Announcements

1. Announcements • First Midterm • March 14thon 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 11th 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! = 1x2x…xn 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 productis 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 numberis 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 <initialization> while (<test>) { <statement1>; ... <statementN>; <update> } for (<initialization>; <test>; <update> ) { <statement1>; ...   <statementN>; }

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 <initialization> do { <statement1>;   ...  <statementN>; <update> } while (<condition>); • 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 < 0 || 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 usebreakin 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: nm=nxnx…xn • Example: 25=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 (intnumstars) // pre: numstars > 0 // post: displays numstars stars in one line { inti; 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 ith 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 <iostream> #include <iomanip> // for setw using namespace std; int main() { inti,k,numlines; const intWIDTH = 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 <iomanip>, so you have to have #include <iomanip> • 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