctolzEgrsf497N7fdUDKyitq3PPjWqwht2NoKKIGehc7pLaSTucAhKWUPQT9S16Huk4+Z/W+WUtZWd1BfL0VqBZqnFx/fO9ksL6LS8ILZ9kzzd1YSaGv3O6HAvxz4SeK7s8y2h1M9Ptdnm+1RvHuA2knS56Hxulsn1uetlTWoS/02ifXbUDtMN8SR8nCNI5De8AKVOUk+lljvoz9XO/ex8bL056ulEejH+EC9XpodVzpVO1nUy118yrtqtIzc0EcHYG8Le6Lekv8LbDCffZDtbaDWXm1stZv0OV/i67uxixmdavNl+PI4trlUWf7f574W5Z+heT66obw/3vgyEd2Ym5ADOhDN5nsy9ceKbYZak79jyk5AXavT8D2GK6eb/Z+dqMu/AkegFst/dtznfdQ+mRwFPBf7/xjUbWsX5MgvQl3PhiLh2LI9am3vBq5BxrKr2feElPL7oqT+/oz95XE9BpFu/PeZZSsFfh8iZ7khx21GI7FeQvWTblui1tl/J30ctE+9itBGkixbmYK6N04yy4tQojWCWkd5LrL70cB1qMtll1k/B7U6R7jqrp5eitQKNE8vPr53pFk+m/JdtG4c1fcC3Ow+8mdzqU18X0XXa3OUmDeyzVMoWH4aJbRx9gbeC/wmsT5vrbyNAnXy3k+keqxwiCPty9+YZfKtJD76c7VzHxsvS3++ttOLEurrU76LaIW4ErRTjWvu4pvrlEFWPuWjA7C3hZtRo1RP7PMQ+iHXQ/WP/mbEjE61+TJ8eYRLLlWW/3e9r0X5Z2gtH+2U98fnfBlrluvRxHG7Jja+B3WteSmxfiQSaZxl6MTvTqz3PYYr/2PqdARqtZuBGodORU+mkm+W2BKNiwNdwO2Qc1uLWjdBc70sQLMaz0KvCT3J7DfiH5Dxj6G25TEphnqMR2PHp8bWXQfcgQxkXmz9FqauS8huFXW9HucjcaxAgj3M1CntPea/N8u9655RazOYytwVd+S0zWB0zZ+n2kZsGYoaNd/BfoiSTb3y1oatrewCXIbGW45DT3PifCT2956oJXkFCiRRr4Az0ORmB6Gul9H4SRfdDaSXIrUCzdOLj++NervskLK/qOvqYPP3E7H/Xew+atBemvH9U+jJzCj0dMN3m7UoQZqGejDejMYx74SenNyJ7ChOEXFkmtnngyggH4uGI10R20+II+3Daei+D0OvnNwLNbwkn/z56M/Vzn1svCz9udoOKOk+Gv3YzJpDsBXiStBONa65i2+uUwb18ikfHYC9LfSbT5zXkc6TQxvKjhmdbPNl+HJwz6XK8v8u97VI/wzN89EN5/3xni8jzPJ01L1mnNnpJ1Cr2N5o7G2cq9BszyPNye2Gxod1o0mzxiTK+xzDlzOROM5CXePPQS1nh1PbwvcpJKKFKBBMNX//S6zM71HL3SHI8Z1jjnFZYl8bSB86tSZlXRbPAT9IrJuDRJgc9/xh1Eq3eoB9ulyPkWhSpieR4MaiCcfSJkJbZ/abHH7VTpyPHOjt6Jrksc33UQNdL35vaDgEGI6C3gqH7WzOJU9t2NrKN1EwOZtaBwwanx1xpFn+gOrhGBuAfzJ1TM72bqs7G70UpRVonl58fG/U/ftU4P2x9ZtQnZy9L/a3q90PM8t1Gd9H64c3uA1oYvivIDucgp5iHIz0NYPa4A35x5FZwCloQshF6Mf6AVT3oIQQR9qF09C1OAXdyztQspp8gOSjPx87vxg3Gy9Tfy62A0pmP0b9IUetEFeCdqpxzV18c50yqJdP+eoA3LVgQ5kxo5Ntvixf7vMb4mKK9/9gf1+L9s/R92X76Ibz/njjS/T3BtRKdi962vAYGvu/Eo03jHermorGSr6I3oKxGD3Jm4aSz75EBXyO4csKZIjdaBKhBcgo08bnzUdPWZKf3kS521CD0hDUEngp1WNKf4G6aD2OespMonaMnQ2LSO96uYLaV39tZZZ/GmCfLtejF3U/3Aw5mn2p3yixFrvW9eUM/JrB+Cc5K3cRnISE/QQSWB7b/DVqMb0ITSLnQ9QN90qHbWzPJU9t9GJnK1G3xSznFGd3s7wr5bulyE/sQCVwuOjORi9FagXs9LKcfLXi43tvMOeyE7q2V6Lur4tQwhH15oiCUh52XyTfQq/snIHOaQv0ZoJnkQ1dmLJNEXHkMrO/zczxk92JQxypz3Jq7f8q8928lO9mWOzTl5HoHo9E12FH1OC2e6JcWbmPj42XhYvtgO5lF+ollkUrxJWgnWpccxefXKcMfHJDW1y1EDGe9HnWyo4Z7W7zZedSrvjmUmX5f9v7WqR/hub56F4azPvjw476zXIhlbcYRbxhdnwUav0cyBiuQE4r2SUsz2PYEH/6dJQ5RpFMQ61txyHHfQoS892olfQhy/30Z6zfQHWDGVRaRIdY7Leo6zEUu5bZZ8ieqDCNP/hVx5oT0I/Jx1EPrrU5bDMYuBo5izM96/WX6K1aK6l9XZ1vvZKUrY3hZrnKomzkXLNaqVejFuThqEXZRXe2einy+tjoJW+t9Juli+99B81hcSrqFn6EqdN8NCnZjabcH/G3++jJyrCM76P1/bF1PtuMR69hnI3OJ+IRlDAtRTHrCmrnuAlxpJpmx5GLqX0S14MmZp1JrX0vaqhWdryIbOsRZEtXU90dvd8sXfTnaufjcbfxsvQXkbftDDfLZsaVoJ0KrrmLT65TBjb5VCM6gHxtoeyY0U423wq5lIut+OZS4ynH/0fY3NfhZlmEf4Y29tHxxpcnzbI/Y8OoxWioxUGiyidnbs7zGANxGJoo5wX0VOpk0scV583V5jMcBZXJqDvVHODjND6fTZKoG9lWdUsVdz0GoXNdZlH28zkcLy9OQS2li1G90oYc+GyzJeoVBdkO/2fmc4nZZ5LoSZDt5HM29YrTDG30m+WHqcwPkkXkWEeiwJnkg4lyYK87G70UeX1s9ZK3Vnx9759RQL8gsX4IGhKwBp3LcPzsPqrXqLQNqLw1Jj4m2Web6G0N81LKv4Gerk9G3XzjjS8hjlTTCnHk4pR1vegH5AzyfV2uK8+hH2w96ClX1PXZR3+udu5j42XpD4qxnX6zbGZcCdqp4Jq7uJYvg1Owy6d8dQDF2EKZMaOdbL4VcikXW/H9DVGW/wf7+9pvlkX5Z2hTHx1vyfwtajHahdqnYlB5imNj7FFXo+QTxDyPUY8DUCBZjMbhPYnGiiXfVlQk/aglf4qpy/spZnKo1ci46p1bkddjZ9Q1eFEO+yqLb6PgugjN9m3T8GK7zVsokUj7LDRl7jX/p/XuGoK6uL5jyuRVr4hmaeN+s9zfomx0ncanfPdRNEnXMtKDXz/1dTeQXoq+Ps3SS96+91A083z0FhJfu48Shf1S6vVe4LMocbi/wW2iWeizumJH6+PdS0McqSbEETs+ZJbxH5M++nO1cx8bL0t/RdlOK8SVoB3hmru4li8Dl3zKRwdQfFzpp/iY0ck2X7Qv982lyvL/Lve1LP8Mbeaj4xf7OfSqpu1Qa0+c/YAvoJOLZvweTfo7ybvR66Ggduyc6zF82At1h19p9vcSmtxwMLVPb/NmArrASaIJmororr4RzRnwASpv2ohT9PWIGtrSWltbkTPRJGoPoxZxm8nIXLZ5Ewk27XOrKTPT/D8rZfuD0djaXzPw5HOu59JMbVyOuq6eiYJWkvis5z83y+9RHUjeg1qlB1GdrLnorp5eyrg+zdKLr+/9i5R99QA/RE94zjfrfO3+GTRJXTdwfOI4U1GMuYbqGep9tvmdWR6DnsLE2R8lGeupvPI5xJFqQhypMIr0btqD0AR9I5AdxceX++jP1c5dbdznGD7bFGk7rRBXgnaES+7iU75oXPMpH+0UZQtlx4xOtvmifblvLlWG/3e9r0X6Z2hjHz04UeB41CVpGnpl0kI0wc0k1Dp9NJUuP19H48fuQcb4Kprg50DUon07umBJXI4RMcl8QF2BQJMZzTB/r0FvHegB/stsP5HK2LEb0divL6OJIiMjzZvZaOKl+9FYwC5zvLHIof+moOP+Cs3B8AXg6dj6Hoq/Hvuh+3ZLA/vwZRJ2dhFxBHrzyDvonE9K2edyqieZ89mmEaJuuD8doJxrvXporjYeR+Myr0CavwW9xm4rpI9XkCMFBYcL0eRhi00dX0cBZFfU6v/D2L5ddZemlx7KuT7N1IuP770TJQOLkY8fbbZ9E80Hk8e8TMehe34pSnqXoAnnJqDurt/NYZsbkR3sa8rORl1LR6Puul3ozQAv03ythDhSLpNwiyMHAOchP7QM2cw2aJLFHZFdTUk5jo/+XOzcxcZ9j+G6TQ/F2k6rxJVO1U4c29zFp/wk3DTqWt43z3PRTg/F2UIzYkYn23zRvtyHov1/D+73tUj/DG3so7v6+vqShbZGr7j6Ehpj9Yo56HlUzzq/D3qz0SepvGq6H3WrucZ8NmZUxPYYEX3o1U9ZPIcM7l7U9Wof4NFEmX3RD4kHqLRC5c2x6MaOQddkvanb9agF8NVY2W6UuM2k8jaMtHVx5qNzS7b0bYqeHCxHogG17hV9PYYhcc+hEujKpI+B7aLboTxooqbxDW6TRbSvKcD0lO9HI2e1EtW73hhol3qVYQu27ImSnnFoDOQaU5/pVCZwjTgUTXw3Br3e+BngOjQDfHwsrIvuoFYvZV2fZusF3H3v6eg+7ITGMK9CTyrPo/o1gfXoo77dA2yLkt8vosC8GgXWqWTPUO+6zSYoaToUPYXZHE2k+CBKPObSGloJccSdXvTWlgm4z1vRh1sc2RXdo73Qk7vhKElcit6IeCnZE5676g/c7NzGxhs9hu02ZWqp2XGlU7UT4ZK7+JTvI99cL4/cMMJGO0Xbgq2tdpMeG7LWR8ynNma0s83nQdG+PI0+6udSRfn/Ru9rEf4Z2thHpzW+BNqTM4Bz0eu6Fg5QNi9ORIIehww4EGgXgl4CgVqCLgIBP4J2Ap1GsPlAO9Ey9hoaX949DEETBD2KhgMUzVDUIrkA+FoJxwsE8iToJRCoJegiEPAjaCfQaQSbD7QTLWOvaTM1B9qT9Wjm+IdInwg5b7rRWN3TBigXCLQiQS+BQC1BF4GAH0E7gU4j2HygnWgZew09XwKBQCAQCAQCgUAgEAgECiT0fAkEAoFAIBAIBAKBQCAQKJDQ+BIIBAKBQCAQCAQCgUAgUCD/Dx4kRymWE3VSAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$\\displaystyle 6561 x^{16} \\sin{\\left(x^{3} \\right)} - 122472 x^{13} \\cos{\\left(x^{3} \\right)} - 694008 x^{10} \\sin{\\left(x^{3} \\right)} + 1360800 x^{7} \\cos{\\left(x^{3} \\right)} + 771120 x^{4} \\sin{\\left(x^{3} \\right)} - 60480 x \\cos{\\left(x^{3} \\right)}$"
      ],
      "text/plain": [
       "      16    ⎛ 3⎞           13    ⎛ 3⎞           10    ⎛ 3⎞            7    ⎛ 3\n",
       "6561⋅x  ⋅sin⎝x ⎠ - 122472⋅x  ⋅cos⎝x ⎠ - 694008⋅x  ⋅sin⎝x ⎠ + 1360800⋅x ⋅cos⎝x \n",
       "\n",
       "⎞           4    ⎛ 3⎞              ⎛ 3⎞\n",
       "⎠ + 771120⋅x ⋅sin⎝x ⎠ - 60480⋅x⋅cos⎝x ⎠"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 0.0$"
      ],
      "text/plain": [
       "0.0"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 19683 x^{18} \\cos{\\left(x^{3} \\right)} + 472392 x^{15} \\sin{\\left(x^{3} \\right)} - 3674160 x^{12} \\cos{\\left(x^{3} \\right)} - 11022480 x^{9} \\sin{\\left(x^{3} \\right)} + 11838960 x^{6} \\cos{\\left(x^{3} \\right)} + 3265920 x^{3} \\sin{\\left(x^{3} \\right)} - 60480 \\cos{\\left(x^{3} \\right)}$"
      ],
      "text/plain": [
       "       18    ⎛ 3⎞           15    ⎛ 3⎞            12    ⎛ 3⎞             9    \n",
       "19683⋅x  ⋅cos⎝x ⎠ + 472392⋅x  ⋅sin⎝x ⎠ - 3674160⋅x  ⋅cos⎝x ⎠ - 11022480⋅x ⋅sin\n",
       "\n",
       "⎛ 3⎞             6    ⎛ 3⎞            3    ⎛ 3⎞            ⎛ 3⎞\n",
       "⎝x ⎠ + 11838960⋅x ⋅cos⎝x ⎠ + 3265920⋅x ⋅sin⎝x ⎠ - 60480⋅cos⎝x ⎠"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle -60480.0$"
      ],
      "text/plain": [
       "-60480.0"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "multiplicity is 10\n"
     ]
    }
   ],
   "source": [
    "def derivative_f(f):\n",
    "    while True:\n",
    "        f = sympy.diff(f)\n",
    "        yield f\n",
    "    \n",
    "f = sympy.sin(sympy.abc.x ** 3) - sympy.abc.x ** 3\n",
    "fx = sympy.lambdify(sympy.abc.x, f)(0)\n",
    "df_generator = derivative_f(f)\n",
    "\n",
    "display(f)\n",
    "display(fx)\n",
    "\n",
    "while fx == 0:\n",
    "    f = next(df_generator)\n",
    "    fx = sympy.lambdify(sympy.abc.x, f)(0)\n",
    "    display(f)\n",
    "    display(fx)\n",
    "\n",
    "print('multiplicity is 10')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(b) Use initial guess x = 0.1 to locate a root. What are the forward and backward errors."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "forward error  = 0.0026072710744117\n",
      "backward error = 0.0000000000000000\n"
     ]
    }
   ],
   "source": [
    "f = lambda x : np.sin(np.power(x, 3)) - np.power(x, 3)\n",
    "root = scipy.optimize.fsolve(f, 0.1)[0]\n",
    "print('forward error  = %.16f' %(np.abs(root)))\n",
    "print('backward error = %.16f' %(np.abs(f(root))))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "3. (a) Use fzero to find the root of $f(x) = 2x\\cos{x} - 2x + \\sin{x^3}$ on $\\left [ -0.1, 0.2 \\right ]$. Report the forward and backward errors."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "root = 0.0001214090717832\n",
      "forward  error : 0.0001214090717832\n",
      "backward error : 0.0000000000000000\n"
     ]
    }
   ],
   "source": [
    "f = lambda x : 2 * x * np.cos(x) - 2 * x + np.sin(np.power(x, 3))\n",
    "root = scipy.optimize.fsolve(f, 0.03)[0]\n",
    "print('root = %.16f' % root)\n",
    "print('forward  error : %.16f' % np.abs(root) )\n",
    "print('backward error : %.16f' % np.abs(f(root)) )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(b) Run the Bisection Method with initial interval $\\left [ -0.1, 0.2 \\right ]$ to find as many correct digits as possible"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$f(x) = 2x\\cos{x} - 2x + \\sin{x^3}$\n",
    "\n",
    "$\\Rightarrow \\frac{2x \\cdot (\\cos^2{x} - 1) + (\\cos{x} + 1) \\cdot \\sin{x^3}}{(\\cos{x} + 1)}$\n",
    "\n",
    "$\\Rightarrow \\frac{-2x\\sin^2{x} + (\\cos{x} + 1) \\cdot \\sin{x^3}}{(\\cos{x} + 1)}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x = -0.0000000238418579\n",
      "forward  error : 0.0000000238418579\n",
      "backward error : 0.0000000000000000\n"
     ]
    }
   ],
   "source": [
    "f = lambda x : (2 * x * -np.power(np.sin(x), 2) + (np.cos(x) + 1) * np.sin(np.power(x, 3))) / (np.cos(x) + 1)\n",
    "root = scipy.optimize.bisect(f, -0.1, 0.2)\n",
    "print('x = %.16f' % root)\n",
    "print('forward  error : %.16f' % np.abs(root))\n",
    "print('backward error : %.16f' % np.abs(f(root)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "4. (a) Use (1.21) to approximate the root near 3 of $f(x) = (1 + \\epsilon)x^3 - 3x^2 + x - 3$ for a constant $\\epsilon$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x = r + Δr = 2.99999730\n",
      "Δr = -0.00000270\n",
      "r = x - Δr = 3.00000000\n"
     ]
    }
   ],
   "source": [
    "epsilon = 1e-6\n",
    "f = lambda x : np.power(x, 3) - 3 * np.power(x, 2) + x - 3\n",
    "df = lambda x : 3 * np.power(x, 2) - 6 * x + 1\n",
    "g = lambda x : np.power(x, 3)\n",
    "f_eps = lambda x, epsilon : f(x) + g(x) * epsilon\n",
    "root = scipy.optimize.fsolve(lambda x : f_eps(x, epsilon), 3)\n",
    "droot = -epsilon * g(3) / df(3)\n",
    "print('x = r + Δr = %.8f' % root)\n",
    "print('Δr = %.8f' % droot)\n",
    "print('r = x - Δr = %.8f' % (root - droot))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(b) Setting $\\epsilon = 10^{-3}$, find the actual root and compare with part (a)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x = r + Δr = 2.99730291\n",
      "Δr = -0.00270000\n",
      "r = x - Δr = 3.00000291\n"
     ]
    }
   ],
   "source": [
    "epsilon = 1e-3\n",
    "f = lambda x : np.power(x, 3) - 3 * np.power(x, 2) + x - 3\n",
    "df = lambda x : 3 * np.power(x, 2) - 6 * x + 1\n",
    "g = lambda x : np.power(x, 3)\n",
    "f_eps = lambda x, epsilon : f(x) + g(x) * epsilon\n",
    "root = scipy.optimize.fsolve(lambda x : f_eps(x, epsilon), 3)\n",
    "droot = -epsilon * g(3) / df(3)\n",
    "print('x = r + Δr = %.8f' % root)\n",
    "print('Δr = %.8f' % droot)\n",
    "print('r = x - Δr = %.8f' % (root - droot))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "5. Use (1.21) to approximate the root of $f(x) = (x - 1)(x - 2)(x - 3)(x - 4) - 10^{-6}x^6$ near $r = 4$. Find the error magnification factor. Use fzero to check your approximation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x = r + Δr = 4.00068251\n",
      "Δr = 0.00068267\n",
      "r = x - Δr = 3.99999984\n",
      "error magnification factor = 170.666667\n"
     ]
    }
   ],
   "source": [
    "epsilon = -1e-6\n",
    "f = lambda x : (x - 1) * (x - 2) * (x - 3) * (x - 4)\n",
    "df = lambda x : 4 * np.power(x, 3) - 30 * np.power(x, 2) + 70 * x - 50\n",
    "g = lambda x : np.power(x, 6)\n",
    "f_eps = lambda x, epsilon : f(x) + g(x) * epsilon\n",
    "root = scipy.optimize.fsolve(lambda x : f_eps(x, epsilon), 4)\n",
    "droot = -epsilon * g(4) / df(4)\n",
    "print('x = r + Δr = %.8f' % root)\n",
    "print('Δr = %.8f' % droot)\n",
    "print('r = x - Δr = %.8f' % (root - droot))\n",
    "emf = np.abs(g(4)) / np.abs(4 * df(4))\n",
    "print('error magnification factor = %.6f' % (emf))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "6. Use the MATLAB command fzero to find the root of the Wilkinson polynomial near $x = 15$ with a relative change of $\\epsilon = 2 \\times 10^{-15}$ in the $x^{15}$ coefficient, making the coefficient slightly more negative. Compare with the prediction made by (1.21)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x = r + Δr = 14.85625953\n",
      "Δr = -0.13999692\n",
      "r = x - Δr = 14.99625645\n"
     ]
    }
   ],
   "source": [
    "def wilkinson(x):\n",
    "    poly = 1\n",
    "    for i in range(1, 21):\n",
    "        poly *= (x - i)\n",
    "    return poly\n",
    "\n",
    "def dW(x):\n",
    "    poly = 1\n",
    "    if 1 <= x <= 20:\n",
    "        for i in range(1, 21):\n",
    "            poly *= (x - i) if i != x else 1\n",
    "    else:\n",
    "        raise ValueError('x must be netween [1, 20]')\n",
    "    return poly\n",
    "\n",
    "epsilon = 2e-15\n",
    "f = lambda x : wilkinson(x)\n",
    "df = lambda x : dW(x)\n",
    "g = lambda x : -1672280820.0 * np.power(x, 15.0)\n",
    "f_eps = lambda x, epsilon : f(x) + g(x) * epsilon\n",
    "root = scipy.optimize.fsolve(lambda x : f_eps(x, epsilon), 15.0)[0]\n",
    "droot = -epsilon * g(15.0) / df(15.0)\n",
    "print('x = r + Δr = %.8f' % root)\n",
    "print('Δr = %.8f' % droot)\n",
    "print('r = x - Δr = %.8f' % (root - droot))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "# 1.4 Newton's Method"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [],
   "source": [
    "def newton_method( f, df, x0, k = 500, tol = 1e-6) :\n",
    "    \"\"\"\n",
    "    Use Newton's method to find the root of the function\n",
    "    \n",
    "    Args:\n",
    "        f (function prototype) : function handle\n",
    "        df (function prototype) : derivative function handle\n",
    "        x0 (real number) : starting guess \n",
    "        k (integer) : number of iteration steps (default : 500)\n",
    "        tol (real number) : tolerance (default : 1e-6)\n",
    "        \n",
    "    Return:\n",
    "        Approximate solution xc\n",
    "        \n",
    "    Raises:\n",
    "        ValueError :\n",
    "            - f or df is None\n",
    "            - k is smaller than 0\n",
    "    \"\"\"\n",
    "    try:\n",
    "        if f is None or df is None :\n",
    "            raise ValueError('Function handle f or df is Null')\n",
    "        \n",
    "        if k <= 0 :\n",
    "            raise ValueError('Iteration k must be larger than 0')\n",
    "        \n",
    "        xc = x0\n",
    "        for _ in range(k) :\n",
    "            xt = xc - f(xc) / df(xc)\n",
    "            if tol > 0 and abs(xt - xc) / abs(xc) < tol :\n",
    "                return xt\n",
    "            else :\n",
    "                xc = xt\n",
    "        return xc\n",
    "    except ValueError as e :\n",
    "        print('ValueError Exception : ', e)\n",
    "        traceback.print_exception()   "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example\n",
    "Find the Newton's Method formula for the equation $x^3 + x - 1 = 0$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.68232780\n"
     ]
    }
   ],
   "source": [
    "f = lambda x : np.power(x, 3) + x - 1\n",
    "df = lambda x : 3 * np.power(x, 2) + 1\n",
    "root = scipy.optimize.newton(f, 0, df)\n",
    "print('{:>.8f}'.format(root))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## DEFINITION 1.10\n",
    "\n",
    "Let $e_i$ denote the error after step $i$ of an iterative method. The iteration is **quadratically convergent** if\n",
    "\n",
    "$$\n",
    "\\large M = \\lim_{i \\rightarrow \\infty}{\\frac{e_{i + 1}}{e_i^2} < \\infty}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## THEOREM 1.11 \n",
    "\n",
    "Let $f$ be twice continuously differentiable and $f(r) = 0$. If $f'(r) = 0$, then Newton’s Method is locally and quadratically convergent to $r$. The error $e_i$ at step $i$ satisfies\n",
    "\n",
    "$$\\large \\lim_{i \\rightarrow \\infty}{\\frac{e_{i + 1}}{e_i^2}} = M $$\n",
    "\n",
    "where\n",
    "\n",
    "$$\\large M = \\frac{f''(r)}{2f'(r)}$$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example\n",
    "Find the Newton's Method formula for the equation $f(x) = x^2$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.00000001\n"
     ]
    }
   ],
   "source": [
    "f = lambda x : np.power(x, 2)\n",
    "df = lambda x : 2 * x\n",
    "root = scipy.optimize.newton(f, 1, df)\n",
    "print('{:>.8f}'.format(root))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## THEOREM 1.12\n",
    "\n",
    "Assume that the $(m + 1)$-times continuously differentiable function $f$ on $[a,b]$ has a multiplicity $m$ root at $r$. Then Newton’s Method is locally convergent to $r$, and the error $e_i$ at step $i$ satisfies\n",
    "\n",
    "$$\\large \\lim_{i \\rightarrow \\infty}{\\frac{e_{i + 1}}{e_i}} = S $$\n",
    "\n",
    "where $ S = (m - 1)/m $."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## THEOREM 1.13\n",
    "\n",
    "If $f$ is $(m + 1)$-times continuously differentiable on $[a,b]$, which contains a root $r$ of multiplicity $m > 1$, then Modified Newton’s Method\n",
    "\n",
    "$$\n",
    "\\large x_{i+1} = x_i - \\frac{mf(x_i)}{f'(x_i)}\n",
    "$$\n",
    "\n",
    "converges locally and quadratically to $r$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example\n",
    "\n",
    "Apply Newton’s Method to $f(x) = 4x^4 - 6x^2 - 11/4$ with starting guess $x_0 = 1/2$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Failed to converge after 50 iterations, value is 0.5.\n"
     ]
    }
   ],
   "source": [
    "try:\n",
    "    f = lambda x : 4 * np.power(x, 4) - 6 * np.power(x, 2) - 11 / 4\n",
    "    df = lambda x : 16 * np.power(x, 3) - 12 * x\n",
    "    root = scipy.optimize.newton(f, 0.5, df)\n",
    "    print('{:>.8f}'.format(root))\n",
    "except RuntimeError as e:\n",
    "    print(e)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "## 1.4 Computer Problems"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1. Each equation has one root. Use Newton’s Method to approximate the root to eight correct decimal places."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " (a) $x^3 = 2x + 2$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.76929235\n"
     ]
    }
   ],
   "source": [
    "f = lambda x : np.power(x, 3) - 2 * x - 2\n",
    "df = lambda x : 3 * np.power(x, 2) - 2\n",
    "root = scipy.optimize.newton(f, 1, df)\n",
    "print('{:>.8f}'.format(root))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(b) $e^x + x = 7$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.67282170\n"
     ]
    }
   ],
   "source": [
    "f = lambda x : np.exp(x) + x - 7\n",
    "df = lambda x : np.exp(x) + 1\n",
    "root = scipy.optimize.newton(f, 1, df)\n",
    "print('{:>.8f}'.format(root))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(c) $e^x + \\sin{x} = 4$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.12998050\n"
     ]
    }
   ],
   "source": [
    "f = lambda x : np.exp(x) + np.sin(x) - 4\n",
    "df = lambda x : np.exp(x) + np.cos(x)\n",
    "root = scipy.optimize.newton(f, 1, df)\n",
    "print('{:>.8f}'.format(root))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "2. Each equation has one real root. Use Newton’s Method to approximate the root to eight correct decimal places."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(a) $x^5 + x = 1$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.75487767\n"
     ]
    }
   ],
   "source": [
    "f = lambda x : np.power(x, 5) + x - 1\n",
    "df = lambda x : 5 * np.power(x, 4) + 1\n",
    "root = scipy.optimize.newton(f, 1, df)\n",
    "print('{:>.8f}'.format(root))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " (b) $\\sin{x} = 6x + 5$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-0.97089892\n"
     ]
    }
   ],
   "source": [
    "f = lambda x : np.sin(x) - 6 * x - 5\n",
    "df = lambda x : np.cos(x) - 6\n",
    "root = scipy.optimize.newton(f, 1, df)\n",
    "print('{:>.8f}'.format(root))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(c) $ln{x} + x^2 = 3$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.59214294\n"
     ]
    }
   ],
   "source": [
    "f = lambda x : np.log(x) + np.power(x, 2) - 3\n",
    "df = lambda x : 1 / x + 2 * x\n",
    "root = scipy.optimize.newton(f, 1, df)\n",
    "print('{:>.8f}'.format(root))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "3. Apply Newton’s Method to find the only root to as much accuracy as possible, and find the root’s multiplicity. Then use Modified Newton’s Method to converge to the root quadratically. Report the forward and backward errors of the best approximation obtained from each method."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(a) $f(x) = 27x^3 + 54x^2 + 36x + 8$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "root = -0.66666226\n",
      "S = 0.666616 ≈ 2/3 implies that multiplicity is 3 according to theorem 1.12\n",
      "forward  error : 0.00000441\n",
      "backward error : 0.00000000\n"
     ]
    }
   ],
   "source": [
    "f = lambda x : 27 * np.power(x, 3) + 54 * np.power(x, 2) + 36 * x + 8\n",
    "df = lambda x : 81 * np.power(x, 2) + 108 * x + 36\n",
    "root = scipy.optimize.newton(f, 1, df)\n",
    "print('root = {:>.8f}'.format(root))\n",
    "\n",
    "e_i = np.abs(scipy.optimize.newton(f, 1, df, maxiter = 10, disp = False) - root)\n",
    "e_ii = np.abs(scipy.optimize.newton(f, 1, df, maxiter = 11, disp = False) - root)\n",
    "print('S = {:.6f} ≈ 2/3 implies that multiplicity is 3 according to theorem 1.12'.format(e_ii / e_i))\n",
    "\n",
    "print('forward  error : {:.8f}'.format(np.abs(- 2 / 3 - root)))\n",
    "print('backward error : {:.8f}'.format(np.abs(f(root))))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(b) $f(x) = 36x^4 - 12x^3 + 37x^2 - 12x + 1$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "root = 0.16666668\n",
      "S = 0.500123 ≈ 1/2 implies that multiplicity is 2 according to theorem 1.12\n",
      "forward  error : 0.00000001\n",
      "backward error : 0.00000000\n"
     ]
    }
   ],
   "source": [
    "f = lambda x : 36 * np.power(x, 4) - 12 * np.power(x, 3) + 37 * np.power(x, 2) - 12 * x + 1\n",
    "df = lambda x : 144 * np.power(x, 3) - 36 * np.power(x, 2) + 74 * x - 12\n",
    "root = scipy.optimize.newton(f, 1, df)\n",
    "print('root = {:>.8f}'.format(root))\n",
    "\n",
    "e_i = np.abs(scipy.optimize.newton(f, 1, df, maxiter = 10, disp = False) - root)\n",
    "e_ii = np.abs(scipy.optimize.newton(f, 1, df, maxiter = 11, disp = False) - root)\n",
    "print('S = {:.6f} ≈ 1/2 implies that multiplicity is 2 according to theorem 1.12'.format(e_ii / e_i))\n",
    "\n",
    "print('forward  error : {:.8f}'.format(np.abs(1 / 6 - root)))\n",
    "print('backward error : {:.8f}'.format(np.abs(f(root))))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "4. Carry out the steps of Computer Problem 3 for\n",
    "\n",
    "(a) $f(x) = 2e^{x-1} - x^2 - 1$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "root = 0.99999328\n",
      "S = 0.666092 ≈ 2/3 implies that multiplicity is 3\n",
      "forward  error : 0.00000672\n",
      "backward error : 0.00000000\n"
     ]
    }
   ],
   "source": [
    "f = lambda x : 2 * np.exp(x - 1) - np.power(x, 2) - 1\n",
    "df = lambda x : 2 * np.exp(x - 1) - 2 * x\n",
    "root = scipy.optimize.newton(f, 0, df)\n",
    "print('root = {:>.8f}'.format(root))\n",
    "\n",
    "e_i = np.abs(scipy.optimize.newton(f, 0, df, maxiter = 10, disp = False) - root)\n",
    "e_ii = np.abs(scipy.optimize.newton(f, 0, df, maxiter = 11, disp = False) - root)\n",
    "print('S = {:.6f} ≈ 2/3 implies that multiplicity is 3'.format(e_ii / e_i))\n",
    "\n",
    "print('forward  error : {:.8f}'.format(np.abs(1.0 - root)))\n",
    "print('backward error : {:.8f}'.format(np.abs(f(root))))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(b) $f(x) = ln(3 - x) + x - 2$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "root = 1.99999999\n",
      "S = 0.494630 ≈ 1 / 2 implies multiplicity is 2\n",
      "forward  error : 0.00000001\n",
      "backward error : 0.00000000\n"
     ]
    }
   ],
   "source": [
    "f = lambda x : np.log(3 - x) + x - 2\n",
    "df = lambda x : 1 - 1 / (3 - x)\n",
    "root = scipy.optimize.newton(f, 0, df)\n",
    "print('root = {:>.8f}'.format(root))\n",
    "\n",
    "e_i = np.abs(scipy.optimize.newton(f, 0, df, maxiter = 20, disp = False) - root)\n",
    "e_ii = np.abs(scipy.optimize.newton(f, 0, df, maxiter = 21, disp = False) - root)\n",
    "print('S = {:.6f} ≈ 1 / 2 implies multiplicity is 2'.format(e_ii / e_i))\n",
    "\n",
    "print('forward  error : {:.8f}'.format(np.abs(2.0 - root)))\n",
    "print('backward error : {:.8f}'.format(np.abs(f(root))))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "5. A silo composed of a right circular cylinder of height $10m$ surmounted by a hemispherical dome contains $400m^3$ of volume. Find the base radius of the silo to four correct decimal places."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Assume that radius is $\\textbf{r}$ and that satisfies\n",
    "$\\normalsize 400 = 10 \\cdot (r^2\\pi) + \\frac{(4/3\\pi r^3)}{2} = 10r^2\\pi + \\frac{2r^3\\pi}{3}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3.2362\n"
     ]
    }
   ],
   "source": [
    "f = lambda x : 10 * np.power(x, 2) * np.pi + 2 * np.power(x, 3) * np.pi / 3 - 400\n",
    "df = lambda x : 20 * x * np.pi + 2 * np.power(x, 2) * np.pi\n",
    "root = scipy.optimize.newton(f, 1, df)\n",
    "print('{:>.4f}'.format(root))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "6. A $10cm$ high cone contains $60cm^3$ of ice cream, including a hemispherical scoop on top. Find the radius of the scoop to four correct decimal places."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The formula for the volume $V$ of a cone is $\\large \\frac{hr^2\\pi}{3}$.\n",
    "\n",
    "Then, $\\normalsize 60 = V + \\frac{(4/3\\pi r^3)}{2} = \\frac{hr^2\\pi}{3} + \\frac{2\\pi r^3}{3}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2.0201\n"
     ]
    }
   ],
   "source": [
    "f = lambda x : 10 * np.power(x, 2) * np.pi / 3 + 2 * np.power(x, 3) * np.pi / 3 - 60\n",
    "df = lambda x : 20 * x * np.pi / 3 + 2 * np.power(x, 2) * np.pi\n",
    "root = scipy.optimize.newton(f, 1, df)\n",
    "print('{:>.4f}'.format(root))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "7. Consider the function $f(x) = e^{\\sin^3{x}} + x^6 - 2x^4 - x^3 - 1$ on the interval $[-2, 2]$. Plot the function on the interval, and find all three roots to six correct decimal places. Determine which roots converge quadratically, and find the multiplicity of the roots that converge linearly."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {},
   "outputs": [
    {
     "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": [
    "f = lambda x : np.exp(np.power(np.sin(x), 3)) + np.power(x, 6) - 2 * np.power(x, 4) - np.power(x, 3) - 1\n",
    "\n",
    "df = lambda x : 3 * np.exp(np.power(np.sin(x), 3)) * np.power(np.sin(x), 2) * np.cos(x) + \\\n",
    "6 * np.power(x, 5) - 8 * np.power(x, 3) - 3 * np.power(x, 2)\n",
    "\n",
    "xpts = np.linspace(-1.4, 1.63, 81)\n",
    "ypts = f(xpts)\n",
    "\n",
    "myplot(xpts, ypts, '$f(x) = e^{\\sin^3{x}} + x^6 - 2x^4 - x^3 - 1$')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(rightmost) root = 1.530134\n",
      "converge quadratically\n"
     ]
    }
   ],
   "source": [
    "# rightmost root\n",
    "root = scipy.optimize.newton(f, 1.5, df)\n",
    "print('(rightmost) root = {:.6f}'.format(root))\n",
    "print('converge quadratically')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(leftmost) root = -1.197624\n",
      "converge quadratically\n"
     ]
    }
   ],
   "source": [
    "# leftmost root\n",
    "root = scipy.optimize.newton(f, -1.5, df)\n",
    "print('(leftmost) root = {:.6f}'.format(root))\n",
    "print('converge quadratically')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "root = 0.000076\n",
      "converge linearly\n",
      "S = 0.749796 ≈ 3 / 4 implies multiplicity is 4\n"
     ]
    }
   ],
   "source": [
    "root = scipy.optimize.newton(f, 1.0, df)\n",
    "e_i = np.abs(scipy.optimize.newton(f, 1.0, df, maxiter = 11, disp = False) - root)\n",
    "e_ii = np.abs(scipy.optimize.newton(f, 1.0, df, maxiter = 12, disp = False) - root)\n",
    "print('root = {:.6f}'.format(root))\n",
    "print('converge linearly')\n",
    "print('S = {:.6f} ≈ 3 / 4 implies multiplicity is 4'.format(e_ii / e_i))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "8. Carry out the steps of Computer Problem 7 for the function\n",
    "$f(x) = 94\\cos^3{x} - 24\\cos{x} + 177\\sin^2{x} - 108\\sin^4{x} - 72\\cos^3{x}\\sin^2{x} - 65$ on the interval $[0, 3]$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {},
   "outputs": [
    {
     "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": [
    "f = lambda x : 94 * np.power(np.cos(x), 3) - 24 * np.cos(x) + 177 * np.power(np.sin(x), 2) - 108 * np.power(np.sin(x), 4) \\\n",
    "- 72 * np.power(np.cos(x), 3) * np.power(np.sin(x), 2) - 65\n",
    "\n",
    "df = lambda x : -282 * np.sin(x) * np.power(np.cos(x), 2) + 24 * np.sin(x) + 354 * np.sin(x) * np.cos(x) \\\n",
    "- 432 * np.power(np.sin(x), 3) * np.cos(x) + 216 * np.power(np.sin(x), 3) * np.power(np.cos(x), 2) \\\n",
    "- 144 * np.sin(x) * np.power(np.cos(x), 4)\n",
    "\n",
    "xpts = np.linspace(0, 2.4, 41)\n",
    "ypts = f(xpts)\n",
    "\n",
    "myplot(xpts, ypts, '$f(x) = 94\\cos^3{x} − 24\\cos{x} + 177\\sin^2{x} − 108\\sin^4{x} − 72\\cos^3{x}\\sin^2{x} − 65$')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(rightmost) root = 2.300524\n",
      "converge quadratically\n"
     ]
    }
   ],
   "source": [
    "# rightmost root\n",
    "root = scipy.optimize.newton(f, 2.2, df)\n",
    "e_i = np.abs(scipy.optimize.newton(f, 2.2, df, maxiter = 4, disp = False) - root)\n",
    "e_ii = np.abs(scipy.optimize.newton(f, 2.2, df, maxiter = 5, disp = False) - root)\n",
    "print('(rightmost) root = {:.6f}'.format(root))\n",
    "print('converge quadratically')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "root = 1.047190\n",
      "converge linearly\n",
      "S = 0.671987 ≈ 2/3 implies multiplicity is 3\n"
     ]
    }
   ],
   "source": [
    "root = scipy.optimize.newton(f, 0.9, df)\n",
    "e_i = np.abs(scipy.optimize.newton(f, 0.9, df, maxiter = 10, disp = False) - root)\n",
    "e_ii = np.abs(scipy.optimize.newton(f, 0.9, df, maxiter = 11, disp = False) - root)\n",
    "print('root = {:.6f}'.format(root))\n",
    "print('converge linearly')\n",
    "print('S = {:.6f} ≈ 2/3 implies multiplicity is 3'.format(e_ii / e_i))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "9. Apply Newton’s Method to find both roots of the function \n",
    "$f(x) = 14xe^{x-2} - 12e^{x-2} - 7x^3 + 20x^2 - 26x + 12$ on the interval $[0, 3]$. For each root, print out the sequence of iterates, the errors $e_i$, and the relevant error ratio $e_{i+1}/e^2_{i}$ or $e_{i+1}/e_{i}$ that converges to a\n",
    "nonzero limit. Match the limit with the expected value $M$ from Theorem 1.11 or $S$ from Theorem 1.12."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {},
   "outputs": [
    {
     "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": [
    "f = lambda x :  14 * x * np.exp(x - 2) - 12 * np.exp(x - 2) - 7 * np.power(x, 3) + 20 * np.power(x, 2) - 26 * x + 12\n",
    "df = lambda x : 14 * x * np.exp(x - 2) + 2 * np.exp(x - 2) - 21 * np.power(x, 2) + 40 * x - 26\n",
    "\n",
    "xpts = np.linspace(0.8, 2, 41)\n",
    "ypts = f(xpts)\n",
    "\n",
    "myplot(xpts, ypts, '$f(x) = 14xe^{x-2} - 12e^{x-2} - 7x^3 + 20x^2 - 26x + 12$')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "root = 0.85714286\n",
      "M = 2.63683075\n"
     ]
    }
   ],
   "source": [
    "root = scipy.optimize.newton(f, 0, df)\n",
    "print('root = {:.8f}'.format(root))\n",
    "\n",
    "e_i = np.abs(scipy.optimize.newton(f, 0, df, maxiter = 6, disp = False) - root)\n",
    "e_ii = np.abs(scipy.optimize.newton(f, 0, df, maxiter = 7, disp = False) - root)\n",
    "M = e_ii / np.power(e_i, 2)\n",
    "print('M = {:.8f}'.format(M))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 92,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "root = 2.00000490\n",
      "S = 0.66664751\n"
     ]
    }
   ],
   "source": [
    "root = scipy.optimize.newton(f, 1.9, df)\n",
    "print('root = {:.8f}'.format(root))\n",
    "\n",
    "e_i = np.abs(scipy.optimize.newton(f, 1.9, df, maxiter = 8, disp = False) - root)\n",
    "e_ii = np.abs(scipy.optimize.newton(f, 1.9, df, maxiter = 9, disp = False) - root)\n",
    "S = e_ii / e_i\n",
    "print('S = {:.8f}'.format(S))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "10. Set $f(x) = 54x^6 + 45x^5 - 102x^4 - 69x^3 + 35x^2 + 16x - 4$. Plot the function on the interval $[-2, 2]$, and use Newton’s Method to find all five roots in the interval. Determine for which roots Newton converges linearly and for which the convergence is quadratic."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "metadata": {},
   "outputs": [
    {
     "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": [
    "f = lambda x :  54 * np.power(x, 6) + 45 * np.power(x, 5) - 102 * np.power(x, 4) - 69 * np.power(x, 3) + 35 * np.power(x, 2) + 16 * x - 4\n",
    "df = lambda x : 324 * np.power(x, 5) + 225 * np.power(x, 4) - 408 * np.power(x, 3) - 207 * np.power(x, 2) + 70 * x + 16\n",
    "\n",
    "xpts = np.linspace(-1.4, 1.19, 101)\n",
    "ypts = f(xpts)\n",
    "\n",
    "myplot(xpts, ypts, '$f(x) = 54x^6 + 45x^5 - 102x^4 - 69x^3 + 35x^2 + 16x − 4$')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 94,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "root = -1.38129848\n",
      "converge quadratically\n"
     ]
    }
   ],
   "source": [
    "root = scipy.optimize.newton(f, -1.5, df)\n",
    "print('root = {:.8f}'.format(root))\n",
    "print('converge quadratically')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 95,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "root = -0.66666668\n",
      "converge linearly\n"
     ]
    }
   ],
   "source": [
    "root = scipy.optimize.newton(f, -0.8, df)\n",
    "print('root = {:.8f}'.format(root))\n",
    "print('converge linearly')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 96,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "root = 0.20518292\n",
      "converge quadratically\n"
     ]
    }
   ],
   "source": [
    "root = scipy.optimize.newton(f, 0, df)\n",
    "print('root = {:.8f}'.format(root))\n",
    "print('converge quadratically')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 97,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "root = 0.50000000\n",
      "converge quadratically\n"
     ]
    }
   ],
   "source": [
    "root = scipy.optimize.newton(f, 0.6, df)\n",
    "print('root = {:.8f}'.format(root))\n",
    "print('converge quadratically')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 98,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "root = 1.17611556\n",
      "converge quadratically\n"
     ]
    }
   ],
   "source": [
    "root = scipy.optimize.newton(f, 1.1, df)\n",
    "print('root = {:.8f}'.format(root))\n",
    "print('converge quadratically')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "11. The ideal gas law for a gas at low temperature and pressure is $PV = nRT$, where $P$ is pressure (in atm), $V$ is volume (in $L$), $T$ is temperature (in $K$), n is the number of moles of the gas, and $R = 0.0820578$ is the molar gas constant. The van der Waals equation\n",
    "\n",
    "$ \\large \\left ( P + \\frac{n^2a}{V^2} \\right ) (V - nb) = nRT $\n",
    "\n",
    "covers the nonideal case where these assumptions do not hold. Use the ideal gas law to compute an initial guess, followed by Newton’s Method applied to the van der Waals equation to find the volume of one mole of oxygen at $320 K$ and a pressure of 15 atm. For oxygen, $a = 1.36 L^2-atm/mole^2$ and $b = 0.003183 L/mole$. State your initial guess and solution with three significant digits."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 99,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "V0 = 1.751, V = 1.701\n"
     ]
    }
   ],
   "source": [
    "vanderWaals = lambda P, V, T, R, n, a, b : (P + np.power(n, 2) * a / np.power(V, 2)) * (V - n * b) - n * R * T\n",
    "dvanderWaals = lambda P, V, n, a, b : P + a * np.power(n, 2) / np.power(V, 2) - 2 * a * np.power(n, 2) * (V - b * n) / np.power(V, 3)\n",
    "f = lambda x : vanderWaals(P = 15.0, V = x, T = 320.0, R = 0.0820578, n = 1.0, a = 1.36, b = 0.003183)\n",
    "df = lambda x : dvanderWaals(P = 15.0, V = x, n = 1.0, a = 1.36, b = 0.003183)\n",
    "V0 = 1 * 0.0820578 * 320.0 / 15.0\n",
    "\n",
    "root = scipy.optimize.newton(f, V0, df)\n",
    "print('V0 = {:.3f}, V = {:.3f}'.format(V0, root))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "12.  Use the data from Computer Problem 11 to find the volume of 1 mole of benzene vapor at 700 $K$ under a pressure of 20 atm. For benzene, $a = 18.0 L^2-atm/mole^2$ and $b = 0.1154 L/mole$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 100,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "V0 = 2.872, V = 2.664\n"
     ]
    }
   ],
   "source": [
    "f = lambda x : vanderWaals(P = 20.0, V = x, T = 700.0, R = 0.0820578, n = 1.0, a = 18.0, b = 0.1154)\n",
    "df = lambda x : dvanderWaals(P = 20.0, V = x, n = 1.0, a = 18.0, b = 0.1154)\n",
    "V0 = 1 * 0.0820578 * 700.0 / 20.0\n",
    "\n",
    "root = scipy.optimize.newton(f, V0, df)\n",
    "print('V0 = {:.3f}, V = {:.3f}'.format(V0, root))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "13. (a) Find the root of the function $f(x) = (1 - 3 / (4x))^{1/3}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 101,
   "metadata": {},
   "outputs": [
    {
     "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"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "root = 0.75000000\n"
     ]
    }
   ],
   "source": [
    "f = lambda x : np.power(1 - 3 / (4 * x), 1 / 3)\n",
    "\n",
    "xpts = np.linspace(0.75, 2, 80)\n",
    "ypts = f(xpts)\n",
    "\n",
    "myplot(xpts, ypts, '$f(x) = (1 - 3 / (4x))^{1/3}$')\n",
    "\n",
    "xpts = np.linspace(-1, -1e-2, 80)\n",
    "ypts = f(xpts)\n",
    "\n",
    "myplot(xpts, ypts, '$f(x) = (1 - 3 / (4x))^{1/3}$')\n",
    "\n",
    "print('root = {:.8f}'.format(3 / 4))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(b) Apply Newton’s Method using an initial guess near the root, and plot the first 50 iterates."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 102,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "root = 0.75000000\n"
     ]
    }
   ],
   "source": [
    "f = lambda x : np.power(1 - 3 / (4 * x), 1 / 3)\n",
    "df = lambda x : 1 / (4 * np.power(x, 2) * np.power(1 - 3 / (4 * x), 2 / 3))\n",
    "root = scipy.optimize.newton(f, 1.0, df)\n",
    "print('root = {:.8f}'.format(3 / 4))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(c) Why are Theorems 1.11 and 1.12 not applicable?\n",
    "\n",
    "$f(x)$ is not differentiable in $[0, 0.75]$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "14. (a) Fix real numbers $a,b > 0$ and plot the graph of $f(x) = a^2x^4 - 6abx^2 - 11b^2$ for your chosen values. Do not use $a = 2,b = 1/2$, since that case already appears in Example 1.15."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 103,
   "metadata": {},
   "outputs": [
    {
     "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": [
    "f = lambda x, a, b : np.power(a, 2) * np.power(x, 4) - 6 * a * b * np.power(x, 2) - 11 * np.power(b, 2)\n",
    "df = lambda x, a, b : 4 * np.power(a, 2) * np.power(x, 3) - 12 * a * b * x\n",
    "xpts = np.linspace(-5, 5, 60)\n",
    "ypts = f(xpts, a = 1, b = 3)\n",
    "myplot(xpts, ypts, '$f(x) = a^2x^4 − 6abx^2 − 11b^2, a = 1, b = 3$')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(b) Apply Newton’s Method to find both the negative root and the positive root of $f(x)$. Then find intervals of positive initial guesses $[d1, d2]$, where $d2 > d1$, for which Newton’s Method: (c) converges to the positive root, (d) converges to the negative root, (e) is defined, but does not converge to any root. Your intervals should not contain any initial guess where $f'(x) = 0$, at which Newton’s Method is not defined."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 104,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[positive] root = 4.73459691\n"
     ]
    }
   ],
   "source": [
    "root = scipy.optimize.newton(lambda x : f(x, a = 1, b = 3), 4, lambda x : df(x, a = 1, b = 3))\n",
    "print('[positive] root = {:.8f}'.format(root))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 105,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[negative] root = -4.73459691\n"
     ]
    }
   ],
   "source": [
    "root = scipy.optimize.newton(lambda x : f(x, a = 1, b = 3), -4, lambda x : df(x, a = 1, b = 3))\n",
    "print('[negative] root = {:.8f}'.format(root))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "15. Solve Computer Problem 1.1.9 using Newton’s Method."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 106,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.635501 m\n"
     ]
    }
   ],
   "source": [
    "f = lambda x : scipy.pi * np.power(x, 2) * (1 - x / 3) - 1\n",
    "df = lambda x : -scipy.pi * np.power(x, 2) / 3 + 2 * scipy.pi * x * (1 - x / 3)\n",
    "\n",
    "root = scipy.optimize.newton(f, 1.0, df)\n",
    "print('{:>.6f} m'.format(root))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "16. Solve Computer Problem 1.1.10 using Newton’s Method.\n",
    "\n",
    "Skip"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "17. Consider the national population growth model $P(t) = (P(0) + \\frac{m}{r})e^{rt} - \\frac{m}{r}$, where m and r are the immigration rate and intrinsic growth rate, respectively, and time t is measured in years."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(a) From 1990 to 2000, the U.S. population increased from 248.7 million to 281.4 million, and the immigration rate was $m = 0.977$ million per year. Use Newton’s Method to find the intrinsic growth rate r during the decade, according to the model."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$f(x) = 32.7 + \\frac{0.977}{r}(1 - e^{10r})$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 107,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "r = 0.20707480\n"
     ]
    }
   ],
   "source": [
    "f = lambda x : 32.7 + 0.977 / x * (1 - np.exp(10 * x))\n",
    "df = lambda x : -9.77 * np.exp(10 * x) / x - 0.977 * (1 - np.exp(10 * x)) / np.power(x, 2)\n",
    "\n",
    "xpts = np.linspace(0.1, 0.3, 40)\n",
    "ypts = f(xpts)\n",
    "myplot(xpts, ypts, r'$f(x) = 32.7 + \\frac{0.977}{x}(1 - e^{10x})$')\n",
    "\n",
    "root = scipy.optimize.newton(f, 0.2, df)\n",
    "print('r = {:>.8f}'.format(root))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(b) The immigration rate from 2000 to 2010 was $m = 1.030$ million per year, and the population in 2010 was 308.7 million. Find the intrinsic growth rate r during the 2000–2010 decade."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$f(x) = 27.3 + \\frac{1.030}{r}(1 - e^{10r})$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 108,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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zdPgNkcJJBSB/ULRoUXr27El6ejrPPfccH374IQkJCXTs2JH09HTf8UQkF6kAJEunnXYa/fv3Z9euXQwePJgFCxZQrVo17rjjDp2URqSQUAHIcZ111lk8+uij7Nq1i169evHPf/6TypUrM3DgQL799lvf8UQkBCoAyZHzzjuPMWPGsG3bNtq2bctTTz1FbGwsI0eO5KeffvIdT0ROggpATkhsbCxTp05lw4YNJCYmcv/993PJJZcwZcoUfvvtN9/xROQEqADkpMTFxbFw4UKWLFlCdHQ03bt357LLLmPRokW+o4lIDnktADM728xmmtk2M0szs3o+88iJu/LKK1m7di3Jycl89913tGjRgquvvpoNGzb4jiYif8H3HMDzwFvOuapALXRS+AKpSJEidOnShbS0NEaPHs2GDRu47LLLuOmmm/jkk098xxORbHgrADMrCTQCJgI454445771lUdCd+qpp9KvXz927drFwIEDeeONN7jkkksYMGCAthgSyYd8zgFUBA4C/zSzDWb2ipmdkXkiM0sysxQzSzl48GDep5QTVrJkSZ544gl27NhBly5dGDlyJJUrV2bMmDEcOXLEdzwRCfJZAEWBy4CXnXO1gR+ABzNP5Jyb4JyLd87FR0dH53VGCcGFF17IpEmTSE1NpVatWvTt25caNWowe/ZsHWxOJB/wWQD7gH3OuQ+Ct2cSKAQpZGrXrs0777zDm2++ySmnnEK7du1o1KgRa9eu9R1NJKJ5KwDn3OfAXjO7JHjXVcBWX3kkvI4ddfTDDz9k/PjxpKenk5iYSNeuXfn00099xxOJSL63AroLmGpmHwFxwBN+40i4FS1alKSkJHbu3MmgQYOYPXs2l1xyCYMGDdJ5ikXymNcCcM5tDC7fv9Q518Y5943PPJJ3SpQowfDhw9m+fTvt2rXj8ccf5+KLL2bixIk6K5lIHvE9ByARLiYmhqlTp7JmzRoqVqzIbbfdRnx8PMuWLfMdTaTQUwFIvpCYmMjKlSuZNm0a33zzDVdeeSVt2rRh586dvqOJFFoqAMk3zIxOnTqRlpbGE088wZIlS6hWrRoDBgzgu+++8x1PpNBRAUi+U7x4cQYOHEh6ejo333wzI0aM0PoBkTBQAUi+Va5cOV599VXWrVtH5cqVue2227j88stZsWKF72gihYIKQPK9+Ph4VqxYQXJyMl9++SUNGzakc+fO2n9AJEQqACkQzIwuXbqwbds2hgwZwty5c6latSrDhg3jf//7n+94IgWSCkAKlDPOOIOhQ4eybds2WrVqxZAhQ6hWrZqOLyRyElQAUiDFxMQwffp0li1bxplnnkm7du1o3rw5aWk6pYRITqkApEBr0qQJGzZsYMyYMaSkpHDppZdyzz33cOjQId/RRPI9FYAUeEWLFqVPnz6kp6dz6623Mnr0aKpUqcI///lPnahe5DhUAFJoREdHM378eNatW0elSpW49dZbadCgAevXr/cdTSRfUgFIoVOnTh1WrFjBpEmT2L17NwkJCfTu3ZtvvtGxBkUyUgFIoVSkSBF69OjB9u3b6dOnD+PGjaNKlSq8+uqrWiwkEqQCkELt7LPP5vnnn2f9+vVUrVqVnj17Ur9+fVJTU31HE/FOBSARoVatWrz33ntMnjyZTz75hISEBO68804tFpKIpgKQiGFmdOvWje3bt9O3b1/Gjx9P1apVmTJlinYik4jkvQDMLMrMNpjZfN9ZJDKULFmS0aNHk5qaSmxsLN27d6dJkyZs2bLFdzSRPOW9AIB+gHbflDwXFxfHypUrmTBhAps2bSIuLo4HH3yQH374wXc0kTzhtQDM7ELgOuAVnzkkchUpUoTbb7+d7du30717d55++mmqVavGf/7zHy0WkkLP9xzAaGAAkO12eWaWZGYpZpZy8ODBPAsmkSU6OpqJEyeyYsUKSpYsSdu2bWndujV79uzxHU0kbLwVgJldD3zpnDvu9njOuQnOuXjnXHx0dHQepZNI1aBBA1JTUxk5ciRLly6lWrVqjBgxgl9++cV3NJFc53MOoAHQ2sw+AaYBV5rZvzzmEQHglFNO4d577yUtLY1mzZoxYMAA4uPjWbNmje9oIrnKWwE45wY65y50zlUAOgNLnXM3+8ojkllMTAxz5sxh9uzZ/Pe//6V+/fr06tWLb7/91nc0kVzhex2ASL7Xpk0btm7dSv/+/ZkwYQJVq1bl3//+t1YSS4GXLwrAObfcOXe97xwi2SlRogTPPvssKSkpxMTE0LVrV6699lo++eQT39FETlq+KACRgqJ27dqsXr2aF154gRUrVlC9enVGjRrF0aNHfUcTOWEqAJETFBUVxV133cXWrVu56qqruO+++0hMTNR5B6TAUQGInKTy5cszZ84cZsyYwf79+0lISOC+++7TnsRSYKgAREJgZnTo0IG0tDRuv/12Ro0aRfXq1Xnrrbd8RxP5SyoAkVxw9tlnM27cON577z2KFy9Oy5Yt6datG1999ZXvaCLZUgGI5KKGDRuyceNGBg8ezPTp06lWrZo2GZV8SwUgkstOPfVUHn30UdavX0/FihXp2rUrrVu3Zt++fb6jifyBCkAkTGrUqMGqVat49tlnfz+u0Lhx43ROYsk3VAAiYRQVFcXdd9/N5s2bSUxMpFevXjRt2pT09HTf0URUACJ5oWLFiixatIhXX32Vjz76iFq1ajFixAjtQCZeqQBE8oiZ8fe//520tDRatmzJgAEDqF+/vk5FKd6oAETyWNmyZZk1axbTp0/n448/pnbt2jz22GM654DkORWAiAdmRseOHdm6dSvt2rXjkUce4fLLL2fjxo2+o0kEUQGIeBQdHc20adN44403OHDgAAkJCQwePJiff/7ZdzSJACoAkXygbdu2bN26lS5dujB8+HDi4+N1cDkJOxWASD5RunRpJk+ezPz58/n666+5/PLLGTx4MEeOHPEdTQopFYBIPnPdddexZcsWunbtyvDhw7VuQMLGWwGYWXkzW2ZmW81si5n185VFJL8pVaoUkydPZs6cOXz++eckJCQwbNgwbSkkucrnHMBR4F7nXDWgLtDbzKp5zCOS77Ru3ZotW7bQsWNHhgwZQmJiIps2bfIdSwqJvywAM7vLzErl9hs75w4459YHrx8G0oALcvt9RAq6c845h6lTp/LGG2/w2WefUadOHZ544gntRSwhy8kcwHnAOjN73cyuMTPL7RBmVgGoDXyQxWNJZpZiZikHDx7M7bcWKTDatm3Lli1baNOmDQ8//DANGzbUMYUkJH9ZAM65QcDFwETgFmCHmT1hZpVyI4CZnQnMAvo7577L4v0nOOfinXPx0dHRufGWIgVWmTJlmD59OsnJyWzfvp24uDjGjh2rI4zKScnROgAXOJvF58HLUaAUMNPMngnlzc3sFAI//lOdc2+E8loikcLM6NKlC5s3b6Zx48bcddddNG/enL179/qOJgVMTtYB9DOzVOAZYCVQ0znXC6gDtD/ZNw4uSpoIpDnnnj3Z1xGJVOeffz4LFixg/PjxrFmzhho1avDaa6/p7GOSYzmZAygNtHPOtXDOzXDO/QLgnPsNuD6E924AdAOuNLONwcu1IbyeSMQxM5KSkn4/xPQtt9xCu3bt+PLLL31HkwLACtJfC/Hx8S4lJcV3DJF86ddff2X06NE8/PDDlCxZkldeeYVWrVr5jiX5gJmlOufiM9+vPYFFComoqCjuvfdeUlJSKFeuHK1btyYpKYnvv//edzTJp1QAIoVMjRo1+OCDDxgwYACvvPIKcXFxrF692ncsyYdUACKF0KmnnsrTTz/N8uXLOXr0KFdccQWPPPKIDiUhf6ACECnEGjVqxEcffUS3bt147LHHqF+/Ptu3b/cdS/IJFYBIIXfWWWcxadIkZs6c+fspKMeNG6fNRUUFIBIp2rdvz6ZNm2jYsCG9evXihhtuQIdXiWwqAJEIUq5cORYuXMjo0aNZtGgRNWvW5K233vIdSzxRAYhEmCJFitCvXz/WrVtHdHQ0LVu2pF+/fvz000++o0keUwGIRKiaNWuydu1a+vbtywsvvEBCQoLONRBhVAAiEax48eI8//zzLFiwgIMHD5KQkMDzzz+vFcQRQgUgIrRs2ZKPPvqI5s2b079/f66//nodTygCqABEBIBzzz2XOXPmMGbMGJYsWUKtWrVYvHix71gSRioAEfmdmdGnTx/Wrl1L6dKlad68OQMGDODIkSO+o0kYqABE5E8uvfRS1q1bxx133MGIESNo0KABO3fu9B1LcpkKQESydPrppzNu3DhmzZrFrl27qF27NlOmTPEdS3KRCkBEjqtdu3Z8+OGH1K5dm+7du9OtWzcOHz7sO5bkAhWAiPyl8uXLs2zZMoYOHUpycjJ16tRhw4YNvmNJiLwWgJldY2bbzWynmT3oM4uIHF9UVBRDhgxh6dKl/PDDD9StW5exY8dqn4ECzFsBmFkU8CLQEqgGdDGzar7yiEjONG7cmA8//JCrr76au+66i/bt2/PNN9/4jiUnweccwOXATufcbufcEWAacIPHPCKSQ2XKlGHevHmMGjWK+fPnExcXx6pVq3zHkhPkswAuAPZmuL0veN8fmFmSmaWYWcrOnTsZOnQoAFWqVCE9PZ3U1FTq1KkDwL333suoUaMAOP/889m/fz/Lly+nSZMmACQlJTFhwgQASpQoweHDh5k3b97vJ87u2rUrycnJx94XgOTkZLp27QpAq1atmDdvHocPH6ZEiRIATJgwgaSkJACaNGnC8uXL2b9/P+effz4Ao0aN4t577wWgTp06pKamkp6eTpUqVQAYOnSoxqQxFcgx3XHHHZx55pmsXLmSvXv30rBhQ3r06MH1119fYMdUGD+nY++dFfO1/M7MOgDXOOduC97uBiQ65/pk95z4+HiXkpKSVxFFJIcOHTpEUlISr7/+Oi1atGDKlClER0f7jiVBZpbqnIvPfL/POYDPgPIZbl8YvE9ECpiSJUsybdo0xo0bx/Lly4mLi+P999/3HUv+gs8CWAdcbGYVzawY0BmY6zGPiITAzLjjjjtYvXo1p59+Ok2bNuWpp57it99+8x1NsuGtAJxzR4E+wNtAGvC6c26Lrzwikjtq165Namoq7dq1Y+DAgbRq1YqvvvrKdyzJgtf9AJxzC5xzVZxzlZxzj/vMIiK556yzzmL69Om8+OKLvPPOO9SuXZuVK1f6jiWZaE9gEQkLM+POO+9k1apVFCtWjMaNGzNixAgtEspHVAAiElZ16tRh/fr1tGnThgEDBtC2bVvtOJZPqABEJOxKlizJjBkzGD16NAsWLPi9FMQvFYCI5Akzo1+/frz33nv88ssv1K9fn3/84x86lpBHKgARyVP16tVj/fr1NGrUiKSkJG655RZ+/PFH37EikgpARPJcdHQ0CxcuZMiQIUyZMoW6deuSnp7uO1bEUQGIiBdRUVEMHTqUhQsXsn//fuLj45k5c6bvWBFFBSAiXrVo0YL169dTrVo1brzxRu677z6OHj3qO1ZEUAGIiHcxMTG899579O7dm1GjRtGsWTO++OIL37EKPRWAiOQLxYoVY+zYsUyePJm1a9dy2WWXsXr1at+xCjUVgIjkK926dWP16tWcdtppNG7cmJdeekmbioaJCkBE8p1atWqRkpJC8+bN6d27Nz169NCmomGgAhCRfKlUqVLMnTuXRx99lH/961/Uq1ePXbt2+Y5VqKgARCTfKlKkCIMHD+bNN99k7969xMfHs3DhQt+xCg0VgIjkey1btiQlJYWYmBiuu+46nnjiCa0XyAUqABEpEGJjY1m1ahWdO3fm4Ycf5sYbb+Tw4cO+YxVoKgARKTDOOOMMpk6dyqhRo5g9ezZ169Zlx44dvmMVWF4KwMxGmNk2M/vIzGab2dk+cohIwWNm3HPPPSxatIgvvviChIQE3nzzTd+xCiRfcwCLgRrOuUuBdGCgpxwiUkBdddVVpKamEhsbS6tWrXjsscd0trET5KUAnHOLgieFB1gDXOgjh4gUbBdddBErV67kpptu4pFHHqFDhw5aL3AC8sM6gFuBbLfrMrMkM0sxs5SDBw/mYSwRKQiKFy/O5MmTee6555g7dy7169dn9+7dvmMVCGErADN7x8w2Z3G5IcM0DwNHganZvY5zboJzLt45Fx8dHR2uuCJSgJkZ/fv356233uKzzz4jISGBJUuW+I6V74WtAJxzzZxzNbK4zAEws1uA64GbnDboFZFc0KxZM9atW0e5cuVo0aIFzz//vPYXOA5fWwFdAwwAWjvndIAPEck1lSpVYvXq1bRu3Zr+/ftz66238vPPP/uOlS/5WgcwFigBLDazjWY2zlMOESmESpQowcyZMxkyZAiTJk2iSZMmHDhwwHesfMfXVkCVnXPlnXNxwcv/85FDRAqvIkWKMHToUGbNmsWmTZuoU6cOa9eu9R0rX8kPWwGJiIRNu3btfj+/QKNGjUhOTvYdKd9QAYhIoVezZk3Wrl1LYmIiN910E4MGDdJOY6gARCRClClThsWLF3Pbbbfx+OOP0759e77//nvfsbxSAYhIxChWrBgTJkzg+eefZ+7cuTRo0IA9e/b4juWNCkBEIoqZ0bdvXxYsWMCePXtISEhg5cqVvmN5oQIQkYjUokUL1qxZQ8mSJWnatCmTJk3yHSnPqQBEJGJVrVqVDz74gEaNGvH3v/+dAQMGRNTKYRWAiES00qVLs3DhQnr16sWIESNo3749P/zwg+9YeUIFICIR75RTTuHFF1/8feVww4YN+eyzz3zHCjsVgIgI/7dyeO7cuezYsYPLL7+c1NRU37HCSgUgIpLBddddx6pVqyhatCiNGjVi9uzZviOFjQpARCSTY3sO16xZk3bt2vH0008XysNKqwBERLJw3nnnsWzZMjp37syDDz5Iz549OXLkiO9Yuaqo7wAiIvlV8eLFSU5OpkqVKgwbNoxPPvmEWbNmUapUKd/RcoXmAEREjsPMePTRR5kyZQorVqygfv36fPzxx75j5QoVgIhIDtx8880sXryYL774gsTERD744APfkUKmAhARyaHGjRuzatUqSpQoQZMmTZg1a5bvSCHxWgBmdq+ZOTMr4zOHiEhOVa1alTVr1hAXF8eNN97IyJEjC+wWQt4KwMzKA82BT31lEBE5GdHR0SxdupQOHTpw//3306tXL44ePeo71gnzuRXQc8AAYI7HDCIiJ6V48eJMmzaN2NhYnn76afbs2cPrr79OiRIlfEfLMS9zAGZ2A/CZc+7DHEybZGYpZpZy8ODBPEgnIpIzRYoU4amnnmLChAksXryYxo0bc+DAAd+xcszCtezKzN4Bymbx0MPAQ0Bz59whM/sEiHfOffVXrxkfH+9SUlJyN6iISC5YsGABHTt25JxzzmHhwoVUq1bNd6TfmVmqcy4+8/1hmwNwzjVzztXIfAF2AxWBD4M//hcC680sq7IQESkQrr32Wt59911+/vlnGjRowLvvvus70l/K80VAzrlNzrlznXMVnHMVgH3AZc65z/M6i4hIbqpTpw5r1qyhbNmyNG/enGnTpvmOdFzaD0BEJBdVqFCBVatWUbduXbp06cIzzzyTbzcT9V4AwTmBv1z+LyJSUJQqVYq3336bTp068cADD9CnTx9+/fVX37H+RAeDExEJg9NOO43k5GRiYmIYMWIEn332Gf/+978pXry472i/8z4HICJSWBUpUoRnnnmGMWPGMHfuXJo1a8Z///tf37F+pwIQEQmzPn368Prrr5OSksIVV1zBp5/mjwMgqABERPJAhw4dWLRoEfv376devXps2rTJdyQVgIhIXmncuDHvv/8+ZsYVV1zB8uXLveZRAYiI5KGaNWuyevVqLrjgAlq0aMGMGTO8ZVEBiIjksfLly7NixQoSEhLo1KkTY8aM8ZJDBSAi4kHp0qVZvHgxN9xwA3379uWhhx7K8x3GVAAiIp4UL16cmTNncscdd/Dkk09y++235+l5BbQjmIiIR1FRUbz88suce+65DB8+nK+//prk5OQ82WFMcwAiIp6ZGcOGDWPMmDHMmTOHa665hkOHDoX9fVUAIiL5RJ8+fUhOTmb16tU0btyYzz8P70GSVQAiIvlI586dmT9/Pjt37qRBgwbs2rUrbO+lAhARyWeaN2/OkiVLOHToEA0aNGDjxo1heR8VgIhIPpSYmMj7779PsWLFaNy4MStXrsz191ABiIjkU3/7299YuXIliYmJXHjhhbn++toMVEQkHytfvjyLFi0Ky2t7mwMws7vMbJuZbTGzZ3zlEBGJVF7mAMysKXADUMs597OZnesjh4hIJPM1B9ALeMo59zOAc+5LTzlERCKWrwKoAjQ0sw/M7F0zS8huQjNLMrMUM0s5ePBgHkYUESncwrYIyMzeAcpm8dDDwfctDdQFEoDXzSzWZXEoPOfcBGACQHx8fN4eKk9EpBALWwE455pl95iZ9QLeCP7grzWz34AygP7EFxHJI74WAf0HaApgZlWAYsBXnrKIiEQkX/sBvAq8amabgSNAj6wW/4iISPhYQfrdNbODwJ6TfHoZIm8uQ2OODBpzZAhlzBc556Iz31mgCiAUZpbinIv3nSMvacyRQWOODOEYs44FJCISoVQAIiIRKpIKYILvAB5ozJFBY44MuT7miFkHICIifxRJcwAiIpKBCkBEJEIVigIws2vMbLuZ7TSzB7N4vJGZrTezo2bWIdNjPcxsR/DSI+9Sh+Zkx2xmcWa2Ongeho/MrFPeJj85oXzGwcfPMrN9ZjY2bxKHLsTvdYyZLTKzNDPbamYV8ix4CEIc8zPB73Wamb1gZpZ3yU9eDsZ8T/Az/MjMlpjZRRkeC+33yzlXoC9AFLALiCVwSIkPgWqZpqkAXApMBjpkuL80sDv431LB66V8jynMY64CXBy8fj5wADjb95jCNd4Mjz8PJANjfY8nL8YMLAeuDl4/Ezjd95jCOWagPrAy+BpRwGqgie8x5dKYmx77/AgcSn968HrIv1+FYQ7gcmCnc263c+4IMI3AyWZ+55z7xDn3EfBbpue2ABY75/7rnPsGWAxckxehQ3TSY3bOpTvndgSv7we+BP60h2A+E8pnjJnVAc4DwnNevfA46TGbWTWgqHNucXC6751zP+ZR7lCE8jk74DQCP6KnAqcAX4Q/cshyMuZlGT6/NcCxkwOH/PtVGArgAmBvhtv7gveF+7k+5UpuM7ucwP8wu3IpV7ic9HjNrAgwCrgvDLnCKZTPuArwrZm9YWYbzGyEmUXlesLcd9Jjds6tBpYRmKM9ALztnEvL9YS570TH3BNYeJLP/ZPCUAByEsysHDAF+Ltz7k9/NRcidwILnHP7fAfJQ0WBhgRKL4HA4oVbfAYKNzOrDPyNwF/HFwBXmllDv6lyl5ndDMQDI3LrNQtDAXwGlM9w+8LgfeF+rk8h5Tazs4A3gYedc2tyOVs4hDLeekAfM/sEGAl0N7OncjdeWIQy5n3AxuBihaMEDr9+We7GC4tQxtwWWBNc3PU9gb+S6+VyvnDI0ZjNrBmBk2m1dsFT6eb0ucdTGApgHXCxmVU0s2JAZ2BuDp/7NtDczEqZWSmgefC+/O6kxxycfjYw2Tk3M4wZc9NJj9c5d5NzLsY5V4HAX8STnXN/2tIiHwrle70OONvMjq3buRLYGoaMuS2UMX8KNDazomZ2CtAYKAiLgP5yzGZWGxhP4Mc/4/nTQ//98r0WPJfWpF8LpBNYlv1w8L5hwX8wCMwG7wN+AL4GtmR47q3AzuDl777HEu4xAzcDvwAbM1zifI8nnJ9xhte4hQKyFVCoYwauBj4CNgGTgGK+xxPOMRPYmmY8gR/9rcCzvseSi2N+h8AK7WP/v87N8NyQfr90KAgRkQhVGBYBiYjISVABiIhEKBWAiEiEUgGIiEQoFYCISIRSAYiIRCgVgIhIhFIBiITAzBKCx2k/zczOCB6PvobvXCI5oR3BREJkZo8ROBRxcWCfc+5Jz5FEckQFIBKi4DFc1gE/AfWdc796jiSSI1oEJBK6cwicdasEgTkBkQJBcwAiITKzuQTO5FQRKOec6+M5kkiOFPUdQKQgM7PuwC/OueTgWbdWmdmVzrmlvrOJ/BXNAYiIRCitAxARiVAqABGRCKUCEBGJUCoAEZEIpQIQEYlQKgARkQilAhARiVD/H7T4x0IufH+hAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "r = 0.17111690\n"
     ]
    }
   ],
   "source": [
    "f = lambda x : 27.3 + 1.03 / x * (1 - np.exp(10 * x))\n",
    "df = lambda x : -10.3 * np.exp(10 * x) / x - 1.03 * (1 - np.exp(10 * x)) / np.power(x, 2)\n",
    "\n",
    "xpts = np.linspace(0.1, 0.2, 40)\n",
    "ypts = f(xpts)\n",
    "myplot(xpts, ypts, r'$f(x) = 27.3 + \\frac{1.030}{r}(1 - e^{10r})$')\n",
    "\n",
    "root = scipy.optimize.newton(f, 0.2, df)\n",
    "print('r = {:>.8f}'.format(root))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "18. A crucial quantity in pipeline design is the pressure drop due to friction under turbulent flow. The pressure drop per unit length is described by the $\\textit{Darcy number} f$, a unitless quantity that satisfies the empirical $\\textit{Colebrook equation}$\n",
    "\n",
    "$$\n",
    "\\large \\frac{1}{\\sqrt{f}} = -2\\log_{10}{\\left [ \\frac{\\epsilon}{3.7D} + \\frac{2.51}{R\\sqrt{f}} \\right ]}\n",
    "$$\n",
    "\n",
    "where $D$ is the inside pipe diameter, $\\epsilon$ is the roughness height of the pipe interior, and $R$ is the Reynolds number of the flow. (Flows in pipes are considered turbulent when $R > 4000$ or so.)\n",
    "\n",
    "(a) For $D = 0.3 m$, $\\epsilon = 0.0002m$, and $R = 105$, use Newton’s Method to calculate the Darcy number $f$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 109,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "f = 0.16539433\n"
     ]
    }
   ],
   "source": [
    "colebrook = lambda f, epsilon, D, R : 1 / np.sqrt(f) + 2 * np.log10(epsilon / (3.7 * D) + 2.51 / (R * np.sqrt(f)))\n",
    "dcolebrook = lambda f, epsilon, D, R : -1 / (2 * np.power(f, 1.5)) - 2.51 / (R * np.power(f, 1.5) * (epsilon / (3.7 * D) + 2.51 / (R * np.sqrt(f))) *  np.log(10))\n",
    "\n",
    "xpts = np.linspace(0.1, 1, 40)\n",
    "ypts = colebrook(xpts, 0.0002, 0.3, 105)\n",
    "myplot(xpts, ypts, 'Colebrook equation')\n",
    "\n",
    "root = scipy.optimize.newton(lambda f : colebrook(f, 0.0002, 0.3, 105), 0.15, lambda f : dcolebrook(f, 0.0002, 0.3, 105))\n",
    "print('f = {:>.8f}'.format(root))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(b) Fix $D$ and $\\epsilon$ as in (a), and calculate the Darcy number for several Reynolds numbers $R$ between $10^4$ and $10^8$. Make a plot of the Darcy number versus Reynolds number, using a log axis for the latter."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 110,
   "metadata": {},
   "outputs": [
    {
     "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": [
    "Rs = np.array(np.power(10, np.linspace(4, 8, 41)))\n",
    "Fs = np.array([])\n",
    "for R in Rs:\n",
    "    Fs = np.append(Fs, scipy.optimize.newton(lambda f : colebrook(f, 0.0002, 0.3, R), 1e-3, lambda f : dcolebrook(f, 0.0002, 0.3, R)))\n",
    "\n",
    "plt.plot(Rs, Fs, color='black')\n",
    "plt.xlabel('Reynolds number R')\n",
    "plt.ylabel('Darcy number f')\n",
    "plt.xscale('log')\n",
    "plt.title('Colebrook equation with fixed $D$ and $\\epsilon$')\n",
    "plt.show()\n",
    "plt.close()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "# 1.5 Root-Finding Without Derivatives"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 111,
   "metadata": {},
   "outputs": [],
   "source": [
    "def secant_method(f, x0, x1, k=500) :\n",
    "    \"\"\"\n",
    "    Use Secant's method to find the root of the formula\n",
    "    \n",
    "    Args:\n",
    "        f (function prototype) : function handle\n",
    "        x0 (real number) : initial guess\n",
    "        x1 (real number) : initial guess \n",
    "        k (integer) : number of iteration steps (default : 500)\n",
    "        \n",
    "    Return:\n",
    "        Approximate solution xc\n",
    "        \n",
    "    Raises:\n",
    "    \n",
    "    \"\"\"\n",
    "    x = np.zeros(k)\n",
    "    x[0] = x0\n",
    "    x[1] = x1\n",
    "    for i in range(1, k - 1):\n",
    "        if f(x[i]) - f(x[i - 1]) == 0 :\n",
    "            return x[i]\n",
    "        x[i + 1] = x[i] - f(x[i]) * (x[i] - x[i - 1])  / (f(x[i]) - f(x[i - 1]))\n",
    "        \n",
    "    return x[k - 1]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example \n",
    "Apply the Secant Method with starting guesses $x_0 = 0,x_1 = 1$ to find the root of $f(x)=x^3 + x + 1$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 112,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "r = 0.68232780\n"
     ]
    }
   ],
   "source": [
    "f = lambda x : np.power(x, 3) + x - 1\n",
    "\n",
    "# No fprime to use secant method instead\n",
    "root = scipy.optimize.newton(f, 0)\n",
    "print('r = {:>.8f}'.format(root))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 113,
   "metadata": {},
   "outputs": [],
   "source": [
    "def false_position_method(f, a, b, max_iter = 500, tol = 1e-8) :\n",
    "    \"\"\"\n",
    "    Use false position method to find the root of the formula f\n",
    "    \n",
    "    Args:\n",
    "        f (function prototype) : function handle\n",
    "        a (real number) : the lowerbound of the bracket of initial guess\n",
    "        b (real number) : the upperbound of the bracket of initial guess\n",
    "        max_iter (integer) : number of iteration steps (default : 500)\n",
    "        tol (real number) : tolerance\n",
    "        \n",
    "    Return:\n",
    "        Approximate solution xc\n",
    "        \n",
    "    Raises:\n",
    "        ValueError:\n",
    "            - f(a) * f(b) must be smaller than 0 (exclude zero)\n",
    "        \n",
    "    \"\"\"\n",
    "    if f(a) * f(b) >= 0 :\n",
    "        raise ValueError('f(a) * f(b) must be < 0')\n",
    "    for _ in range(max_iter) :\n",
    "        _tol = 0\n",
    "        c = (b * f(a) - a * f(b)) / (f(a) - f(b))\n",
    "        if f(c) == 0 :\n",
    "            return c\n",
    "        if f(a) * f(c) < 0 :\n",
    "            _tol = np.abs(b - c)\n",
    "            b = c\n",
    "        else :\n",
    "            _tol = np.abs(a - c)\n",
    "            a = c\n",
    "        if _tol <= tol:\n",
    "            break\n",
    "    return c"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example\n",
    "Apply the Method of False Position on interval $[-1,1]$ to find the root $r = 0$ of $f(x) = x^3 - 2x^2 + \\frac{3}{2}x$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 114,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 1.62008024991969 \\cdot 10^{-8}$"
      ],
      "text/plain": [
       "1.620080249919692e-08"
      ]
     },
     "execution_count": 114,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = lambda x : np.power(x, 3) - 2 * np.power(x, 2) + 1.5 * x\n",
    "false_position_method(f, -1, 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 115,
   "metadata": {},
   "outputs": [],
   "source": [
    "def inverse_quadratic_interpolation(f, x0, x1, x2, k):\n",
    "    a = x0\n",
    "    b = x1\n",
    "    c = x2\n",
    "    for _ in range(k) :\n",
    "        q = f(a) / f(b)\n",
    "        r = f(c) / f(b)\n",
    "        s = f(c) / f(a)\n",
    "        \n",
    "        denominator = (q - 1) * (r - 1) * (s - 1)\n",
    "        if denominator == 0 :\n",
    "            break\n",
    "        \n",
    "        tmp = c - (r * (r - q) * (c - b) + (1 - r) * s * (c - a)) / denominator\n",
    "        a = b\n",
    "        b = c\n",
    "        c = tmp\n",
    "        \n",
    "        \n",
    "    return c"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Additional Examples\n",
    "\n",
    "1. Apply two steps of the Secant Method on the interval with initial guesses $x_0 = 1$ and $x_1 = 2$ to ﬁnd the approximate root of $f(x) = 2x^3 - x - 7$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 116,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "r = 1.62788925\n"
     ]
    }
   ],
   "source": [
    "root = scipy.optimize.newton(lambda x : 2 * np.power(x, 3) - x - 7, 1)\n",
    "print('r = {:>.8f}'.format(root))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2. Write a MATLAB program that uses the Secant Method to ﬁnd both roots of $f(x) = 8x^6 - 12x^5 + 6x^4 - 17x^3 + 24x^2 - 12x + 2$. Is the Secant Method superlinearly convergent to both roots?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 117,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "r1 = 0.50000000\n",
      "r2 = 1.25992105\n"
     ]
    }
   ],
   "source": [
    "f = lambda x : 8 * np.power(x, 6) - 12 * np.power(x, 5) + 6 * np.power(x, 4) - 17 * np.power(x, 3) + 24 * np.power(x, 2) - 12 * x + 2\n",
    "\n",
    "root = scipy.optimize.newton(f, 0.5)\n",
    "print('r1 = {:>.8f}'.format(root))\n",
    "\n",
    "root = scipy.optimize.newton(f, 1.5)\n",
    "print('r2 = {:>.8f}'.format(root))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "## 1.5 Computer Problems"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1. Use the Secant Method to ﬁnd the (single) solution of each equation in Exercise 1.\n",
    "\n",
    "- Initial guesses $x_0 = 1$ and $x_1 = 2$.\n",
    "\n",
    "(a) $x^3 = 2x + 2$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 118,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.76929235\n"
     ]
    }
   ],
   "source": [
    "f = lambda x : np.power(x, 3) - 2 * x - 2\n",
    "root = scipy.optimize.newton(f, 1)\n",
    "print('{:>.8f}'.format(root))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(b) $e^x + x = 7$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 119,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.67282170\n"
     ]
    }
   ],
   "source": [
    "f = lambda x : np.exp(x) + x - 7\n",
    "root = scipy.optimize.newton(f, 1)\n",
    "print('{:>.8f}'.format(root))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(c) $e^x + sin(x) = 4$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 120,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.12998050\n"
     ]
    }
   ],
   "source": [
    "f = lambda x : np.exp(x) + np.sin(x) - 4\n",
    "root = scipy.optimize.newton(f, 1)\n",
    "print('{:>.8f}'.format(root))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "2.  Apply two steps of the Method of False Position with initial bracket $[1, 2]$ to the equations of Exercise 1."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(a) $x^3 = 2x + 2$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 121,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.76929235\n"
     ]
    }
   ],
   "source": [
    "f = lambda x : np.power(x, 3) - 2 * x - 2\n",
    "root = false_position_method(f, 1, 2)\n",
    "print('{:>.8f}'.format(root))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(b) $e^x + x = 7$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 122,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.67282170\n"
     ]
    }
   ],
   "source": [
    "f = lambda x : np.exp(x) + x - 7\n",
    "root = false_position_method(f, 1, 2)\n",
    "print('{:>.8f}'.format(root))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(c) $e^x + sin(x) = 4$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 123,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.12998050\n"
     ]
    }
   ],
   "source": [
    "f = lambda x : np.exp(x) + np.sin(x) - 4\n",
    "root = false_position_method(f, 1, 2)\n",
    "print('{:>.8f}'.format(root))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "3. Use Inverse Quadratic Interpolation to ﬁnd the solution of each equation in Exercise 1.\n",
    "\n",
    "Here we use Brent’s method instead."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(a) $x^3 = 2x + 2$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 124,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.76929235\n"
     ]
    }
   ],
   "source": [
    "f = lambda x : np.power(x, 3) - 2 * x - 2\n",
    "root = scipy.optimize.brentq(f, 1, 2)\n",
    "print('{:>.8f}'.format(root))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(b) $e^x + x = 7$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 125,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.67282170\n"
     ]
    }
   ],
   "source": [
    "f = lambda x : np.exp(x) + x - 7\n",
    "root = scipy.optimize.brentq(f, 1, 2)\n",
    "print('{:>.8f}'.format(root))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(c) $e^x + sin(x) = 4$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 126,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.12998050\n"
     ]
    }
   ],
   "source": [
    "f = lambda x : np.exp(x) + np.sin(x) - 4\n",
    "root = scipy.optimize.brentq(f, 1, 2)\n",
    "print('{:>.8f}'.format(root))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "4. Set $f(x) = 54x^6 + 45x^5 - 102x^4 - 69x^3 + 35x^2 + 16x - 4$. Plot the function on the interval $[-2,2]$, and use the Secant Method to ﬁnd all ﬁve roots in the interval. To which of the roots is the convergence linear, and to which is it superlinear?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 127,
   "metadata": {},
   "outputs": [
    {
     "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": [
    "f = lambda x : 54 * np.power(x, 6) + 45 * np.power(x, 5) - 102 * np.power(x, 4) - 69 * np.power(x, 3) + 35 * np.power(x, 2) + 16 * x - 4\n",
    "\n",
    "xpts = np.linspace(-1.4, 1.2, 120)\n",
    "ypts = f(xpts)\n",
    "myplot(xpts, ypts, r'$f(x) = 54x^6 + 45x^5 - 102x^4 - 69x^3 + 35x^2 + 16x - 4$')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 128,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-1.38129848\n",
      "-0.66666668\n",
      "0.20518292\n",
      "0.50000000\n",
      "1.17611556\n"
     ]
    }
   ],
   "source": [
    "root = scipy.optimize.newton(f, -1.5)\n",
    "print('{:>.8f}'.format(root))\n",
    "\n",
    "root = scipy.optimize.newton(f, -0.8)\n",
    "print('{:>.8f}'.format(root))\n",
    "\n",
    "root = scipy.optimize.newton(f, 0.1)\n",
    "print('{:>.8f}'.format(root))\n",
    "\n",
    "root = scipy.optimize.newton(f, 0.4)\n",
    "print('{:>.8f}'.format(root))\n",
    "\n",
    "root = scipy.optimize.newton(f, 1.2)\n",
    "print('{:>.8f}'.format(root))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "5. In Exercise 1.1.6, you were asked what the outcome of the Bisection Method would be for $f(x) = 1/x$ on the interval $[-2,1]$. Now compare that result with applying **fzero** to the problem.\n",
    "\n",
    "In python, we use **scipy.optimize.fsolve** to represent **fzero** instead."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 129,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-2.64035519e+83\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "c:\\users\\jim\\appdata\\local\\programs\\python\\python38\\lib\\site-packages\\scipy\\optimize\\minpack.py:175: RuntimeWarning: The number of calls to function has reached maxfev = 400.\n",
      "  warnings.warn(msg, RuntimeWarning)\n"
     ]
    }
   ],
   "source": [
    "root = scipy.optimize.fsolve(lambda x : 1 / x, -1.5)[0]\n",
    "print('{:>.8e}'.format(root))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "6. What happens if **fzero** is asked to ﬁnd the root of $f(x) = x^2$ near 1 (do not use a bracketing interval)? Explain the result."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 130,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.00000000\n"
     ]
    }
   ],
   "source": [
    "root = scipy.optimize.fsolve(lambda x : np.power(x, 2), 1)[0]\n",
    "print('{:>.8f}'.format(root))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(b) Apply the same question to $f(x) = 1 + \\cos{x}$ near -1."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 131,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-3.14159264\n"
     ]
    }
   ],
   "source": [
    "root = scipy.optimize.fsolve(lambda x : 1 + np.cos(x), -1)[0]\n",
    "print('{:>.8f}'.format(root))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "## MIT License\n",
    "\n",
    "Copyright (c) Jim00000\n",
    "\n",
    "Permission is hereby granted, free of charge, to any person obtaining a copy\n",
    "of this software and associated documentation files (the \"Software\"), to deal\n",
    "in the Software without restriction, including without limitation the rights\n",
    "to use, copy, modify, merge, publish, distribute, sublicense, and/or sell\n",
    "copies of the Software, and to permit persons to whom the Software is\n",
    "furnished to do so, subject to the following conditions:\n",
    "\n",
    "The above copyright notice and this permission notice shall be included in all\n",
    "copies or substantial portions of the Software."
   ]
  }
 ],
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