1. Announcements First Midterm March 14 th on Monday next week at 19:40 (100 minutes) Exam classes: if (LastName <= "Çoruh") cout << " FENS G077 " << endl; else if ("Demir" <= LastName <= "Kantoğlu") cout << " FMAN 1099 " << endl; else if ("Kaplan" <= LastName <= "Örs") cout << " FASS G062 " << endl; else if ("Öz" <= LastName <= "Soylu") cout << " FENS L045 " << endl; else if ("Şahin" <= LastName <= "Tüysüz") cout << " FENS G032 " << endl; else if ("Uçar" <= LastName) cout << " FENS G035 " << endl; This week recitations, we will solve sample midterm questions. Detailed email about the midterm was sent with previous years’ exams. Homework 3 is due on Friday, March 11 th this week

2. Developing Loops Some loops are easy to develop, others are not Sometimes the proper loop test and body are hard to design Practice helps, but remember: Good design comes from experience, experience comes from bad design

3. Number Crunching Number crunching is a CS term that means a computing operation that requires several (and sometimes complex) arithmetic operations It was the job of early computers Numeric Analysis classical sub-discipline of computer science Today implicitly or explicitly, all operations are numeric Now we will see some mathematical applications factorial calculation prime number testing

4. Factorial n! = 1 x 2 x … x n is “n factorial”; used in math, statistics long factorial(long n) // pre: 0 <= n // post: returns n! (1 x 2 x … x n) Similar to sum, but this time we will calculate a product within the loop. At the end we will return the final product. The loop will iterate n times, multiplying by 1, 2, …, n Suppose we use a variable called product to hold the result, then product is n! when the loop terminates. Then we will return it at the end.

5. Factorial long Factorial(int num) // precondition: num >= 0 // postcondition returns num! (1 x 2 x … x num) { long product = 1; int count = 0; while (count < num) { count += 1; product *= count; } return product; } Issues Why did we use long? What happens if we use int instead? What happens if we initialize count to 1? Let’s see fact.cpp

6. Factorial (Cont’d) – Using BigInt class What is the problem of the previous program? integer overflow even long is not sufficient (actually there is no difference between long and int for 32-bit computers like ours) 12! is 479,001,600 so what happens with 13! ? The type BigInt, accessible via #include "bigint.h" can be used like an int, but gets as big as you want it to be Really arbitrarily large? No, limited to computer memory, but computers most likely run out of time before running out of memory Disadvantages of using BigInt compared to int? processing speed is lower uses up more memory Use BigInt if you really need it Do not forget to add bigint.cpp to your project, BigInt is a Tapestry class Download wincode.zip file from http://cs.duke.edu/csed/tapestry

7. Factorial using BigInt class See bigfact.cpp

8. Determining if a number is prime Prime number is a natural number which has only two divisors: 1 and itself Some Cryptographic algorithms depend on prime numbers Determining if a number is prime must be “easy” Actually factoring a number must be “hard” “hard” in the sense that it must be computationally infeasible to factorize in a reasonable amount of time RSA Cryptosystem Rivest, Shamir, Adleman based on the factorization problem of large numbers has been utilized by several security products and services PGP (Pretty Good Privacy) – e-mail security WWW security using SSL protocol Sophisticated mathematics used for fast prime-testing, we’ll do basic prime testing Since our algorithm is based on factorization, it will be really slow for large numbers

9. Determining Primeness (continued) 1 is NOT prime, 2 is prime, 3 is prime, 5 is prime, 17 is prime, … 137, 193? We do not need to check even numbers other than 2 (2 is a special case) To check 193, divide it by 3, 5, 7, 9, 11, 13 Note that 14x14 = 196, so 13 largest potential factor? We will use modulus operator to check divisibility We’ll check odd numbers as potential divisors Watch out for 2, it is a special case How far should we go to check potential divisors? up to and including sqrt(number) + 1 If there was a bigger factor, a smaller factor would exist. And this smaller one must have been checked before. So we do not need to go beyond this limit. +1 is there to make sure that there will be no problems with precision See primes.cpp for code

10. Primeness Check – Details Special even number check is added before the loop to eliminate even numbers to be checked in the loop In order to make the code more efficient int limit = int(sqrt(n) + 1); To assign a double value to an int, a typecast is used, to tell the compiler that the loss of precision is intentional Make typecasts explicit to tell the compiler you know what you are doing Compiler warnings are avoided We will see typecast in more detail later

11. for loop compared with while while ( ) { ;... ; } for ( ; ; ) { ;... ; }

12. Example Rewrite the while loop of main of primes.cpp using for k = low; while (k <= high) { if (IsPrime(k)) { cout << k << endl; numPrimes += 1; } k += 1; } for (k = low; k <= high; k += 1) { if (IsPrime(k)) { cout << k << endl; numPrimes += 1; } }

13. Shorthand for increment/decrement Lots of code requires incrementing a variable by one Three methods, using = and +, using +=, and using ++ effectively they are same num = num + 1; num += 1; num++; // post increment It is also possible to write ++num; pre-increment These differ on when the increment is performed, but this difference doesn’t matter when used as an abbreviation for the statement n += 1; in a single statement Similarly there are post-decrement (and pre-decrement) num = num - 1; num -= 1; num--;

14. The do-while loop Similar to while loop, but the test is after the execution of the loop body The while loop may never execute, do-while loop executes at least once do { ;... ; } while ( ); Example: Prompt for a number between 0 and 100, loop until such a number is entered ( user should enter at least one number) do { cout << "enter number in range [0..100] "; cin >> num; } while (num 100 ); Don’t forget

15. Priming Priming: reading an initial value before the loop do not get confused with prime numbers; this is something else Problem: enter numbers, add them up, stop when -1 entered int sum = 0; int num; cin >> num; // prime the loop while (num != -1) { sum += num; cin >> num; } cout << " total = " << sum << end; Code duplication exists here: input (and perhaps prompt) code is repeated before the loop and in the loop

16. Pseudo infinite solution using break To avoid repeating code, include it in the body of the loop only, use a test to break out of the loop break statement exits (inner-most) loop I don’t prefer this kind of loops (I’d prefer code duplication) Because the loop condition is not clear, hence prevents readability Try not to use break in this course Save it for later when you develop really complicated loops int sum = 0; int num; while (true) // seemingly infinite loop { cin >> num; if (num == -1) { break; // get out of loop } sum += num; } cout << "total = " << sum << end;

17. Fence Post Problem The problem that occurs when one or more operations of the loop body are executed one less then the others. Example: Display integers between 1 and 10 separated by comma 1,2,3,4,5,6,7,8,9,10 no comma after 10; no comma before 1. for (n=1; n <= 10; n++) { cout << n << ","; } Problem: comma after 10 for (n=1; n < 10; n++) { cout << n << ","; } cout << n; No problem, but code duplicates Think of other solutions! (see page 175 of Tapestry)

18. Downward-counting loop Calculate n to the power of m: n m =nxnx…xn Example: 2 5 =2x2x2x2x2=32 int power = 1; int n, m; cin >> n >> m; for (int i = m; i <= 1; i--) { power = power * n; }

19. Nested loops Sometimes one loop occurs in another Generating 2-dimensional tabular data multiplication table Sorting vectors (which will be studied much later) Display some geometric figures using character * (or any other character) display rectangles, triangles Although other loops can be nested as well, most of the time, for loops are used in nested manner

20. Nested loops - Example Write a function to display a rectangle of stars (height and width are parameters) e.g. if height is 4 and width is 7, the output should look like ******* for (i=1; i<= height; i++) { for (j=1; j<=width; j++) // inner loop prints 1 line of stars { cout << "*"; } cout << endl; // end of line is put to the end of each line } See drawfigures.cpp for the complete function and its use in main

21. Nested loops - Example Write a function to display a perpendicular isosceles triangle of stars (perpendicular side length is parameter) e.g. if side length is 6, the output should look like * ** *** **** ***** ****** for (i=1; i <= side; i++) { for (j=1; j<=i; j++) // inner loop prints 1 line of stars { cout << "*"; } cout << endl; // end of line is put to the end of each line } See drawfigures.cpp for the complete function and its use in main

22. Same loop: downward-counting for (i=1; i <= side; i++) { for (j=1; j<=i; j++) { cout << "*"; } cout << endl; } for (i=1; i <= side; i++) { for (j=i; j>=1; j--) { cout << "*"; } cout << endl; }

23. Drawfigures – Other Considerations What about having a function to display a line of stars (number of stars is a parameter) useful for both rectangle and triangle void PrintLine (int numstars) // pre: numstars > 0 // post: displays numstars stars in one line { int i; for (i=1; i<= numstars; i++) { cout << "*"; } cout << endl; // end of line is put to the end of the line } in rectangle function, inner loop is replaced by a function call for (i=1; i<=height ; i++) { PrintLine(width); } use of PrintLine in triangle function is similar

24. Example – Multiplication Table On i th line print, i*1, i*2, i*3,..., i*i Total number of lines is an input. Display lines starting with 1. See multiply.cpp #include #include // for setw using namespace std; int main() { int i,k,numlines; const int WIDTH = 4; cin >> numlines; for (i=1; i <= numlines; i++) { for (k=1; k <= i; k++) { cout << setw(WIDTH) << i*k; } cout << endl; } return 0; }

25. Constants Sometimes very useful provides self documentation re-use the same value across the program avoid accidental value changes like variables, but their value is assigned at declaration and can never change afterwards declared by using const before the type name (any type is OK) const double PI = 3.14159; const string thisclass = "CS201" const int WIDTH = 4; later you can use their value cout << (PI * 4 * 4); but cannot change their value PI = 3.14; causes a syntax error

26. Formatting Output We use stream manipulator setw to specify the total number of spaces that the next output will use setw(field length) written in cout and affects only the next output value not the whole cout line output is displayed using field length spaces in right justified manner (any empty space is on the left) defined in header file, so you have to have #include Example cout << setw(9) << "cs201"; output shown is four blanks and cs201

27. Example using robot class (see rectangularscan.cpp) Write a program in which the robot starts at 0,0 and searches a rectangular space that covers n*n cells n is input (in the example below, n is 8) during this journey the robot should pick or put things on the cells so that all visited cells occupy one thing