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The discrete binary logarithm is computed to be the number of times the input number can be divided by 2 before reaching 1. So the for loop initializes i to n and then divides i by 2 once on each iteration. The counter c counts the number of iterations. So when the loop terminates, c contains the value of the discrete binary logarithm of n. In addition to using the setw() function that is defined in the <iomanip> header, this program also uses the log() function that is defined in the <cmath> header. That function returns the natural
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(base e) logarithm of n: log(n) = log e n = lnn. It is used in the expression log(n)/log(2) to compute the binary (base 2) logarithm of n: log 2 n = lg n = (ln n)/(ln2). The printed results compare the discrete binary logarithm with the continuous binary logarithm. The former is equal to the latter truncated downward to its nearest integer (the floor of the number). The loop invariant in this example is the condition 2^d <= n/i < 2*2^d (i.e., 2d n/i < 2 2d). It is tested by printing the values of the three expressions p2d, n, and 2*p2d, where the quantity p2d is computed with the power function pow() that is defined in the <cmath> header. We can prove that this for loop will always compute the discrete binary logarithm correctly. When it starts, d = 0 and i = n, so 2d = 20 = 1, n/i = n/n = 1, and 2 2d = 2 1= 2; thus 2d n/i < 2 2d. On each iteration, d increments and i is halved, so n/i is doubled. Thus the condition 2d n/i < 2 2d remains invariant; i.e., it is true initially and it remains true throughout the life of the loop. When the loop terminates, i = 1, so the condition becomes 2d n/1 < 2 2d, which is equivalent to 2d n < 2d+1. The logarithm of this expression is d = lg (2d) lg n < lg (2d+1) = d+1, so d is greatest integer lgn.
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4.5 THE break STATEMENT We have already seen the break statement used in the switch statement. It is also used in loops. When it executes, it terminates the loop, breaking out of the iteration at that point. EXAMPLE 4.20 Using a break Statement to Terminate a Loop
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This program has the same effect as the one in Example 4.1 on page 60. It uses a break statement to control the loop: int main() { int n, i=1; cout << "Enter a positive integer: "; cin >> n; long sum=0; while (true) { if (i > n) break; sum += i++; } cout << "The sum of the first " << n << " integers is " << sum; } Enter a positive integer: 8 The sum of the first 8 integers is 36 As long as (i <= n), the loop will continue, just as in Example 4.1. But as soon as i > n, the break statement executes, immediately terminating the loop.
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The break statement provides extra flexibility in the control of loops. Normally a while loop, a do..while loop, or a for loop will terminate only at the beginning or at the end of the complete sequence of statements in the loop s block. But the break statement can be placed anywhere among the other statements within a loop, so it can be used to terminate a loop anywhere from within the loop s block. This is illustrated by the following example. EXAMPLE 4.21 Controlling Input with a Sentinel
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This program reads a sequence of positive integers, terminated by 0, and prints their average:
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