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  {
   "cell_type": "markdown",
   "metadata": {},
   "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": [
    "# 15.2. Solving equations and inequalities"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "from sympy import *\n",
    "init_printing()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "var('x y z a')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Use the function solve to resolve equations (the right hand side is always 0)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "solve(x**2 - a, x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can also solve inequations. You may need to specify the domain of your variables. Here, we tell SymPy that x is a real variable."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "x = Symbol('x')\n",
    "solve_univariate_inequality(x**2 > 4, x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Systems of equations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This function also accepts systems of equations (here a linear system)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "solve([x + 2*y + 1, x - 3*y - 2], x, y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Non-linear systems are also supported."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "solve([x**2 + y**2 - 1, x**2 - y**2 - S(1)/2], x, y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Singular linear systems can also be solved (here, there are infinitely many equations because the two equations are colinear)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "solve([x + 2*y + 1, -x - 2*y - 1], x, y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, let's solve a linear system using matrices with symbolic variables."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "var('a b c d u v')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We create the augmented matrix, which is the horizontal concatenation of the system's matrix with the linear coefficients, and the right-hand side vector."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "M = Matrix([[a, b, u], [c, d, v]]); M"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "solve_linear_system(M, x, y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This system needs to be non-singular to have a unique solution, which is equivalent to say that the determinant of the system's matrix needs to be non-zero (otherwise the denominators in the fractions above are equal to zero)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "det(M[:2,:2])"
   ]
  },
  {
   "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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