Exams/Midterm-Sample-Copy1.ipynb

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   "cells": [
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Instructions\n",
      "----\n",
      "\n",
      "This is a \"closed book\" examination - in particular, you are not to use any resources outside of this notebook (except possibly pen and paper). You may consult help from within the notebook using ? but not any online references. You should turn wireless off or set your laptop in \"Airplane\" mode prior to taking the exam. \n",
      "\n",
      "You have 2 hours to complete the exam."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import os\n",
      "import sys\n",
      "import glob\n",
      "import matplotlib.pyplot as plt\n",
      "import numpy as np\n",
      "import pandas as pd\n",
      "%matplotlib inline\n",
      "plt.style.use('ggplot')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "np.set_printoptions(formatter={'float': '{: 0.3f}'.format})"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%load_ext rpy2.ipython"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 3
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "**Question 1 (10 points)**. \n",
      "\n",
      "Euclid's algorithm for finding the greatest common divisor of two numbers is\n",
      "\n",
      "```python\n",
      "gcd(a, 0) = a\n",
      "gcd(a, b) = gcd(b, a modulo b)\n",
      "```\n",
      "\n",
      "1. Write a function to find the greatest common divisor in Python (4 poinst)\n",
      "2. What is the greatest common divisor of 17384 and 1928? (1 point)\n",
      "3. Write a function to calculate the least common multiple (4 points)\n",
      "4. What is the least common multiple of 17384 and 1928? (1 point)"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def gcd(a, b):\n",
      "    if b == 0:\n",
      "        return a\n",
      "    else:\n",
      "        return gcd(b, a % b)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 4
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "gcd(17384, 1928)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 5,
       "text": [
        "8"
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def lcm(a, b):\n",
      "    return (a*b)/gcd(a, b)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 6
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "lcm(17384, 1928)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 7,
       "text": [
        "4189544"
       ]
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%%file dummy.csv\n",
      "humpty,dumpty,sat,on,a,wall\n",
      "humpty,dumpty,had,a,great,fall\n",
      "all,the,king's,horses\n",
      "and,all,the,king's,men\n",
      "couldn't,put,humpty,dumpty,together,again"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Writing dummy.csv\n"
       ]
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "**Question 2 (10 points)**. \n",
      "\n",
      "1. Use Unix commands to display only the last two lines (2 points)\n",
      "2. Using Python, read the `dummy.csv` file, and print a list of each word togehter with its frequency (4 points)\n",
      "3. What is the average frequency of words that begin wtih a vowel (aeiou)? (2 points)\n",
      "4. Create a Pandas DataFrame with 2 columns ('Word', 'Frequency') such that each row consists of a word in the nursery rhyme and its frequency. (2 points)"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "! cat dummy.csv | tail -n 2 "
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "and,all,the,king's,men\r\n",
        "couldn't,put,humpty,dumpty,together,again"
       ]
      }
     ],
     "prompt_number": 11
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "words = []\n",
      "for line in open('dummy.csv'):\n",
      "    words.extend(line.strip().split(','))\n",
      "\n",
      "c = {word: words.count(word) for word in set(words)}\n",
      "    \n",
      "for k in sorted(c):\n",
      "    print k, c[k]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "a 2\n",
        "again 1\n",
        "all 2\n",
        "and 1\n",
        "couldn't 1\n",
        "dumpty 3\n",
        "fall 1\n",
        "great 1\n",
        "had 1\n",
        "horses 1\n",
        "humpty 3\n",
        "king's 2\n",
        "men 1\n",
        "on 1\n",
        "put 1\n",
        "sat 1\n",
        "the 2\n",
        "together 1\n",
        "wall 1\n"
       ]
      }
     ],
     "prompt_number": 41
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "np.array([c[word] for word in c if word[0] in 'aeiou']).mean()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 42,
       "text": [
        "1.3999999999999999"
       ]
      }
     ],
     "prompt_number": 42
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "df = pd.DataFrame(np.column_stack((c.keys(), c.values())))\n",
      "df.columns = ('Word', 'Frequency')\n",
      "df.head()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "html": [
        "<div style=\"max-height:1000px;max-width:1500px;overflow:auto;\">\n",
        "<table border=\"1\" class=\"dataframe\">\n",
        "  <thead>\n",
        "    <tr style=\"text-align: right;\">\n",
        "      <th></th>\n",
        "      <th>Word</th>\n",
        "      <th>Frequency</th>\n",
        "    </tr>\n",
        "  </thead>\n",
        "  <tbody>\n",
        "    <tr>\n",
        "      <th>0</th>\n",
        "      <td>      a</td>\n",
        "      <td> 2</td>\n",
        "    </tr>\n",
        "    <tr>\n",
        "      <th>1</th>\n",
        "      <td>     on</td>\n",
        "      <td> 1</td>\n",
        "    </tr>\n",
        "    <tr>\n",
        "      <th>2</th>\n",
        "      <td>  great</td>\n",
        "      <td> 1</td>\n",
        "    </tr>\n",
        "    <tr>\n",
        "      <th>3</th>\n",
        "      <td> humpty</td>\n",
        "      <td> 3</td>\n",
        "    </tr>\n",
        "    <tr>\n",
        "      <th>4</th>\n",
        "      <td>  again</td>\n",
        "      <td> 1</td>\n",
        "    </tr>\n",
        "  </tbody>\n",
        "</table>\n",
        "</div>"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 55,
       "text": [
        "     Word Frequency\n",
        "0       a         2\n",
        "1      on         1\n",
        "2   great         1\n",
        "3  humpty         3\n",
        "4   again         1"
       ]
      }
     ],
     "prompt_number": 55
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "**Question 3 (10 points)**.\n",
      "\n",
      "1. What is the gradient and Hessian of the function $f(x, y) = 3x^2y^2 + 2x^2y + xy^2 + 4y + 7$? (4 points)\n",
      "2. What is the condition number of the Hessian at (1,1)? (4 points)\n",
      "3. Is the Hessaian positive semi-definite? (2 points)"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from sympy import symbols, diff, hessian, Function, init_printing\n",
      "init_printing()\n",
      "\n",
      "x, y = symbols('x y')\n",
      "f = Function('f')\n",
      "f = 3*x**2*y**2 + 2*x**2*y + x*y**2 + 4*y + 7\n",
      "grad = [diff(f, s) for s in (x, y)]\n",
      "hess = hessian(f, (x, y))\n",
      "grad"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\begin{bmatrix}6 x y^{2} + 4 x y + y^{2}, & 6 x^{2} y + 2 x^{2} + 2 x y + 4\\end{bmatrix}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
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       "prompt_number": 59,
       "text": [
        "\u23a1     2            2     2        2            \u23a4\n",
        "\u23a36\u22c5x\u22c5y  + 4\u22c5x\u22c5y + y , 6\u22c5x \u22c5y + 2\u22c5x  + 2\u22c5x\u22c5y + 4\u23a6"
       ]
      }
     ],
     "prompt_number": 59
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "hess"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}6 y^{2} + 4 y & 12 x y + 4 x + 2 y\\\\12 x y + 4 x + 2 y & 6 x^{2} + 2 x\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAATQAAAAzBAMAAADr3emgAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n/U9TS/zTcffOm/+nY+JftbtHDf+q/RuYCeipEWyrNAAAAABJRU5ErkJggg==\n",
       "prompt_number": 60,
       "text": [
        "\u23a1       2                              \u23a4\n",
        "\u23a2    6\u22c5y  + 4\u22c5y      12\u22c5x\u22c5y + 4\u22c5x + 2\u22c5y\u23a5\n",
        "\u23a2                                      \u23a5\n",
        "\u23a2                           2          \u23a5\n",
        "\u23a312\u22c5x\u22c5y + 4\u22c5x + 2\u22c5y      6\u22c5x  + 2\u22c5x    \u23a6"
       ]
      }
     ],
     "prompt_number": 60
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "hess.subs(x, 1).subs(y, 1).condition_number().evalf()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$2.99385088029409$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAJ8AAAAPBAMAAAAIUwCQAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nYII=\n",
       "prompt_number": 67,
       "text": [
        "2.99385088029409"
       ]
      }
     ],
     "prompt_number": 67
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Not positive semi-definite - e.g.\n",
      "\n",
      "[e.evalf() for e in hess.subs(x,1).subs(y,1).eigenvals()]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\begin{bmatrix}27.0277563773199, & -9.02775637731995\\end{bmatrix}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAXcAAAAZBAMAAAA4Z3NTAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nYII=\n",
       "prompt_number": 78,
       "text": [
        "[27.0277563773199, -9.02775637731995]"
       ]
      }
     ],
     "prompt_number": 78
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "**Question 4 (20 points)**. \n",
      "\n",
      "Consider the following matrix $A$ with dimensions (4,6), to be interpreted as 4 rows of the measurements of 6 features.\n",
      "```python\n",
      "np.array([[5, 5, 2, 6, 2, 0],\n",
      "          [8, 6, 7, 8, 9, 7],\n",
      "          [9, 5, 0, 4, 6, 8],\n",
      "          [8, 7, 9, 3, 6, 1]])\n",
      "```\n",
      "\n",
      "1. Add 1 to the first row, 2 to the second row, 3 to the third row and 4 to the fourth row using a vector `v = np.array([1,2,3,4])` and broadcasting. (2 points)\n",
      "2. Normalize A so that its row means are all 0 and call it A1. (2 points)\n",
      "3. What are the singular values of A1? (2 points)\n",
      "4. What are the eigenvalues of the covariance matrix of A1? (2 points)\n",
      "5. Find the least squares solution vector $x$ if $Ax = y$ where `y = np.array([1,2,3,4]).T` (2 points)"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A = np.array([[5, 5, 2, 6, 2, 0],\n",
      "          [8, 6, 7, 8, 9, 7],\n",
      "          [9, 5, 0, 4, 6, 8],\n",
      "          [8, 7, 9, 3, 6, 1]])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 79
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "v = np.array([1,2,3,4])\n",
      "\n",
      "A + v[:, None]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 80,
       "text": [
        "array([[ 6,  6,  3,  7,  3,  1],\n",
        "       [10,  8,  9, 10, 11,  9],\n",
        "       [12,  8,  3,  7,  9, 11],\n",
        "       [12, 11, 13,  7, 10,  5]])"
       ]
      }
     ],
     "prompt_number": 80
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A1 = A - A.mean(1)[:, None]\n",
      "A1"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 95,
       "text": [
        "array([[ 1.667,  1.667, -1.333,  2.667, -1.333, -3.333],\n",
        "       [ 0.500, -1.500, -0.500,  0.500,  1.500, -0.500],\n",
        "       [ 3.667, -0.333, -5.333, -1.333,  0.667,  2.667],\n",
        "       [ 2.333,  1.333,  3.333, -2.667,  0.333, -4.667]])"
       ]
      }
     ],
     "prompt_number": 95
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import scipy.linalg as la\n",
      "\n",
      "la.svd(A1)[1]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 99,
       "text": [
        "array([ 8.317,  5.975,  4.635,  2.268])"
       ]
      }
     ],
     "prompt_number": 99
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "np.real_if_close(la.eig(np.cov(A1))[0])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 100,
       "text": [
        "array([ 13.833,  7.141,  4.297,  1.029])"
       ]
      }
     ],
     "prompt_number": 100
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "y = np.array([1,2,3,4]).T\n",
      "x = la.lstsq(A, y)[0]\n",
      "print x\n",
      "print A.dot(x)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[ 0.360  0.209  0.057 -0.336  0.028 -0.014]\n",
        "[ 1.000  2.000  3.000  4.000]\n"
       ]
      }
     ],
     "prompt_number": 108
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "**Question 5 (10 points)**.\n",
      "\n",
      "1. Prove that $e^{x^2 + y^2}$ is a convex function. (5 points)\n",
      "2. Using `scipy.optimize`, find the values of $x$ and $y$ that minimize $e^{x^2 + y^2}$ in the unconstrained case and in the presence of the constraint that $x + y = 3$. Use (1,1) as a starting guess (5 points)"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let $z = x^2 + y^2$. Then $z \\ge 0$, and \n",
      "\n",
      "$$\n",
      "\\frac{d^2}{dz^2} e^z = e^z\n",
      "$$\n",
      "\n",
      "So the second derivative is positive everywhere and we have a convex function."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import scipy.optimize as opt\n",
      "\n",
      "def f(x):\n",
      "    return np.exp(x[0]**2 + x[1]**2)\n",
      "\n",
      "opt.minimize(f, [1,1])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 123,
       "text": [
        "   status: 0\n",
        "  success: True\n",
        "     njev: 10\n",
        "     nfev: 40\n",
        " hess_inv: array([[ 0.750, -0.250],\n",
        "       [-0.250,  0.750]])\n",
        "      fun: 1.0000000000000018\n",
        "        x: array([-0.000, -0.000])\n",
        "  message: 'Optimization terminated successfully.'\n",
        "      jac: array([-0.000, -0.000])"
       ]
      }
     ],
     "prompt_number": 123
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "cons = {'type': 'eq', 'fun': lambda x: x[0] + x[1] - 3}\n",
      "opt.minimize(f, [1,1], constraints=cons)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 124,
       "text": [
        "  status: 0\n",
        " success: True\n",
        "    njev: 9\n",
        "    nfev: 44\n",
        "     fun: 90.017131300560507\n",
        "       x: array([ 1.500,  1.500])\n",
        " message: 'Optimization terminated successfully.'\n",
        "     jac: array([ 270.051,  270.051,  0.000])\n",
        "     nit: 9"
       ]
      }
     ],
     "prompt_number": 124
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "**Question 6 (10 points)**.\n",
      "\n",
      "Given the following covariance matrix\n",
      "```python\n",
      "A = np.array([[2,1],[1,4]])\n",
      "```\n",
      "\n",
      "1. What is the vector representing the first principal component direction? (3 points)\n",
      "2. Find $A^{10}$ without repeated matrix multiplication or the `matrix_power` function. (4 points)\n",
      "3. What is the explained variance if we keep only the projection onto the first principal component? (3 points)"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A = np.array([[2,1],[1,4]])\n",
      "A"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 143,
       "text": [
        "array([[2, 1],\n",
        "       [1, 4]])"
       ]
      }
     ],
     "prompt_number": 143
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "M = np.cov(A)\n",
      "M"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 144,
       "text": [
        "array([[ 0.500, -1.500],\n",
        "       [-1.500,  4.500]])"
       ]
      }
     ],
     "prompt_number": 144
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "e, v = la.eig(A)\n",
      "e = np.real_if_close(e)\n",
      "v[:, e.argmax()]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 149,
       "text": [
        "array([-0.383, -0.924])"
       ]
      }
     ],
     "prompt_number": 149
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "v.dot(np.diag(e**10).dot(v.T))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 156,
       "text": [
        "array([[ 411437.000,  993054.000],\n",
        "       [ 993054.000,  2397545.000]])"
       ]
      }
     ],
     "prompt_number": 156
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "e"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "ename": "NameError",
       "evalue": "name 'e' is not defined",
       "output_type": "pyerr",
       "traceback": [
        "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m\n\u001b[0;31mNameError\u001b[0m                                 Traceback (most recent call last)",
        "\u001b[0;32m<ipython-input-1-9ffbf43126e3>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0me\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
        "\u001b[0;31mNameError\u001b[0m: name 'e' is not defined"
       ]
      }
     ],
     "prompt_number": 1
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "**Question 7 (10 points)**.\n",
      "\n",
      "A milkmaid is at point A and needs to get to point B. However, she also needs to fill a pail of water from the river en route from A to B. The equation of the river's path is shown in the figure below. What is the minimum distance she has to travel to do this?\n",
      "\n",
      "1. Solve using `scipy.optimize` and constrained minimization.\n",
      "2. Solve without using `scipy.optimize`. Hint: Use Lagrange multipliers.\n",
      "\n",
      "![Milkmaid problem](milkmaid.png)"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x = np.linspace(0, 10, 100)\n",
      "plt.xlim([0, 10])\n",
      "plt.plot(x, 10.0/(1+x), 'b', linewidth=5)\n",
      "plt.scatter([2,8], [8,4], c=[0,1], s=100)\n",
      "plt.text(8, 5, 'B (8,4)', fontsize=16)\n",
      "plt.text(2, 9, 'A (2,8)', fontsize=16)\n",
      "plt.text(0.5, 2, 'River', fontsize=16)\n",
      "plt.text(0.5, 0.5, 'g(x y) = y - 10/(1+x)', fontsize=16);"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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dOx0ZONCHa9cea5SoEFWOZ/PmtJw4scDtPT74AJf69U1XoEquyqxRWlbS0hRM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\nF03CWkAAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x11857d1d0>"
       ]
      }
     ],
     "prompt_number": 22
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def f(x, A, B):\n",
      "    return la.norm(x-A) + la.norm(x-B)\n",
      "\n",
      "x0 = np.array([4, 4])\n",
      "A = np.array([2,8])\n",
      "B = np.array([8,4])\n",
      "cons = {'type': 'eq', 'fun': lambda x: x[1] - 10.0/(1 + x[0])}\n",
      "\n",
      "opt.minimize(f, x0, args=(A, B), constraints=cons)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 161,
       "text": [
        "  status: 0\n",
        " success: True\n",
        "    njev: 31\n",
        "    nfev: 163\n",
        "     fun: 10.792179145264615\n",
        "       x: array([ 4.399,  1.852])\n",
        " message: 'Optimization terminated successfully.'\n",
        "     jac: array([-0.495, -1.444,  0.000])\n",
        "     nit: 31"
       ]
      }
     ],
     "prompt_number": 161
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    }
   ],
   "metadata": {}
  }
 ]
}