438 lines
97 KiB
Plaintext
438 lines
97 KiB
Plaintext
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{
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"nbformat": 4,
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"nbformat_minor": 0,
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"metadata": {
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"colab": {
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"provenance": []
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},
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"kernelspec": {
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"name": "python3",
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"display_name": "Python 3"
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},
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"language_info": {
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"name": "python"
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}
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},
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"cells": [
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{
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"cell_type": "code",
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"source": [
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"import numpy as np\n",
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"from numba import njit\n",
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"import time"
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],
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"metadata": {
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"id": "Qaq--qm2_Qfm"
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},
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"execution_count": 19,
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"outputs": []
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},
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{
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"cell_type": "markdown",
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"source": [
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"## The Hofstadter-Conway sequence $a(n)$: \n",
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"\n",
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"$a(1) = 1, a(2) = 1, ... , a(n) = a(a(n-1)) + a(n - a(n - 1))$"
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],
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"metadata": {
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"id": "9A0VfT5yePm5"
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}
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},
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{
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"cell_type": "code",
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"source": [
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"n = int(1e3)\n",
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"a = np.ones(n, dtype = int)"
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],
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"metadata": {
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"id": "l_UpCyuqQBwt"
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},
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"execution_count": 20,
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"outputs": []
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},
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{
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"cell_type": "markdown",
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"source": [
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"## **First implemination:** pre-allocating `numpy` array and filling it using `for` loop."
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],
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"metadata": {
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"id": "GkeqAQzceyOD"
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}
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},
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{
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"cell_type": "code",
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"source": [
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"def HC_func1(array, n):\n",
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" if n < 1:\n",
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" raise Exception(\"n must be positive\")\n",
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" elif n == 1:\n",
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" array[0] = 1\n",
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" else:\n",
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" array[0], array[1] = 1, 1\n",
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"\n",
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" for idx in range(2, n):\n",
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" array[idx] = array[array[idx - 1] - 1] + array[idx - array[idx - 1]]"
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],
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"metadata": {
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"id": "i1B9t7PwPrB-"
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},
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"execution_count": 21,
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"outputs": []
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},
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{
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"cell_type": "code",
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"source": [
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"%time HC_func1(a, n)\n",
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"%time HC_func1(a, n)\n",
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"%time HC_func1(a, n)"
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],
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"metadata": {
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"id": "MlymyYQDPscK",
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"colab": {
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"base_uri": "https://localhost:8080/"
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},
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"outputId": "c7f93f12-16f3-4b18-af01-b7c85ebcc5c9"
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},
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"execution_count": 22,
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"outputs": [
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{
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"output_type": "stream",
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"name": "stdout",
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"text": [
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"CPU times: user 1.73 ms, sys: 0 ns, total: 1.73 ms\n",
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"Wall time: 1.74 ms\n",
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"CPU times: user 1.62 ms, sys: 104 µs, total: 1.73 ms\n",
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"Wall time: 1.73 ms\n",
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"CPU times: user 1.72 ms, sys: 0 ns, total: 1.72 ms\n",
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"Wall time: 1.73 ms\n"
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]
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}
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]
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},
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{
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"cell_type": "markdown",
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"source": [
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"## **Second implemination:** cumulatively appending `python` list and converting it to `numpy` array."
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],
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"metadata": {
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"id": "mtt236qrfDnI"
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}
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},
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{
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"cell_type": "code",
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"source": [
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"def HC_func2(n):\n",
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" if n < 1:\n",
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" raise Exception(\"n must be positive\")\n",
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" elif n == 1:\n",
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" result = [1]\n",
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" else:\n",
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" result = [1, 1]\n",
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" for idx in range(2, n):\n",
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" elem = result[result[idx - 1] - 1] + result[idx - result[idx - 1]]\n",
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" result.append(elem)\n",
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" return np.array(result)"
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],
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"metadata": {
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"id": "LY9YUjFFNIs2"
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},
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"execution_count": 23,
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"outputs": []
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},
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{
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"cell_type": "code",
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"source": [
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"%time HC_func2(n)\n",
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"%time HC_func2(n)\n",
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"%time HC_func2(n)\n",
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"\"\""
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],
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"metadata": {
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"id": "9X9tMCeYVYhm",
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"colab": {
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"base_uri": "https://localhost:8080/",
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"height": 140
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},
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"outputId": "957c3a66-7e73-4997-8f7c-aead3217beab"
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},
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"execution_count": 24,
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"outputs": [
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{
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"output_type": "stream",
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"name": "stdout",
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"text": [
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"CPU times: user 779 µs, sys: 0 ns, total: 779 µs\n",
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"Wall time: 785 µs\n",
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"CPU times: user 720 µs, sys: 13 µs, total: 733 µs\n",
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"Wall time: 737 µs\n",
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"CPU times: user 736 µs, sys: 0 ns, total: 736 µs\n",
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"Wall time: 740 µs\n"
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]
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},
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{
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"output_type": "execute_result",
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"data": {
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"text/plain": [
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"''"
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],
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"application/vnd.google.colaboratory.intrinsic+json": {
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"type": "string"
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}
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},
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"metadata": {},
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"execution_count": 24
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}
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]
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},
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{
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"cell_type": "markdown",
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"source": [
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"## **Second implemination:** compiled (`jit`) version."
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],
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"metadata": {
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"id": "tKz26-9RfUex"
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}
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},
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{
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"cell_type": "code",
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"execution_count": 25,
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"metadata": {
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"id": "7h5Woys4-Z3S"
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},
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"outputs": [],
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"source": [
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"@njit\n",
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"def HC_func3(array, n):\n",
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" if n < 1:\n",
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" raise Exception(\"n must be positive\")\n",
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" elif n == 1:\n",
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" array[0] = 1\n",
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" else:\n",
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" array[0], array[1] = 1, 1\n",
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"\n",
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" for idx in range(2, n):\n",
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" array[idx] = array[array[idx - 1] - 1] + array[idx - array[idx - 1]]"
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]
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},
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{
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"cell_type": "code",
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"source": [
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"%time HC_func3(a, n)\n",
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"%time HC_func3(a, n)\n",
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"%time HC_func3(a, n)"
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],
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"metadata": {
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"id": "6rhqO950CzUg",
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"colab": {
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"base_uri": "https://localhost:8080/"
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},
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"outputId": "06e1ac2d-11f2-49a2-e3a9-7b8b165e42b0"
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},
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"execution_count": 26,
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"outputs": [
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{
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"output_type": "stream",
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"name": "stdout",
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"text": [
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"CPU times: user 235 ms, sys: 19.7 ms, total: 255 ms\n",
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"Wall time: 298 ms\n",
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"CPU times: user 23 µs, sys: 0 ns, total: 23 µs\n",
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"Wall time: 26.7 µs\n",
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"CPU times: user 19 µs, sys: 0 ns, total: 19 µs\n",
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"Wall time: 22.6 µs\n"
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]
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}
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]
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},
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{
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"cell_type": "markdown",
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"source": [
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"## Let's compare execution time"
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],
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"metadata": {
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"id": "6WubSLEUfcLW"
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}
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},
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{
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"cell_type": "code",
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"source": [
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"def comp_time_res():\n",
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" res1, res2, res3 = np.array([]), np.array([]), np.array([])\n",
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" for k in list(map(int, [1e1, 1e2, 1e3, 1e4, 1e5])):\n",
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" data = np.ones(k, dtype = int)\n",
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"\n",
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" start_time = time.time()\n",
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" HC_func1(data, k)\n",
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" end_time = time.time()\n",
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" res1 = np.append(res1, end_time - start_time)\n",
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"\n",
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" start_time = time.time()\n",
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" HC_func2(k)\n",
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" end_time = time.time()\n",
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" res2 = np.append(res2, end_time - start_time)\n",
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"\n",
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" start_time = time.time()\n",
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" HC_func3(data, k)\n",
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" end_time = time.time()\n",
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" res3 = np.append(res3, end_time - start_time)\n",
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" return (res1, res2, res3)"
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],
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"metadata": {
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"id": "LELxb0HRoWvh"
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},
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"execution_count": 27,
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"outputs": []
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},
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{
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"cell_type": "code",
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"source": [
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"import matplotlib.pyplot as plt\n",
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"first_func, second_func, third_func = comp_time_res()\n",
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"x = np.array([1e1, 1e2, 1e3, 1e4, 1e5], dtype=int)\n",
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"\n",
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"plt.plot(x, first_func)\n",
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"plt.plot(x, second_func)\n",
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"plt.plot(x, third_func)\n",
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"\n",
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"plt.xlabel(\"n\", fontsize=14)\n",
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"plt.ylabel(\"comp time\", fontsize=14)\n",
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"plt.legend([\"HC_func_1\", \"HC_func_2\", \"HC_func_3\"]);"
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],
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"metadata": {
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"colab": {
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||
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"base_uri": "https://localhost:8080/",
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"height": 455
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},
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"id": "4MPOWTsXtZhU",
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||
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"outputId": "7299eddf-166f-4e26-9058-716eaefe6168"
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},
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"execution_count": 28,
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"outputs": [
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{
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"output_type": "display_data",
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"data": {
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"text/plain": [
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"<Figure size 640x480 with 1 Axes>"
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],
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"image/png": "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
|
||
|
|
},
|
||
|
|
"metadata": {}
|
||
|
|
}
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"source": [
|
||
|
|
"## The option with a `jit` compiler turned out to be the fastest. Let's use it to calculate $a(10^8)$:"
|
||
|
|
],
|
||
|
|
"metadata": {
|
||
|
|
"id": "KDEaQcaifhfn"
|
||
|
|
}
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"source": [
|
||
|
|
"n = int(1e8)\n",
|
||
|
|
"a = np.ones(n, dtype = int)\n",
|
||
|
|
"\n",
|
||
|
|
"%time HC_func3(a, n)\n",
|
||
|
|
"%time HC_func3(a, n)\n",
|
||
|
|
"%time HC_func3(a, n)"
|
||
|
|
],
|
||
|
|
"metadata": {
|
||
|
|
"colab": {
|
||
|
|
"base_uri": "https://localhost:8080/"
|
||
|
|
},
|
||
|
|
"id": "dhgtJB8X0fkO",
|
||
|
|
"outputId": "e01fe169-8137-4a64-88f9-e08d57600fcd"
|
||
|
|
},
|
||
|
|
"execution_count": 29,
|
||
|
|
"outputs": [
|
||
|
|
{
|
||
|
|
"output_type": "stream",
|
||
|
|
"name": "stdout",
|
||
|
|
"text": [
|
||
|
|
"CPU times: user 586 ms, sys: 4.05 ms, total: 590 ms\n",
|
||
|
|
"Wall time: 588 ms\n",
|
||
|
|
"CPU times: user 586 ms, sys: 4.75 ms, total: 590 ms\n",
|
||
|
|
"Wall time: 586 ms\n",
|
||
|
|
"CPU times: user 587 ms, sys: 2.11 ms, total: 589 ms\n",
|
||
|
|
"Wall time: 584 ms\n"
|
||
|
|
]
|
||
|
|
}
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"source": [
|
||
|
|
"x = np.arange(1, n + 1, 1, dtype=int)\n",
|
||
|
|
"plt.plot(x, a);\n",
|
||
|
|
"plt.title(\"$a(10^8)$\", fontsize=14);"
|
||
|
|
],
|
||
|
|
"metadata": {
|
||
|
|
"colab": {
|
||
|
|
"base_uri": "https://localhost:8080/",
|
||
|
|
"height": 476
|
||
|
|
},
|
||
|
|
"id": "fggJi6PlgQOv",
|
||
|
|
"outputId": "7a4d3f18-b3d8-4002-d4b5-38cae766af9f"
|
||
|
|
},
|
||
|
|
"execution_count": 30,
|
||
|
|
"outputs": [
|
||
|
|
{
|
||
|
|
"output_type": "display_data",
|
||
|
|
"data": {
|
||
|
|
"text/plain": [
|
||
|
|
"<Figure size 640x480 with 1 Axes>"
|
||
|
|
],
|
||
|
|
"image/png": "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
|
||
|
|
},
|
||
|
|
"metadata": {}
|
||
|
|
}
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"source": [
|
||
|
|
"## Finally we display the graph $f(n) = a(n)/n$ for $n = 1, 2, ..., 2000$"
|
||
|
|
],
|
||
|
|
"metadata": {
|
||
|
|
"id": "pH6dsdzmhBmv"
|
||
|
|
}
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"source": [
|
||
|
|
"n = int(2000)\n",
|
||
|
|
"a = np.ones(n, dtype = int)\n",
|
||
|
|
"HC_func3(a, n)\n",
|
||
|
|
"\n",
|
||
|
|
"x = np.arange(1, n + 1, 1, dtype=int)\n",
|
||
|
|
"y = a * np.array([1 / (i + 1) for i in range(n)])\n",
|
||
|
|
"plt.plot(x, y);\n",
|
||
|
|
"plt.title(\"a(n)/n\", fontsize=14);"
|
||
|
|
],
|
||
|
|
"metadata": {
|
||
|
|
"colab": {
|
||
|
|
"base_uri": "https://localhost:8080/",
|
||
|
|
"height": 453
|
||
|
|
},
|
||
|
|
"id": "tm9UvLrD9wfW",
|
||
|
|
"outputId": "e0b72afe-3cd1-444b-8849-744c827a7f75"
|
||
|
|
},
|
||
|
|
"execution_count": 31,
|
||
|
|
"outputs": [
|
||
|
|
{
|
||
|
|
"output_type": "display_data",
|
||
|
|
"data": {
|
||
|
|
"text/plain": [
|
||
|
|
"<Figure size 640x480 with 1 Axes>"
|
||
|
|
],
|
||
|
|
"image/png": "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
|
||
|
|
},
|
||
|
|
"metadata": {}
|
||
|
|
}
|
||
|
|
]
|
||
|
|
}
|
||
|
|
]
|
||
|
|
}
|