1. Carrano and Prichard: Page 41, Self-Test Exercises 1 and 2

a) What is the loop invariant for the following?

// precondition: n >= 0, loop invariant: sum = item [0] + item [1] + … + item [index]
int index = 0;
int sum = item [0];
while (index < n)
{
      index ++;
      // index <= n and sum = item [0] + … item [index – 1]
      sum += item [index];
      // index <= n and sum = item [0] + … item [index]
} // while loop

b) Write specifications for a method that computes the sum of the first five postitive integers in an array of N arbitrary integers.

 public void sumFivePositives ()
 {
      // Number of positive entries in array, item, is >= 5.  sum = sum of first 5 entries > 0.
      int [] item = {0, -1, 1, -2, 2, -3, 3, -4, 4, -5, 5, -6, 6};
      int N = 13;
      int sum = 0, index = 0, count = 0;
      while ((count < 5) && (index < N))
      {
           if (item [index] > 0)
           {
                sum += item [index];
                count ++;
           }
           index ++;
      }
      if (count < 5) System.out.println ("Too few positive entries.");
      else System.out.println ("Sum: " + sum);
 } // sumFivePositives

2. Write a method to calculate x n for x any double and n an interger >= to 0.  Write the pre- and post-conditions for the method and the loop invariant.

// precondition: (n >= 0), loop invariant is product = x count
public double power ( double x, int n)
{
    int count = 0; double product = 1;

    // product = x n
    while (count < n) do
     {
          // (count < n) and (product = x n)
         product = x * product;
          // (count < n) and (product = x (n+1))
          count ++;
        // (count <= n) and (product = x n)
     }
 // (count = n) and (product = x n)
return product;
} // method power
// postcondition: x n  returned

3. Write a method to calculate x ! (the factorial function).  (See page 32 of the text.)  Then write the pre- and post-conditions and the loop invariant.

// pre-condition: n >= 1, loop invariant: product = 1 * 2 * … * count.
 public int factorial (int n)
 {
      int product = 1, count = 1;
      while (count < n)
      {
            count ++;
            product *= count;
      }
      return product;
 } // method factorial
 // post-condition: count = n, product = 1 * 2 * … * n