Name CptS 111, Spring 2018 Programming Assignment #5 Date Name of program Brief description and/or purpose of the program; include sources

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1 CptS 111 PA #5 Due Tuesday, Mar. 20, 2018 for-loops, FIBONACCI, AND A STATISTICS PROGRAM This assignment has two parts and isn t nearly as formidable as it looks (although you shouldn t wait until the last minute to start it). In the first part you ll have some fun with the Fibonacci sequence, and in the second part you ll actually create a statistics program you can use instead of your calculator (just joking). You must include a docstring in each function, use comments, and make your program readable by using whitespace effectively. You will be graded on all these aspects, not just on whether your programs run correctly. Header information. Your program must include the following information in the header. Name CptS 111, Spring 2018 Programming Assignment #5 Date Name of program Brief description and/or purpose of the program; include sources The Fibonacci Sequence and the Golden Ratio Believe it or not, the golden ratio has been used to predict trends in the stock market and for other applications in financial analysis. The golden ratio and the Fibonacci sequence are related, and both seem to reflect a basic law of nature which you can read about here. The relationship is given by F(n+1) lim = φ n F(n) wheref(n) is thenth term in the Fibonacci sequence and φ is the golden ratio given by φ = For large values of n, you can find the approximate value of the next term in the Fibonacci sequence by multiplying the current term by the golden ratio. You may recall that the exact Fibonacci sequence is obtained by adding two successive terms to get the next term: i.e.,1+1 = 2, 1+2 = 3,2+3 = 5, and so on. The Fibonacci spiral shown here: 1

2 is constructed by representing each term of the Fibonacci sequence by a square with sides equal to the value of the term. First you draw two squares with sides of length 1 next to each other; you add a square with sides of length 2 adjacent to the original two squares; next you add a square with sides of length 3 adajacent to the three existing squares, and so on. After you have as many boxes as you wish, you draw quarter-circle arcs through each adjacent square and voila! Approximations of Fibonacci spirals show up in nature all the time as seen in the next two images of a snail shell and Hurricane Irene: For part 1 of this assignment, we re going to approximate the golden ratio using successive terms of the Fibonacci sequence. To begin we must first convert the sequence shown above into a mathematical formula. We write: x n+1 = x n +x n 1 (1) Don t be intimidated by this formula. It simply states that the next value in the sequence relative to thenth term is the sum of thenth term and then 1th term. We can seed the sequence by starting with the first two terms to obtain the third: x 2 = x 1 +x 0 2 = 1+1 2

3 How do you write an algorithm to implement Eq. (1) and then generate a given number of terms in the Fibonacci sequence? Imagine there s a window just wide enough for you to see spaces for three terms in the sequence as shown in the figure below. When you align this window with the three spaces, the first and second numbers are known, and it s easy to determine the third number: simply add the first and second numbers. Next push the window to the right one space so that the new first number previously was the second number and the new second number previously was the third. Once again the first and second numbers in the window are known. Add them to obtain the third number. You can repeat this to determine as many number of terms as you want. first second third 1, 1,?,... first second third 1, 1, 2,?,... first second third 1, 1, 2, 3,?,... Next, consider the following code. It prints the first 10 numbers of the Fibonacci sequence 11 with the starting valuex 0. 1 # seed the starting values 2 first = 1 # x_0 3 print(first) # print the starting value, x_0 4 second = 1 # x_1 5 for i in range(10): # execute loop 10 times 6 print(second) # print current second value 7 third = first + second # set third to the next number in the series 8 first = second # set first to previous second value 9 second = third # set second to previous third value The window we imagined above is represented by three variables: first, second, and third. We set first and second to one. After we ve initialized first, we print its value before entering the for-loop. In the for-loop, the value of second is printed, and the next number in the sequence is calculated and assigned to the variable third. Next the first and second variables are reset, 3

4 which is just like moving our window to the right one space. (This can be coded more pythonically using simultaneous assignment which is when you use a, b = x, y.) Now look at the code and the window figure and make sure you understand both before continuing on to the actual programming assignment! Program Requirements For part 1, you must write a single program calledgold ratio.py with two functions: fib(): Takes a single integer argumentnum terms greater than or equal to 1 and for each of thenum terms terms of the Fibonacci sequence prints 1) the number of the term i.e., 1, 2, 3..., 2) the corresponding Fibonacci number, and 3) the ratio of the current and previous Fibonacci numbers as shown below. The ratio, printed only forx 1 and higher, is an approximation of the golden ratioφ = main(): Prompts the user for the number of terms s/he wants to find and calls the void function fib(). The program gives the following output (with input in boldface font): 1 Enter the number of terms you want to find: Enter the number of terms you want to find: The last line of your program should be a call to main(). Don t forget to includedocstrings! How many terms are needed to obtain an approximation to the golden ratio accurate to 4 decimal places? A Statistics Program You ve probably used a calculator to find the average of a list of numbers and perhaps you ve also calculated the standard deviation. Averages, known as means in statistics talk, and standard deviations are used all the time and in all walks of life, basically whenever groups of numbers are of interest. 4

5 The average on your first exam was 80.1 and the standard deviation was This means that about 68% of you scored between 64.2 and 96.0, i.e., within 1 standard deviation of the mean (you did better than last fall s class!). In Lab #7 you defined a function calledget floats(). For part 2 of this assignment you ll add four other functions, and your program will then be capable of prompting a user for numbers, summing their values, and finding their average and standard deviation. Program Requirements For part 2, you need to write a program calledstats.py with five functions. get floats(): Copy this function from Lab #7. It take a single integer argument and returns a list of floats. summer(): This non-void function takes a single list argument, and returns the sum of the list. However, it does not use the sum() function to do so. See Summing Values the Hard Way below for instructions on how to implement this function. average(): This non-void function uses the list of numbers returned by get floats() as its argument and returns the average value, a float, to the calling program. It calls summer() to sum the values in the list. Its body can actually be a single line: areturn statement followed by the appropriate expression. See the notes below on Calculating Averages. std dev(): This non-void function takes two arguments, the list of numbers obtained by get floats() and the average obtained using average(), and returns the standard deviation, a float value. The function should initialize an accumulator to zero and then use a for-loop in which the average is subtracted from the list value and the result is squared and added to the accumulator. After thefor-loop has finished looping, the accumulator should be divided by the number of elements in the list, the square root taken, and the result returned to the calling function. See the notes below on Calculating Standard Deviations. main(): Finally, write the void function main(). It should prompt a user for the number of elements desired in the list and should call the non-void functions get floats(), average(), andstd dev(). Then it should print the average and standard deviation to the screen. Use theround() function to round off the standard deviation to two decimal places. Summing Values the Hard Way 1 summer() takes a single list argument. First you need to initialize an accumulator to zero, then use a for-loop to add the value of each element in the list, and finally return the total sum to the calling program. The proper behavior ofsummer() is demonstrated below. 1 The easy way is to use the built-in functionsum(). 5

6 1 >>> x = [9, 3, 21, 15] 2 >>> summer(x) >>> summer([31.12, , 24.9, 82.0, 14.0]) Calculating Averages Recall that the average of a list of numbers is simply the sum of the values divided by the number of values. Mathematically, for a set of numbersx 1, x 2,, x N, the average is given by x = 1 N (x 1 +x 2 + +x N ) where N is the number of terms in the list. The average, or mean, is often indicated by an overbar i.e., x indicates the average of a set of numbersx k. The average is also given by x = 1 N N x k k=1 where the uppercase Greek letter Sigma (Σ) means perform a sum. The expression to the right of Sigma is what is to be summed. The expression below Sigma initializes a summation index (in this casek) and the term above Sigma indicates the final value of the summation index. Calculating Standard Deviations The standard deviation is a measure of how much fluctuation or deviation there is in a set of numbers after the average has been subtracted from each of the values 2. If all the values in a list are equal to the average, then the standard deviation is zero. If the standard deviation is large, then it means many values deviate considerably from the average. The standard deviation is represented by the lowercase Greek letter sigma (σ). For a set of numbersx 1, x 2,, x N, the standard deviation is given by σ = 1 N [(x 1 x) 2 +(x 2 x) 2 + +(x N x) 2 ] where x is the average. Recall that in Python we can square a value z using z ** 2 and we can calculate the square root usingz ** 0.5. The following three examples demonstrate the results you should obtain for three different sets of input for your stats.py program. Each of these has the same mean but an increasing standard deviation. In the first, all the values are the same and hence the standard deviation is zero. 1 Enter the number of list elements: 5 2 Enter float 1: 50 2 For further discussion of standard deviation, see< 6

7 3 Enter float 2: 50 4 Enter float 3: 50 5 Enter float 4: 50 6 Enter float 5: 50 7 Average = Standard deviation = 0.0 Here the values are spread out, spanning the range 30 to 70: 1 Enter the number of list elements: 5 2 Enter float 1: 30 3 Enter float 2: 40 4 Enter float 3: 50 5 Enter float 4: 60 6 Enter float 5: 70 7 Average = Standard deviation = In the final example the values range between 10 and 90: 1 Enter the number of list elements: 5 2 Enter float 1: 10 3 Enter float 2: 30 4 Enter float 3: 50 5 Enter float 4: 70 6 Enter float 5: 90 7 Average = Standard deviation = Don t forget to includedocstrings in all your functions! Submission information. Use Blackboard to submit both your programs. Submit a zipped file called <first initial+lastname> pa5.zip. If you don t know how to zip a file, be sure to ask your TA before your PA is due! Also, be sure it s not empty by checking to see how large it is before submitting it! 7