CSE100 Principles of Programming with C++

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1 1 Instructions You may work in pairs (that is, as a group of two) with a partner on this lab project if you wish or you may work alone. If you work with a partner, only submit one lab project with both of your names in the source code file to Blackboard for grading; you will each earn the same number of points. What to hand in, and by when, is discussed in Section 7; read it now. 2 Lab Objectives After completing this assignment the student should be able to, Complete all of the objectives of the previous lab projects. Write programs using while and for loops. Write programs consisting of functions defined in multiple source code files. Write function prototypes in header files and include header files when required. 3 Background 3.1 Newton's Method [Ref: Wikipedia: Newton's Method] Newton's method is a mathematical algorithm that can be used to find the roots of a real-valued function, i.e., the roots are real numbers. For example, consider the 2nd-degree polynomial function p(x) = 2.1x x and its plot, Clearly p(x) has two real roots: the first one, root 1, is around 0.45 and the second one, root 2, is near 6.5. Since p(x) is a second degree polynomial equation, i.e., a quadratic, we could use the well-known quadratic formula to find the roots, but we could also find the roots using Newton's method. Newton's method is an iterative method, i.e., it involves a loop. Here is the pseudocode for Newton's method which finds a root of f(x) where f(x) is any real-valued function, function newtons_method (f(x) is a real-valued function, ε is a double) double x i guess a value for the root we are trying to find x i+1 x i - f(x i ) / f'(x i ) -- evaluate f(x) at x i and evaluate the first derivative, f'(x), at x i while x i+1 - x i ε do x i x i+1 x i+1 x i - f(x i ) / f'(x i ) end while return x i+1 The parameter ε is a small number which controls how accurate the returned value is. For example, if we desire the returned value to be correct to three digits after the decimal point, then we would pass = as ε. If we passed 10-9 = for ε, the returned value would be accurate to at least eight digits after the decimal point. In general, if ε is 10 -p, the accuracy will be p - 1 correct digits after the decimal point. The smaller ε is, the more accurate the result, but this accuracy comes at the expense of performing more iterations to find the root, i.e., it takes longer. Note that in C++, we can represent real numbers in exponential notation, which is similar to scientific notation. For example, would be represented as E-78, where the E represents "times 10 to the" and the exponent is written after the E. Consequently, if we wished to use for ε, we would write 1E- 16. (c) Kevin R. Burger :: Computer Science & Engineering :: Arizona State University Page 1

2 Note that Newton's method starts by initializing x i to some value that is basically a guess for where we think the root is. The more accurate this guess, the more quickly Newton's method will converge to the correct answer. For a function with more than one root, e.g., our p(x) function from above, this initial guess will affect which root (the one near 0.45 or the one near 6.5) is found. For exam - ple, if x i were initialized to 0 then Newton's method is going to converge on the root near On the other hand, if we guessed 100 for x i then it will converge to the root near 6.5. The loop iterates until the absolute value of the difference between the prior value of x (x i) and the current value of x (x i+1 ) is less than ε. Each time through the loop, a new value (x i+1 ) for the root we are searching for is calculated by subtracting from the prior value (x i) the quotient of dividing f(x) evaluated at x i by the first derivative of f(x) evaluated at x i. (If you want to understand the mathematics behind Newton's method, i.e., why it works, refer to the Wikipedia page referenced above or any calculus text book). In my day, Newton's method was learned in Calculus II. We are going to use Newton's method in this lab project to write a function that computes and returns the a th root of a real number n, i.e., a n We will discuss how the algorithm works, using the square root of n = 17 (a = 2) as the example. If x = n then, x 2 = n which we can write as, x 2 n = 0 If we define f(x) = x 2 - n, then we can use Newton's method to find the positive root of f(x), which will be the square root of n. For example, let's trace newtons_method() for f(x) = x 2-17 and ε = 1E-4 = , i.e., we are going to find the square root of n = 17 with at least three digits of accuracy after the decimal point. Note that f'(x) = 2x. Let's use n / 2 as our starting guess because the square root of n will be between 1 and n / 2 (accurate digits of x i+1 are underlined), x i x i+1 x i+1 - x i Difference is > ε so keep looping Difference is > ε so keep looping Difference is > ε so keep looping Difference is > ε so keep looping x i+1 - x i < ε = so return x i+1 Notice how each new value of x i+1 is closer to the root than the previous value x i. If Newton's method is going to converge, the absolute value of the difference of x i+1 and x i will become smaller and smaller, until it is less than ε (in theory, the difference will go to 0 but the double data type cannot represent real numbers with more than digits of accuracy). Consequently, newtons_ method() would return (the underlined digits are all accurate) for the square root of 17. The correct value to 25-digits is (the correct value is a non-terminating fraction, i.e., it goes on forever) and the absolute value of the difference between the correct value to 25-digits and what newtons_method() returns is approximately , which is our error. Therefore, newtons_method() returned a value that is correct to = 13 digits after the decimal point and it only took four iterations of the while loop to do it. In general, to find the a th root of n, a 2, we define f(x) = x a - n where f'(x) = ax a-1. The pseudocode for our function (named ath_root) is, function ath_root(n 0 is a double, a 2 is an integer, ε is a double) double x_i n / 2 -- initial guess x_i_plus_1 x_i - (x_i a - n) / (a x_i a-1 ) while x_i_plus_1 - x_i ε do -- notice this is a sentinel loop with the sentinel being a value less than ε x_i x_i_plus_1 x_i_plus_1 x_i - (x_i a - n) / (a x_i a-1 ) end while return x_i_plus_1 (c) Kevin R. Burger :: Computer Science & Engineering :: Arizona State University Page 2

3 We have to make a guess for the initial value of x i so we will use n divided by 2. I believe, but have not verified, that this initial value will cause Newton's Method to always converge to the a th root. 3.2 Fibonacci Numbers [Ref: Wikipedia: Fibonacci Numbers] The Fibonacci numbers (also referred to as the Fibonacci sequence) is this sequence of integers, where, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377,... 1 f 0 = 0 f 1 = 1 f n = f n f n - 1 when n > 1 That is, the first two Fibonacci numbers are 0 and 1 and each successive number in the sequence is the sum of the two previous numbers. We can write a function to compute and return the n th Fibonacci number (n 0), i.e., fib n by employing a for loop. Here is the pseudocode for the function, function nth_fib(n 0 is an long) long fib_n n if n > 1 then fib_n_minus_2 0 fib_n_minus_1 1 for i 2 to n do -- notice this is a counting loop that executes n - 1 times fib_n fib_n_minus_2 + fib_n_minus_1 -- fib n is the sum of fib n-2 + fib n-1 fib_n_minus_2 fib_n_minus_1 fib_n_minus_1 fib_n end for end if return fib_n I suggest you trace this function for various values of n to convince yourself that you understand how it works. Note: C++ has more than one basic data type for representing integers. We have been using int all semester, but there is also an integer data type named long. The difference between the int and long data types is that the size of a long value, i.e., the number of binary bits that are allocated to represent the long integer, is guaranteed to be either the same size as int or larger. If 32-bits are allocated for int (as it commonly is), then we can represent decimal integers in the range [-2 31, ], which is [-2,147,483,648 to 2,147,483,647]. Depending on the word size of your computer and the C++ compiler being used (a 64-bit compiler versus a 32-bit compiler) a long integer will generally either be 32- or 64-bits. Using 64-bits to represent signed integers gives us a range of [-2 63, ], which is approximately ± x 10 18, which is around [-9,223,372,037,000,000,000 to 9,223,372,037,000,000,000]. 4 Lab Exercise Software Requirements The program shall display a menu with three menu items, Choice? 1 and prompt the user to select menu item 1, 2, or 3. If the user selects menu item 1, the program shall prompt the user to enter values for n and a, Your choice is to compute the a-th root of n. Enter n (>= 0): Enter a (>= 2): 5 The 5-th root of is Note, the starting values for the Fibonacci sequence are not always listed as 0 and 1. I have seen books refer to the sequence 1, 1, 2, 3, 5,... Also, the first Fibonacci number is not always denoted by f 0. Hence, an alternative sequence is f 1 = 1, f 2 = 1, f 3 = 2,... (c) Kevin R. Burger :: Computer Science & Engineering :: Arizona State University Page 3

4 and it shall compute and display the a th root of n (configure cout to display real number in fixed notation with 16 digits after the decimal point). After displaying the result, the program shall display the menu again, Choice? 2 and prompt the user to select menu item 1, 2, or 3. If the user selects menu item 2, it shall prompt the user to enter a value for n, Your choice is to compute the n-th Fibonacci number. Enter n (>= 0): 22 The 22-th Fibonacci number is and shall display the n th Fibonacci number. After displaying the result, the program shall display the menu again. It shall continue this process of displaying the menu, performing the selected operation when menu items 1 or 2 are selected. When the user selects menu item 3, the program shall terminate, Choice? 2 Your choice is to compute the n-th Fibonacci number. Enter n (>= 0): 22 The 22-th Fibonacci number is Choice? 2 Your choice is to compute the n-th Fibonacci number. Enter n (>= 0): 46 The 46-th Fibonacci number is Choice? 3 Bye. 5 Software Design The program shall consist of two.cpp files: Lab12.cpp and MyMath.cpp and a header file named MyMath.hpp. The contents of Lab12.cpp shall be implemented using the pseudocode shown below. Note that we use a sentinel loop in main() to continue displaying the menu and reading the user's choice, until the user selects menu item 3 to quit. This means that MENU_ITEM_3_QUIT is the sentinel value for the loop. Remember, a sentinel loop always has two read operations: the first read operation occurs before the loop begins and reads the first value from the sequence, i.e., the first selected menu item; the second read operation occurs at the end of the body of the loop and reads the next value from the sequence, i.e., the next selected menu item. define integer constant MENU_ITEM_1_ROOT 1 define integer constant MENU_ITEM_2_FIB 2 define integer constant MENU_ITEM_3_QUIT 3 define real constant EPSILON 1E E-16 means , where the E stands for "times ten to the". function compute_ath_root() nothing send "Your choice is to compute the a-th root of a number." to cout n get_double("enter n (>= 0): ") (c) Kevin R. Burger :: Computer Science & Engineering :: Arizona State University Page 4

5 a get_int("enter a (>= 2): ") root ath_root(n, a, EPSILON) send "The " a "-th root of " n " is " root to cout function compute_nth_fib() nothing send "Your choice is to compute the n-th Fibonacci number." to cout n get_long("enter n (>= 0): " fib_n nth_fib(n) send "The " n "-th Fibonacci number is " fib_n to cout function get_double(prompt is a string) double send prompt to cout read a double from the keyboard into n return n function get_int(prompt is a string) int send prompt to cout read an int from the keyboard into n return n function get_long(prompt is a string) long send prompt to cout read a long from the keyboard into n return n function menu() int send "" to cout send "" to cout send "" to cout send "" to cout choice get_int("choice? ") return choice function main() int configure cout to display real numbers in fixed notation with 16 digits after the decimal point choice menu() -- this read operation reads the first value in the sequence while choice = MENU_ITEM_3_QUIT do if choice = MENU_ITEM_1_ROOT then compute_ath_root() else -- choice must be MENU_ITEM_2_FIB compute_nth_fib() end if choice menu() -- this read operation reads the next value in the sequence (c) Kevin R. Burger :: Computer Science & Engineering :: Arizona State University Page 5

6 end while send "Bye." to cout return 0 The contents of MyMath.cpp shall be the two functions ath_root() and nth_fib() for which the pseudocode was presented earlier. Because we call these two functions from functions in Lab12.cpp, we must have prototypes in Lab12.cpp for the two functions; the prototypes are required to give the compiler the information it needs to know about the functions so it can compile the lines of code where we call the functions. If we place the prototypes in a header file named MyMath.hpp, then we can #include MyMath.hpp in Lab12.cpp, which would make the prototypes available to the compiler. Read the comments in MyMath.hpp to implement it. 6 Additional Programming Requirements 1. Add a comment header block at the top of each source code file with your name (and your partners name if applicable), address (and your partner's address if applicable), the lab number, your lab date/time, and your lab TA. 2. Carefully format your code and follow the indentation of the text as shown in the example programs of the textbook. 7 What to Submit for Grading and by When When you are finished with the program, create a new empty folder named Lab12. Copy Lab12.cpp, MyMath.hpp, and MyMath.cpp to this folder; there should only be these three files in this folder. Then compress this folder creating a zip archive named cse100-s17- lab12-auriteid.zip where asuriteid is your ASU user name that you use to log in to Blackboard, e.g., mine is kburger2. If you worked with a partner then name the zip archive cse100-s17-lab12-auriteid1-asurite2.zip where asurite1 and asurite2 are the user names of both partners. Then upload the.zip archive file to Blackboard using the lab submission link by the deadline. If your program does not compile or run correctly, upload what you have completed for grading anyway (you will generally receive some partial credit for effort). The deadline for the complete lab project is 4:00am Sat 29 Apr. Consult the online syllabus for the late and academic integrity policies. 8 Grading Rubric 1. Lab Exercise Program (0 to 5 pts) Testing the Program: Compile and run the student's program using these test cases: Test Case 1: The cube root of is Test Case 2: The 5th root of 173 is Test Case 3: The 18 root of is Test Case 4: fib(0) is 0. Test Case 5: fib(1) is 1. Test Case 6: fib(5) is 5. Test Case 7: fib(41) is a. If the submitted program does, or does not, compile and the student completed less than 50% of the required code correctly, assign +2 pts. b. If the submitted program does not compile and the student completed more than 50% of the required code correctly, assign +3 pts. c. If the submitted program compiles and the student completed more than 50% of the required code correctly, assign +4 pts. d. If the submitted program compiles and is implemented perfectly, or close to perfect with only one or two minor mistakes, assign +5 pts. 2. Deadline was 4:00am Sat 29 Apr. a. Assign 20% bonus calculated on the earned pts for a submission prior to 4:00am Thu 27 Apr. b. Assign 10% bonus calculated on the earned pts for a submission between 4:00am Thu 27 Apr and 3:59am Fri 28 Apr. c. Deduct 10% (-0.5 pts) for a submission between 4:00am Sat 29 Apr and 3:59am Sun 30 Apr. d. Deduct 20% (-1.0 pt) for a submission after 4:00am Sun 30 Apr. (c) Kevin R. Burger :: Computer Science & Engineering :: Arizona State University Page 6