{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "0",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "source": [
    "# Play around area"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "1",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [],
   "source": [
    "from sympy import *\n",
    "x, y, z = symbols(\"x y z\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "2",
   "metadata": {},
   "outputs": [],
   "source": [
    "expr = cos(x) + 1\n",
    "expr.subs(x, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "3",
   "metadata": {},
   "outputs": [],
   "source": [
    "expr.subs(x, 0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "4",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [],
   "source": [
    "expr = x**y\n",
    "expr\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "5",
   "metadata": {},
   "outputs": [],
   "source": [
    "expr = expr.subs(y, x**y)\n",
    "expr\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "6",
   "metadata": {},
   "outputs": [],
   "source": [
    "expr = expr.subs(y, x**x)\n",
    "expr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "7",
   "metadata": {},
   "outputs": [],
   "source": [
    "expr = sin(2*x) + cos(2*x)\n",
    "expand_trig(expr)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "8",
   "metadata": {},
   "outputs": [],
   "source": [
    "expr.subs(sin(2*x), 2*sin(x)*cos(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "9",
   "metadata": {},
   "outputs": [],
   "source": [
    "expr = cos(x)\n",
    "expr.subs(x, 0)\n",
    "expr\n",
    "x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "10",
   "metadata": {},
   "outputs": [],
   "source": [
    "expr = x**4 - 4*x**3 + 4*x**2 - 2*x + 3\n",
    "replacements = [(x**i, y**i) for i in range(5) if i % 2 == 0]\n",
    "expr.subs(replacements)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "11",
   "metadata": {},
   "source": [
    "To evaluate a numerical expression into a floating point number, use `evalf`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "12",
   "metadata": {},
   "outputs": [],
   "source": [
    "expr = sqrt(8)\n",
    "expr.evalf()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "13",
   "metadata": {},
   "outputs": [],
   "source": [
    "pi.evalf(100)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "14",
   "metadata": {},
   "outputs": [],
   "source": [
    "pi.evalf(10000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "15",
   "metadata": {},
   "outputs": [],
   "source": [
    "expr = cos(2*x)\n",
    "expr.evalf(subs={x: 2.4})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "16",
   "metadata": {},
   "outputs": [],
   "source": [
    "one = cos(1)**2 + sin(1)**2\n",
    "(one - 1).evalf()\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "17",
   "metadata": {},
   "outputs": [],
   "source": [
    "(one - 1).evalf(chop=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "18",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "19",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy \n",
    "a = numpy.arange(10) \n",
    "expr = sin(x)\n",
    "f = lambdify(x, expr, \"numpy\") \n",
    "f(a) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "20",
   "metadata": {},
   "outputs": [],
   "source": [
    "f = lambdify(x, expr, \"math\")\n",
    "f(0.1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "21",
   "metadata": {},
   "outputs": [],
   "source": [
    "def mysin(x):\n",
    "    \"\"\"\n",
    "    My sine. Note that this is only accurate for small x.\n",
    "    \"\"\"\n",
    "    return x\n",
    "f = lambdify(x, expr, {\"sin\":mysin})\n",
    "f(0.1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "22",
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import init_printing\n",
    "init_printing() "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "23",
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import init_session\n",
    "init_session() "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "24",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "from __future__ import division\n",
    "from sympy import *\n",
    "x, y, z, t = symbols('x y z t')\n",
    "k, m, n = symbols('k m n', integer=True)\n",
    "f, g, h = symbols('f g h', cls=Function)\n",
    "init_printing() # doctest: +SKIP\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "25",
   "metadata": {},
   "outputs": [],
   "source": [
    "Integral(sqrt(1/x),x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "26",
   "metadata": {},
   "outputs": [],
   "source": [
    "simplify(sin(x)**2 + cos(x)**2)\n",
    "simplify((x**3 + x**2 - x - 1)/(x**2 + 2*x + 1))\n",
    "simplify(gamma(x)/gamma(x - 2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "27",
   "metadata": {},
   "outputs": [],
   "source": [
    "simplify(x**2 + 2*x + 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "28",
   "metadata": {},
   "outputs": [],
   "source": [
    "expand((x + 1)**2)\n",
    "expand((x + 2)*(x - 3))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "29",
   "metadata": {},
   "outputs": [],
   "source": [
    "expand((x + 1)*(x - 2) - (x - 1)*x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "30",
   "metadata": {},
   "outputs": [],
   "source": [
    "factor(x**3 - x**2 + x - 1)\n",
    "factor(x**2*z + 4*x*y*z + 4*y**2*z)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "31",
   "metadata": {},
   "outputs": [],
   "source": [
    "factor_list(x**2*z + 4*x*y*z + 4*y**2*z)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "32",
   "metadata": {},
   "outputs": [],
   "source": [
    "expand((cos(x) + sin(x))**2)\n",
    "factor(cos(x)**2 + 2*cos(x)*sin(x) + sin(x)**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "33",
   "metadata": {},
   "outputs": [],
   "source": [
    "expr = x*y + x - 3 + 2*x**2 - z*x**2 + x**3\n",
    "expr\n",
    "collected_expr = collect(expr, x)\n",
    "collected_expr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "34",
   "metadata": {},
   "outputs": [],
   "source": [
    "collected_expr.coeff(x, 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "35",
   "metadata": {},
   "outputs": [],
   "source": [
    "cancel((x**2 + 2*x + 1)/(x**2 + x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "36",
   "metadata": {},
   "outputs": [],
   "source": [
    "expr = 1/x + (3*x/2 - 2)/(x - 4)\n",
    "expr\n",
    "cancel(expr)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "37",
   "metadata": {},
   "outputs": [],
   "source": [
    "expr = (x*y**2 - 2*x*y*z + x*z**2 + y**2 - 2*y*z + z**2)/(x**2 - 1)\n",
    "expr\n",
    "cancel(expr)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "38",
   "metadata": {},
   "outputs": [],
   "source": [
    "factor(expr)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "39",
   "metadata": {},
   "outputs": [],
   "source": [
    "expr = (4*x**3 + 21*x**2 + 10*x + 12)/(x**4 + 5*x**3 + 5*x**2 + 4*x)\n",
    "expr\n",
    "apart(expr)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "40",
   "metadata": {},
   "outputs": [],
   "source": [
    "acos(x)\n",
    "cos(acos(x))\n",
    "asin(1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "41",
   "metadata": {},
   "outputs": [],
   "source": [
    "trigsimp(sin(x)**2 + cos(x)**2)\n",
    "trigsimp(sin(x)**4 - 2*cos(x)**2*sin(x)**2 + cos(x)**4)\n",
    "trigsimp(sin(x)*tan(x)/sec(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "42",
   "metadata": {},
   "outputs": [],
   "source": [
    "trigsimp(cosh(x)**2 + sinh(x)**2)\n",
    "trigsimp(sinh(x)/tanh(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "43",
   "metadata": {},
   "outputs": [],
   "source": [
    "expand_trig(sin(x + y))\n",
    "expand_trig(tan(2*x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "44",
   "metadata": {},
   "outputs": [],
   "source": [
    "trigsimp(sin(x)*cos(y) + sin(y)*cos(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "45",
   "metadata": {},
   "outputs": [],
   "source": [
    "x, y = symbols('x y', positive=True)\n",
    "a, b = symbols('a b', real=True)\n",
    "z, t, c = symbols('z t c')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "46",
   "metadata": {},
   "outputs": [],
   "source": [
    "sqrt(x) == x**Rational(1, 2)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "47",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "source": [
    "## Power"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "48",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [],
   "source": [
    "powsimp(x**a*x**b)\n",
    "powsimp(x**a*y**a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "49",
   "metadata": {},
   "outputs": [],
   "source": [
    "powsimp(t**c*z**c)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "50",
   "metadata": {},
   "outputs": [],
   "source": [
    "powsimp(t**c*z**c, force=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "51",
   "metadata": {},
   "outputs": [],
   "source": [
    "(z*t)**2\n",
    "sqrt(x*y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "52",
   "metadata": {},
   "outputs": [],
   "source": [
    "powsimp(z**2*t**2)\n",
    "powsimp(sqrt(x)*sqrt(y))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "53",
   "metadata": {},
   "outputs": [],
   "source": [
    "expand_power_exp(x**(a + b))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "54",
   "metadata": {},
   "outputs": [],
   "source": [
    "expand_power_base((x*y)**a)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "55",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "source": [
    "## Derivative"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "56",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [],
   "source": [
    "diff(cos(x), x)\n",
    "diff(exp(x**2), x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "57",
   "metadata": {},
   "outputs": [],
   "source": [
    "diff(x**4, x, x, x)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "58",
   "metadata": {},
   "outputs": [],
   "source": [
    "diff(x**4, x, 3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "59",
   "metadata": {},
   "outputs": [],
   "source": [
    "expr = exp(x*y*z)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "60",
   "metadata": {},
   "outputs": [],
   "source": [
    "diff(expr, x, y, y, z, z, z, z)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "61",
   "metadata": {},
   "outputs": [],
   "source": [
    "diff(expr, x, y, 2, z, 4)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "62",
   "metadata": {},
   "outputs": [],
   "source": [
    "diff(expr, x, y, y, z, 4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "63",
   "metadata": {},
   "outputs": [],
   "source": [
    "expr.diff(x, y, y, z, 4)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "64",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "source": [
    "## unevaluated derivative"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "65",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "source": [
    "These unevaluated objects are useful for delaying the evaluation of the derivative, or for printing purposes."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "66",
   "metadata": {},
   "outputs": [],
   "source": [
    "deriv = Derivative(expr, x, y, y, z, 4)\n",
    "deriv"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "67",
   "metadata": {},
   "outputs": [],
   "source": [
    "deriv.doit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "68",
   "metadata": {},
   "outputs": [],
   "source": [
    "m, n, a, b = symbols('m n a b')\n",
    "expr = (a*x + b)**m\n",
    "expr.diff((x, n))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "69",
   "metadata": {},
   "source": [
    "Integral"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "70",
   "metadata": {},
   "outputs": [],
   "source": [
    "integrate(cos(x), x)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "71",
   "metadata": {},
   "source": [
    "Note that SymPy does not include the constant of integration. If you want it, you can add one yourself, or rephrase your problem as a differential equation and use dsolve to solve it, which does add the constant (see Solving Differential Equations)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "72",
   "metadata": {},
   "outputs": [],
   "source": [
    "integrate(exp(-x), (x, 0, oo))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "73",
   "metadata": {},
   "outputs": [],
   "source": [
    "integrate(exp(-x**2 - y**2), (x, -oo, oo), (y, -oo, oo))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "74",
   "metadata": {},
   "outputs": [],
   "source": [
    "expr = integrate(x**x, x)\n",
    "print(expr)\n",
    "expr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "75",
   "metadata": {},
   "outputs": [],
   "source": [
    "expr = Integral(log(x)**2, x)\n",
    "expr\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "76",
   "metadata": {},
   "outputs": [],
   "source": [
    "expr.doit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "77",
   "metadata": {},
   "outputs": [],
   "source": [
    "integ = Integral((x**4 + x**2*exp(x) - x**2 - 2*x*exp(x) - 2*x -\n",
    "    exp(x))*exp(x)/((x - 1)**2*(x + 1)**2*(exp(x) + 1)), x)\n",
    "integ\n",
    "integ.doit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "78",
   "metadata": {},
   "outputs": [],
   "source": [
    "integ = Integral(sin(x**2), x)\n",
    "integ\n",
    "integ.doit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "79",
   "metadata": {},
   "outputs": [],
   "source": [
    "integ = Integral(x**y*exp(-x), (x, 0, oo))\n",
    "integ\n",
    "integ.doit()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "80",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "source": [
    "## Limit"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "81",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [],
   "source": [
    "limit(sin(x)/x, x, 0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "82",
   "metadata": {},
   "outputs": [],
   "source": [
    "expr = x**2/exp(x)\n",
    "expr.subs(x, oo)\n",
    "limit(expr, x, oo)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "83",
   "metadata": {},
   "outputs": [],
   "source": [
    "expr = Limit((cos(x) - 1)/x, x, 0)\n",
    "expr\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "84",
   "metadata": {},
   "outputs": [],
   "source": [
    "expr.doit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "85",
   "metadata": {},
   "outputs": [],
   "source": [
    "limit(1/x, x, 0, '+')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "86",
   "metadata": {},
   "outputs": [],
   "source": [
    "limit(1/x, x, 0, '-')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "87",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "source": [
    "## Series expansion"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "88",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [],
   "source": [
    "expr = exp(sin(x))\n",
    "expr.series(x, 0, 4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "89",
   "metadata": {},
   "outputs": [],
   "source": [
    "x + x**3 + x**6 + O(x**4)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "90",
   "metadata": {},
   "outputs": [],
   "source": [
    "x*O(1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "91",
   "metadata": {},
   "outputs": [],
   "source": [
    "expr.series(x, 0, 4).removeO()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "92",
   "metadata": {},
   "outputs": [],
   "source": [
    "exp(x - 6).series(x, x0=6)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "93",
   "metadata": {},
   "outputs": [],
   "source": [
    "f, g = symbols('f g', cls=Function)\n",
    "differentiate_finite(f(x)*g(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "94",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "source": [
    "## Eqaution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "95",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [],
   "source": [
    "Eq(x, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "96",
   "metadata": {},
   "outputs": [],
   "source": [
    "solveset(Eq(x**2, 1), x)\n",
    "solveset(Eq(x**2 - 1, 0), x)\n",
    "solveset(x**2 - 1, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "97",
   "metadata": {},
   "outputs": [],
   "source": [
    "solveset(x**2 - x, x)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "98",
   "metadata": {},
   "outputs": [],
   "source": [
    "solveset(x - x, x, domain=S.Reals)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "99",
   "metadata": {},
   "outputs": [],
   "source": [
    "solveset(sin(x) - 1, x, domain=S.Reals)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "100",
   "metadata": {},
   "outputs": [],
   "source": [
    "solveset(exp(x), x)     # No solution exists\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "101",
   "metadata": {},
   "outputs": [],
   "source": [
    "solveset(cos(x) - x, x)  # Not able to find solution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "102",
   "metadata": {},
   "outputs": [],
   "source": [
    "linsolve([x + y + z - 1, x + y + 2*z - 3 ], (x, y, z))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "103",
   "metadata": {},
   "outputs": [],
   "source": [
    "linsolve(Matrix(([1, 1, 1, 1], [1, 1, 2, 3])), (x, y, z))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "104",
   "metadata": {},
   "outputs": [],
   "source": [
    "M = Matrix(((1, 1, 1, 1), (1, 1, 2, 3)))\n",
    "system = A, b = M[:, :-1], M[:, -1]\n",
    "linsolve(system, x, y, z)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "105",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "source": [
    "## Solving differential equation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "106",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [],
   "source": [
    "f, g = symbols('f g', cls=Function)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "107",
   "metadata": {},
   "outputs": [],
   "source": [
    "f(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "108",
   "metadata": {},
   "outputs": [],
   "source": [
    "f(x).diff(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "109",
   "metadata": {},
   "outputs": [],
   "source": [
    "diffeq = Eq(f(x).diff(x, x) - 2*f(x).diff(x) + f(x), sin(x))\n",
    "diffeq"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "110",
   "metadata": {},
   "outputs": [],
   "source": [
    "dsolve(diffeq, f(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "111",
   "metadata": {},
   "outputs": [],
   "source": [
    "dsolve(f(x).diff(x)*(1 - sin(f(x))) - 1, f(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "112",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "source": [
    "## Matrices"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "113",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [],
   "source": [
    "init_printing(use_unicode=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "114",
   "metadata": {},
   "outputs": [],
   "source": [
    "Matrix([[1, -1], [3, 4], [0, 2]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "115",
   "metadata": {},
   "outputs": [],
   "source": [
    "Matrix([1, 2, 3])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "116",
   "metadata": {},
   "outputs": [],
   "source": [
    "Matrix([[1], [2], [3]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "117",
   "metadata": {},
   "outputs": [],
   "source": [
    "Matrix([[1, 2, 3]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "118",
   "metadata": {},
   "outputs": [],
   "source": [
    "Matrix([[1, 2, 3],[4,5,6]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "119",
   "metadata": {},
   "outputs": [],
   "source": [
    "M = Matrix([[1, 2, 3], [3, 2, 1]])\n",
    "N = Matrix([0, 1, 1])\n",
    "M*N"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "120",
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import shape\n",
    "M = Matrix([[1, 2, 3], [-2, 0, 4]])\n",
    "M\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "121",
   "metadata": {},
   "outputs": [],
   "source": [
    "shape(M)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "122",
   "metadata": {},
   "outputs": [],
   "source": [
    "M.row(0)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "123",
   "metadata": {},
   "outputs": [],
   "source": [
    "M.col(-1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "124",
   "metadata": {},
   "outputs": [],
   "source": [
    "M.col_del(0)\n",
    "M\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "125",
   "metadata": {},
   "outputs": [],
   "source": [
    "M.row_del(1)\n",
    "M"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "126",
   "metadata": {},
   "outputs": [],
   "source": [
    "M = M.row_insert(1, Matrix([[0, 4]]))\n",
    "M\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "127",
   "metadata": {},
   "outputs": [],
   "source": [
    "M = M.col_insert(0, Matrix([1, -2]))\n",
    "M"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "128",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": [
     "raises-exception"
    ]
   },
   "outputs": [],
   "source": [
    "M = Matrix([[1, 3], [-2, 3]])\n",
    "N = Matrix([[0, 3], [0, 7]])\n",
    "M + N\n",
    "M*N\n",
    "3*M\n",
    "M**2\n",
    "M**-1\n",
    "N**-1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "129",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [],
   "source": [
    "eye(3)\n",
    "eye(4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "130",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [],
   "source": [
    "ones(3, 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "131",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [],
   "source": [
    "diag(1, 2, 3)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "132",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [],
   "source": [
    "diag(-1, ones(2, 2), Matrix([5, 7, 5]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "133",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [],
   "source": [
    "M = Matrix([[1, 0, 1], [2, -1, 3], [4, 3, 2]])\n",
    "M\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "134",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [],
   "source": [
    "M.det()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "135",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [],
   "source": [
    "M = Matrix([[1, 0, 1, 3], [2, 3, 4, 7], [-1, -3, -3, -4]])\n",
    "M\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "136",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [],
   "source": [
    "M.rref()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "137",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [],
   "source": [
    "M = Matrix([[1, 2, 3, 0, 0], [4, 10, 0, 0, 1]])\n",
    "M\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "138",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [],
   "source": [
    "M.nullspace()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "139",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [],
   "source": [
    "M = Matrix([[1, 1, 2], [2 ,1 , 3], [3 , 1, 4]])\n",
    "M\n",
    "M.columnspace()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "140",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [],
   "source": [
    "M = Matrix([[3, -2,  4, -2], [5,  3, -3, -2], [5, -2,  2, -2], [5, -2, -3,  3]])\n",
    "M\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "141",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [],
   "source": [
    "M.eigenvals()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "142",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [],
   "source": [
    "M.eigenvects()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "143",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [],
   "source": [
    "P, D = M.diagonalize()\n",
    "P\n",
    "D\n",
    "P*D*P**-1\n",
    "P*D*P**-1 == M"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "144",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [],
   "source": [
    "lamda = symbols('lamda')\n",
    "p = M.charpoly(lamda)\n",
    "factor(p.as_expr())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "145",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [],
   "source": [
    "q = Symbol(\"q\", positive = True)\n",
    "m = Matrix([\n",
    "[-2*cosh(q/3),      exp(-q),            1],\n",
    "[      exp(q), -2*cosh(q/3),            1],\n",
    "[           1,            1, -2*cosh(q/3)]])\n",
    "m.nullspace() "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "146",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [],
   "source": [
    "import warnings\n",
    "def my_iszero(x):\n",
    "    try:\n",
    "        result = x.is_zero\n",
    "    except AttributeError:\n",
    "        result = None\n",
    "    # Warnings if evaluated into None\n",
    "    if result is None:\n",
    "        warnings.warn(\"Zero testing of {} evaluated into None\".format(x))\n",
    "    return result\n",
    "m.nullspace(iszerofunc=my_iszero) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "147",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [],
   "source": [
    "def my_iszero(x):\n",
    "    try:\n",
    "        result = x.rewrite(exp).simplify().is_zero\n",
    "    except AttributeError:\n",
    "        result = None\n",
    "    # Warnings if evaluated into None\n",
    "    if result is None:\n",
    "        warnings.warn(\"Zero testing of {} evaluated into None\".format(x))\n",
    "    return result\n",
    "m.nullspace(iszerofunc=my_iszero) "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "148",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "source": [
    "## Expression Trees"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "149",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [],
   "source": [
    "x, y, z = symbols('x y z')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "150",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [],
   "source": [
    "expr = x**2 + x*y\n",
    "srepr(expr)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "151",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [],
   "source": [
    "x = symbols('x')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "152",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [],
   "source": [
    "x = Symbol('x')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "153",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [],
   "source": [
    "srepr(x**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "154",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [],
   "source": [
    "Pow(x, 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "155",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [],
   "source": [
    "expr = sin(x*y)/2 - x**2 + 1/y\n",
    "srepr(expr)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "156",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [],
   "source": [
    "srepr(x - y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "157",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [],
   "source": [
    "expr = x/y\n",
    "srepr(expr)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "158",
   "metadata": {},
   "outputs": [],
   "source": [
    "import graphviz\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "159",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": [
     "raises-exception"
    ]
   },
   "outputs": [],
   "source": [
    "import graphviz\n",
    "\n",
    "# Create a new Digraph object\n",
    "dot = graphviz.Digraph(comment='The Visualization of x**2 + xy')\n",
    "\n",
    "# Adding nodes\n",
    "dot.node('A', 'Add')\n",
    "dot.node('B', 'Pow')\n",
    "dot.node('C', 'x', shape='ellipse')\n",
    "dot.node('D', '2', shape='ellipse')\n",
    "dot.node('E', 'Mul')\n",
    "dot.node('F', 'x', shape='ellipse') # Reused label 'x', but it's a different node in the graph context\n",
    "dot.node('G', 'y', shape='ellipse')\n",
    "\n",
    "# Adding edges\n",
    "dot.edges(['AB', 'AE'])\n",
    "dot.edge('B', 'C', label='0,0')\n",
    "dot.edge('B', 'D', label='0,1')\n",
    "dot.edges(['EC', 'EG'])\n",
    "\n",
    "# Generate the diagram\n",
    "dot_path = '/mnt/data/expr_graphviz.svg'\n",
    "dot.render(dot_path, format='svg', view=False)\n",
    "\n",
    "\n",
    "dot_path"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "160",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "outputs": [],
   "source": []
  }
 ],
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