Lecture-10A-Taylors-Series.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## In-Class Demonstration and Visualization"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import numpy as np\n",
    "import sympy as sp\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# You can change the default figure size to be a bit larger if you want,\n",
    "# uncomment the next line for that:\n",
    "plt.rc('figure', figsize=(10, 6))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The function `plot_taylor_approximations` included here was written by Fernando Perez and was part of work on the original `IPython` project.  Although attribution seems to have been lost over time, we gratefully acknowledge FP and thank him for this code!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "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 = sp.Symbol('x')\n",
    "    # Make a numpy-callable form of the original function for plotting\n",
    "    fx = func(xs)\n",
    "    f = sp.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, sp.numbers.Number):\n",
    "            y = np.zeros_like(x)\n",
    "            y.fill(app.evalf())\n",
    "        else:\n",
    "            fa = sp.lambdify(xs, app, modules=['numpy'])\n",
    "            y = fa(x)\n",
    "        tex = sp.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)\n",
    "    \n",
    "# For an expression made from elementary functions, we must first make it into\n",
    "# a callable function, the simplest way is to use the Python lambda construct.\n",
    "# plot_taylor_approximations(lambda x: 1/sp.cos(x), 0, [2,4,6,8], (0, 2*sp.pi), (-5,5))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 720x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_taylor_approximations(sp.sin, 0, [2, 4, 6, 8], (0, 2*sp.pi), (-2,2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Lecture 10:  Taylor's Series and Discrete Calculus"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Background\n",
    "\n",
    "It is common in physics and engineering to represent transcendental functions and other nonlinear expressions using a few terms from a Taylor series.  This series provides a fast and efficient way to compute quantities such as $\\mathrm{sin}(x)$ or $e^x$ to a prescribed error.  Learning how to calculate the series representation of these functions will provide practical experience with the Taylor series and help the student understand the results of Python methods designed to accelerate and simplify computations.  The series can be written generally as:\n",
    "\n",
    "$$f(x) = f{\\left (0 \\right )} + x \\left. \\frac{d}{d x} f{\\left (x \\right )} \\right|_{\\substack{ x=0 }} + \\frac{x^{2}}{2} \\left. \\frac{d^{2}}{d x^{2}}  f{\\left (x \\right )} \\right|_{\\substack{ x=0 }} + \\frac{x^{3}}{6} \\left. \\frac{d^{3}}{d x^{3}}  f{\\left (x \\right )} \\right|_{\\substack{ x=0 }} + \\frac{x^{4}}{24} \\left. \\frac{d^{4}}{d x^{4}}  f{\\left (x \\right )} \\right|_{\\substack{ x=0 }} + \\frac{x^{5}}{120} \\left. \\frac{d^{5}}{d x^{5}}  f{\\left (x \\right )} \\right|_{\\substack{ x=0 }} + \\mathcal{O}\\left(x^{6}\\right)$$\n",
    "\n",
    "Of equal importance the Taylor series permits discrete representation of derivatives and is a common way to perform numerical integration of partial and ordinary differential equations.  Expansion of a general function $f(x)$ about a point, coupled with algebraic manipulation, will produce expressions that can be used to approximate derivative quantities.  Although any order of derivative can be computed, this lesson will focus on first and second derivatives that will be encountered in the diffusion equation.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## What Skills Will I Learn?\n",
    "\n",
    "You will practice the following skills:\n",
    "\n",
    "* Defining and determining the limits of infinite sequences, series and power series.\n",
    "* Define the Taylor series and write the general form about any point and to any order.\n",
    "* Derive the central and forward difference formulae for numerical derivatives using the Taylor's series."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## What Steps Should I Take?\n",
    "\n",
    "1.  Learn to use Sympy to define and find the limits of sequences are series.\n",
    "1.  Learn how to approximate transcendental functions about center points of your choosing.\n",
    "1.  Differentiate an explicit series representation of a function to see that the coefficients of such a series can be determined algebraically.\n",
    "1.  Use Sympy to compute a power series symbolically\n",
    "1.  Derive the finite difference expressions for the first and second derivatives.\n",
    "1.  Read the relevant pages from Hornbeck's text on numerical methods.\n",
    "1.  Generate a list of values that approximate the function $f(x)=x^8$ on the domain $\\{x | 0 \\leq x \\leq 1\\}$.  Using these values, numerically compute the derivative at your selected grid points and compare it to the analytical solution.  Using this technique, examine how the observed error changes as the number of grid points is varied.  Visualize and explain the results.\n",
    "1.  Prepare a new notebook (not just modifications to this one) that describes your approach.\n",
    "\n",
    "Optional challenge:  A _list_ is one of the fundamental data structures within Python.  Numpy (a Python library) and other parts of Python libraries use _vectorized_ computations.  From Wikipedia, vectorization is \"a style of computer programming where operations are applied to whole arrays instead of individual elements.\"  \n",
    "\n",
    "With this in mind, we certainly _can_ iterate over our list of points and apply the function that you will soon write in an element by element fashion, however, it is a more common practice in Python and other modern languages to write vectorized code.  If this is your first exposure to vectorized computation, I recommend two initial strategies:  write out your algorithms and use \"classic\" flow control and iteration to compute the results.  From that point you will more easily see the strategy you should use to write vectorized code.  \n",
    "\n",
    "Using the discrete forms of the first and second derivatives (based on central differences) can you devise a vectorized operation that computes the derivative without looping in Python?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## A Sucessful Jupyter Notebook Will\n",
    "\n",
    "* Present a description of the essential elements of Taylor's series and how to compute numerical derivatives;\n",
    "* Identify the audience for which the work is intended;\n",
    "* Run the code necessary to compute and visualize the error associated with the second order approximation and the changes in grid point spacing;\n",
    "* Provide a narrative and equations to explain why your approach is relevant to solving the problem;\n",
    "* Provide references and citations to any others' work you use to complete the assignment;\n",
    "* Be checked into your GitHub repository by the due date (one week from assignment).\n",
    "\n",
    "A high quality communication provides an organized, logically progressing, blend of narrative, equations, and code that teaches the reader a particular topic or idea.  You will be assessed on:\n",
    "* The functionality of the code (i.e. it should perform the task assigned).\n",
    "* The narrative you present.  I should be able to read and learn from it.  Choose your audience wisely.\n",
    "* The supporting equations and figures you choose to include.\n",
    "\n",
    "If your notebook is just computer code your assignment will be marked incomplete."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Reading and Reference\n",
    "\n",
    "* Essential Mathematical Methods for Physicists, H. Weber and G. Arfken, Academic Press, 2003\n",
    "* Advanced engineering Mathematics, E. Kreyszig, John wiley and Sons, 2010\n",
    "* Numerical Recipes, W. Press, Cambridge University Press, 1986\n",
    "* Numerical Methods, R. Hornbeck, Quantum Publishers, 1975"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Infinite Sequences\n",
    "----\n",
    "\n",
    "Ideas relating to sequences, series, and power series are used in the formulation of integral calculus and in the construction of polynomial representations of functions.  The limit of functions will also be investigated as boundary conditions for differential equations.  For this reason understanding concepts related to sequences and series are important to review.   \n",
    "\n",
    "A sequence is an ordered list of numbers.  A list such as the following represents a sequence:\n",
    "\n",
    "$$a_1, a_2, a_3, a_4, \\dots, a_n, \\dots $$\n",
    "\n",
    "The sequence maps one value $a_n$ for every integer $n$.  It is typical to provide a formula for construction of the nth term in the sequence.  While _ad-hoc_ strategies could be used to develop sequences using SymPy and lists in Python, SymPy has a sequence class that can be used.  A short demonstration is provided next:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "import sympy as sp\n",
    "x, y, z, t, a = sp.symbols('x y z t a')\n",
    "k, m, n = sp.symbols('k m n', integer=True)\n",
    "f, g, h = sp.symbols('f g h', cls=sp.Function)\n",
    "sp.var('a1:6')\n",
    "sp.init_printing()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It is important to read about `SymPy` symbols at this time."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can generate a sequence using `SeqFormula`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ 0, \\  1, \\  4, \\  9, \\  16, \\  25\\right]$"
      ],
      "text/plain": [
       "[0, 1, 4, 9, 16, 25]"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a1 = sp.SeqFormula(n**2, (n,0,5))\n",
    "list(a1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "if we want the limit of the sequence at infinity:\n",
    "\n",
    "$$[0, 1, 4, 9, \\ldots ]$$\n",
    "\n",
    "we can use `limit_seq`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\infty$"
      ],
      "text/plain": [
       "∞"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.limit_seq(a1.formula, n)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### DIY:  Determine if the following sequences are convergent or divergent.  If convergent, what is the limit?\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "a_n = & \\frac{1}{n} \\\\\n",
    "a_n = & 1 - (0.2)^n \\\\\n",
    "a_n = & \\frac{1}{2n+1}\n",
    "\\end{aligned}\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Your code here."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Infinite Series\n",
    "----\n",
    "\n",
    "A series is the sum of a sequence.  An infinite series will converge if the partial sums of the series has a finite limit.  For example, examine the partial sums of the series:\n",
    "\n",
    "$$\n",
    "\\sum^{\\infty}_{n=1} \\frac{1}{2^n}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\sum_{n=0}^{1} 2^{- n}$"
      ],
      "text/plain": [
       "  1      \n",
       " ___     \n",
       " ╲       \n",
       "  ╲    -n\n",
       "  ╱   2  \n",
       " ╱       \n",
       " ‾‾‾     \n",
       "n = 0    "
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a2 = sp.Sum(1/2**n, (n,0,1))\n",
    "a2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{3}{2}$"
      ],
      "text/plain": [
       "3/2"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a2.doit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\sum_{n=0}^{5} n^{2}$"
      ],
      "text/plain": [
       "  5     \n",
       " ___    \n",
       " ╲      \n",
       "  ╲    2\n",
       "  ╱   n \n",
       " ╱      \n",
       " ‾‾‾    \n",
       "n = 0   "
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a4 = sp.Sum(n**2, (n,0,5))\n",
    "a4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\sum_{k=1}^{m} k^{2}$"
      ],
      "text/plain": [
       "  m     \n",
       " ___    \n",
       " ╲      \n",
       "  ╲    2\n",
       "  ╱   k \n",
       " ╱      \n",
       " ‾‾‾    \n",
       "k = 1   "
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a5 = sp.Sum(k**2, (k, 1, m))\n",
    "a5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABMAAAAPCAYAAAAGRPQsAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABCElEQVQ4Ea1T0Q2CMBBF4gDEEXADcARXcAVHkE/41Q1wBHUDdATcQEcwbIDvQa+5NNIaY5PHXV9fr9e7EvV9H5VlmdFqgEuA1OG8unk0jltVVQncu5nT58hHY79eXWxkL9gnkAEMdAFyHNDB6uHXmWue9XWmfFzbq5PM9Ok/+38NJg2IUJ8tUpLCL+HX4KQhNlufTjJjkBOEBwL+Hmjhr22U0fHrPMVuUPDH1Lrw0FidZOYkMEz5VFJkl35aVJzVxRA3QKsWXXeo4zc6ZrYCpPA60IITBJEmBHUMdsQGds8dLP5VkUEdg/EJ1GoTs9mZ+UbxQd2MXcFmFrkwG3k9/oMF+M5wgwnp3jhP/DKUzESwAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$\\displaystyle 55$"
      ],
      "text/plain": [
       "55"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a4.doit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{m^{3}}{3} + \\frac{m^{2}}{2} + \\frac{m}{6}$"
      ],
      "text/plain": [
       " 3    2    \n",
       "m    m    m\n",
       "── + ── + ─\n",
       "3    2    6"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a5.doit()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A power series is of the form:\n",
    "\n",
    "$$\n",
    "\\sum_{n=0}^{\\infty} M_{n} x^{n} = M_0 + M_1 x + M_2 x^2 + \\cdots\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "M = sp.IndexedBase('M')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\sum_{n=0}^{m} x^{n} {M}_{n}$"
      ],
      "text/plain": [
       "  m          \n",
       " ___         \n",
       " ╲           \n",
       "  ╲    n     \n",
       "  ╱   x ⋅M[n]\n",
       " ╱           \n",
       " ‾‾‾         \n",
       "n = 0        "
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.Sum(M[n]*x**n, (n,0,m))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can define the series about the point $a$ as follows:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\sum_{n=0}^{m} \\left(- a + x\\right)^{n} {M}_{n}$"
      ],
      "text/plain": [
       "  m                 \n",
       " ___                \n",
       " ╲                  \n",
       "  ╲           n     \n",
       "  ╱   (-a + x) ⋅M[n]\n",
       " ╱                  \n",
       " ‾‾‾                \n",
       "n = 0               "
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.Sum(M[n]*(x-a)**n, (n,0,m))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "SymPy has a function that can take SymPy expressions and represent them as power series:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle f{\\left(0 \\right)} + x \\left. \\frac{d}{d x} f{\\left(x \\right)} \\right|_{\\substack{ x=0 }} + \\frac{x^{2} \\left. \\frac{d^{2}}{d x^{2}} f{\\left(x \\right)} \\right|_{\\substack{ x=0 }}}{2} + \\frac{x^{3} \\left. \\frac{d^{3}}{d x^{3}} f{\\left(x \\right)} \\right|_{\\substack{ x=0 }}}{6} + \\frac{x^{4} \\left. \\frac{d^{4}}{d x^{4}} f{\\left(x \\right)} \\right|_{\\substack{ x=0 }}}{24} + \\frac{x^{5} \\left. \\frac{d^{5}}{d x^{5}} f{\\left(x \\right)} \\right|_{\\substack{ x=0 }}}{120} + O\\left(x^{6}\\right)$"
      ],
      "text/plain": [
       "                             ⎛  2      ⎞│         ⎛  3      ⎞│         ⎛  4   \n",
       "                           2 ⎜ d       ⎟│       3 ⎜ d       ⎟│       4 ⎜ d    \n",
       "                          x ⋅⎜───(f(x))⎟│      x ⋅⎜───(f(x))⎟│      x ⋅⎜───(f(\n",
       "                             ⎜  2      ⎟│         ⎜  3      ⎟│         ⎜  4   \n",
       "         ⎛d       ⎞│         ⎝dx       ⎠│x=0      ⎝dx       ⎠│x=0      ⎝dx    \n",
       "f(0) + x⋅⎜──(f(x))⎟│    + ────────────────── + ────────────────── + ──────────\n",
       "         ⎝dx      ⎠│x=0           2                    6                    24\n",
       "\n",
       "   ⎞│         ⎛  5      ⎞│           \n",
       "   ⎟│       5 ⎜ d       ⎟│           \n",
       "x))⎟│      x ⋅⎜───(f(x))⎟│           \n",
       "   ⎟│         ⎜  5      ⎟│           \n",
       "   ⎠│x=0      ⎝dx       ⎠│x=0    ⎛ 6⎞\n",
       "──────── + ────────────────── + O⎝x ⎠\n",
       "                  120                "
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.series(f(x), x, x0=0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### DIY:  Use SymPy to determine series approximations to $e^x$, $sin(x)$, and $cos(x)$ about the point $x=0$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Your code here."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Taylor's Series\n",
    "----\n",
    "\n",
    "Below we present a derivation of Taylor's series and small algebraic argument for series representations of functions.  In contrast to the ability to use `sympy` functions without any deeper understanding, these presentations are intended to give you insight into the origin of the series representation and the factors present within each term.  While the algebraic presentation isn't a general case, the essential elements of a general polynomial representation are visible.\n",
    "\n",
    "The function $f(x)$ can be expanded into an infinite series or a finite series plus an error term.  Assume that the function has a continuous nth derivative over the interval $a \\le x \\le b$.  Integrate the nth derivative n times:\n",
    "\n",
    "$$\\int_a^x f^n(x) dx = f^{(n-1)}(x) - f^{(n-1)}(a)$$\n",
    "\n",
    "The power on the function $f$ in the equation above indicates the order of the derivative.  Do this n times and then solve for f(x) to recover Taylor's series.  One of the key features in this derivation is that the integral is definite.  This derivation is outlined on [Wolfram’s Mathworld](http://mathworld.wolfram.com/TaylorSeries.html).\n",
    "\n",
    "As a second exercise, assume that we wish to expand sin(x) about x=0.  First, assume that the series exists and can be written as a power series with unknown coefficients.  As a first step, differentiate the series and the function we are expanding.  Next, let the value of x go to the value of the expansion point and it will be possible to evaluate the coefficients in turn:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\sin x = A+Bx+Cx^2+Dx^3+Ex^4\n",
    "$$\n",
    "\n",
    "We can choose an expansion point (e.g. $x = 0$) and differentiate to get a set of simultaneous equations permitting determination of the coefficients.  The computer algebra system can help us with this activity:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "import sympy as sp\n",
    "sp.init_printing()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "x, A, B, C, D, E = sp.symbols('x, A, B, C, D, E')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To help us get our work done we can use `sympy`'s `diff` function.  Testing this function with a known result, we can write:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\cos{\\left(x \\right)}$"
      ],
      "text/plain": [
       "cos(x)"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.diff(sp.sin(x),x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A list comprehension is used to organize the results.  In each iteration the exact function and the power series are differentiated and stored as an element of a list.  The list can be inspected and a set of simultaneous equations can be written down and solved to determine the values of the coefficients.  Casting the list as a `sympy` `Matrix` object clarifies the correspondance between entries in the list."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\cos{\\left(x \\right)}, \\  B + 2 C x + 3 D x^{2} + 4 E x^{3}\\right]$"
      ],
      "text/plain": [
       "⎡                         2        3⎤\n",
       "⎣cos(x), B + 2⋅C⋅x + 3⋅D⋅x  + 4⋅E⋅x ⎦"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "orderOfDifferentiation = 1\n",
    "powerSeries = A+B*x+C*x**2+D*x**3+E*x**4\n",
    "# Differentiate, element by element, the list [sp.sin(x),powerSeries]\n",
    "[sp.diff(a,x,orderOfDifferentiation) for a in [sp.sin(x),powerSeries]]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A list comprehension can be used to organize and extend the results further.  We can wrap the list above into another list that changes the order of differentiation each iteration."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\left[ \\sin{\\left(x \\right)}, \\  A + B x + C x^{2} + D x^{3} + E x^{4}\\right], \\  \\left[ \\cos{\\left(x \\right)}, \\  B + 2 C x + 3 D x^{2} + 4 E x^{3}\\right], \\  \\left[ - \\sin{\\left(x \\right)}, \\  2 \\left(C + 3 D x + 6 E x^{2}\\right)\\right], \\  \\left[ - \\cos{\\left(x \\right)}, \\  6 \\left(D + 4 E x\\right)\\right], \\  \\left[ \\sin{\\left(x \\right)}, \\  24 E\\right]\\right]$"
      ],
      "text/plain": [
       "⎡⎡                     2      3      4⎤  ⎡                         2        3⎤\n",
       "⎣⎣sin(x), A + B⋅x + C⋅x  + D⋅x  + E⋅x ⎦, ⎣cos(x), B + 2⋅C⋅x + 3⋅D⋅x  + 4⋅E⋅x ⎦\n",
       "\n",
       "  ⎡           ⎛                 2⎞⎤                                          ⎤\n",
       ", ⎣-sin(x), 2⋅⎝C + 3⋅D⋅x + 6⋅E⋅x ⎠⎦, [-cos(x), 6⋅(D + 4⋅E⋅x)], [sin(x), 24⋅E]⎦"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "maximumOrder = 5\n",
    "funcAndSeries = [[sp.diff(a,x,order) for a in [sp.sin(x),powerSeries]] for order in range(maximumOrder)]\n",
    "funcAndSeries"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Casting the results as a `sympy` `Matrix` object the list is more easily viewed in the Jupyter notebook:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\sin{\\left(x \\right)} & A + B x + C x^{2} + D x^{3} + E x^{4}\\\\\\cos{\\left(x \\right)} & B + 2 C x + 3 D x^{2} + 4 E x^{3}\\\\- \\sin{\\left(x \\right)} & 2 \\left(C + 3 D x + 6 E x^{2}\\right)\\\\- \\cos{\\left(x \\right)} & 6 \\left(D + 4 E x\\right)\\\\\\sin{\\left(x \\right)} & 24 E\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡                      2      3      4⎤\n",
       "⎢sin(x)   A + B⋅x + C⋅x  + D⋅x  + E⋅x ⎥\n",
       "⎢                                     ⎥\n",
       "⎢                          2        3 ⎥\n",
       "⎢cos(x)   B + 2⋅C⋅x + 3⋅D⋅x  + 4⋅E⋅x  ⎥\n",
       "⎢                                     ⎥\n",
       "⎢              ⎛                 2⎞   ⎥\n",
       "⎢-sin(x)     2⋅⎝C + 3⋅D⋅x + 6⋅E⋅x ⎠   ⎥\n",
       "⎢                                     ⎥\n",
       "⎢-cos(x)         6⋅(D + 4⋅E⋅x)        ⎥\n",
       "⎢                                     ⎥\n",
       "⎣sin(x)               24⋅E            ⎦"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.Matrix(funcAndSeries)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### DIY:  Determine the coefficients in the above power series.\n",
    "----\n",
    "\n",
    "You don't necessarily need to write code to complete this DIY problem.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Your code here if you feel you need it."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "`Your markdown here.`"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Computing a Taylor's Series Symbolically\n",
    "---\n",
    "\n",
    "Using `sympy` the Taylor's series can be computed symbolically."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle f{\\left(c \\right)} + \\left(- c + x\\right) \\left. \\frac{d}{d \\xi_{1}} f{\\left(\\xi_{1} \\right)} \\right|_{\\substack{ \\xi_{1}=c }} + \\frac{\\left(- c + x\\right)^{2} \\left. \\frac{d^{2}}{d \\xi_{1}^{2}} f{\\left(\\xi_{1} \\right)} \\right|_{\\substack{ \\xi_{1}=c }}}{2} + O\\left(\\left(- c + x\\right)^{3}; x\\rightarrow c\\right)$"
      ],
      "text/plain": [
       "                                              ⎛  2        ⎞│                  \n",
       "                                            2 ⎜ d         ⎟│                  \n",
       "                                    (-c + x) ⋅⎜────(f(ξ₁))⎟│                  \n",
       "                                              ⎜   2       ⎟│                  \n",
       "                ⎛ d        ⎞│                 ⎝dξ₁        ⎠│ξ₁=c    ⎛        3\n",
       "f(c) + (-c + x)⋅⎜───(f(ξ₁))⎟│     + ──────────────────────────── + O⎝(-c + x) \n",
       "                ⎝dξ₁       ⎠│ξ₁=c                2                            \n",
       "\n",
       "        \n",
       "        \n",
       "        \n",
       "        \n",
       "       ⎞\n",
       "; x → c⎠\n",
       "        "
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sympy import init_printing, symbols, Function\n",
    "init_printing()\n",
    "\n",
    "x, c = symbols(\"x,c\")\n",
    "f = Function(\"f\")\n",
    "\n",
    "f(x).series(x, x0=c, n=3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "One of the major uses of Taylor's series in computation is for the evaluation of derivatives.  Take note of the fact that the derivatives of a function appear in the evaluation of the series."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Computing Derivatives of Discrete Data\n",
    "---\n",
    "\n",
    "It may be straightforward to compute the derivative of some functions.  For example:\n",
    "\n",
    "$$f(x) = x^2$$\n",
    "$$f'(x) = 2x$$\n",
    "\n",
    "In numerical computing situations there is no analytical solution to the problem being solved and therefore no function to integrate or differentiate.  The approximate solution is available as a list of discrete points in the domain of the problem's independent variables (e.g. space, time).  The values could be represented as a list of numbers:\n",
    "\n",
    "$$\\{f(x_0), f(x_1), f(x_2), ...\\}$$\n",
    "\n",
    "The neighboring points $f(x_0)$ and $f(x_1)$ are seperated by a distance $\\Delta x = x_1 - x_0$ in the independent variable.  Although this will not be apparent from the values, it is implicit in the structure of the data.  Taylor's series can be used to compute approximate derivatives of the unknown function directly from the list of points in this situation.  We are going to compute a series expansion for an unknown function $f(x)$ in the vicinity of the point $c$ and then examine the relationship between that function and it's derivative quantities at a point $c\\pm h$.  The goal of the activity is to see if we can find expressions for the derivatives using the data point of interest ($c$) and its neighbors ($c \\pm h$).  We are going to use the idea of _forward_ and _backward_ differences.\n",
    "\n",
    "1. Forward differences are computed by expanding an unknown function in a Taylor series about a point “x=c” and then letting x go to c+h.\n",
    "1. Then, for backward differences, let x go to c-h."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Symbolically Computing Forward and Backward Differences\n",
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the figure below we indicate the following:\n",
    "\n",
    "* the unknown function $f(x)$ as a dashed line\n",
    "* the point about which the unknown function is expanded at $x=c$\n",
    "* the distance between successive points is shown as $h$\n",
    "* the approximate values of the function given at the filled squares"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![Approximation](./images/derivative.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Imagine that we take the above series expansion and use it to compute the value of the function near the point $c$.  Let us evaluate this series by adding and subtracting to the independent varable the quantity $h$.  To accomplish this we write down the series expansion for our function about the point $c$, then we let the independent variable $x \\rightarrow c+h$ and $c-h$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [],
   "source": [
    "x, h, c = sp.symbols(\"x,h,c\")\n",
    "f = sp.Function(\"f\")\n",
    "\n",
    "# the .subs() method replaces occurences of 'x' with something else\n",
    "taylorExpansionPlus = f(x).series(x, x0=c, n=3).removeO().subs(x,c+h)\n",
    "taylorExpansionMinus = f(x).series(x, x0=c, n=3).removeO().subs(x,c-h)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{h^{2} \\left. \\frac{d^{2}}{d \\xi_{1}^{2}} f{\\left(\\xi_{1} \\right)} \\right|_{\\substack{ \\xi_{1}=c }}}{2} + h \\left. \\frac{d}{d \\xi_{1}} f{\\left(\\xi_{1} \\right)} \\right|_{\\substack{ \\xi_{1}=c }} + f{\\left(c \\right)}$"
      ],
      "text/plain": [
       "   ⎛  2        ⎞│                                 \n",
       " 2 ⎜ d         ⎟│                                 \n",
       "h ⋅⎜────(f(ξ₁))⎟│                                 \n",
       "   ⎜   2       ⎟│                                 \n",
       "   ⎝dξ₁        ⎠│ξ₁=c     ⎛ d        ⎞│           \n",
       "───────────────────── + h⋅⎜───(f(ξ₁))⎟│     + f(c)\n",
       "          2               ⎝dξ₁       ⎠│ξ₁=c       "
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "taylorExpansionPlus"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Meaning that:\n",
    "\n",
    "$$\n",
    "f(c+h) = \\frac{h^{2}}{2} \\left. \\frac{d^{2}}{d \\xi_{1}^{2}}  f{\\left (\\xi_{1} \\right )} \\right|_{\\substack{ \\xi_{1}=c }} + h \\left. \\frac{d}{d \\xi_{1}} f{\\left (\\xi_{1} \\right )} \\right|_{\\substack{ \\xi_{1}=c }} + f{\\left (c \\right )}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\frac{h^{2} \\left. \\frac{d^{2}}{d \\xi_{1}^{2}} f{\\left(\\xi_{1} \\right)} \\right|_{\\substack{ \\xi_{1}=c }}}{2} - h \\left. \\frac{d}{d \\xi_{1}} f{\\left(\\xi_{1} \\right)} \\right|_{\\substack{ \\xi_{1}=c }} + f{\\left(c \\right)}$"
      ],
      "text/plain": [
       "   ⎛  2        ⎞│                                 \n",
       " 2 ⎜ d         ⎟│                                 \n",
       "h ⋅⎜────(f(ξ₁))⎟│                                 \n",
       "   ⎜   2       ⎟│                                 \n",
       "   ⎝dξ₁        ⎠│ξ₁=c     ⎛ d        ⎞│           \n",
       "───────────────────── - h⋅⎜───(f(ξ₁))⎟│     + f(c)\n",
       "          2               ⎝dξ₁       ⎠│ξ₁=c       "
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "taylorExpansionMinus"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Meaning that:\n",
    "\n",
    "$$\n",
    "f(c-h) = \\frac{h^{2}}{2} \\left. \\frac{d^{2}}{d \\xi_{1}^{2}}  f{\\left (\\xi_{1} \\right )} \\right|_{\\substack{ \\xi_{1}=c }} - h \\left. \\frac{d}{d \\xi_{1}} f{\\left (\\xi_{1} \\right )} \\right|_{\\substack{ \\xi_{1}=c }} + f{\\left (c \\right )}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Solving for First and Second Derivatives\n",
    "----\n",
    "\n",
    "Inspection of the results shows that the signs on the terms containing the first derivative are different between the two expressions.  We can use this to our advantage in solving for the derivative terms explicitly.  Note that each grouped expression is equal to zero as is the default in `sympy`."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "_Find the first derivative in this expression:_"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - 2 h \\left. \\frac{d}{d \\xi_{1}} f{\\left(\\xi_{1} \\right)} \\right|_{\\substack{ \\xi_{1}=c }} - f{\\left(c - h \\right)} + f{\\left(c + h \\right)}$"
      ],
      "text/plain": [
       "      ⎛ d        ⎞│                          \n",
       "- 2⋅h⋅⎜───(f(ξ₁))⎟│     - f(c - h) + f(c + h)\n",
       "      ⎝dξ₁       ⎠│ξ₁=c                      "
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(taylorExpansionMinus-f(c-h))-(taylorExpansionPlus-f(c+h))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Remember that `sympy` expressions are zero by default.  So this is true:\n",
    "\n",
    "$$\n",
    "- 2 h \\left. \\frac{d}{d \\xi_{1}} f{\\left (\\xi_{1} \\right )} \\right|_{\\substack{ \\xi_{1}=c }} - f{\\left (c - h \\right )} + f{\\left (c + h \\right )} = 0\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "_Find the second derivative in this expression:_"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle h^{2} \\left. \\frac{d^{2}}{d \\xi_{1}^{2}} f{\\left(\\xi_{1} \\right)} \\right|_{\\substack{ \\xi_{1}=c }} + 2 f{\\left(c \\right)} - f{\\left(c - h \\right)} - f{\\left(c + h \\right)}$"
      ],
      "text/plain": [
       "   ⎛  2        ⎞│                                   \n",
       " 2 ⎜ d         ⎟│                                   \n",
       "h ⋅⎜────(f(ξ₁))⎟│     + 2⋅f(c) - f(c - h) - f(c + h)\n",
       "   ⎜   2       ⎟│                                   \n",
       "   ⎝dξ₁        ⎠│ξ₁=c                               "
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(taylorExpansionMinus-f(c-h))+(taylorExpansionPlus-f(c+h))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Similarly:\n",
    "\n",
    "$$\n",
    "h^{2} \\left. \\frac{d^{2}}{d \\xi_{1}^{2}}  f{\\left (\\xi_{1} \\right )} \\right|_{\\substack{ \\xi_{1}=c }} + 2 f{\\left (c \\right )} - f{\\left (c - h \\right )} - f{\\left (c + h \\right )} = 0\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Solve each of the equations above for the derivative quantites and you will have recovered the formula for discrete derivatives.\n",
    "\n",
    "Use these expressions to complete your assignment as outlined above."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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