{ "cells": [ { "cell_type": "markdown", "id": "21ad97c3-de27-4eef-a332-ba0740ab8a10", "metadata": {}, "source": [ "## Lecture 13, 18 September 2025" ] }, { "cell_type": "markdown", "id": "7ca08ef5-1e38-4c8a-87c3-b77962562059", "metadata": {}, "source": [ "- Recursive definition of factorial\n", "- Add diagnostic `print()` to trace recursive calls" ] }, { "cell_type": "code", "execution_count": 1, "id": "4df745e9-4ed5-42ef-99d0-8471516c0f43", "metadata": {}, "outputs": [], "source": [ "def fact(n):\n", " print(\"evaluating factorial of\",n)\n", " if n == 0:\n", " return(1)\n", " else:\n", " return(n*fact(n-1))" ] }, { "cell_type": "code", "execution_count": 2, "id": "096a6b66-5367-4c88-92c7-210fc8e13cdc", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "evaluating factorial of 8\n", "evaluating factorial of 7\n", "evaluating factorial of 6\n", "evaluating factorial of 5\n", "evaluating factorial of 4\n", "evaluating factorial of 3\n", "evaluating factorial of 2\n", "evaluating factorial of 1\n", "evaluating factorial of 0\n" ] }, { "data": { "text/plain": [ "40320" ] }, "execution_count": 2, "metadata": {}, "output_type": "execute_result" } ], "source": [ "fact(8)" ] }, { "cell_type": "markdown", "id": "4a179f2e-56f8-4137-b267-7ed9847a66a9", "metadata": {}, "source": [ "- A negative argument creates an unending sequence of recursive calls\n", "- Python has a built in `recursion limit` to abort such cases\n", " - This limit may deny some legitimate computation -- for instance `factorial(3000)` will also abort\n", " - We will see later that we can increase this limit" ] }, { "cell_type": "code", "execution_count": 3, "id": "29b076b8-7518-4f3c-b678-ff914398904d", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "evaluating factorial of -1\n", "evaluating factorial of -2\n", "evaluating factorial of -3\n", "evaluating factorial of -4\n", "evaluating factorial of -5\n", "evaluating factorial of -6\n", "evaluating factorial of -7\n", "evaluating factorial of -8\n", "evaluating factorial of -9\n", "evaluating factorial of -10\n", "evaluating factorial of -11\n", "evaluating factorial of -12\n", "evaluating factorial of -13\n", "evaluating factorial of -14\n", "evaluating factorial of -15\n", "evaluating factorial of -16\n", "evaluating factorial of -17\n", "evaluating factorial of -18\n", "evaluating factorial of -19\n", "evaluating factorial of -20\n", "evaluating factorial of -21\n", "evaluating factorial of -22\n", "evaluating factorial of -23\n", "evaluating factorial of -24\n", "evaluating factorial of -25\n", "evaluating factorial of -26\n", "evaluating factorial of -27\n", "evaluating factorial of -28\n", "evaluating factorial of -29\n", "evaluating factorial of -30\n", "evaluating factorial of -31\n", "evaluating factorial of -32\n", "evaluating factorial of -33\n", "evaluating factorial of -34\n", "evaluating factorial of -35\n", "evaluating factorial of -36\n", 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-2932\n", "evaluating factorial of -2933\n", "evaluating factorial of -2934\n", "evaluating factorial of -2935\n", "evaluating factorial of -2936\n", "evaluating factorial of -2937\n", "evaluating factorial of -2938\n", "evaluating factorial of -2939\n", "evaluating factorial of -2940\n", "evaluating factorial of -2941\n", "evaluating factorial of -2942\n", "evaluating factorial of -2943\n", "evaluating factorial of -2944\n", "evaluating factorial of -2945\n", "evaluating factorial of -2946\n", "evaluating factorial of -2947\n", "evaluating factorial of -2948\n", "evaluating factorial of -2949\n", "evaluating factorial of -2950\n", "evaluating factorial of -2951\n", "evaluating factorial of -2952\n", "evaluating factorial of -2953\n", "evaluating factorial of -2954\n", "evaluating factorial of -2955\n", "evaluating factorial of -2956\n", "evaluating factorial of -2957\n", "evaluating factorial of -2958\n", "evaluating factorial of -2959\n", "evaluating factorial of -2960\n", "evaluating factorial of -2961\n", "evaluating factorial of -2962\n", "evaluating factorial of -2963\n", "evaluating factorial of -2964\n", "evaluating factorial of -2965\n", "evaluating factorial of -2966\n", "evaluating factorial of -2967\n", "evaluating factorial of -2968\n", "evaluating factorial of -2969\n", "evaluating factorial of -2970\n", "evaluating factorial of -2971\n", "evaluating factorial of -2972\n", "evaluating factorial of -2973\n", "evaluating factorial of" ] }, { "ename": "RecursionError", "evalue": "maximum recursion depth exceeded", "output_type": "error", "traceback": [ "\u001b[31m---------------------------------------------------------------------------\u001b[39m", "\u001b[31mRecursionError\u001b[39m Traceback (most recent call last)", "\u001b[36mCell\u001b[39m\u001b[36m \u001b[39m\u001b[32mIn[3]\u001b[39m\u001b[32m, line 1\u001b[39m\n\u001b[32m----> \u001b[39m\u001b[32m1\u001b[39m \u001b[43mfact\u001b[49m\u001b[43m(\u001b[49m\u001b[43m-\u001b[49m\u001b[32;43m1\u001b[39;49m\u001b[43m)\u001b[49m\n", "\u001b[36mCell\u001b[39m\u001b[36m \u001b[39m\u001b[32mIn[1]\u001b[39m\u001b[32m, line 6\u001b[39m, in \u001b[36mfact\u001b[39m\u001b[34m(n)\u001b[39m\n\u001b[32m 4\u001b[39m \u001b[38;5;28;01mreturn\u001b[39;00m(\u001b[32m1\u001b[39m)\n\u001b[32m 5\u001b[39m \u001b[38;5;28;01melse\u001b[39;00m:\n\u001b[32m----> \u001b[39m\u001b[32m6\u001b[39m \u001b[38;5;28;01mreturn\u001b[39;00m(n*\u001b[43mfact\u001b[49m\u001b[43m(\u001b[49m\u001b[43mn\u001b[49m\u001b[43m-\u001b[49m\u001b[32;43m1\u001b[39;49m\u001b[43m)\u001b[49m)\n", "\u001b[36mCell\u001b[39m\u001b[36m \u001b[39m\u001b[32mIn[1]\u001b[39m\u001b[32m, line 6\u001b[39m, in \u001b[36mfact\u001b[39m\u001b[34m(n)\u001b[39m\n\u001b[32m 4\u001b[39m \u001b[38;5;28;01mreturn\u001b[39;00m(\u001b[32m1\u001b[39m)\n\u001b[32m 5\u001b[39m \u001b[38;5;28;01melse\u001b[39;00m:\n\u001b[32m----> \u001b[39m\u001b[32m6\u001b[39m \u001b[38;5;28;01mreturn\u001b[39;00m(n*\u001b[43mfact\u001b[49m\u001b[43m(\u001b[49m\u001b[43mn\u001b[49m\u001b[43m-\u001b[49m\u001b[32;43m1\u001b[39;49m\u001b[43m)\u001b[49m)\n", " \u001b[31m[... skipping similar frames: fact at line 6 (2970 times)]\u001b[39m\n", "\u001b[36mCell\u001b[39m\u001b[36m \u001b[39m\u001b[32mIn[1]\u001b[39m\u001b[32m, line 6\u001b[39m, in \u001b[36mfact\u001b[39m\u001b[34m(n)\u001b[39m\n\u001b[32m 4\u001b[39m \u001b[38;5;28;01mreturn\u001b[39;00m(\u001b[32m1\u001b[39m)\n\u001b[32m 5\u001b[39m \u001b[38;5;28;01melse\u001b[39;00m:\n\u001b[32m----> \u001b[39m\u001b[32m6\u001b[39m \u001b[38;5;28;01mreturn\u001b[39;00m(n*\u001b[43mfact\u001b[49m\u001b[43m(\u001b[49m\u001b[43mn\u001b[49m\u001b[43m-\u001b[49m\u001b[32;43m1\u001b[39;49m\u001b[43m)\u001b[49m)\n", "\u001b[36mCell\u001b[39m\u001b[36m \u001b[39m\u001b[32mIn[1]\u001b[39m\u001b[32m, line 2\u001b[39m, in \u001b[36mfact\u001b[39m\u001b[34m(n)\u001b[39m\n\u001b[32m 1\u001b[39m \u001b[38;5;28;01mdef\u001b[39;00m\u001b[38;5;250m \u001b[39m\u001b[34mfact\u001b[39m(n):\n\u001b[32m----> \u001b[39m\u001b[32m2\u001b[39m \u001b[38;5;28;43mprint\u001b[39;49m\u001b[43m(\u001b[49m\u001b[33;43m\"\u001b[39;49m\u001b[33;43mevaluating factorial of\u001b[39;49m\u001b[33;43m\"\u001b[39;49m\u001b[43m,\u001b[49m\u001b[43mn\u001b[49m\u001b[43m)\u001b[49m\n\u001b[32m 3\u001b[39m \u001b[38;5;28;01mif\u001b[39;00m n == \u001b[32m0\u001b[39m:\n\u001b[32m 4\u001b[39m \u001b[38;5;28;01mreturn\u001b[39;00m(\u001b[32m1\u001b[39m)\n", "\u001b[36mFile \u001b[39m\u001b[32m~/python-venv/lib/python3.13/site-packages/IPython/core/interactiveshell.py:3056\u001b[39m, in \u001b[36mInteractiveShell._tee..write\u001b[39m\u001b[34m(data, *args, **kwargs)\u001b[39m\n\u001b[32m 3054\u001b[39m \u001b[38;5;28;01mif\u001b[39;00m \u001b[38;5;129;01mnot\u001b[39;00m data:\n\u001b[32m 3055\u001b[39m \u001b[38;5;28;01mreturn\u001b[39;00m result\n\u001b[32m-> \u001b[39m\u001b[32m3056\u001b[39m execution_count = \u001b[38;5;28;43mself\u001b[39;49m\u001b[43m.\u001b[49m\u001b[43mexecution_count\u001b[49m\n\u001b[32m 3057\u001b[39m output_stream = \u001b[38;5;28;01mNone\u001b[39;00m\n\u001b[32m 3058\u001b[39m outputs_by_counter = \u001b[38;5;28mself\u001b[39m.history_manager.outputs\n", "\u001b[36mFile \u001b[39m\u001b[32m~/python-venv/lib/python3.13/site-packages/traitlets/traitlets.py:687\u001b[39m, in \u001b[36mTraitType.__get__\u001b[39m\u001b[34m(self, obj, cls)\u001b[39m\n\u001b[32m 685\u001b[39m \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[38;5;28mself\u001b[39m\n\u001b[32m 686\u001b[39m \u001b[38;5;28;01melse\u001b[39;00m:\n\u001b[32m--> \u001b[39m\u001b[32m687\u001b[39m \u001b[38;5;28;01mreturn\u001b[39;00m t.cast(G, \u001b[38;5;28;43mself\u001b[39;49m\u001b[43m.\u001b[49m\u001b[43mget\u001b[49m\u001b[43m(\u001b[49m\u001b[43mobj\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;28;43mcls\u001b[39;49m\u001b[43m)\u001b[49m)\n", "\u001b[36mFile \u001b[39m\u001b[32m~/python-venv/lib/python3.13/site-packages/traitlets/traitlets.py:666\u001b[39m, in \u001b[36mTraitType.get\u001b[39m\u001b[34m(self, obj, cls)\u001b[39m\n\u001b[32m 664\u001b[39m \u001b[38;5;28;01mraise\u001b[39;00m TraitError(\u001b[33m\"\u001b[39m\u001b[33mUnexpected error in TraitType: default value not set properly\u001b[39m\u001b[33m\"\u001b[39m) \u001b[38;5;28;01mfrom\u001b[39;00m\u001b[38;5;250m \u001b[39m\u001b[34;01me\u001b[39;00m\n\u001b[32m 665\u001b[39m \u001b[38;5;28;01melse\u001b[39;00m:\n\u001b[32m--> \u001b[39m\u001b[32m666\u001b[39m \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[43mt\u001b[49m\u001b[43m.\u001b[49m\u001b[43mcast\u001b[49m\u001b[43m(\u001b[49m\u001b[43mG\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mvalue\u001b[49m\u001b[43m)\u001b[49m\n", "\u001b[31mRecursionError\u001b[39m: maximum recursion depth exceeded" ] } ], "source": [ "fact(-1)" ] }, { "cell_type": "markdown", "id": "2a0740e9-a98f-4a3d-ac9b-8b2360831189", "metadata": {}, "source": [ "- We can also induct on the structure of a datastructure\n", "- We can break up a list into the first element (the *head*) and the rest (the *tail*) as a smaller list\n", "- Here is a recursive definition of `len()`" ] }, { "cell_type": "code", "execution_count": 4, "id": "36e28393-683b-41dc-8867-df53a92e4acd", "metadata": {}, "outputs": [], "source": [ "def mylen(l):\n", " print(\"Call with\",l)\n", " if l == []:\n", " return(0)\n", " else:\n", " return(1+mylen(l[1:]))" ] }, { "cell_type": "code", "execution_count": 5, "id": "12f90b4f-3935-45b2-963b-e7e934f4ae30", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Call with [7, 18, 2, -1, 3]\n", "Call with [18, 2, -1, 3]\n", "Call with [2, -1, 3]\n", "Call with [-1, 3]\n", "Call with [3]\n", "Call with []\n" ] }, { "data": { "text/plain": [ "5" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "mylen([7,18,2,-1,3])" ] }, { "cell_type": "markdown", "id": "202a6a09-c232-4a16-88ed-eecc626e7d9d", "metadata": {}, "source": [ "- A similar definition for `sum()`" ] }, { "cell_type": "code", "execution_count": 6, "id": "6c5300b6-c923-4a1d-9338-c63232fee712", "metadata": {}, "outputs": [], "source": [ "def mysum(l):\n", " print(\"Call with\",l)\n", " if l == []:\n", " return(0)\n", " else:\n", " return(l[0]+mysum(l[1:]))" ] }, { "cell_type": "code", "execution_count": 7, "id": "08dc5821-a797-4d6d-81d0-9162abb0a3b5", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Call with [7, 18, 2, -1, 3]\n", "Call with [18, 2, -1, 3]\n", "Call with [2, -1, 3]\n", "Call with [-1, 3]\n", "Call with [3]\n", "Call with []\n" ] }, { "data": { "text/plain": [ "29" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "mysum([7,18,2,-1,3])" ] }, { "cell_type": "markdown", "id": "4d418cce-1585-438f-8dbe-61b870003d6b", "metadata": {}, "source": [ "- Insert a value `v` into a list `l` sorted in ascending order" ] }, { "cell_type": "code", "execution_count": 8, "id": "0105623f-5068-463d-9bc5-22d2db65a682", "metadata": {}, "outputs": [], "source": [ "def sortedinsert(v,l):\n", " if l == []:\n", " return([v])\n", " if v <= l[0]:\n", " return([v]+l)\n", " else:\n", " return(l[:1] + sortedinsert(v,l[1:]))" ] }, { "cell_type": "code", "execution_count": 9, "id": "5dfc31eb-3baa-4941-83f2-5882df1cf98c", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[1, 3, 5, 7, 9, 11, 13, 14, 15, 17, 19]" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sortedinsert(14,list(range(1,20,2)))" ] }, { "cell_type": "markdown", "id": "0d1d0fdd-8e1c-4b2e-903f-cca01ea8b1b7", "metadata": {}, "source": [ "- The built in recursion limit causes this function to fail on moderate size lists" ] }, { "cell_type": "code", "execution_count": 10, "id": "bdb96410-7c62-4f20-ad8d-caba91998759", "metadata": {}, "outputs": [ { "ename": "RecursionError", "evalue": "maximum recursion depth exceeded", "output_type": "error", "traceback": [ "\u001b[31m---------------------------------------------------------------------------\u001b[39m", "\u001b[31mRecursionError\u001b[39m Traceback (most recent call last)", "\u001b[36mCell\u001b[39m\u001b[36m \u001b[39m\u001b[32mIn[10]\u001b[39m\u001b[32m, line 1\u001b[39m\n\u001b[32m----> \u001b[39m\u001b[32m1\u001b[39m \u001b[43msortedinsert\u001b[49m\u001b[43m(\u001b[49m\u001b[32;43m3000\u001b[39;49m\u001b[43m,\u001b[49m\u001b[38;5;28;43mlist\u001b[39;49m\u001b[43m(\u001b[49m\u001b[38;5;28;43mrange\u001b[39;49m\u001b[43m(\u001b[49m\u001b[32;43m3000\u001b[39;49m\u001b[43m)\u001b[49m\u001b[43m)\u001b[49m\u001b[43m)\u001b[49m\n", "\u001b[36mCell\u001b[39m\u001b[36m \u001b[39m\u001b[32mIn[8]\u001b[39m\u001b[32m, line 7\u001b[39m, in \u001b[36msortedinsert\u001b[39m\u001b[34m(v, l)\u001b[39m\n\u001b[32m 5\u001b[39m \u001b[38;5;28;01mreturn\u001b[39;00m([v]+l)\n\u001b[32m 6\u001b[39m \u001b[38;5;28;01melse\u001b[39;00m:\n\u001b[32m----> \u001b[39m\u001b[32m7\u001b[39m \u001b[38;5;28;01mreturn\u001b[39;00m(l[:\u001b[32m1\u001b[39m] + \u001b[43msortedinsert\u001b[49m\u001b[43m(\u001b[49m\u001b[43mv\u001b[49m\u001b[43m,\u001b[49m\u001b[43ml\u001b[49m\u001b[43m[\u001b[49m\u001b[32;43m1\u001b[39;49m\u001b[43m:\u001b[49m\u001b[43m]\u001b[49m\u001b[43m)\u001b[49m)\n", "\u001b[36mCell\u001b[39m\u001b[36m \u001b[39m\u001b[32mIn[8]\u001b[39m\u001b[32m, line 7\u001b[39m, in \u001b[36msortedinsert\u001b[39m\u001b[34m(v, l)\u001b[39m\n\u001b[32m 5\u001b[39m \u001b[38;5;28;01mreturn\u001b[39;00m([v]+l)\n\u001b[32m 6\u001b[39m \u001b[38;5;28;01melse\u001b[39;00m:\n\u001b[32m----> \u001b[39m\u001b[32m7\u001b[39m \u001b[38;5;28;01mreturn\u001b[39;00m(l[:\u001b[32m1\u001b[39m] + \u001b[43msortedinsert\u001b[49m\u001b[43m(\u001b[49m\u001b[43mv\u001b[49m\u001b[43m,\u001b[49m\u001b[43ml\u001b[49m\u001b[43m[\u001b[49m\u001b[32;43m1\u001b[39;49m\u001b[43m:\u001b[49m\u001b[43m]\u001b[49m\u001b[43m)\u001b[49m)\n", " \u001b[31m[... skipping similar frames: sortedinsert at line 7 (2974 times)]\u001b[39m\n", "\u001b[36mCell\u001b[39m\u001b[36m \u001b[39m\u001b[32mIn[8]\u001b[39m\u001b[32m, line 7\u001b[39m, in \u001b[36msortedinsert\u001b[39m\u001b[34m(v, l)\u001b[39m\n\u001b[32m 5\u001b[39m \u001b[38;5;28;01mreturn\u001b[39;00m([v]+l)\n\u001b[32m 6\u001b[39m \u001b[38;5;28;01melse\u001b[39;00m:\n\u001b[32m----> \u001b[39m\u001b[32m7\u001b[39m \u001b[38;5;28;01mreturn\u001b[39;00m(l[:\u001b[32m1\u001b[39m] + \u001b[43msortedinsert\u001b[49m\u001b[43m(\u001b[49m\u001b[43mv\u001b[49m\u001b[43m,\u001b[49m\u001b[43ml\u001b[49m\u001b[43m[\u001b[49m\u001b[32;43m1\u001b[39;49m\u001b[43m:\u001b[49m\u001b[43m]\u001b[49m\u001b[43m)\u001b[49m)\n", "\u001b[31mRecursionError\u001b[39m: maximum recursion depth exceeded" ] } ], "source": [ "sortedinsert(3000,list(range(3000)))" ] }, { "cell_type": "markdown", "id": "da6a268e-4a26-48f0-94bc-2e268241b06d", "metadata": {}, "source": [ "- Can manually increase the recursion limit\n", "- Maximum is $2^{31}-1$, the largest `int` value that can be stored in 32 bits" ] }, { "cell_type": "code", "execution_count": 11, "id": "8f492d02-9ac7-469d-b2ee-5153a30f2bca", "metadata": {}, "outputs": [], "source": [ "import sys\n", "sys.setrecursionlimit(2**31-1)" ] }, { "cell_type": "code", "execution_count": 12, "id": "03701de0-80b5-4ab4-8ec9-e5cb9ceee962", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "3001" ] }, "execution_count": 12, "metadata": {}, "output_type": "execute_result" } ], "source": [ "len(sortedinsert(3000,list(range(3000))))" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3 (ipykernel)", "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": "3.13.5" } }, "nbformat": 4, "nbformat_minor": 5 }