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"Write two Python functions to find the $F_n$ Fibonacci number, one using an iterative method and the other one using recursion. The only argument is `n` and the $n$-th number in the Fibonacci sequence should be retuned.\n",
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"\n",
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"***Wikipedia source***: In mathematics, the Fibonacci numbers, commonly denoted $F_n$, form a sequence, called the Fibonacci sequence, such that each number is the sum of the two preceding ones, starting from 0 and 1. That is,\n",
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"\n",
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"$$F_{0}=0,\\quad F_{1}=1,$$\n",
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"and\n",
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"$$F_{n}=F_{n-1}+F_{n-2}$$\n",
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"for $n > 1$. \n",
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"The beginning of the sequence is thus:\n",
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"$$0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55,...$$\n",
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"\n",
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"<ins>For example</ins>: The Fibonacci number $F_7$ equals $13$, so the functions should return this value."
"Write a Python program to create a dictionary from a string. The dictionary should contain as keys the characters of the string and as values the number of times the character is repeated.\n",
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"\n",
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"***Note 1***: Track the count of the letters from the string. \n",
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"***Note 2***: To check if a character is in a string just type `\"char\" in str1` \n",
"Define two functions, one for $f(x)$ and the other one must return the integral solution given $a$, $b$ and $n$.\n",
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"\n",
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"***Note 2***: You can call a function inside another function."
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"id": "JtjyQi2ujlzw"
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},
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"source": [
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"**SOLUTION**:"
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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": 41,
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"metadata": {
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"id": "ByVIxIpPjo32"
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},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"0.5226816058039061\n"
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]
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}
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],
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"source": [
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"#the function to be integrated:\n",
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"def f(x):\n",
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" return x ** 4 * (1 - x) ** 4 / (1 + x ** 2)\n",
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"\n",
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"#define a function to do integration of f(x) btw. 0 and 1:\n",
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"def trapezium(a, b, n):\n",
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" h = (b-a) / n\n",
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" intgr = 0\n",
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" for i in range(0, n+1):\n",
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" intgr = intgr + h * f(a + i * h)\n",
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" return intgr\n",
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"\n",
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"print(trapezium(1, 2, 10000))"
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]
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}
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],
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"metadata": {
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"colab": {
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"collapsed_sections": [],
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"name": "SesionThree_Exercises.ipynb",
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"provenance": [],
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"toc_visible": true
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},
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"kernelspec": {
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"display_name": "Python 3",
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"language": "python",
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"name": "python3"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.8.5"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 1
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}
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{"cells":[{"cell_type":"markdown","source":["# PYTHON COURSE FOR SCIENTIFIC PROGRAMMING \n","**Contributors:** \\\n","Artur Llabrés Brustenga: Artur.Llabres@e-campus.uab.cat \\\n","Gerard Navarro Pérez: Gerard.NavarroP@e-campus.uab.cat \\\n","Arnau Parrilla Gibert: Arnau.Parrilla@e-campus.uab.cat \\\n","Jan Scarabelli Calopa:Jan.Scarabelli@e-campus.uab.cat \\\n","Xabier Oyanguren Asua: Xabier.Oyanguren@e-campus.uab.cat\n","\n","Course material can be found at: https://llacorp.github.io/Python-Course-for-Scientific-Programming/ "],"metadata":{"id":"P4hDyWJnlkkV"}},{"cell_type":"markdown","metadata":{"id":"-rYMKOqDez5v"},"source":["## <ins>Exercise 1</ins>:"]},{"cell_type":"markdown","metadata":{"id":"0-NaBG9XhgU8"},"source":["Write a Python function to check whether a number is perfect or not. It should return `True` or `False`.\n","\n","***Wikipedia source***: \"*A perfect number is a positive integer that is equal to the sum of its positive divisors excluding the number itself.*\n","\n","*Equivalently, a perfect number is a number that is half the sum of all of its positive divisors (including itself).*\" \n","\n","<ins>For example</ins>: \n","\n","The first perfect number is 6, because 1, 2, and 3 are its positive divisors excluding itself, and 1 + 2 + 3 = 6.\n","\n","Equivalently, the number 6 is equal to half the sum of all its positive divisors: ( 1 + 2 + 3 + 6 ) / 2 = 6.\n","\n","\n","The next perfect number is 28 = 1 + 2 + 4 + 7 + 14. This is followed by the perfect numbers 496 and 8128."]},{"cell_type":"markdown","metadata":{"id":"tR4v1HGeiApl"},"source":["**SOLUTION**:"]},{"cell_type":"code","execution_count":null,"metadata":{"colab":{"base_uri":"https://localhost:8080/"},"id":"4Iciqs4Sh-ar","outputId":"f423d936-ccad-401d-ef14-e115c40708cb"},"outputs":[{"name":"stdout","output_type":"stream","text":["True\n"]}],"source":["def perfect_number(n):\n"," sum = 0\n"," for x in range(1, n):\n"," if n % x == 0:\n"," sum += x\n"," return sum == n\n"," \n","print(perfect_number(6))"]},{"cell_type":"markdown","metadata":{"id":"0IZIsOxNjbhw"},"source":["## <ins>Exercise 2</ins>:"]},{"cell_type":"markdown","metadata":{"id":"PVlUe6mHjpeU"},"source":["Write two Python functions to find the $F_n$ Fibonacci number, one using an iterative method and the other one using recursion. The only argument is `n` and the $n$-th number in the Fibonacci sequence should be retuned.\n","\n","***Wikipedia source***: In mathematics, the Fibonacci numbers, commonly denoted $F_n$, form a sequence, called the Fibonacci sequence, such that each number is the sum of the two preceding ones, starting from 0 and 1. That is,\n","\n","$$F_{0}=0,\\quad F_{1}=1,$$\n","and\n","$$F_{n}=F_{n-1}+F_{n-2}$$\n","for $n > 1$. \n","The beginning of the sequence is thus:\n","$$0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55,...$$\n","\n","<ins>For example</ins>: The Fibonacci number $F_7$ equals $13$, so the functions should return this value."]},{"cell_type":"markdown","metadata":{"id":"XKhg1tVkjkto"},"source":["**SOLUTION**:"]},{"cell_type":"code","execution_count":null,"metadata":{"id":"pKC9NZDYjnkg"},"outputs":[],"source":["def F_iter(n):\n"," if (n == 0):\n"," return 0\n"," elif (n == 1):\n"," return 1\n"," elif (n > 1):\n"," fn = 0\n"," fn1 = 1\n"," fn2 = 2\n"," for i in range(3, n):\n"," fn = fn1 + fn2\n"," fn1 = fn2\n"," fn2 = fn\n"," return fn\n","\n","def F(n):\n"," if (n == 0):\n"," return 0\n"," elif (n == 1):\n"," return 1\n"," elif (n > 1):\n"," return (F(n-1) + F(n-2))"]},{"cell_type":"code","execution_count":null,"metadata":{"colab":{"base_uri":"https://localhost:8080/"},"id":"vmuuU6Qb4tHO","outputId":"bc611b30-0594-4b6c-9efb-f6ff2c8507f0"},"outputs":[{"name":"stdout","output_type":"stream","text":["13\n","13\n"]}],"source":["print(F_iter(7))\n","print(F(7))"]},{"cell_type":"markdown","metadata":{"id":"bcsyVRrLjeNA"},"source":["## <ins>Exercise 3</ins>:"]},{"cell_type":"markdown","metadata":{"id":"mq5VNFVZjqdm"},"source":["Write a Python program to create a dictionary from a string. The dictionary should contain as keys the characters of the string and as values the number of times the character is repeated.\n","\n","***Note 1***: Track the count of the letters from the string. \n","***Note 2***: To check if a character is in a string just type `\"char\" in str1` \n","<ins>For example</ins>: The string 'scn2python' should output: `{'s': 1, 'c': 1, 'n': 2, '2': 1, 'p': 1, 'y': 1, 't': 1, 'h': 1, 'o': 1}`"]},{"cell_type":"markdown","metadata":{"id":"w0UpFVuBjlXc"},"source":["**SOLUTION**:"]},{"cell_type":"code","execution_count":null,"metadata":{"colab":{"base_uri":"https://localhost:8080/"},"id":"enTtP8kljoZa","outputId":"3dfbcf73-0805-4853-a8c6-1c5949ea70a3"},"outputs":[{"name":"stdout","output_type":"stream","text":["{'s': 1, 'c': 1, 'n': 2, '2': 1, 'p': 1, 'y': 1, 't': 1, 'h': 1, 'o': 1}\n"]}],"source":["str1 = 'scn2python' \n","my_dict = {}\n","for letter in str1:\n"," if (my_dict.get(letter) == None): #or letter not in my_dict.keys()\n"," my_dict[letter] = 1\n"," else:\n"," my_dict[letter] += 1\n","print(my_dict)"]},{"cell_type":"markdown","metadata":{"id":"ngPDJ-pSjgGb"},"source":["## <ins>Exercise 4</ins>:"]},{"cell_type":"markdown","metadata":{"id":"oMSJ_bvjjrNC"},"source":["Write a Python program to perform numerical integration using rectangles.\n","\n","To test the program use the function\n","\n","$$f(x)=\\frac{x^4(1-x^4)}{1+x^2}dx$$\n","\n","and integrate it between $1$ and $2$. The result should be about $0.522...$ for $n=10000$.\n","\n","***Note 1***: Remember that we can approximate an integral using tiny rectangles given a step $\\Delta x$ with\n","$$\\int_{a}^{b} f(x)dx = \\sum_{i=0}^{n} \\Delta xf(a+i\\Delta x)$$\n","\n","We will define our $\\Delta x=\\frac{b-a}{n}$. For higher values of $n$ (*i.e.* smaller $\\Delta x$), the approximation is more precise\n","\n","<img src=\"https://i.stack.imgur.com/BJPUl.png\" alt=\"Drawing\" style=\"width: 500px;\"/>\n","\n","Define two functions, one for $f(x)$ and the other one must return the integral solution given $a$, $b$ and $n$.\n","\n","***Note 2***: You can call a function inside another function."]},{"cell_type":"markdown","metadata":{"id":"JtjyQi2ujlzw"},"source":["**SOLUTION**:"]},{"cell_type":"code","execution_count":null,"metadata":{"id":"ByVIxIpPjo32","outputId":"e2199d33-3fdd-4b3f-ea8b-b8dbe43f82d8"},"outputs":[{"name":"stdout","output_type":"stream","text":["0.5226816058039061\n"]}],"source":["#the function to be integrated:\n","def f(x):\n"," return x ** 4 * (1 - x) ** 4 / (1 + x ** 2)\n"," \n","#define a function to do integration of f(x) btw. 0 and 1:\n","def trapezium(a, b, n):\n"," h = (b-a) / n\n"," intgr = 0\n"," for i in range(0, n+1):\n"," intgr = intgr + h * f(a + i * h)\n"," return intgr\n"," \n","print(trapezium(1, 2, 10000))"]}],"metadata":{"colab":{"collapsed_sections":[],"name":"Exercises_Lecture_03.ipynb","provenance":[],"toc_visible":true},"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":"3.8.5"}},"nbformat":4,"nbformat_minor":0}
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