Ex08_Intro_Sympy.ipynb

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   "cell_type": "markdown",
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   "source": [
    "# Exercise 8: Introduction to SymPy\n",
    "\n",
    "SymPy is a Python package for symbolic computation. Instead of working only with variable values (as we did before in our programming exercises), in symbolic computation we actually work with mathematical objects, for example symbolic descriptions of functions. This enables us to do incredible things: we can, for example, use symbolic computations to perform complicated differentiation or integration operations symbolically! If you used Mathematica or Maple before, then you are probably familiar with this type of computation. \n",
    "\n",
    "Here a description from the [SymPy homepage](http://docs.sympy.org/dev/tutorial/intro.html):\n",
    "\n",
    "> \"Symbolic computation deals with the computation of mathematical objects symbolically. This means that the mathematical objects are represented exactly, not approximately, and mathematical expressions with unevaluated variables are left in symbolic form.\"\n",
    "\n",
    " \n",
    "\n",
    "You might not be impressed - but wait and see..."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Taylor series expansion\n",
    "\n",
    "We will first discuss the Taylor series expansion as a way to formally determine the differential of a property. As a brief reminder, the Taylor series can be used to approximate the value of a function $f(x)$ at a location $f(x +h)$ with the help of a polynomial series:\n",
    "\n",
    "$$f(x+h) = \\sum_{i=1}^n \\frac{h^n}{n!} \\frac{d^nf}{dx^n}(h)$$\n",
    "\n",
    "As an example, we can approximate the sine function at several locations. This is pretty easy as the derivatives of $f(x) = \\sin(x)$ can easily be calculated.\n",
    "\n",
    "However, this is not always the case. Think about a tiny bit more complex function: $f(x) = \\sin(x^2)$. Determining higher-order differentials quickly gets quite tricky (i.e. dirty) - try it out, to convince yourself:\n",
    "\n",
    "$$f(x) = \\sin(x^2)$$\n",
    "\n",
    "$$\\frac{df}{dx} = 2x \\cos(x^2)$$\n",
    "\n",
    "$$\\frac{d^2f}{dx^2} = ?$$\n",
    "\n",
    "$$\\frac{d^3f}{dx^3} = ??$$\n",
    "\n",
    "Creating more differentials is a very systematic process and maybe not difficult - but requires quickly a lot of bookkeeping and is prone to errors - and it becomes very repetitive if you have to do it very often...\n",
    "\n",
    "As soon as you read things in descriptions like: \"requires bookkeeping\", \"not difficult systematic process\" and \"repetitive\", I hope you think by now: **we should do that in Python**! Yes, correct!\n",
    "\n",
    "So luckily, someone did the hard work of coding all these things into Python for us before: it is implemented in the sympy package for symbolic computation. \n",
    "\n",
    "We'll have a look at some of the basic features in this notebook - and then apply it to determine Taylor Series Expansions, and finally to the accuracy estimation that we will see again in the lecture.\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "#from IPython.core.display import HTML\n",
    "#css_file = 'nre2_style.css'\n",
    "#HTML(open(css_file, \"r\").read())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import sympy as sym\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# to get LaTeX function rendereing:\n",
    "sym.init_printing()\n",
    "#sym.init_printing(use_unicode=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# writing function as sympy\n",
    "sym.var('x')\n",
    "f = sym.sin(x**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "x_vals = np.arange(0, 2*np.pi, 0.01)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0xbd73e10>]"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
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wvWyt8Pzzt2L6dODSS4HLL2/GpEnNrqeUSnyKdI4e5T8eU+0OikmT4Osqy3Qt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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0xbc01e48>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# plot the function\n",
    "Fx = sym.lambdify(x,f,'numpy') # keyword numpy required to create vectorised function\n",
    "plt.plot(x_vals, Fx(x_vals))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Creating derivatives with SymPy\n",
    "\n",
    "The first way to make the evaluation of the function with a Taylor series easier is, of course, to let SymPy do the hard work of doing the differentiation of the function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEMAAAAcBAMAAADIPB9iAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nTkSuQmCC\n",
      "text/latex": [
       "$$\\sin{\\left (x^{2} \\right )}$$"
      ],
      "text/plain": [
       "   ⎛ 2⎞\n",
       "sin⎝x ⎠"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAF4AAAAcBAMAAAAabBU6AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nmCiBlRCk0KkAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$2 x \\cos{\\left (x^{2} \\right )}$$"
      ],
      "text/plain": [
       "       ⎛ 2⎞\n",
       "2⋅x⋅cos⎝x ⎠"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dfdx = f.diff()\n",
    "dfdx"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "d2fdx2 = dfdx.diff()\n",
    "d3fdx3 = d2fdx2.diff()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAOQAAAAcBAMAAACDuA7KAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMpm7du8iZolU\nq0RaI+fpAAAACXBIWXMAAA7EAAAOxAGVKw4bAAADpElEQVRIDb1VS2gTQRj+Ntk03SbGpQcvImoU\nPYlvRQ8aiqgXbRTUg5fFg+hB3B6kFRWKD1TwURHFW6sivhCiUEQFjeATRFufKFZzUxBpxba+G/+Z\nndmdMdttTw5s9nv83/w7s48AlSO1ekWlqChD+UqpCqNi1/BULa3AQ/kVAU+Iis1DfUtYbLkQmZ8M\nKwjXzJKn82nd8JoFaBVVmm86gjI/06F5UWSxEsPnwSqf2CFOLBDJvx5SES6N92WKpdp8poO5OvXY\nmkAkPx5cQKCHoYTrq2za6T4jYOYlS0lk7pUSYH3xMfOTskaq1qr1OYnVc6stGZ92i2TsHHfoh48z\nEpi/JAL86wCYb/UEFkdjkBr4R+K03Rf5tK0+JdDkCJYpZHICKqfqoiSeP1/QtOuBTmCfrFDPByTx\nYvE2yemyTzmC1D26529G4Nf7muevFpZsuQt4nwvKJTL9++HFEo50aONikhwul4VsLQr8cz70/CbB\nZcupdmjLmmaZ82JJnwPnqaXR+PrDVVFibj877vgf1O1ZsoNv3UrSVb9brFq2JHshScba9iJGN94E\ny5OWoB81lvFXDcOllqNx2p1BdWyMAqZhCjDHtX4wvp4O1a/PMRUIWmbYw3aiaO4356HK5XmgytFj\nZi8L8ZECtbyMW/ZDIXy04WITsAH4ySTqrvnyCxW0jHdQSRbp3ngBZi/PAzES1WmtAdTOZGMi7rCW\nNiax2fmI928Fa7kO+MaE2XSofmuRhFQ2O2FnNst6AavoMProp74I/OJ5evXyesz6TQV8GAXWUszu\nKVfKedaSNpS3ZKtU/YpV8o9DhtU+KQH9Bst7q1RjtEox0l1d7w+6tB9SwBHUDFS0VHz5b+Nv7F1Y\nLcEq+3hetFRiJn8wRJdqB7eTPWgQ9D7wRm35knTV77a9QtmSvpfJFnYvka7OIzPA88AIR48pTyww\n0sn8ruqx2ryZ8MDGRrarcmPpPdR8+Z7KlkdvtD+j6Ik2XDN3I1bgefoWN+ux4D2lJ2Hzn5OvGrZc\nEB3x6ezF0rJy57Ly83vfO0ikVRmq/+/XZ2q5/JXKjBfbWnDsxjueJ06r0mIJh8RhjuqSXjhZULlK\n3Q3YoQAyFHN1HsUSBc015FfEoBsYNdbq5lhb51HMbNbcmg6NDk6W6tYlnUazt5odz2l0cDKiqHmz\nNDYEqdX8ixqLIMY41azRb49qheB0XhEtRyHR8K5qH1fJ0PixUpIY/lOQLim5FQr+f/AvOnwJjCTz\nvdQAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$- 4 x^{2} \\sin{\\left (x^{2} \\right )} + 2 \\cos{\\left (x^{2} \\right )}$$"
      ],
      "text/plain": [
       "     2    ⎛ 2⎞        ⎛ 2⎞\n",
       "- 4⋅x ⋅sin⎝x ⎠ + 2⋅cos⎝x ⎠"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d2fdx2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAPoAAAAcBAMAAAC637+RAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$- 8 x^{3} \\cos{\\left (x^{2} \\right )} - 12 x \\sin{\\left (x^{2} \\right )}$$"
      ],
      "text/plain": [
       "     3    ⎛ 2⎞           ⎛ 2⎞\n",
       "- 8⋅x ⋅cos⎝x ⎠ - 12⋅x⋅sin⎝x ⎠"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d3fdx3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# If you did the differentiation by hand before then you will surely appreciate how much SymPy can make your life easier...\n",
    "\n",
    "However, for the determination of the full Taylor expansion, it even gets better:\n",
    "\n",
    "## Taylor expansion up to order $n$\n",
    "\n",
    "SymPy provides a direct method to determine the coefficients of a Taylor series for the polynomials up to order $n$. In addition, it is possible to directly plot and evaluate results:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "taylor = sym.series(f, x0 = 1., n=6).removeO()\n",
    "evaltaylor = sym.lambdify(x, taylor, modules=['numpy'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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UFFFRUSn3hwwZ4t0SuMaYfwGLrLU10vm591vm06bBp5/KYs1KKZUdf/0l4863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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0xc15b6d8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "func = sym.sin(x)/x\n",
    "taylor = sym.series(func, n=4).removeO()\n",
    "\n",
    "evalfunc = sym.lambdify(x, func, modules=['numpy'])\n",
    "evaltaylor = sym.lambdify(x, taylor, modules=['numpy'])\n",
    "\n",
    "t = np.linspace(-5.5, 5.5, 100)\n",
    "plt.plot(t, evalfunc(t), 'b', label='sin(x)/x')\n",
    "plt.plot(t, evaltaylor(t), 'r', label='Taylor')\n",
    "plt.legend(loc='best')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAARYAAAAwBAMAAAAm4UjjAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$1 - \\frac{x^{2}}{6} + \\frac{x^{4}}{120} - \\frac{x^{6}}{5040} + \\mathcal{O}\\left(x^{8}\\right)$$"
      ],
      "text/plain": [
       "     2     4     6         \n",
       "    x     x     x      ⎛ 8⎞\n",
       "1 - ── + ─── - ──── + O⎝x ⎠\n",
       "    6    120   5040        "
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.series(func, n=8)  #.removeO()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "f1 = sym.series(sym.sin(x), x0=sym.pi/4, n=4).removeO()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "f1_np = sym.lambdify('x', f1, modules=['numpy'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0xc5e1a90>"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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SFwlj6enwf/8H55wDBQUwZ47fEUmkUblGJEK89x5ceCFMmuSNWS/dm2ryIlHoq69g7Fhv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\niWJK8iIiUUxJXkQkiinJi4hEsf8H9+qZkWlFkcoAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0xc3a9c88>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "xvals = np.arange(0,2*np.pi,0.01)\n",
    "\n",
    "plt.plot(xvals, f1_np(xvals), label = 'approximation')\n",
    "plt.plot(xvals, np.sin(xvals), label = 'sin(x)')\n",
    "plt.legend(loc='best')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-1.0 + 0.405897075109648*pi\n"
     ]
    }
   ],
   "source": [
    "x0 = np.pi/4.\n",
    "dx = np.pi/8.\n",
    "sym.var('x')\n",
    "\n",
    "f1 = sym.series(sym.sin('x'), x0=0, n=2).removeO()\n",
    "f1_np = sym.lambdify('x', f1, modules=['numpy'])\n",
    "\n",
    "approx_val = f1_np(sym.pi/4. + sym.pi/8.)\n",
    "true_val = np.sin(np.pi/4. + np.pi/8.)\n",
    "error = (approx_val - true_val)/true_val\n",
    "\n",
    "print(error)\n",
    "\n",
    "#print(approx_val)\n",
    "\n",
    "#print(f1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Complete function for Taylor series analysis\n",
    "\n",
    "Here a little more complex example: a function that can be used to take any SymPy function as an input and then performes a plot of Taylor series for specified orders and ranges.\n",
    "\n",
    "The function is from the IPython docs examples page (github repository on: https://github.com/ipython/ipython/blob/3607712653c66d63e0d7f13f073bde8c0f209ba8/docs/examples/notebooks/sympy.ipynb)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def plot_taylor_approximations(func, x0=None, orders=(2, 4), xrange=(0,1), yrange=None, npts=200):\n",
    "    \"\"\"Plot the Taylor series approximations to a function at various orders.\n",
    "\n",
    "    Parameters\n",
    "    ----------\n",
    "    func : a sympy function\n",
    "    x0 : float\n",
    "      Origin of the Taylor series expansion.  If not given, x0=xrange[0].\n",
    "    orders : list\n",
    "      List of integers with the orders of Taylor series to show.  Default is (2, 4).\n",
    "    xrange : 2-tuple or array.\n",
    "      Either an (xmin, xmax) tuple indicating the x range for the plot (default is (0, 1)),\n",
    "      or the actual array of values to use.\n",
    "    yrange : 2-tuple\n",
    "      (ymin, ymax) tuple indicating the y range for the plot.  If not given,\n",
    "      the full range of values will be automatically used. \n",
    "    npts : int\n",
    "      Number of points to sample the x range with.  Default is 200.\n",
    "    \"\"\"\n",
    "    if not callable(func):\n",
    "        raise ValueError('func must be callable')\n",
    "    if isinstance(xrange, (list, tuple)):\n",
    "        x = np.linspace(float(xrange[0]), float(xrange[1]), npts)\n",
    "    else:\n",
    "        x = xrange\n",
    "    if x0 is None: x0 = x[0]\n",
    "    xs = sym.Symbol('x')\n",
    "    # Make a numpy-callable form of the original function for plotting\n",
    "    fx = func(xs)\n",
    "    f = sym.lambdify(xs, fx, modules=['numpy'])\n",
    "    # We could use latex(fx) instead of str(), but matploblib gets confused\n",
    "    # with some of the (valid) latex constructs sympy emits.  So we play it safe.\n",
    "    plt.plot(x, f(x), label=str(fx), lw=2)\n",
    "    # Build the Taylor approximations, plotting as we go\n",
    "    apps = {}\n",
    "    for order in orders:\n",
    "        app = fx.series(xs, x0, n=order).removeO()\n",
    "        apps[order] = app\n",
    "        # Must be careful here: if the approximation is a constant, we can't\n",
    "        # blindly use lambdify as it won't do the right thing.  In that case, \n",
    "        # evaluate the number as a float and fill the y array with that value.\n",
    "        if isinstance(app, sym.numbers.Number):\n",
    "            y = np.zeros_like(x)\n",
    "            y.fill(app.evalf())\n",
    "        else:\n",
    "            fa = sym.lambdify(xs, app, modules=['numpy'])\n",
    "            y = fa(x)\n",
    "        tex = sym.latex(app).replace('$', '')\n",
    "        plt.plot(x, y, label=r'$n=%s:\\, %s$' % (order, tex) )\n",
    "        \n",
    "    # Plot refinements\n",
    "    if yrange is not None:\n",
    "        plt.ylim(*yrange)\n",
    "    plt.grid()\n",
    "    plt.legend(loc='best').get_frame().set_alpha(0.8)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here an example:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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3L/9OTeXZU6fYtHmz7XRawcn8fF46c4bPu3bFu5bpUU3yc7cjrqJT4T4ox2xD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\nJjCrCahqaWqyff7y559/xq233oqvv/4ab7zxBgDg5ZdfVn3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      "text/plain": [
       "<matplotlib.figure.Figure at 0xc162ef0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_taylor_approximations(sym.sin, sym.pi/4, [0,2, 4], (-1, 2*np.pi), (-2,2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Pretty neat - and actually quite useful to get a quick idea about the radius of convergence for an expansion of an equation about a specific point."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Accuracy estimation of Taylor Series Expansion\n",
    "\n",
    "We know from theory that the accuracy of an approximation with a Taylor Series increses when we consider higher orders. This is actually really important for spatial discretisations of partial differential equations, as we will see in the lecture.\n",
    "\n",
    "To get an understanding about the decreasing error, take the sine function approximations from above and write a function to determine the accuracy:\n",
    "\n",
    "1. Define a location $x_0$ where we want to determine the gradient and a step size $\\Delta x$. \n",
    "\n",
    "2. Determine the correct answer easily by differentiating the function. Based on that correct value (let's call it $y_{true}$, we can determine the error of the approximation with the value $y_{approx}$ by:\n",
    "\n",
    "    $$\\epsilon = \\frac{y_{approx} - y_{true}}{y_{true}}$$\n",
    "\n",
    "3. Generate several approximations for different locations $x_0$ and step sizes $\\Delta x$. \n",
    "\n",
    "4. Analyse the decrease in the approximation error $\\epsilon$ (for example in a plot).\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "## Your code here:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 2",
   "language": "python",
   "name": "python2"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.11"
  }
 },
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}