ramanujan_pi_approx.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"https://pbs.twimg.com/media/EWjp0c-XQAEqnNV?format=jpg&name=large\">"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Ramanujan proved the above identity of pi\n",
    "# Going to see how fast it converges\n",
    "\n",
    "# initialize\n",
    "initial_state = \"3 + 1**2\""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = initial_state"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "\"\"\"\n",
    "My first attempt relies on parsing the string version of the equation.\n",
    "It's inefficient and OOMs around 46 levels deep. We can do better.\n",
    "\n",
    "The problem is that it compiles the string and then below I\n",
    "evaluate it in one go, which for a sufficient amount of nesting \n",
    "causes an OOM; more scalable to compute iteratively and then return the \n",
    "final value, done below.\n",
    "\"\"\"\n",
    "\n",
    "def get_final_term(eq): \n",
    "    return eq.rsplit(\" \")[-1].replace(\")\",\"\")\n",
    "\n",
    "def increment(eq, n):\n",
    "    \"\"\"\n",
    "    Starts with the initial equation and returns a Ramanujan \n",
    "    equation n levels deep.\n",
    "    \n",
    "    e.g. 3**2 --> (3**2 / (6 + 5**2)) would be 1 level deeper.\n",
    "    \"\"\"\n",
    "    for x in range(n):\n",
    "        last_term = get_final_term(eq)\n",
    "        new = \"(\" + last_term + \" / \" + \"(6 + \" + str(eval(last_term.split(\"**\")[0])+2) + \"**2))\"\n",
    "        new_eq = eq.replace(last_term, new)\n",
    "        eq = new_eq\n",
    "    \n",
    "    return eq"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "# this works until n = 46, then OOMs\n",
    "pi_values = []\n",
    "for i in range(1,47):\n",
    "    pi_values.append(eval(increment(x,i)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# See how it converges...\n",
    "\n",
    "import matplotlib\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "\n",
    "# Going deeper into the levels\n",
    "y = pi_values[14:]\n",
    "x = range(15,47)\n",
    "\n",
    "fig, ax = plt.subplots()\n",
    "ax.plot(x,y)\n",
    "\n",
    "ax.set(xlabel='iteration', ylabel='pi',\n",
    "       title='Ramanujan pi approximation')\n",
    "ax.grid()\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3 + (1**2 / (6 + (3**2 / (6 + (5**2 / (6 + (7**2 / (6 + (9**2 / (6 + (11**2 / (6 + (13**2 / (6 + (15**2 / (6 + (17**2 / (6 + (19**2 / (6 + (21**2 / (6 + (23**2 / (6 + (25**2 / (6 + (27**2 / (6 + (29**2 / (6 + (31**2 / (6 + (33**2 / (6 + (35**2 / (6 + (37**2 / (6 + (39**2 / (6 + (41**2 / (6 + (43**2 / (6 + (45**2 / (6 + (47**2 / (6 + (49**2 / (6 + (51**2 / (6 + (53**2 / (6 + (55**2 / (6 + (57**2 / (6 + (59**2 / (6 + (61**2 / (6 + (63**2 / (6 + (65**2 / (6 + (67**2 / (6 + (69**2 / (6 + (71**2 / (6 + (73**2 / (6 + (75**2 / (6 + (77**2 / (6 + (79**2 / (6 + (81**2 / (6 + (83**2 / (6 + (85**2 / (6 + (87**2 / (6 + (89**2 / (6 + (91**2 / (6 + 93**2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))\n"
     ]
    }
   ],
   "source": [
    "# Example of the string it's constructing, which is then evaluted...\n",
    "\n",
    "print(increment(initial_state,46))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'\\nLevels\\n======\\n1/ 3 + 1**2\\n2/ 3 + (1**2 / (6 + 3**2))\\n3/ 3 + (1**2 / (6 + (3**2 / (6 + 5**2))))\\n4/ 3 + (1**2 / (6 + (3**2 / (6 + (5**2 / (6 + 7**2))))))\\n5/ 3 + (1**2 / (6 + (3**2 / (6 + (5**2 / (6 + (7**2 / (6 + 9**2))))))))\\netc.\\n'"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\"\"\"\n",
    "Levels (i.e. depth)\n",
    "======\n",
    "1/ 3 + 1**2\n",
    "2/ 3 + (1**2 / (6 + 3**2))\n",
    "3/ 3 + (1**2 / (6 + (3**2 / (6 + 5**2))))\n",
    "4/ 3 + (1**2 / (6 + (3**2 / (6 + (5**2 / (6 + 7**2))))))\n",
    "5/ 3 + (1**2 / (6 + (3**2 / (6 + (5**2 / (6 + (7**2 / (6 + 9**2))))))))\n",
    "etc.\n",
    "\"\"\""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.142021107132669"
      ]
     },
     "execution_count": 76,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\"\"\"\n",
    "Second, proper attempt.\n",
    "This is far more efficient. Computes the function recursively. \n",
    "You tell it what depth you want the function computed to.\n",
    "It starts at the \"deepest\" level and then recurses its way \"up\".\n",
    "\"\"\"\n",
    "\n",
    "def recursive_method(d):\n",
    "    pi = 0\n",
    "    x = 0\n",
    "    \n",
    "    # depth must be odd number\n",
    "    if (d % 2 == 0):\n",
    "        d += 1\n",
    "    \n",
    "    # compute the deepest, most \"nested\" parens\n",
    "    result = (d**2 / (6 + (d+2)**2))\n",
    "    \n",
    "    # core loop; iterate \"upwards\" from the innermost parens\n",
    "    while (d>=5):\n",
    "        x = (d - 2)**2 / (6 + result)\n",
    "        d -= 2\n",
    "        result = x\n",
    "\n",
    "    pi = 3 + (1**2 / (6 + result))\n",
    "    return pi\n",
    "\n",
    "recursive_method(15)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 77,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "## Test\n",
    "recursive_method(7) == 3 + (1**2 / (6 + (3**2 / (6 + (5**2 / (6 + (7**2 / (6 + 9**2))))))))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {},
   "outputs": [],
   "source": [
    "pis = []\n",
    "iteration = range(0,20000,2)\n",
    "for x in (range(0,20000,2)):\n",
    "    pis.append(recursive_method(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# See how it converges...\n",
    "\n",
    "import matplotlib\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "\n",
    "# Going deeper into the levels\n",
    "y = pis[5000:6000]\n",
    "x = iteration[5000:6000]\n",
    "\n",
    "fig, ax = plt.subplots()\n",
    "ax.plot(x,y)\n",
    "\n",
    "ax.set(xlabel='iteration', ylabel='pi',\n",
    "       title='Ramanujan pi approximation')\n",
    "ax.grid()\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 102,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-0.00586080586080584\n",
      "-0.023736263736263474\n",
      "0.00932249322493206\n",
      "-0.00457998618734301\n",
      "0.002581410874563872\n",
      "-0.001596390768696665\n",
      "0.001055484751117497\n",
      "-0.0007339743911121133\n",
      "0.0005309391074743708\n",
      "-0.0003964379971712617\n",
      "0.00030381578154203837\n",
      "-0.00023795514188984512\n",
      "0.0001898442312375792\n",
      "-0.0001538821552613534\n",
      "0.0001264625117838314\n",
      "-0.00010519136490083625\n",
      "8.843699202243371e-05\n",
      "-7.506106089438802e-05\n",
      "6.425294838541262e-05\n",
      "-5.5424724838548656e-05\n",
      "4.81429109266962e-05\n",
      "-4.208313014819609e-05\n",
      "3.699936389045533e-05\n",
      "-3.2702724168220954e-05\n",
      "2.9046553088551974e-05\n",
      "-2.5915804000575804e-05\n",
      "2.3219368264104645e-05\n",
      "-2.0884459245529285e-05\n",
      "1.8852453197659713e-05\n",
      "-1.7075775213371003e-05\n",
      "1.551554383327769e-05\n",
      "-1.4139772532395511e-05\n",
      "1.2921984224867344e-05\n",
      "-1.184013505950432e-05\n",
      "1.0875771943474888e-05\n",
      "-1.0013368202166362e-05\n",
      "9.2397960882451e-06\n",
      "-8.543905225533166e-06\n",
      "7.916183640599428e-06\n",
      "-7.348483613611734e-06\n",
      "6.833798730454532e-06\n",
      "-6.366081624520348e-06\n",
      "5.940094239154092e-06\n",
      "-5.551284225191466e-06\n",
      "5.195682450498396e-06\n",
      "-4.869817653574415e-06\n",
      "4.5706450766402895e-06\n",
      "-4.295486554894978e-06\n",
      "4.041980036006976e-06\n",
      "-3.8080368889303884e-06\n",
      "3.591805677771731e-06\n",
      "-3.391641319794303e-06\n",
      "3.206078742490348e-06\n",
      "-3.0338103136351435e-06\n",
      "2.873666451908008e-06\n",
      "-2.724598916259424e-06\n",
      "2.585666368570827e-06\n",
      "-2.4560218672142753e-06\n",
      "2.3349020055185576e-06\n",
      "-2.2216174548894685e-06\n",
      "2.115544710967754e-06\n",
      "-2.0161188754030945e-06\n",
      "1.9228273293592224e-06\n",
      "-1.8352041748492809e-06\n",
      "1.7528253390963755e-06\n",
      "-1.6753042593187217e-06\n",
      "1.6022880697796893e-06\n",
      "-1.5334542231570936e-06\n",
      "1.4685074920528507e-06\n",
      "-1.407177309786789e-06\n",
      "1.3492153994043576e-06\n",
      "-1.2943936589238092e-06\n",
      "1.2425022766215932e-06\n",
      "-1.1933480408288233e-06\n",
      "1.1467528273634287e-06\n",
      "-1.1025522441698854e-06\n",
      "1.060594410962068e-06\n",
      "-1.0207388623229008e-06\n",
      "9.828555596058663e-07\n",
      "-9.468239978716042e-07\n",
      "9.125324011982627e-07\n",
      "-8.798769939311057e-07\n",
      "8.48761339433679e-07\n",
      "-8.190957432319124e-07\n",
      "7.907967085607481e-07\n",
      "-7.637864412046724e-07\n",
      "7.379924027439699e-07\n",
      "-7.133468993281156e-07\n",
      "6.897867077526598e-07\n",
      "-6.672527370632508e-07\n",
      "6.45689714584563e-07\n",
      "-6.250459017032028e-07\n",
      "6.052728340755209e-07\n",
      "-5.863250813753496e-07\n",
      "5.681600279139332e-07\n",
      "-5.507376719116053e-07\n",
      "5.340204398684989e-07\n",
      "-5.179730164783791e-07\n",
      "5.025621878651521e-07\n"
     ]
    }
   ],
   "source": [
    "\"\"\"\n",
    "The Cauchy convergence criterion:\n",
    "\n",
    "a sequence whose elements become arbitrarily close to each other as the sequence progresses.\n",
    "More precisely, given any small positive distance, all but a finite number of elements of the \n",
    "sequence are less than that given distance from each other.\n",
    "\n",
    "A s\n",
    "\n",
    "Informally: if the difference between successive terms is getting smaller, the sequence \n",
    "is Cauchy convergent, which we can see from the below.\n",
    "\"\"\"\n",
    "\n",
    "for x in range(3,200,2):\n",
    "    print(recursive_method(x)-recursive_method(x-2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "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": "3.7.0"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}