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Algorithm 1:   Even(positive integer k)

Input: k , a positive integer

Output: k-th even natural number (the first even being 0)

Algorithm:

if k = 1, then return 0;

else return Even(k-1) + 2 .

Here the computation of Even(k) is reduced to that of Even for a smaller input value, that is Even(k-1). Even(k) eventually becomes Even(1) which is 0 by the first line. For example, to compute Even(3), Algorithm Even(k) is called with k = 2. In the computation of Even(2), Algorithm Even(k) is called with k = 1. Since Even(1) = 0, 0 is returned for the computation of Even(2), and Even(2) = Even(1) + 2 = 2 is obtained. This value 2 for Even(2) is now returned to the computation of Even(3), and Even(3) = Even(2) + 2 = 4 is obtained.

As can be seen by comparing this algorithm with the recursive definition of the set of nonnegative even numbers, the first line of the algorithm corresponds to the basis clause of the definition, and the second line corresponds to the inductive clause.

By way of comparison, let us see how the same problem can be solved by an iterative algorithm.

Algorithm 1-a:   Even(positive integer k)

Input: k, a positive integer

Output: k-th even natural number (the first even being 0)

Algorithm:

int   i, even;

i := 1;

even := 0;

while( i<k ) {

          even := even + 2;

          i := i + 1;

}

return even .

Example 2: Algorithm for computing the k-th power of 2

Algorithm 2   Power_of_2(natural number k)

Input: k , a natural number

Output: k-th power of 2

Algorithm:

if k = 0, then return 1;

else return 2*Power_of_2(k - 1) .

By way of comparison, let us see how the same problem can be solved by an iterative algorithm.

Algorithm 2-a   Power_of_2(natural number k)

Input: k , a natural number

Output: k-th power of 2

Algorithm:

int   i, power;

i := 0;

power := 1;

while( i<k ) {

          power := power * 2;

          i := i + 1;

}

return power .

The next example does not have any corresponding recursive definition. It shows a recursive way of solving a problem.

Example 3: Recursive Algorithm for Sequential Search

Algorithm 3   SeqSearch(L, i, j, x)

Input: L is an array, i and j are positive integers, i ≤j, and x is the key to be searched for in L.

Output: If x is in L between indexes i and j, then output its index, else output 0.

Algorithm:

if i ≤j , then

{

   if L(i) = x, then return i ;

   else return SeqSearch(L, i+1, j, x)

}

else return 0.

Recursive algorithms can also be used to test objects for membership in a set.

Example 4: Algorithm for testing whether or not a number x is a natural number

Algorithm 4   Natural(a number x)

Input: A number x

Output: "Yes" if x is a natural number, else "No"

Algorithm:

if x<0,   then return "No"

else

    if x = 0,   then return "Yes"

    else return Natural( x - 1 )

Example 5: Algorithm for testing whether or not an expression w is a proposition (propositional form)

Algorithm 5   Proposition( a string w )

Input: A string w

Output: "Yes" if w is a proposition, else "No"

Algorithm:

if w is 1(true), 0(false), or a propositional variable, then return "Yes"

else if w = ~w1, then return Proposition(w1)

Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
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what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
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2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
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you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
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"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
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progressive wave
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Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
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Source:  OpenStax, Discrete structures. OpenStax CNX. Jan 23, 2008 Download for free at http://cnx.org/content/col10513/1.1
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