nbs/01-Symbolic-mathematics-with-Sympy.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Symbolic mathematics with Sympy\n",
    "\n",
    "[Sympy](http://www.sympy.org/en/index.html) is described as a:\n",
    "\n",
    "> \"... Python library for symbolic mathematics.\"\n",
    "\n",
    "This means it can be used to:\n",
    "\n",
    "- Manipulate symbolic expressions;\n",
    "- Solve symbolic equations;\n",
    "- Carry out symbolic Calculus;\n",
    "- Plot symbolic function.\n",
    "\n",
    "It has other capabilities that we will not go in to in this handbook. But you can read more about it here: http://www.sympy.org/en/index.html\n",
    "\n",
    "## Manipulating symbolic expressions\n",
    "\n",
    "Before we can start using the library to manipulate expressions, we need to import it."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import sympy as sym"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The above imports the library and gives us access to it's commands using the shortand `sym` which is conventially used.\n",
    "\n",
    "If we wanted to get Python to check that $x - x = 0$ we would get an error if we did not tell Python what $x$ was:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "ename": "NameError",
     "evalue": "name 'x' is not defined",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mNameError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-2-e2d9e13b7cb9>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mx\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;31mNameError\u001b[0m: name 'x' is not defined"
     ]
    }
   ],
   "source": [
    "x - x"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is where Sympy comes in, we can tell Python to create $x$ as a symbolic variable:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "x = sym.symbols('x')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we can calculate $x - x$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x - x"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can create and manipulate expressions in Sympy. Let us for example verify:\n",
    "\n",
    "$$(a + b) ^ 2 = a ^ 2 + 2ab + b ^2$$\n",
    "\n",
    "First, we create the symbolic variables $a, b$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "a, b = sym.symbols('a, b')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now let's create our expression:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(a + b)**2"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr = (a + b) ** 2 \n",
    "expr"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Note** we can get Sympy to use LaTeX so that the output looks nice in a notebook:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "sym.init_printing()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEQAAAAbBAMAAAA35TSjAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left(a + b\\right)^{2}$$"
      ],
      "text/plain": [
       "       2\n",
       "(a + b) "
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let us expand our expression:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHYAAAAWBAMAAADnfXV4AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpmJdu8QRM1mu90y\nVKvMIHo8AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABu0lEQVQ4EZWSP0vDUBTFbxvyp7Qp0Q+gBUWk\ng9S6OAjWLq5OojgYBReXFsRNUBwkIIr4CaSDODhUVNxKcXKzg5NQ2qGDboqQIijxJffd21AV2jfk\nnpxzf7mP9wIQLCU9hqLnZ7Zky95JeOmZChoVK7kniSt4tPqC41biXQJ3MFzsjy1G2gw0bJa9CdPl\nvhtWPYp4jhqVFClZB9OXUm2udEVoZNk9ZYVCa8IsncBxVwa+EUuRq2f0KuonLEkb1AWZfstq5kLG\nFuzIt4f6s41SsmoZkl/o6HQoxPpG4qh+LtlVz5NKsobLrFKWEbG+oXreG0BkZHRRhn6RrFBxFzOj\nla8FDcSyAduWIb7Aq8PuZjArHGj4CxHLhrYM0TMmw3MrgJn4az6CBmLZiLrQueVpx7l1HLyTaAow\nGwdNsIrjHC45TlN8BQ0h1DMoVEWlxXvOU7YPOt4SzWWjkIEGcX4l1kzBAGaf/nx/EctGoQYXphVk\nwYPYe4A5zNqwcRJExLJh5GIVpYPS3Mj11EQZs1coYU4sG1prZmj+N2t4nlfGbD2PY3nPbISwQNKe\nu33xTnP/iNBa+zcBLXQsftcPGl9+pvSkZG4AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$a^{2} + 2 a b + b^{2}$$"
      ],
      "text/plain": [
       " 2            2\n",
       "a  + 2⋅a⋅b + b "
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr.expand()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that we can also get Sympy to produce the LaTeX code for future use:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'a^{2} + 2 a b + b^{2}'"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.latex(expr.expand())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "**EXERCISE** Use Sympy to verify the following expressions:\n",
    "\n",
    "- $(a - b) ^ 2 = a ^ 2 - 2 a b + b^2$\n",
    "- $a ^ 2 - b ^ 2 = (a - b) (a + b)$ (instead of using `expand`, try `factor`)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Solving symbolic equations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can use Sympy to solve symbolic expression. For example let's find the solution in $x$ of the quadratic equation:\n",
    "\n",
    "$$a x ^ 2 + b x + c = 0$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# We only really need to define `c` but doing them all again.\n",
    "a, b, c, x = sym.symbols('a, b, c, x')  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The Sympy command for solving equations is `solveset`. The first argument is an expression for which the root will be found. The second argument is the value that we are solving for."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAb0AAAAyBAMAAAA6kRjOAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nVdPV8Bp5s+nU+g/zBWOkT6CRDwAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left\\{- \\frac{b}{2 a} - \\frac{1}{2 a} \\sqrt{- 4 a c + b^{2}}, - \\frac{b}{2 a} + \\frac{1}{2 a} \\sqrt{- 4 a c + b^{2}}\\right\\}$$"
      ],
      "text/plain": [
       "⎧           _____________             _____________⎫\n",
       "⎪          ╱           2             ╱           2 ⎪\n",
       "⎨   b    ╲╱  -4⋅a⋅c + b       b    ╲╱  -4⋅a⋅c + b  ⎬\n",
       "⎪- ─── - ────────────────, - ─── + ────────────────⎪\n",
       "⎩  2⋅a         2⋅a           2⋅a         2⋅a       ⎭"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.solveset(a * x ** 2 + b * x + c, x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "**EXERCISE** Use Sympy to find the solutions to the generic cubic equation:\n",
    "\n",
    "$$a x ^ 3 + b x ^ 2 + c  x + d = 0$$\n",
    "\n",
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It is possible to pass more arguments to `solveset` for example to constrain the solution space. Let us see what the solution of the following is in $\\mathbb{R}$:\n",
    "\n",
    "$$x^2=-1$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAASBAMAAACQmVJ4AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAu1SJds0i3ZlEEO9m\nMqs+5asqAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAa0lEQVQIHWNgYGBgBGImE0MgacLgysDAVsAg\n6MDA0sBQWcDA6MDZ9YuB+wDHrL8M/AfYGX4y8CskMnwCsn2AJPeBKKAIu1cB118GPiMD3g8MPHIX\neAQYGMoZWCcwMKhnmQDNtJmTACQVgBgAel4WPlvNWewAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\emptyset$$"
      ],
      "text/plain": [
       "∅"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.solveset(x ** 2 + 1, x, domain=sym.S.Reals)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "**EXERCISE** Use Sympy to find the solutions to the following equations:\n",
    "\n",
    "- $x ^ 2 == 2$ in $\\mathbb{N}$;\n",
    "- $x ^ 3 + 2 x = 0$ in $\\mathbb{R}$.\n",
    "\n",
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Symbolic calculus\n",
    "\n",
    "We can use Sympy to compute limits. Let us calculate:\n",
    "\n",
    "$$\\lim_{x\\to 0^+}\\frac{1}{x}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABMAAAALBAMAAABv+6sJAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEKvvZom7mXYyzVQi\n3UQ6SGZXAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAaklEQVQIHWNgYBBgAAIQwaj82YGBIayogYGB\nbQLHLwapDQxTGRg8GRj2J6xkYGA5wACUYP0LJBgcQEyGfBDRAGYm/wNqd2BwZGDgiDE+wMBxgIGd\ngSGcYb4dgytQolxtAwNjvXEAUDncNgBJUBUwaYAbUgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\infty$$"
      ],
      "text/plain": [
       "∞"
      ]
     },
     "execution_count": 68,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.limit(1/x, x, 0, dir=\"+\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "**EXERCISE** Compute the following limits:\n",
    "\n",
    "1. $\\lim_{x\\to 0^-}\\frac{1}{x}$\n",
    "2.  $\\lim_{x\\to 0}\\frac{1}{x^2}$\n",
    "\n",
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can use also Sympy to differentiate and integrate. Let us experiment with differentiating the following expression:\n",
    "\n",
    "$$x ^ 2 - \\cos(x)$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGYAAAAUBAMAAACNG1WbAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAElFTkSuQmCC\n",
      "text/latex": [
       "$$2 x + \\sin{\\left (x \\right )}$$"
      ],
      "text/plain": [
       "2⋅x + sin(x)"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.diff(x ** 2 - sym.cos(x), x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Similarly we can integrate:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGgAAAAvBAMAAAD9b5JGAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n1+jGaUH0UyZOcJ3mYZ9xNYh+P8qwRLc2bXTDRdfrVgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{x^{3}}{3} - \\sin{\\left (x \\right )}$$"
      ],
      "text/plain": [
       " 3         \n",
       "x          \n",
       "── - sin(x)\n",
       "3          "
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.integrate(x ** 2 - sym.cos(x), x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can also carry out definite integrals:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAIYAAAAqBAMAAACEiT43AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nOZrjGtTv0TKC0dkPK4bj/QtIgKcpZ5vQfAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$- \\sin{\\left (5 \\right )} + \\frac{125}{3}$$"
      ],
      "text/plain": [
       "-sin(5) + 125/3"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.integrate(x ** 2 - sym.cos(x), (x, 0, 5))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "\n",
    "**EXERCISE** Use Sympy to calculate the following:\n",
    "\n",
    "1. $\\frac{d\\sin(x ^2)}{dx}$\n",
    "2. $\\frac{d(x ^2 + xy - \\ln(y))}{dy}$\n",
    "3. $\\int e^x \\cos(x)\\;dx$\n",
    "4. $\\int_0^5 e^{2x}\\;dx$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Plotting with Sympy\n",
    "\n",
    "Finally Sympy can be used to plot functions. Note that this makes use of another Python library called [matplotlib](http://matplotlib.org/). Whilst Sympy allows us to not directly need to make use of matplotlib it could be worth learning to use as it's a very powerful and versatile library.\n",
    "\n",
    "Before plotting in Jupyter we need to run a command to tell it to display the plots directly in the notebook:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let us plot $x^2$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXcAAAEFCAYAAAAYKqc0AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nsPzRAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fb87f5cf438>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "expr = x ** 2\n",
    "p = sym.plot(expr);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can directly save that plot to a file if we wish to:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXcAAAEFCAYAAAAYKqc0AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fb87f45b748>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "p.save(\"x_squared.pdf\");"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "**EXERCISE** Plot the following functions:\n",
    "\n",
    "- $y=x + cos(x)$\n",
    "- $y=x ^ 2 - e^x$ (you might find `ylim` helpful as an argument)\n",
    "\n",
    "Experiment with saving your plots to a file.\n",
    "\n",
    "---\n",
    "\n",
    "## Summary\n",
    "\n",
    "This section has discussed using Sympy to:\n",
    "\n",
    "- Manipulate symbolic expressions;\n",
    "- Calculate limits, derivates and integrals;\n",
    "- Plot a symbolic expression.\n",
    "    \n",
    "This just touches the surface of what Sympy can do.\n",
    "\n",
    "Let us move on to using [Numpy](02 - Linear algebra with Numpy.ipynb) to do Linear Algebra."
   ]
  }
 ],
 "metadata": {
  "anaconda-cloud": {},
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.5.1"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}