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   "source": [
    "> This is one of the 100 recipes of the [IPython Cookbook](http://ipython-books.github.io/), the definitive guide to high-performance scientific computing and data science in Python.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 15.1. Diving into symbolic computing with SymPy"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "SymPy is a pure Python package for symbolic mathematics."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First, we import SymPy, and enable rich display LaTeX-based printing in the IPython notebook (using the MathJax Javascript library)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "from sympy import *\n",
    "init_printing()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "With NumPy and the other packages we have been using so far, we were dealing with numbers and numerical arrays. With SymPy, we deal with symbolic variables. It's a radically different shift of paradigm, which mathematicians may be more familiar with."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To deal with symbolic variables, we need to declare them."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "var('x y')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The var function creates symbols and injects them into the namespace. This function should only be used in interactive mode. In a Python module, it is better to use the symbol function which returns the symbols."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "x, y = symbols('x y')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can create mathematical expressions with these symbols."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "expr1 = (x + 1)**2\n",
    "expr2 = x**2 + 2*x + 1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Are these expressions equal?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "expr1 == expr2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "These expressions are mathematically equal, but not syntactically identical. To test whether they are equal, we can ask SymPy to simplify the difference algebraically."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "simplify(expr1-expr2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A very common operation with symbolic expressions is substitution of a symbol by another symbol, expression, or a number."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "expr1.subs(x, expr1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "expr1.subs(x, pi)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A rational number cannot be written simply as \"1/2\" as this Python expression evaluates to 0. A possibility is to use a SymPy object for 1, for example using the function S."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "expr1.subs(x, S(1)/2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Exactly-represented numbers can be evaluated numerically with evalf:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "_.evalf()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can transform this *symbolic* function into an actual Python function that can be evaluated on NumPy arrays, using the `lambdify` function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "f = lambdify(x, expr1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "f(np.linspace(-2., 2., 5))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "> You'll find all the explanations, figures, references, and much more in the book (to be released later this summer).\n",
    "\n",
    "> [IPython Cookbook](http://ipython-books.github.io/), by [Cyrille Rossant](http://cyrille.rossant.net), Packt Publishing, 2014 (500 pages)."
   ]
  }
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