MATH 1MP3 Homework #4 Due: 11:59pm, Wednesday, March 6.

Transcription

1 MATH 1MP3 Homework #4 Due: 11:59pm, Wednesday, March 6. Important notes: To start the assignment, download the Jupyter notebook file assignment 4 template.ipynb found here: ipynb You might need to copy the above url and paste it into a browser to download the file. Your assignment must be submitted as a Jupyter notebook file called yourmacid assignment 4.ipynb, where yourmacid is replaced with your macid from your McMaster address. Since my McMaster address is valeriot@mcmaster.ca then the file that I would submit would be valeriot assignment 4.ipynb. You must submit this file to the homework #4 Dropbox on the MATH 1MP3 Avenue to Learn site. There are four separate questions in this assignment, but your solutions will be contained in the single Jupyter notebook file that you will submit. The solution to each question consists of python code that implements the functions described in each question, plus appropriate docstrings. To complete your assignment, place your code for each question as indicated in the template. Do not alter any other part of the assignment 4 template.ipynb file and do not add python code in any other parts of the file that you submit. 1

2 To see an example of what is expected, look over the solutions to homework assignments #2 and #3. The code that you enter cannot contain any import statements. The template already has import math and import string statements in cells where they might be needed (but you don t need to use the functions and constants from these modules). The functions should use the return statement to return the result of the function call, so print statements should not occur in the code that you produce. While developing and testing your code, it might be helpful to use print statements, but they should be removed before submitting your solution. Note that the template contains several print statements that when executed will test your code. Any file that is submitted for grading that does not conform to the above specifications will lead to a grade of 0 on the assignment. Before submitting your solution, you should make sure that it runs properly. To do this, you can open it in the Jupyter notebook server that comes installed with Anaconda, or you can use the McMaster Jupyter server, by going to the website ca/. You may want to use the Spyder IDE, or some other IDE to develop your code before entering it into the Jupyter notebook that you submit. The file assignment 4 template.ipynb contains six cells, one for each part of the assignment questions. At the start of each cell is the function definition, followed by a placeholder docstring entry. At the end of each cell are some print statements that when executed will print out the results of running your code on a few test cases. Do not remove, alter, or add to these print statements. For each question, you should include a suitable docstring in the appropriate place. Your docstrings should have the same format as those that appear in the solutions for assignments #2 and #3. This is the numpy docstring format. Note that for functions that have filenames as arguments, then you do not need to provide examples in your docstrings. 2

3 Carefully read over each of the following four questions. Once you have produced code for a given question, you should run it in the Jupyter server or on your IDE on the test cases provided to make sure that it is working properly. You should also try out your code on other test cases of your own design. Your grade for each question will depend on whether or not your code correctly handles not just the test cases provided in the template, but other test cases that will be run on your solutions. Points will also be awarded based on the quality of the docstrings that you provide for each function. Do not leave this until the last minute, since you might encounter computer/internet/avenue issues. Late assignments will not be accepted. All work submitted for grading must be your own. You may discuss homework problems and related material with other students, but you must not submit work copied from others or from the internet. Question #1 a) Write a function long strings that has one argument: file name that is a string that is the name of a text file. The file will consist of a number of lines of text, and each line will consist of a number of strings, separated by one or more space characters. The command long strings(file name) should return a set that contains all of the longest strings that are found in the file. Note that the strings found in the lines of the file may consist of any nonwhitespace characters, including letters, digits, and punctuation symbols. If the file only consists of lines that contain spaces, then the function should return the empty set. For example, if the file question 1.txt consists of the following five lines of text: If there are two arguments, they must be strings of equal length, and in the resulting dictionary, each character!! in x will be mapped to the characters at the same #positi%on! in y. If there is a third arg,!ument, it 3

4 must be a string, whose characters will be mapped to None in the result. then long strings( question 1.txt ) should return the set { #positi%on!, arg,!ument,, character!!, dictionary, }, since each of the strings in this set have length 11 and no string in the file has a longer length. To test your code on the file question 1.txt you can download it here: Be sure that this file, or any other text file that you test your code on is located in the same folder/directory as your jupyter notebook or python code. Your code should close any files that it opens before returning any values. Hint: You will probably want to use the string functions.strip() and.split() to help identify the strings in the file that should be considered. Question #1 b) This question is similar to the previous one and you should be able to modify your code from the previous question for this one. Write a function long words that has one argument: file name that is a string that is the name of a text file. The file will consist of a number of lines of text, and each line will consist of a number of strings, separated by one or more space characters. The command long words(file name) should return a set that contains all of the longest words that are obtained from strings found in the file. For the purposes of this exercise, a word is obtained from a string by stripping away any trailing punctuation symbols, as long as the only remaining characters are alphabetic, upper or lower case. For example, after stripping away the trailing punctuation symbols the string "abc!$,?" gives rise to the word "abc". Since the string "$a,b!c??" contains non-alphabetic characters, even after stripping away the trailing punctuation symbols, then it does not give rise to a word, in this sense. If the file only consists of lines that contain spaces or does not contain any words, then the function should return the empty set. For example, long words( question 1.txt ) should return the set { characters, characters, dictionary }, where the file question 1.txt is from the previous question. This is because both of these words were obtained by stripping away any trailing punctuation symbols from strings found in the file and no other words obtained from the file are longer than these 4

5 three. You should consider strings that consists of the same sequence of letters, but with different capitalizations, to be different words. For example, in the test file, the words characters and characters should both appear in the set. To test your code on the file question 1.txt you can download it here: Be sure that this file, or any other text file that you test your code on is located in the same folder/directory as your jupyter notebook or python code. Hint: You should consider using the string string.punctuation. It consists of all of the punctuation characters recognized by python. Question #2 Write a function grade stats that has two arguments: file name that is a string that is the name of a text file that contains the grades for the students in some course, and col that is a column number, with default value of 1. Each line of the file begins with the (first) name of a student, followed by a sequence of strings that are supposed to represent integers that lie between 0 and 100, inclusive. The elements of each line are separated by blank space " ". The function should compute the average and median values of the valid numbers that appear in the specified column. Any string that appears in the specified column that does not represent an integer that lies between 0 and 100 inclusive, should be disregarded. If there are no valid numbers in the specified column, then the function should return no value. The statement return None will accomplish this. Note that you should not return the string "None", but rather the python None object. If the specified column contains strings that represent integers between 0 and 100, then the function should return a tuple of length 3 that consists of, in order: the average of the numbers in the column, rounded to 2 decimal places, the median of the numbers in the column, and the highest grade in the column. To test your code, you can use the file grades.txt that can be downloaded here: Since the first column of this file does not contain any valid integer entries, then grade stats( grades.txt, 0) should return no value. On the other hand, grade stats( grades.txt ) should return the tuple (71.14, 78.0, 5

6 96) and grade stats( grades.txt, 5) should return the tuple (41.85, 48, 60). Both columns 1 and 5 contain some non-valid entries. Note: To compute the average of a non-empty list of numbers, you can first sum the entries in the list (using the sum function) and then divide by the length of the list. The median of a sorted list of numbers of odd length is equal to the value of the middle entry of the list, while the median of a sorted list of numbers of even length is equal to the average of the two middle entries of the list. Question #3 Produce a function prime factors dict that has one integer variable n that is assumed to be greater than 1. The function will return a dictionary with keys the integers from 2 to n, inclusive, and with values the set of prime factors of each key. So, the command prime factors dict(8) will return the dictionary {2 : {2}, 3 : {3}, 4 : {2}, 5 : {5}, 6 : {2, 3}, 7 : {7}, 8 : {2}} Note that even though the prime number 2 appears twice in the prime factorization of 4, it only appears once in the set of its prime divisors. The function smallest divisor is available to use in your code. This function has one integer argument n that is greater than 1. It will return the smallest integer divisor of n that is greater than 1. So smallest divisor(35) will return the integer 5 and smallest divisor(23) will return 23. Note that in all cases, this function will return a prime number and that smallest divisor(n) will return the number n if and only if n is a prime number. The code for this function appears at the top of the question #3 cell. You may find it useful in solving this question, but you are not required to use it. Hint: If n is the product of the integers c and d, then the set of prime divisors of n is equal to the union of the set of prime divisors of c and the set of prime divisors of d. Question #4 Write a function power function that has two arguments k, a number, and tol, a number with default value 1e-6. This function will return a function 6

7 of one variable f(x) that approximates the value of the number (1 + x) k within the specified tolerance, for values of x between 0 and 1, excluding 1. The approximation should be produced using the series expansion that Sir Isaac Newton discovered in He showed that for k a real number and x a number with 0 x < 1, (1 + x) k is equal to 1+kx+ k(k 1) x 2 + 2! k(k 1)(k 2) x ! k(k 1)(k 2) (k n + 1) x n + n! For details on this discovery, consult Section of Dr. James Stewart s Calculus text or just look up the Binomial Theorem on the internet. Note that the above series actually holds for all x strictly between -1 and 1. Your function should compute, in sequence, the terms of the above series and keep a running total of them. Once a term s size is less than or equal to the specified tolerance, then the current total will be an accurate enough approximation to (1 + x) k, and this is the value that f(x) should return. So, the statement f = power function(1/3) will assign to the variable f the one argument function that computes (1 + x) (1/3), accurate to So f(0.5) will return the value , the cube root of 1.5. Note: You cannot just make use of the built in exponentiation function in your code to directly compute the value of (1 + x) k. No credit will be given for doing this. Of course, you may compute x n for integer values of n with the expression x ** n. You do not need to worry about the behaviour of the function produced for values of x that do not lie in the interval [0, 1). Hint: Study the example of the sine function that was worked out in class. The code for this question can be closely modelled on the sine function code. It can be downloaded from the week #5 section of the course website. Note that to compute the nth term in the series from the previous term, you just need to multiply it by (k n+1)x. So, you need not directly compute factorials n in your code (but you may). 7